Tokamak Edge Plasma Modeling Using an Improved Onion-Skin Method

Wojciech R Fundamenski

A thesis submitted in conformity with the requirements for the degree of doctor of philosophy (Ph.D.)

Graduate Department of the Institute of Aerospace Studies University of Toronto

Q Copyright by Wojciech R. Fundamenski (1999)

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Abstract:

Boundary (edge) plasma phenornena play a crucial role in the dtvelopment of controlled

thermonuclear fiision via magnetic confinement. In this work the edge plasma arc modeied by a coupled

set of plasma/neutrai transport equations; the neutral transport equations are solved by Monte-Carlo

techniques (NIMBUS code). while the plasma transport equations are solved using an improved onion-skin

method (OSM) based on the numerical techniques of computational fluid dynamics. The OSM approach, as

defined in this thesis. consists of replacing the anomalous cross-field flux terms by (ofien simplified,

always adjustable) cross-field source te-, and taking advantage of available diagnostic data (typically

radiai profiles of î and Te from target Langmuir probes). of varying these cross-field sources in order to

minimize the error betwten the solution and the diagnostic data. The OSM approach was shown capable of

capturing the transition h m attacheci to detachcd conditions in a tokamak divenor plasma, an important

and timely aspect of present fusion research. The other applications presented in the thesis include a study

of the sensitivity of modeling assumptions, a codecode comparison between the OSM and a standard 2 0

approach (represented by the EDGE2D code), and a codecxperiment comparison with reciprocating probe

data for Ohmic. L-mode and H-mode shots on the JET tokamak

Acknowledgments:

1 would like to thank my supervisor, Dr. Peter Stangeby, whose inexhaustible enthusiasm and

wide knowledge were invaluable fiom beginning to end. 1 feel greatly obliged to David Elder for fiequent

discussions and daily assistance with numerical prograauning, to Dr. Zingg for his skilled introduction to

cornputational fluid dynamics, to Dr's Haasz and DeLeeuw for providing yearly reviews as part of the

doctoral assessrnent cornmittee, to D.G.Porter for acting as an external examiner and providing a number

of valuable suggestions, and finally to everyone at the fusion lab at UTIAS for their constant

companionship, The work was supported in part by the National Science and Engineering Research

Council of Canada (NSERC). and by the Canadian Fusion Fuels and Technology Project (CFFïP), which

is gratefull y ac knowledged.

Table of Contents:

1. Introduction 2. Tokamak Edge Plasma Model

2.1 Plasma Transport II B 2.2 Plasma-Surface Interaction 2.3 Plasma-Neutra1 Interaction 2.4 Tokamak Edge Plasma

2.4.1 Limiter edge plasma 2.4.2 Divertor edge plasma 2.4.3 Edge plasma modeling and the onion-skin method

3. Numerical Solution of Model Equations: the 0SM2 code 3.1 MHD Equations: Magnetic Grid 3.2 Plasma Equations: Computational Fluid Dynamics 3.3 Neutrai Equations: Monte-Carlo simulation 3.4 Iterative stability: Source Relaxation 3.5 Cross-field socltces: Diagnostic Variance Minimization

4. Modeling Results: Application of the OSM2/NIMBUS code 4.1 Default model: transition from attachai to detached conditions 4.2 Sensitivity to modeling assumptions

4.2.1 Effect of cross-field sources, QL : the QJU] ansua 4.2.2 Effect of neuuai sources, 9, 4.2.3 Effect of plasma sources, Q, 4.2.4 Effcct of flux expansion sources, Qg

4.2.5 Effect of kinetic boundary conditions 4.3 Comparison with the EDGE2D/NIMBUS code 4.4 Comparison with experimental data on the JET tokamak

5. Conclusions 6. References

Appendices:

A. 1 Energy in the XXI Century and Nuclear Fusion A.2 Plasma Physics

A 2 1 Charged Particle Dynamics A.2.2 Statistical Mechanics A.2.3 Kinetic Theory

A.3 Braginskii and CGL Equations A.4 Electrostatic sheath probIem A S Tokamak Core Plasma

AS. 1 MHD equilibrium A.5.2 MHD instabilities A.5.3 Kinetic Instabilities and Transport I B

List of Tables:

4 1.1 Target and upstream conditions for the attacheddetached transition results

4.1.2 Comparison of 0SM2 and 2PM correction pradictions

4.1.3 Comparison of 0SM2 and 2PM upsueam predictions

4.2.1 Summary of key resulw for the parametric sensitivity study

A.4.1 Sheath parameters compared in models

List of Figures (results section excepted):

Charged particle orbits and drifts

Plasma-surface and neutrai-surface contact

Near sheath regions

H recycling

H atom spectnim

H molecule specuurn

Pd vs. Te ,Z

AtomiJMolecular Cross-sections

Limiter SOL

Divertor SOL

Typical poloidal grid

Close up of grid near the X-point

Charged particle orbits and drifts

Rosenbluth potentials

S hapes of fi(v)

Tokamak schematic

Tokamak CO-ordinates and flux-surface geornetry

Typical tokamak fields and currents

Magnetic island structure

1. Introduction

The objective of the work prcscntcd here was to develop an improved onion-skin method (OSM) of

solving the plasma transport equations in magnetized edge plasmas with the specific aim of studying the

phenomenon of divertor plasma detachment in tokamaks. OSM is an approach at interpretive twodimensional

modeling of a tokamak edge plasma; in ihis contcxt the term interprerive is taken to mean well suited to

interprerarion of experimcnts and closely linked to diagnostic &m. Onion-skin interpretive modeling irnplies an

intention of incorporating (as input) as much diagnostic data as possible, while making the fewest possible

assumptions about unknown (anondous) cross-field coefficients, It is based on a standard set of plasma

transport equations (used by such 2-D edge plasma codes as U E D G E ~ ~ ' . ~ 2 ~ ~ . ~ ~ ' ~ , and E D G E ~ D ~ ~ ~ ~ ~ ~ ~ I )

which are solved on each flux surfiace with tinctic boundary conditions at the targets; the cross-field sources are

iteratively adjusted to best match the specified (diagnostic) target quantities. An improved onion-skin algorithm

capable of capturing plasma dctachment was developed based on standard numerical methods of computational

fluid d y ~ m i c s (CFD). The formulation, implementation and successful application of this improved anion-skin

merhod, taken tagether, constitutc the contribution to knowledge containeci in the thesis. Original material may

be found in 32.1, 92.4.3, 83.2, 53.4, 93.5 and 94; when presented alongside previous work, original equations

will be markcd with an asterix, eg. (2.1.17*).

Previously, the onion-skin merhod had been applied successfully to mode1 arrached plasma

b e h a v i o ~ r [ ~ ~ , but has been incapable of mdeling divertor plasma chme ment- In addition. due to the steady-

state parabolic form of the plasma transport equations required by the previous numerical scheme, this approach

could not be made to include parallel viscosity, i o n ~ l e c u o n energy exchange o r tmly continuos uansonic mid-

fiow transition. In the work presented here, these problems were overcome by using the CFD m e t h o d o l o ~ to

evolve the time-dependent plasma transport quations in parabolic f m until convergence (which is equivalent

to solving these equations in a steady-state hyperbolidelliptic formml); as a result of this modification, stable

derached plasma solutions were obtained.

in addition, a general method of adjusting the cross-field sources in order to find the best match to

selccted diagnostic data was developed bascd on variarional techniques. In the present context. this mcthod will

bc referred to as Diagnostic Variance Minimizarion (DVM). It consists of a variational schernc to adjust the

cross-field particle, momentum and energy sources in order to best match selected diagnostic data (to minimize

the statisticaI variance between selected diagnostic and numerically calculated quantities). The DVM approach

providcs a frmework in which the onion-skin merhod could in the future be cmployed with various types of and

multiple sets of diagnostic data (this data need not be located at the target - as for the earlier OSM - provided

that it includes radial profiles across the edge plasma, eg. fkom reciprocating probes, Li-beam diagnostic,

Thomson scattering arrays, hydrogen spectroscopy).

Modeling is by definition a theontical task and must not shy away fiom mathematical ngor when

appropriate; however, excessive attention to detail runs the risk of obscuring rather than clarifying the physical

Modeling is by definition a theoretical task and must not shy away from mathematical rigor when

appropriate; however, excessive attention to detail nins the risk of obscwing rather than clarifying the

physical content of the model. In addition, the rtlationship of modeling to experiment generally varies

widely from intimate to non-existent. The intention of the OSM-DVM interpretive approach was to achieve

a balance between the theoretical and experimental aspects of tokamak edge plasma research. to aid the

experimentalist by placing the large amount of diagnostic data in a unifying framework, and hopefully offer

insight into these diagnostic measurements. By providing a fast method of estimating the 2-D edge plasma

dismbution (as compared with the standard 2-D fluid codes which r e q u i . lengthy parameter xarchesWm)

the OSM-DVM approach is intended to supplement the more laborious 2-D calculations by offering the

experimental fusion community an alternative interpretive tool for routine edge plasma modeiing, Aside

from significant savings in CPU time (ranging from one to two orders of magnitude, with typical solution

times of 1 to 10 hours on an IBM RS6000 wcdcstation), the OSM-DVM approach makes fewer

assumptions about the physics of cross-field transport, sidestepping the greatest uncertainty in edge plasma

modeling.

The presentation which foUows is subdivided into sections dealing with theory, numerical

methods and cornputational results. The level of pttsentation assumes graduate level knowledge of physics

and mathematical analysis and some familiarity with piasma physics, fusion research and computational

analysis. For the sake of conciseness, only the physics directly relevant to plasma edge modeling is

discussed; a number of excellent monograms on plasma physi~s[OY60. Rd. kh921, plasma-surface

interactions-.S090i, plasma-neueal i n t e r a c ~ o n s " ~ ~ ~ ~ , hision t e c h n o l ~ ~ ~ ~ ' ~ ~ ~ and tolramaksIwUm, are

recommended to the interested reader for further scope in any one of these areas. In addition, the

appendices contain brief summaries of the necessary background material and are frequently referred to for

the basic concepts and equations. The discussion of numerical rnethods is broken down into five blocks,

corresponding to successive stages in the iterative computational loop, most importandy the CFD and

Monte Carlo (MC) methods. The results section deals with severat applications of the ncwly developed

0SM2 code (iteratively coupled with the NIMBUS hydrogenic code; this computational loop will be

dcnoted by OSM2NIMBUS). including an investigation of the attached-detached uansition, a sensitivity

study with respect to various modeling assumptions, a cornparison with an existing 2D edge plasma fluid

code EDGE2D/NIMBUS, and a direct confrontation with experimental data from the JET tokamak.

Conclusions are drawn from the above results and assessed in light of our present understanding of

tokamak edge plasma behaviour.

2.1 Plasma Tronport H B

The kinctic equation for the single-particle distribution function fdx,v.t) may be written asm1,

= ~ C , + S , a E [i, e. nl ma ha k1i.e.n 1

where Da is a phase space derivative for species 'a', Cd represents the change to fAx,v,t) due to collisions

with species 'b', and Sa is a volumetnc source of species 'a '; this equation is valid for electrons (2, = 4).

ions (2, 2 1) and neutrals (2, = O) provided appropriate collision te- are used. Starting with (2.1.1).

fluid-like plasma transport cquations may bc dcrïvcd. The standard approach dapted in neutral gas

dynamics consists of multiplying the kinetic quation by powers of velocity and integrating over velocity

space; the resulting velocity moment hierarchy is usually closed on the level of the third moment (the heat

flux)[-']. This procedure has also been applied to the plasma kinetic equation with the Fokker-Planck

collision tetmm, yielding expressions for transport coefficients dong and across the magnetic field in

terms of the h a and electron characteristic times. The final set of moment equations, named afier

Braginskii who carrîed out the most complete denvationrBml, may be found in OA.3; these are the most

widely used equations in eüge plasma modeling.

In a stroogly magnetized plasma, the above procedure introduces inaccuracies related to the

anisouopy of fdv). which may be signifiant if collisionality is ~eak*"~~ ' . In the collisionless limit, the

Vlasov equation may be expressed in terms of integrals of motion and integrated to yield the Chew-

Goldberg-Low (CGL) or double adiabatic equati~nd-~'. also given in OA.3;. The Braginskii and CGL

moment equations rnay be considered as the asymptotes for high and low collisionality. It is also possible

to derive moment cquations valid for d l degrees of coUisionality, reducing to the Bragïnskii and CGL

expressions in their respective limits. This is done by first transforming the kinetic equation into magnetic

CO-ordinates, next averaging out the rapid gyraung motion ro form the drift-kinetic equation, and finally

integrating the drih-kinetic equation over powers of velocity. The procedure has first been carried out by

Zawaideh. Najmabadi and Conn ( Z N C ) ~ ' ~ . ~ ' ; the following calculation is closely guided by their original

paper, but since several minor extensions are introduced. it is prrsented in some detriil.

We first inuoduce orthogonal magnetic coordinates ( s l l , a P ) and velocities (v,,, v, v8).

where cp is the gyration angle and il,. i, and ig are orthogonal unit vectors. The CO-variant phase-space

derivative D, may be expressed in terms of thtse variables, ie. for a distributionf,(s, a: J v,,, v1 ,p,t). as

where the dot denotes a convective time derivative. We would next like to average the kinetic equation

over a gyration period of a particle, which in a strong B field is more rapid than any collision process. Such

an average may be camied out for a helical trajectory of a charged panicle in a strong magnetic field with

slowly varying electric and magnetic fields, 5A.2.1, yielding the guiding center drift velocity; the averaging

is valid subject to the adiabatic field constraints, 6, << I defined in (A.2.6). The convective time

derivatives are found by expanding (A.2.3) in powers of ô,,, retaining the lowest order terms,

We take the B-field itself as constant in time, so that (dfd){sl~,afl'=O; drift terms paralle1 to B are small

and can be neglected. Inserting the above into the operator D, and averaging D& over one gyration period,

jokd9 /2x. we obtain the drift Linetic equationmdl. For weak drifts. v&fih - O(&,,,), the dependence on a

and /3 rnay be neglected, fd@) - O(&:). Inuoducing the relative velocities, Vil = vil - u, VL = v, where

u, = c v p , is the average along-B velocity, the phasespace derivative of fAsIl, KI, VLt) becomes,

a a a a Da =-+il, --+y, --+v, a-

at asl I 341 a h

where O,, = a/as,,. Equation (2.1.1) formed with this operator, valid in the weak drift approximation vdnfi/v

- O(&), is accurate to first order in 6, Multiplying the drift kinetic equation (2.1.5) by m n ~ ~ ~ ~ ~ l and

integrating over velocity space, dV = 2nV&'ildVL, we arrive at the (k,j)-moment equationFw881

where the 'a' index was dropped for simplicity and didt = a /& + u,VI was introduced. We will denote these

moments by the u s d fluid-like variables: density p=nm, velocity u, pressure p, heat flux q, tempetature T,

Similarly. we define symbols for collision and source inte@,

From now on, we will neglect cross-field fluxes as fmt order in G, and concentrate on parailel transport

alone. There are two incentives for adapting this approach. Firstiy, parallel transport is faster by many

orders of magnitude, sce 5k5 .3 and (A.5.15). Secondly, it is found experimentally chat cross-field fluxes

differ sharply from classical predictions, due to a combination of a several physical mechanismc, including

elcctrostatic and magnetic plasma turbulence. and plasma-neutrai interactionwa. In fact, no satisfactory

mode1 ewts to predict their dependence on local plasma cond~t ions '~~~ ' . We will revisit bis question in

52.4.3.

Using the fluid-variable notation, we write down the moment equations up to order rhree.

dl1 ,O dr+ V,, pi , -(pli - Pi )R-'v~,B = ZenE,, + R + rtr~F

where once again the index 'a' is implied, n E {i.e}. These equations are not closed. since 4& moments

appear in the 3* moment equatioos; closure requùcs that these be rclated to lower moments (in gas

dynamics, this procedure is usually terminated on die level of the 2* rather than the 3* in

our case, the 3d moment equations are included in tbe b o p of increasing the accuracy of the heat flux

expressions). At this point we can make use of the hierarchy of collision times derived in SA.2.3, r, cc qi

cc T&. which suggests that a bi-Maxwcllian distributi~nf,~(s, VII. vbt; lits TA a exp(-fii(~I-u)2 -PL(v/] ,

finiRT, should provide a good approximation tof, for the purpose of evaluating higher moments; withf, =

hm, 4h moments become

Pli PL cq,%q ~ 2 - p2

To the same accuracy, 3d order collision terms may be approximated by a BGK rn~del[~*~,

where .th are classical collision times (A-2.37) and a, is a constant to be determined by comparison with

the cIassical expression for q, in the highly collisional limir, Q., + O. Thus, by assuming a bi-Maxwellian

disuibution for the evolution of the 4' moments we have effectively closed the 3" moment equations

(2. I .9) which are now valid over the entire range of collisionality subject to the weak drift approximation.

The 2"* order (rnergy exchange) collision terms. Qu and Q, contain contributions from like-

particle (i-i, e-e) and unlike-particle (i-e) collisions. Once again, based on T, CC Tii CC T~,, we expect these

processes to proceed on different time scales. allowing us to separate [hem by writing,

Relaxation times may be calculated by isolating each proçess in tum. The calculation of energy exchange

due to like-particle collisions has b e n carried out for fa = faw, and is valid for any degree of

a n i s o t r ~ ~ ~ ' " ~ ~ ~ ~ .

where g(&) = { tan -'& x>O: tanh " d x x<O/. and sa refers to the characteristic tirne (A.2.37) evaluated

a t the effective temperature Tc . The caiculation of ion-electron energy exchange for bi-Maxwellian

elecaons,f, = f , 2 M (f; does not effect the result) yields classicai collisional times,

The final, la order collision term, vanishes for like particle (i-i, e e ) coilisions; hence, Ri = - R, = - RCG a consequence of momentum conservation. W e find R, by neglecting terms of order (m/m,) fiom the lu

moment equation for electrons. which we assume are nearly isotropic (6pe « p,),

In the above, we made use of plasma quasi-neutraiity, ne = Z#,; we can thus replace Z,en,Ell + R,, which

occur together in the ion equation, by the clectron pressure gradient; in the highly collisionai regime,

Braginskii's result for R, may be used, 5A.3. In al1 the collision t e m evaiuated above. we neglected the

neutrd conmbution, which will be added separately at a lacer stage, 92.4.3

As a next step, we f o m a linear combination of the parallel and perpendicular equations, effecting

a transformation to isotropic and non-isutropic quantities, (y, ,yL) + (r, 6 ~ ) , \YE (p,q),

where we used {3~.3&yl'2) = { y + 2 y ~ ,y" -y,) and {yk y,) = (yr + âv, - 8 W ) ; an expression for

d/dt(6q) was omitted for the sake of brevity, since it is not needed below- No information was lost in

transforrning from {y, ,vl) to (v, OW); both sets of equations are valid for any anisotropy in f,(v,,v~,

However, by forming isotropic expressions, which domînate for high collisionality (tyi+vL+yt, &4r+O),

the non-isotropie contributions may be ueated as perturbations to the isotropic equations. For example. a

quasi-stationary 6p can be obtained by neglecting non-isotropie gradients and extemal sources in (2.1.16).

where we introduced local gradient scale lengths, Lv,= w / ~ I ; ~ w , and expressed our result in the 'viscous*

form. The non-isotropie nature of the parallel 'viscosity' q, ofien obscured in the Braginskii formulation, is

here made evident. Our collisional limit, q = (4~)f "p = I . I I f p is somewhat higher than the classical

values of Braginskii (obtained by expanding the pressure equation in terms of the mean-fiee-path to

gradient scale length ratio. A / Lv,): f l = 0 . 9 6 5 ~ ~ = 0.73?#,. In the oppsite limit i + W. the

'viscous* flux 6p saturates to a value of - pp , where p = (4/LFL + UZ&(~/ZA, - 4 1 2 ~ ) with p + 4/7 as LII/Z + 0, and p -+ -IR as L d n , +O. Transport coefficients of this type are said to be 'flux limited'.

At this point we would like to write down moment equations for ions and electrons separately,

stopping with 2d order moments. We simplify our analysis by assuming quasi-neutrality, n, = Zni, no net

charge production (ZS, = S, = So), negligible parallel currents (ijll I= lZeniui - en& << Ien,u,l, .-. ui = u, =

u). and neglecting ternis of order i(z, - ( r n / r n ~ ' ~ or smaller (thus. C$P~ = &-O(m/m,)" will be neglected.

while opi will follow fiom (2.1.17); in other words we cxpect anisotropy in fJVII, VJ to be much smaller

than inf;(V,,, VJ ). This allows us to add the and l* moment equations for ions and eIectrons.

Wc note that there is no net momentum exchange between ions and elecuons: (Zn, - n,)eEII vanishes by

quasi-neutrality, while Ri + R, = O by symmetry of the collision operator. To calculate the heat fluxes qi

and q.. we use the isotropic form of the 3m moment equations dong with a quasi-stationary. source-free

solution of the non-isotropic 2d moment equations. For electrons, we may combine (2.1.16a) and (2.1.16~).

and eliminate & and Vau using (2.1.15) and (2.1.18a),

We find four contributions to q,: eiecmni heat conduction. themo-electric effect. ion-eiectron coilisionai

rnomentum and eneigy transfers ( t y p i d y u cc ce and the 2d term vanishes). We note that in the

collisionai limit (4 << Lpv), Re = Rej = - 0.71n. KT,; comparing wilh the classical electron heat flux q." =

na-' FIT, where = 3.2Ter; /me, we can estimate &(" - 1.12%. In the opposite lunit (& r> Lpd, qe

saturates to a &ee-streaming value q ) - k # ~ ~ , with k, - 1 for X, < LvB, and k, 1 for & > Lm. A simple

flux-limited expression for qe. approachiiig the above two limits, l/qe = l/q," + Uq?, shows remarkably

good agreement with the more accurate expression (2.1.18). as long as &AL& L @) is appropriately

c h o ~ e n ~ " ~ ~ ~ . We define a flux-limited clecaon hcat diffkivity in analogy with (2.1.17),

A similar procedure may be repeated for tbc ions. Before commencing. we note that ;G" 1 and qi" 1 q:' - ~ ( m J r n ; ) ' ~ and that the ion contribution to the total heat flux is small (ZNC ignore q; entirely). However,

in a warm, stagnant plasma (uccc,), when Ti and Te are sufficiently de-coupled, ion heat conduction may

affect Ti(sR). and should be included. As beforc. we combine (2.1.16a) and (2.1.16~). eliminating & and Vau

us ine (2.1.15) and (2.1.18a) and neglcct *, as an order higher in smallness,

The main difference in cornparison to q, aside from the smailer magnitude, is the fact that the first terrn,

representing ion heat conduction, is strongly reduced for tra~sonic flow, u - çei. Evaluating (2.1.21) for

and comparing with qdi = aJCiC'~I~i whem = 3.9Tig /W. we estimate - 1.174. The flux-

lirnited ion heat diffusivity K, is defined in complete analogy to (2.1.20).

W e next transform thc lower moment equations from primitive (p, U, pi, p,) to conservative (p, pu,

pet p ~ ) variables, better suitcd to numerical solution, and display the final equation set in vector form,

where U contain the state variables, Fu the convective flues, are the viscous/conductive fluxes, Q, the

pIasma source terms (including the collisions between the electron and ion species and their interaction

with the net electric field), Q the neutral source terms (which in addition to sources associated with the

creation and destruction of particles, shown in (2.1 -22). would also include the change in momentum and

energy due to plasma-neutral interactions, not explicitiy shown above; the complete neutral sources will be

defined in $2.3, cf. (2.3.5)). and a the rnagnetic flux expansion terms; 6pi, qi, q,, are given by (2.1.17)

and (2.1.20). We expect the above set of equations to reduce to those of Braginskii in the high-collisionality

limit. r, -, O. Omitting al1 cross-field terms from (A.3.1). which we have neglected at the outset. we find

that the two equation sets are ncarly identical, asidc from flux limited expressions for qi and q, and thc an-

isotropic energy exchange term (boîh reduce, with slight transport coefficient corrections. to the Braginskii

expressions as T, -, O). The flux expansion terms, QB, are found to agree if the metrïc is made to reflect the

magnetic geometry. In the region of intermediate collisionality. the anisouopy of fi(vii.V,) becornes more

pronounced affecting. Spi, qi, and G'". The two equation sets progressiwly diverge in this regirnr. wiih

differences in their solutions becoming significant beyond A, / LvB > 0.1. Finally, in the low collisionality

limit (T. -, -), the 2d moment equations (2.1.9cd) d u c e to the CGL equatïons'0"s61, see 5A.3.

We can summarize the various plasma equations by the following schematic:

Boltzmann / Fokker-Phck (2.1.1)

I drifi-kinetic (2-1.5)

kinetic

It is worth emphasizing rhat the ZNC moment description does not presuppose a perturbed Maxwellian

velocity distribution- This assumption, appropriate for high collisionality, is only introduced to close the

Braginskii equations. In contrast, the collisionless CGL equations, make no such assumption; the fluid-like

primitive variables p. II, p represent velocity moments over potentially non-Maxwellian Jiv). The domain

of vdidity of the ZNC or drift-kinetic moment equations presented in this section, which fiom now on will

be referred to simply as the (parallel) plasma transport equations, is given beIow:

a) quasi-neutrality, n, = Zni

b) small parallel currents, I Z e n i ~ - en& « lenul

C) weak drifts, vdrift << Ivl,vll

d) slowly varying fields (A.2.6). 6, cc 1 (rL cc Lvg . O, CC 'tas)

e) a11 collisionality regimes, r i , r, E [OP]

The extensions to include parallel currents and finite poloidaf drifts are straight fonvard. although non-

t r i ~ i a l ' ~ ~ . ~ ~ ~ ] ; since they do not involve cross-field plasma transport. they will not affect our conclusions

regarding the validity of the OSM approach. We shall see presentiy that in the vicinity of a solid surface.

some of the above conditions may be violated: consequcntly, chat region will be considered separately. with

the aim of deriving appropriate boundary conditions for the plasma transport equations (2.1.22).

2.2 Plasma Surface Interaction

A solid surface has a very differtnt effect on a plasma than on a neutral gas; for the latter, it is a

reflecting surface. for the former. an absorbing one (plasma ions and electrons recombine on the surface,

and re-enter the plasma volume a neutralsW1). In both cases, massive particles moving towards the

surface have an average velocity on the order of cheu respective sound speeds c. - (~drn~? a r [ i n ] .

Neutra1 particles moving away fiom the surface have the same average velocity as those moving towards it,

producing a zero net flow. There ac no plasma ions moving away fiom the absorbing surface, fi (v,>O) = 0.

hence the plasma flow velocity in front of the surface is equal to the average ion velocity moving towards

the surface. Plasma transport equaions based on a weakiy sbifted Maxwellian fi ( v a are thus invalid in the

near surface region; theu inaccuracy may be estimated by a cornparison with a h l l y kinetic solution. The

problem of plasma-solid contact is m e of the oldcst in plasma physicsm. and it is worth considering it in

sorne detaiI. We take the x = O plant as the location of the absorbing surface, assume symmetry in the y and

z directions (a/& = a/ay = O, = <v> = O). and zero magnetic field; the x domain is closed by

requiring vanishing gradients at infinity, L » b, (a/axh = O. The problem is described by kinetic

equations (2.1.1) for ions, electrom and neutrals integrated over dv@, and by Poisson's equation for the

electrostatic potential$ (we assume Maxweiiian distributions in v, and v,, normalized to unity),

a e z ap a -+v .----- Cc, +si ar '-'ax mi a x a v I j

&/ira l

together with the boundary conditions of no net curent and no back flow at x = 0,

The neutrals have been included expiicitly in the problem, since they enter both the source and collision

terms in the ion and elecuon e q h n s ; they do not, however, interact with the electric field. Plasma-

neutral interactions will be d i s c u s d in the next section, 82.3.

W e begin by o b s e ~ n g that since ce >> Ci . an initidly neutrai surface would quickly bccorne

negatively charged. <p(x=O) = q+, CO. The negative floating potential opposes electron flow. until at stcady

state the elecuon and ion currents to the surface arc equal, Zcv,i> = <v+. The extent and magnitude of

this potential may be estimated by considering the Poisson equation in the near surface region. The

srnallness of the electron mass, and the consequent high electron self-collisionality, dows us to neglect the

fmt two terms in the elecuon finetic equation which leads to (2.1.15); the solution f;(v) is easily obtained

frorn (2.2.1 ) as a shifted Maxwellian dismbution reduced by the Bolumann facto?'.

The ion distribution becomes entirely forward shifted at some location x,, fi (Oorc~, . vLi>O) = 0; the region

x r [O,x*) will be refemd to as the clectrostatic s h a h . Since we expect x, to be of the order . which is

much shorter than any other length in the plasma, including collisionai mean free paths. colIisions and

sources may be neglected in the sheath, Ca = S, = O. At this point, the kinetic equations may be analyzed in

various degrees of sophisticationrSri901; for our analysis. we will adapt an approximate method. which none

the less. leads to the same resdt as the more involved and much lengthier ueatmentsw'. We begin by

noting that any trajectory satisfies the collisionless kinetic cquationml, and that conservation of energy

gives one such û-ajectory, vx(x): 4 ~ i ~ v ~ ( x ) ' + Zecp(x) = !4rn,vx((x3' + Zecp(x3; based on chis result we can

construct a solution to fi(x,v3 from v,(x). In the absence of sources, particle continuity requires that fi(vl)vI

= f2(vT)v2, where vl = v,(xl) and v-vX(x2) are related by v,(x),

inserting rq(x) and ni (x) into Poisson's equation and expanding to fmt order in AQ = M x ) - q&), we get

Assuming quasi-neutrality at the sheath edge, the condition for a non-oscillatory solution becomes,

This inequality is referred to as the generalized Bohm ~ r i t e r i o n ~ ~ " ~ ~ . Physically it means that the sheath

potential increases until the charge separation within the sheath is reduced sufficiently; this requires (via

continuity) that the ion velocity incrcase to the order of (Zï& )ln. The Bohm criterion was 6nt obtained

for cold ionsw1. Ti = O with f,(x.va = Q (x)ô(vX - u(x)). for which it becornes u(xJ > (ZT ~ m , ) ' ~ . For cool

ions. desctibed by a shified Maxwellian. fi (x.v J = n , ( ~ ) ( B / n ) ' ~ e x ~ ( - ~ ( v ~ - ~(x))'). where $ = m/2T&),

such that f,(vpû) = O. and fi(v.4) is a b d e n e û delta hiaction of width ci = ~ l ; i /mlR the Bohm criterion

becomer 1 / (u2(xJ - c:) < (zTJ~)-'"; this may k written as u(&) > ((ZT.+T& / m. )ln = c, . where c,

is the parallel plasma sound speed. The flow velocity at the entrance into the sheach is increased by

roughly the spread of the ion velocities parallel to the flow. Defining a local Mach number as M = u/c, , the

Bohm criterion is often stated (with some inaccuracy, since Tb need not equal Ti) as.

From (2.2.5) we see chat x, - O&), confirming our collisionless ansatz.

We can estimate the floating potenual at the wall by assuming zero current flow to the wall,

sonic ion flow at the sheath edge, M ( G = 1, and Maxwellian electrons throughout,

We thus obtain an expression for the potential drop across the sheath.

The potential at the sheath edge depends on the flow in the pre-sheath region, x > n, . but the pre-sheath

potencial is generally smail in comparison with the potential drop across the sheathlsu8''; typical values for a

hydrogenic plasma with Ti - Te are -cp,fïc - In, and - (c~o-9~) me - 3.

We would also like to estimate the ion and electron power flowing into the sheath, and the total

power depositcd onto the surface. which may include chernical energics. Thc electron encrgy flux at .u = O

is casily calculated with a cut-off-Maxwellian distribution. f,(v,>O) = O, whilc the flux at the sheath edgc x,

musc be Iarger by the value of the sheath potential drop (which is repukive for the electrons).

If we assume a shifted Maxwellian distribution with M(%) = 1 for the ions, we obtain the sanie result as for

the electrons. except the potential drop is reversed, qb = 2TiTi , q i o = qis + Zelv, - cpol. In reality the

presence of the E-field in the pre-sheath leads to ion flow which distorts fi(&,vd, inuoducing enors to the

above estimate. To progress further the system (2.2.1) would have Co be solved on the entire domain.

XE [O,L]. At present, no such complete solution has yet been obtained (not even when plasma-neutml

interactions arc neglectcd); however. a variety of simplified cases are present in the literaturewl. These

are summarized in 5A.4, where they are cornparcd in terms of the predicted normalid fluxes and

potentials, Table A.4.1, and the profiles of the ion distributions fi (v) at the sheath edge, Fig. A.4.1; the

estimated values are strikingly robust with respect to a wide range of assumptions.

In a magnetizcd plasrna. the governing quations (2.2.1) are modified by vxB terms entering the

ion and electron kinetic equations. If B is perpendicular to the solid surface, the analysis as presented so far

remains valid; otherwise ExB # O and the geometry of the problem becomes more complicated, The

oblique field problem has been considered in s o m detail[-'; the trajectories of the ions and elecuons in

the near surface region were followed numencally, with the Efield determined self-consistently from the

Poisson equation using the particle-in-celI method, and the B-field taken as constant at an angle 0 to the

surface normal. The resulting flow is charactenzed by two d e lengths: a) within a few Debye lengths of

the surface x E [O, Sb], the usual electrostatic sheath is formed with flow normai to the surface and strong

potential gradients, b) further out, up to a few ion Larmor radii at the local sound speed = c, 4, x E

[5b. 5&J, a quasi-neutral magnetic pre-sheath region appears in which the net flow is parallel to B. The

Bohm criterion with an oblique B-field becomes,

where va is a velocity parallel to B. Non-magnetized analysis remains accurate (and insensitive to IY), if x is

replaced by sfl and the sheath edge x, is taken as the point of enuy into the magnetic pre-sheath &. We began the plasma-surface analysis with the express aim of obtaining boundary conditions for

the plasma transport equations denved in 92.1 ; to accomplish this final step we consider the various iength

scales in the near surface region. These are: the Debye lengh & . the ion Larmor radius r~ and the ion

collisional rnean-free-path A, The evolution of the ion and electron velocity distributions in these regions

and the validity of the kinetic and moment equations are shown in Fig. 2 - 2 2 The fact that kinetic solutions

are not vaIid for s,, >> A, expresses the inadequacy of neglecting source and collision terms. For the purpose

of rnodcling, the morncnt equations are a more efficient tool. with Iittle loss of accuracy for s,, >> hD . Thcy

must however be constrained by the kinetic solution at some location in the region where both solutions are

valid, ie. 5b < s, < 5% . The sheath edge, taken roughly as the location where Bohm criterion is satisfied,

may be chosen as a convenient iocation to effeçt this matching of solutions; it is both close enough to the

surface that any sources or collisions within the sheath may be neglected, and far enough for quasi-

oeutrality and moment equations to be vaiid- In reaiity, some fi(v) distribution exists at this location which

changes continuously into the forward shifted sheath fi(v) for g < Sb , and into gradually less distorted

Maxwellian fi(v) for > ; this distribution is unknown to us. W e have instead an approximation to f-,(v)

in the sheath region and again an approximation to its moments in the pre-sheath region. If we denote the

former by f ( v ) and the latter by f(v), we seek the matching conditions for a E (i.e). which assure that P(v) - F(V) = ~ ( a ~ > ~ - d*). These requirements may be expressed as

The right-hand side averages rnay be related to the density n, and fluxes Fa + from the moment

equations, (2.1.22); the Ieft-hand side averages require the howledge of f:(v) nom the kinetic solution.

Typicall y, the energy fluxes are matched by introducing energy sheath transmission coefficients for ions yi

and electrons y,, defined as

The momentum flux is usually constrained by imposing the Bohm criterion on plasma velocity (since this

criterion contains an inequality, the constraint wilI prevent sub-sonic flow, while allowing supersonic

flow). We will adapt both these methods of imposing the kinetic requirements ont0 the fluid-like moment

equations. The effect of boundary conditions on the plasma solution will be studied in $4.2.4

2.3 Plasma-Neutral Interaction

Due to proximity of a solid surface, the edge plasma normally contains a significant concentration

of neutral species with which it interacts. In this section, we briefly examine the relevant atomic-molecular

physics, determine the dominant processes on the bais of their reaction cross-sections. and construct an

interaction source term for the moment equations.

Plasma-neutral interactions are normally classified according to the interacting spccies (neutrals.

ions, elecuons, photons) and the reaction type (scattering. excitation / emission, dissociation / molecular

formation. ionization / recombination. charge exchange, More generally. we can define Y(n) to

be a set of al1 possible. neutral. n-atom molecules XuiJu2, ... Xun, (XZ denotes an atom of nuclear charge

2). Y. to be a set of al1 neutral molecules of any size, Y+ to be a set of al1 molecules (neutral or ionized),

and 3 to be a set of molecules, elecuons and photons,

Using the above, we cm write down al1 possible Cbody encounters by taking k-1 successive set products

of 3 with itself, 3t = 9 @ 9 @ , . .@ 9. Each cncountcr can lead to several interactions with a probability

dependent on the relative energy of the reâctants. In the context of fusion research, we are usually

interesteci in hydrogenic plasmas containhg s d l quantities (40%) of low-Z impurities, originating due to

plasma-surface contact at the vesse1 wall (Be, B, C, O, .. .) or injected with the aim of increasing radiation

from the plasma (N, Ne, Ar, Xe, .. .). We will thus divide ow discussion into hydrogen (251) and impurity

(Z>l) reactions. The dominant hydrogen reactions. 3 = {hv. e. H , H', H2, H z ), are summarized

be 1ow[1=951:

Aromic hydrogen reactions: 3 = (hv, e, H, H')

H + H + H + H

H + e + H* + e

H* + H + h v

H + e + W + 2e

W + e + ~ * + h v

H' + 2e - H . + e

H + H ' + W + H

H + H 3 H 2 + h v

Molecular hydrogen reacnons:9 = {hv. e. Hz, Hz4)

atomic scanering

aromic excitation

aromic radiative de-exciriution

atomic ioniaztion

rodiarive (nvo-body) recombimtion

dielecrronic (three- body) recombi~rion

atomic charge exchange

molecular formation

molecular scatte ring

molecular ionitation

dissociative ionizarion

dissociative double ioniarion

dissociarive arrachnzerit

ntolecrrla r dissociation

niolecular excirarion

molecular radiative de-excirarion

radia rive dissocia rion

niolecirlar exciration

ionized molecule radiarive dissociation

ionized molecule dissociation

dissociative recombinarion

ionized molecule dissociative ionimtion

Hydrogen atomic and molccular reaction cross sections have been determinecl experimentally over a wide

energy range, and are generaily in good agreement with quantum mechanical scattering predictions. Semi-

empirical formulations are employed to express experimental data as well as extrapolate to lower energies.

For the purpose of the present work, al1 relevant atomic and moIecular data was extracted from the ADAS

database[s"mw. The dominant atomic/molecular interaction cross-sections are s h o w in Fig. 2.35.

Let us briefly consider the impact of plasma-neutral interactions on the plasma transport equations.

Plasma particle balance is govenicd by ionization, dissociation and recombination processes (occurring

either volumetrically or at the solid surface) leading in the simplest form to the well known Saha equation

for the degree of ionization of a gadplasrna mixture, which is obtained by estimating the ionization and

recombination cross sections for hydrogen a t o m ~ ' ~ ~ ' ~ ,

In the above b, ni, II,, are the ion, electron and neutral particle densities, E, is the ionization potentiai, and

go, g, are statistical factors. For Te >> E, a hydrogenic plasma becornes fully ionized. A more complete

chain of hydrogen reactions including transport and recombination at the vesse1 walls. is shown

schematicalty in Fig. 2.3.1. In practice, such calculations are carried out numerkally using Monte-Carlo

transport codes such as E I R E N E ~ ] , NIMBUS~-I, or DEGAS-'. x e 53.3.

The energy spectrum of the hydrogen atom in a field free environment is described by the

quantum numbers (n,I,s), which detennine the orbital electron energies and photon energies for allowed

transitions[-. Fig. 2.3.2. The hydrogen atom is most readily ionized by electron impact (E, = 13.6 eV).

Atomic recombination can occur as either a two-body (radiative) or a three-body (dielecuonic) process,

with the latter becoming significant at higher electron densities. Charge exchange between hydrogen atorns

and ions transfer both momentum and energy, but does not effect the particle balance; since the charge

exchange cross section is far Iarger than the ionization and recombination cross sections (Sn >> S,,. Sm).

increasingly so as the elecuon temperature is reduced below - 5 eV, this process is important in

detennining both neutral and ionized hydrogen transport within the plasma (it is also a key mechanism

involved in plasma deta~hment[~'"@~~). Neuual-neutral scattering coliisions begin to effect the plasma

momentum balance at higher neuual densities, Â, < L.

Reactions involving photons generally remove energy from the plasma, producing a local energy

sink. provided that the rnean-free-path for photon re-absorption is large in cornparison with the sizc of the

systern, A, > L (which fails to be true for a sufficiently dense plasmas). A hydrogen atom emits several

photons before k i n g ionized. the number increasing for lower temperatures: typical energy radiated pet

ionization event varies from - 30 eV at Te > 20 eV to - 150 eV at Te = 2 evUd1 (this is usually the

dominant cooling process in hydrogenic plasmas).

The energy spectmm of an Hz molecule is a function of the inter-nuclear distance, electronic

configuration, motecular vibration and rotationrC0hn-Ha*7i , Fig. 2.3.3. Dissociation of Hz produces two

atoms which share - 4 eV of energy (Franck-Condon n e u t r a l ~ ~ * ~ ~ ) ; these becorne important for colder

plasma (Ti < 2 eV) regions in providing a source of warm neutrals which can heat the plasma and oppose

further recombination.

Impurity (2 > 1) atomic reactions generally involve more than one orbital electron. which adds

complexity to the analysis, and the reactant set 9 soon becomes intractably large. Dissociation and

ionization normaily occur in a stepwise manner (X~~++X~*++X~~++. . .). Each ionization state (except

for the fuIly stripped nucleus) can emit line radiation following excitation. The sum over dl states has been

calculated as a function of Te under the assumption of local thermal equilibriumlP"w, Fig. 2.3.4. Radiation

is seen to gradually increase with atomic number. falling off sharply below some critical Te. The latter

effect is responsible for a radiation instability encountered in tokamak edge plasmas, leading to MARFE

f o r m a t i ~ n ~ ~ ~ ~ ' . The above treatment neglect the transpon of impurities. which is of crucial importance for

core plasma performance (radiative cmling and plasma dilution'se1, see (A.5.3) and edge plasma

behaviour (impurity radiative cooling is widely acccpted as a key mechanism in plasma detachmentmm).

Impurity transport rnay be simulated by Monte-Carlo codes such as DIVJMP'~~', or by multi-fluid plasma

codes such as ~ 2 ~ ~ ~ ~ " ~ UEDGE-]. or EDGE~D['-~I; we will r e m to this topic in (3.2.

In h e context of edge plasma modeling, we are interested in the plasma-neutral interactions as

sources of mass, momenturn and energy in the plasma transport equations (2.2.1); for simpIicity we will

refer to these simply as neutral sources. Using the vector notation of (2.1.22), we can write down the

dominant neuual sources as follows,

where Si, . S, and Scx are respectively the local ionization, recombination and charge exchange rates, u is

the pIasma flow velocity, vll.~ is the along-B neutral vcIocity, v~ is the ncutral birth velocity. Q,, and Q,, are

the energy sources for ions and electrons due to plasma-neutral interactions, and Qz and Q, are simiIar

sources due to plasma-impurity interactions. The default Q,, contains energy gain due photon cmission

following excitation (-XE&), atomic ionization (-13.6eV-Sia. fast dissociation (-10.4eV-Sfd), slow

dissociation (-S.OeV-Sd). dissociative molecular ionization (-8.0eV.Sdmi~. and three-body recombination

(+13.6eV.S,,). The default Qqi is set to zero. in kceping with a sirnilar assumption in the EDGE2D code.

As an option, Qqi wiii contain energy gain due to at0mk ionization (+fHEHSiz* where fH = n~ / (nH + nH2) is

the atomic fraction of the neutrai density), moiecular ionization (+fH2EmSitr with EH2 - 3 eV which is the

Frank-Condon dissociation energy'JMgs'). recombination (-&Sm. with Ei = 3T/2). and charge exchnnge

(-(E,-E~)S,-(~D)~U*S,. where S, = n e 6 - d . The effect of the various energy temis on the edge

plasma solution will be discussed in 54.2.2.

2.4 T o k p d edge plasma

In the pïevious sections we examined the general properties of plasmas. as well as their interaction

with a solid surface and seutral species. Kinetic and moment equations for ions and electrons were derived,

boundary conditions were selected, and dominant source terms were constructed from atomic-molecuJar

considerations. We are now ready to apply these eqmtions to the modeling of edge plasmas, particularly in

tokamak experiments. A mader u n f d i a r wich tokamak research may wish to consult gA.5 or [Gi181,

Wes971 for an overview of the tokamak core plasma before continuing further with the presmt section.

The distinction between cort and edge (boundary) plasmas is simple and yet profound: magnetic

flux surfaces in the core are topologically closed, while those in the edge are open, in the sense that t k y

penetrate a solid surface (V-B = O is satisfied by closure of field Iines within the solid). in other words, the

edge plasma is in direct contact with a solid surface and is subject to the behaviour discussed in 62.2. For

an overview of edge plasma phenomna in tokamaks the readcr is refend to a recent review articler-.

The present section draws a distinction between limiter and divertor plasmas, and proçeeds to discuss Lhe

various methods emplo ycd in theu mûde ling. Historicd y limiters prtceded divenors, consequentl y the y

are considered fïrst.

2.4.1 Limiter edge plasma

By the term limiter we mean a solid object protruding into the plasma and determining the l .

closedflux suvace (LCFS) at the point of furthest penetration. r = r- , Fig. 2.4.1. Particles crossing the

LCFS carry mass, momentum and energy into the edge plasma,

(va {m,mv . f mv2 }

a€ { i v e )

which in the absence of volumemc sinks are deposited on the limiter targets. Cross-field fluxes decay

radially away from the LCFS: for that reason the edge region adjacent to the LCFS is known as the scrape-

ofl-loyer (SOL), The radial density decay iength within the SOL may be çstimatcd by ncglccting dl

volumeuic sources within that region and equating the total cross-field particle flow across the separrttrix to

the total paraIlel particle flow reaching the targetdsugo',

where DL is a cross-field diffusion coefficient and La is a parailel connection length between the targets; in

deriving (2.4.2) we have assurned that c,, 4 and DL are independent of r. Similar expressions may be

obtained for Ti@), Te(r) and qdr). with decay len-. ksoL hem' . p. The practical hplications are

clear: the limiter guards the vessel wall h m contact with the hot core plasma 0, > IkeV), provided that

rms - rvd >> &'OL (this is especiaily imponant during high power transients such as disruptions or run-

away elecirons); it also localizes plasma-surface interactions to target plates, which are quickly stripped of

film and oxides. and appear as pure substrate to the impinging ionswL1. There are several disadvantages

of the limiter configuration, al1 linked to the intimate contact of the solid limiter with the core plasma:

a) high heat fiuxes deposited onto the targets near the LCFS lead to melting and sublimation; for this

reason, targets are typicaily made from heat load resistant materials. such as graphite or

r ungstenlGdl;

b) charge exchangc collisions between H neutrals and core plasma ions give nse to a flux of energetic H

neutrais onto the vessel wails, leading to atomk sputtering and influx of wall material to the

C) sputtering of limiter target atoms by plasma ions. with severai consequences: i) erosion of target

materiai, ii) impwity (mid-Z) atoms leaving the target as neutnls, penetrating directly into the core

plasma. and degrading tokamak performance by increasing &. thus diluting the ions (lower Pf,) and

increasing radiation from the core Pd , see (A- 1.6); iii) cooling of the outer core, which has been

linked to the disruptive instabi~it~[-'~. Target sputtering is the main reason why lirniters have been

replaced by divertors in most tokamaksmm1.

2.4.2 Divertor edge plasma

The divertor configuration, as the name indicates, diverts the flow in the edge plasma away from

the core region. This is accomplished by aitering the magnetic equilibrïum with external current carrying

coils (magnetic flux surfaces are modified by toroidal coil currents, in such a way as to create a nul1 in the

poioidal B-field at some location in the plasma. called the X-point; the separatrïx flux surface passes

through the X-point and intersects the targets which are some distance removed from the X-point. and

therefore not in direct contact with the core plasma, Fig. 2.4.2. By modulating the coil current, the

separatrix may be periodically swept across the target surface. producing a more uniform heat flux profile).

The divenot design emerged in direct response to the problcms assaiiated with the limiter configuration,

above ail impurity influx from the targets into the core It was hoped that by removing the

targets away from the core, the influx of impurities would be attenuated by ionization of impurity neuuals

in the interposing edge layer and by impurity-plasma friction sweeping the impurities back towards the

target (an expectation largely fiilfilleci), that magnetic sweeping and active cooling of target components

would allow higher heat fluxes, and that cryogenic pumping of neutrals in the divenor region would

facilitate impurity (as well as helium ash) removalm. The introduction of a divenor configuration into

the tokamak environment had several un forseen ~ o n s e ~ u e n c e s ~ ~ :

a) the appearance of an enhanced confinemenî regime (H-mode), due to a formation of a core transport

barrier near the LCFS; H-mode increased confinement by a factor of 2 to 4, leading to higher values of

core and T, aad facilitated the fmt ever rhievement of exapolated energy breakeven ai

b) the appearaace of edge localized modes (ELMs), which are transient bursts of particles and energy into

the edge layer[Adegl; two types of ELMs were discovered and linked to steep radial pressure gradients

formed by the H-mode transport barrier and to MHD magnetic perturbations; the nature of ELMs is

still unclear, but high mode numbers (m - 10) inferred and dp(r)/dr estimated suggest a ballooning

instability. Alchough, the transient nature of ELM activity increases the peak target heat flux to

potentially dangerous levels, ELMs are also viewed as 2 desirable stabilizing mechanism against

disruptions;

c) the appearance of a rudiative instubility (MARFE), driven by the invened temperature dependence of

line-radiative power for mid-Z elements (C, O, N, ...), Fig. 2.3.4, producing a region of cold. dense,

highly radiating plasma in the vicinity of the X-pointm1. The instability saturates for sufficiently low

Te, when parailel energy transport from the core into the M A R E region can compensate for radiative

losses; the resulting state is quasi-stable. Aside fiom degrading core plasma performance, it norrnalIy

l a d s to the disruptive instability and is regarded as an undesirable effect;

d) the appearance of gradud plasma detachnient from the target in which the plasma target temperature

Tc,-, and target plasma flow To progressively decrease as the upsmam density II,, - n m s increases'lh*"

(in contrast to the behaviour when the plasma is atrached to the target, in which both Tco and To

increase rapidly with nu, see the discussion of the two-point mode1 in the following section. 52.4.3).

Another diagnostic signature of detachment is the increase of hydrogenic line radiation linked to the

divertor neutral density n ~ . Geomeny of the divenor chamber exerts a strong influence on detachment,

which seerns to underline the importance of plasma-neutral interactions in detached The

onset of detachment occurs when TcVo drops to - 5 eV, this is usually accomplished by raising the

upstrearn density or lowering the core power. Comparison of target and upsueam conditions reveals a

significant drop of the total plasma pressure. p , , = p, + p, + pu' = (p,+p,)( l + ~ ' ) , indicating volumctric

paralIcI momentum losses; significant departurcs from total pressure uniforrnity d o n g a flux surface

are the preferred condition for defining detachment; when the pressure drop correction is included, the

observed decrease in Te.o and To may be explaincd even by a relatively simple two-point mode!. see

53.4.3. In the generally acceptcd explanation, charge-exchange collisions bctween ion and ncutral

atoms remove parailel momentum from the plasma flow, leading to the observed pressure drop, while

hydrogenic excitation/emission remove most of the Detachment is currently viewed. along

with the concept of an outer core radiative rnantle, as providing the most appealing solution to the

problem of reducing the prohibitively high target heat flux in the eventual tokamak fusion reactor. eg.

ITER. to values required by material ~ i m i t s ~ ~ . Frorn among the proposed designs. spanning a range

of geometries and pumping a s s e m b k , the gas target divertor, in which plasma flow is entirely

extinguished by the neutrai gas beforc reaching the target, rnay be viewed as an asymptote of the

divenor concept (the leakage of neutral gas atoms into the core plasma would clearly pose the main

constraint in this case). Some uncertainty still remains about the role of volumeaic recombination

(clearly observeci for TCo < 1 eV, but too weak to account for detachment for TCo - 5 eV), cross-field

transport (how are DI and affected by plasma-neutral coIlisions?), neutral-neutral and photon-

neutral interactions (short mean fke paths requin transport c a l c ~ l a t i o n s ) ~ ~ .

A recent review of divenor experimenu is recommended for huther detailsmmi.

2.4.3 Edge plasma modeling

In modeling the tokamak edge plasma we would like to simulate, as best as possible, al1 the

phenomena o b s e ~ c d in limiter and divertor experiments. This is a very ambitious task since much of the

underlying physics remahs unknown (cross-field plasma transport, low energy plasmdatomic/molecular

collisions, electronatic shuth effoas. chemicd spuncring and other plasma-surfixe interactions1sYg01).

Even the much simple; task of simulating quasi-stationary edge plasma behaviow is far from

t r i ~ i a l ~ ~ ~ , . In this section, we focus on the quasi-stationary phenornenon of divenor plasma

detachment, and introduce the improved onion-skin mcthad wbich lies at the very hem of this thesis.

In practice, modeling has an added aim of either prediction or interpretation. The difference is one

of approach: predictive modeling relies on establishcd theory to calculate the behaviour of the systern

(often extrapolating beyond experimentally studied regimes); interpretive modeling places the emphasis on

experimental data, which is employed in the modeling framework to supplement available physics and

explore the 'gaps* in the theory. Clearly, a theory must be sufficiently incomplete for the latter approach to

warrant consideration; in edge plasma physics, the most visible 'gap' is offered by a n o d o u s (turbulent)

cross-field transport. Based on these considerations, the onion-skin method of solving the plasma transport

equations may be considered as an approach at interpretive modeling of the edge plasma.

The simplest description of the edge plasma. the so called two point model, relates upsueam and

t a r p quantities. denoted by the subscripts 't* and 'u', dong a single flux tube. It rnay be obtained from the

paraIIel transpon equations (2.1 -22) by retaining only the dominant transport processes: rnass and

momentum transport by convection (no parallel viscosity), and energy transport by electron thermal

conduction. This amounts to neglecting Q,, G / ~ ' . F:". F,;~' in (2.1.22). d d i n g the two energy equations.

adapting a classical estimate for && = = & T ~ * ~ , where & - 2000 W/(eVm) and using y = y, + .I, as target boundary conditions. In addition the two point model neglects volumeuic momentum losses Q ~ ' "

and flux expansion terms QB. The simplified set of transport equations may be written adsugo1

:. nu = nu, + I ~ " d i (

where Qn is the particle source, @ is the sum of ion and electron energy sources. and the bottorn

expressions are a quadrature form of the simplificd transport quations. Evaluating the latter at the target

and rnid-point locations, ski = O and %u = b / 2, we obtain the above mentioned two-point model,

7 9ilsll.u ( z + M , ~ ) ~ , T , = ( I + M ~ )n,,T, ru7/ -rt7l2 = --

4 Kea

which is supplemented by the Bohm criterion, M, 2 1. AU the energy was assumed to enter the flux-tube

uniformly (if al1 the power entercd at the upstrcam point, the factor qfi, above is doubled); due to the Q a

~2~ scaiing, upstream quantities are oniy weakly sensitive to îhe energy source distribution Q~(s& and due

to the absence of convective power, they are independent of the particle source distribution Qn(sp). It is

instructive to trcat (ql, nJ as independent variables (these quantities are usually controlled experirnentally,

although qu rnust still be related to the total power deposited into the SOL, PsoÙ and (n,, T,, Tu) as

dependent variables. Reananging the above equuions under the large VIT assumption T : ~ « T:~, we

Target quantities appear to be very sensitive to upstream quantities. especially to the upstream density (this

sensitivity arises due to Coulomb scattering (A.2.37) which leads to n,.~, = T:" ); in contrasi. the upsuearn

temperaturc Tu dcpends only weakly on qll and is highly insensitive to the flux-surface distribution of input

power ~ ~ ( s , , ) . We can estirnate the effect of volurneuic power losses, mornentum tosses and convective

power flow by inuoducing three new factors 6 , ~ [O, 11, QE [O, 11, and Sq E [O, I ] in10 (2.4.4).

These factors propagate accordingly into (2.4.5),

Target quantities are highly sensitive to volumemc momenturn losses & and volumetric power losses 69, and somewhat sensitive to the convective conmbution 4. The upstream temperatme Tu is independent of

&,, or 6q , and is oniy weakly sensitive to %. The modified two-point model (2.4.6) is sufficient to explain the basic trends of edge plasma

behaviour, typically classified in terms of four experimental regimes: sheath limited, conducticm Iimited.

high recycling, and detached-. These regimes correspond. in the above order. to an ascending degree of

plasma collisionality (11% a flCU2). d s o represented by the nonnalized collisional mean-free-plib &,& =

T:/&. Since collisiondity depends on both density and temperature, the above regimes can be traversed by

varying either of these quantitics. In keeping with the two-point model, we will consider tbt gradua1

increase of upstream density n, for constant input power (the comrnon experimentaf situation). which by

(2.45) decreases T, and increases q and Tt (note that = T:/% = 1111.' at the target).

In the sheath limited ftgime, which occurs at the highest target temperahues. the collisioeal rnean-

free-path is of the order of the flux tube length (a - l), which leads to negligible dong-B gradients. In

tenns of the two-point m d e l , the electron conductivity K, is so high chat small Te gradients are sufficient to

carry d l the power along the flux tube, leading to flat Te profiles; in addition, atomic i n t e d o n s are

ineffective at such high Te in removing either momentum or energy from the plasma (4- 1, %- 1, $- 1). As

a resul t, upstream conditions am strongly linked to target (or sheath) conditions.

As temperature and the collisional mean-free-path decrease (A& cc 1) the plasma eaâers the

conductively limited regime, chanicterized by larger dong-B gradients; with efectron conduction bacoming

progressively less effective, VIT, increases in order to carry the same power (most pronounced in the near

target, colder regions) leading to steeper Te profiles. Volurnetric energy losses begin to appear as the first

corrections to (2.4.4) while tord pressure is conserved (6,- 1.4- 1.8, < 1).

At su11 lower Te (or higher nu) the particle flux to the target, also called the recyciïng flux,

becornes Iarge enough to dorninaie the power balance, ie. in the vicinity of the target most of the power is

carried by convection, which consequently reduces VilTe over this region. Upstream conditions are thus

separated into a convectively Iimited (transonic, near target) region. and a conductively limited (stagnant.

upstream) region. This behaviour is typically referred to as the high recycling regime, emphasizing the role

of convection (% < 1) although the dividing iine between the regimes is somewhat arbitrary.

Finally, for Te < 5 eV another physical mechanism bccornes active, namely ion-neutral charge

exchange. This mass conserving interaction. removes both cncrgy and, morc importantly, momentum from

the plasma flow, crcating a net pressure drop along the flux tube (6, < 1). The effect on target conditions is

visible frorn (2.4.6): the recycling flux and target density are reduced with increasing nu as long as tk ratio

6,n, decreases. This phenomenon is traditionally referred to as plasma 'detachment', for obvious B o n s .

Two different approaches are uscd to quanti& the degree of plasma detachment: a ) the ratio of the rctual T,

to that predicted by the two point mode1 (Tt = n.') bawd on some reference T, and nu in the attachai (high-

recycling) regime is used when only target data are availabk, and rcsts on the tenuous assumption, recently

challenged by experimental rcsults, that n, scales linearly with the average core n, b) total pressure ratio

p,u/p,' measures dong-B momentum losses, which are the physicd mechanism at the heart of

detachment; this is the preferred quantity when reliable upsîream measurements are available, and it will be

employed in our study (somewhat arbitrarily we introduce the criterion for detachment as pWu/p,' > 2).

Since in the coarse of detachment, the contact between the plasma and the target becomes progressively

weaker. another useful quantity is the position of the plasma density pcak and the ratio of peak to

target densities (nwlnJ; in the limit of complete detachment (or gas target formation). n, and T, are

reduced to zero as the ions and electron recombine volumetrically before reaching the target (much as a

candle fiame is extinguished by the cold gas into which it flows).

It is worth emphasizing the role of neutral partick penetration in the above transition. As the near

target plasma cools, the meutrai mean-free-path with rcspect to ionization inaeases; this pushes the

ionization front fùrthcr away n o m the targtt, thcreby extending the region of strong plasma fiow, which in

turn increases convection and flattens the temperature proale. By ihe same argument, larger target density

CT,,t held constant) would d u c e the neutral peaetration kagth, causing steeper near target profiles.

The simptified transport equations (2.4.3) may be inkgratcd ftom the target to an arbitrary s,

location, provided target values {Tc& ro] and the spatial source distribution (Qn(+), Q~(S,)] are specified;

this gives the simplest 1 -D edge plasma description, U = U(s). The original onion-skin m e t M (OSM) of

estimating 2-D edge plasma profiles. U = U(r,su), consisted of a collection of such 1-D solutions based on

radiai target information {Teo(r), T&)) across ttie SOL. Tbe quadrature procedure of (2.4.3) remains valid

even if simplifications regarding convection, single temperature and momentum losses are r e m o ~ e d [ ~ ~ ~ ~ ~ ;

numerical quadrature may be performed using predictorestimator methods, such as the fourth order

Runge-Kutta technique[Hin4J. In this extended form, OSM was previously combined with neutral transport

codes, which r e m Q3[U(r,q,)]. and iterated until convergence. Attached plasma conditions were modeled

successfully, producine good agreement with experimental data and E D G E 2 D m B U S code r e s u l t ~ ~ ~ ~ ~ ' ~ .

Unfortunately, this earlier version of OSM suffers from serious limitations: a) atthough it has been used

successfully to mode1 high temperature plasmas. Te > IO eV (conductively dominated and attached). it

becomes unstable for colder targct conditions (convectively dorninated and detached). b) the quadrature

approach rcquires that the steady-state fluid equations appear in parabolic form, which rneans thrtt ion

viscosity can not be i n c l ~ d e d ~ ~ ! c) ion-electron enugy exchange terms are destabilizinp to the

quadrature method and must be neglected. d) sonic transition appears as a singularity in the steady-statc

parabolic form causing a discontinuity in a quadrature soIurion, e) Roating boundary conditions. such as the

super-sonic target velocity M, > 1, require laborious iterations. For these reasons, the steady-state parabolic

form was abandoned, in this thesis, in favour of the steady-sraic hyperbolic-elliptic f ~ r r n [ ~ ' ' ~ l (or

equivalently the standard time-dependent parabolic form of the Navier-Stokes and the

quadrature method in favour of implicit relaxation grid-based methods; this basic change in the numerical

approach was undertaken only after sufficicnt familiarity with CFD reveakd the reasons for the limitations -1 (b,c,d) of the quadrature method .

The nature of the onion-skin method is made clearest by a cornparison with the standard 2-D

modeling approach. Plasma transport equations (A.3.1) may be written in magnetic co-ordinates as

where U represents the plasma fluid variables, F the convective fluxes, G the diffusive fluxes expressed in

terms of diffusive transport coefficients of mas, momentum and energy in the parallel. radial and

diamagnetic directions, (41. DL, DA, q ~ , q~ ,q,. X I , X I , &) and Q the interaction source terms. Asswning

toroidal syrnrnetry, a/% = O, they may be recast into poloidal CO-ordinates (r,s), introduced in $AS. 1,

where F,, Fr, G,, G, are the cornponents of F an C in the poloidal and radiai directions; poloidal fluxes

contain the parailel and diamagnetic fluxes with appropriate projections, see (A.3.1). These are the

equations, usually written in primitive variables (n, II, p,. pi), which are most ofien ernployed in edge

phsrna modelling. They are typicdly solved (using CFD-type numerical techniques) on a poloidal dornain

bounded by the LCFS, the target and the wall, with boundary conditions of the type discussed in 92.2 at thc

cargets, and the following inputcRri921: a) cross-field flow of power across the separamx, b) density on the

separatrix or particle flow across the separate, c) anomalous cross-field transport coefficients for p h c l e s

and energy (DL, Xi11 xCI). The interaction sources Q[Ul are computed in tandem with the plasma variables

U[Q] using a hydrogenic neutral transport code, usually (but not aiways) based on Monte-Carlo techniques.

It is customary to refer to such computational models by the names of the two codes which make up the

iterative ~OOP, eg. B ~ / F I R E + J E [ ~ ~ ~ ~ M ] , EDGEZD/NIMB U S [ ~ ~ ~ ~ L ~ U Q ~ ( ~ , UEDGWDEGAS[~"~". A

discussion of the physics and nurnerics employed in these codes. and the status of code relaied research.

may be found in [ReigSJ.

Reccntly an assessrnent of divertor edge plasma modelling (using 2D codes dcscribed above) has

been carried out based on both code-code and code-experiment comparisons~a971, finding a large rneasure

of success in modelling quasi-stationary phenornena (divertor plasma detachment. geometry effects.

M A R E S ) and some success with the-dependent phenomcna. notabiy ELMs. It is wonh strcssing thrit the

diagnostic signatures of plasma detachment, $2.4.2, have been replicated numerically; the dominant

processes have k e n identified as hydrogenic and impurity radiation, charge-exchange and. indirectly,

recombination (it is anticipated that impurity radiation would be necessary as an edge plasma coofing

mechanism in high-power tokamaks, such as ITER). Some discrepancy remains in the absolute levels of

plasma quantities, especially Te which is calculated at - 1-4 eV vs. Langmuir probe data of 3-6 eV (this

difference has a pronounced eHect on the reombination raterJ-). Several assumptions of the 2D modeb

are supported by comparison with experiments:

parallel electron energy transport with classical heat-diffusivity f i , cf. (A.3.2).

diffusive noss-field transport with constant (DL, xiL. xeL) in low recycling regimes (Bohm-like DL in

the private flux region, and inwafd pinch velocity vl in the high recycling regime; the double peaked

structure of To in this regime is not reproduced and is thought to be linked to classical diamagnetic

drifts); typical values of (DL, iWL) range over (0.14.3.0.5-2.0) m2/s for ohmic discharges, (0.1-05,

1 .O-5.0) m2/s for L-mode dirharges, and (0.05-0.2. O. 1-03 ) m21s for H-mode discharges), xe (A5.3.

The transport of impurities using mufti-fluid approaches has reinforced the impurity force models, the

need to includt chernical spunering of carbon at low target ternperatures, and the observed

phenomenon of impurity compression in the divertor (the absolute impurity levels are typically

calibrated using linc radiative intcnsities fiom the divertor region)-.

Significant differences between the various codes have been observed. and linked to the unccrtainties in:

a) determining the position of the separaaix (different methods used in various codes),

b) measuring upstrcam Ti (therefore the ion encrgy flow across the separatrix),

C) determining the recycling coefficient at the targets.

Several areas requiring improvemenis have been identi fiedbm: edge plasma measurements (especially Ti

and impurity/hydrogen radiative losses); more accurate physical models of recombination, charge-

exchange, cross-field transport and classical drifts; neutrai-neutral (gas dynamic) effects and coupling core

transport and edge transport codes.

The 2D code approach, despite showing much success in modelling expenmental results, suffers

from pracrical disadvancages for interpreting edge dam F i t of alI, it relies on the knowledge of cross-field

coefficients (Di, qL, which are aaomalous (probably governed by turbulence, see gA.5.3) and therefore

remain Iargely unlcno~d-~; they are treated as parameters and are adjusted to match experimental

r e s u l t ~ ~ ~ ' ~ ~ ~ . Secondly, diagnostic measurements in the edge ( n o d l y near the target) are difficult to

impose as boundary conditions, at the risk of ill-posedness (over-constraining the system of

equations)rRC'g2'. Instead, upstream boundary conditions (normally the density n m s and cross-field power

qLLCFS) are imposed; finding the closest match to diagnostic data requires lcngthy paramctcr scarchcs in [Loa971 which both (n. qdmS and (DL, q,, are adjusted . Cross-field transport coefficients are probably

the least known quantities in a tokamak plasma, and the investigation of mechanisms involved in cross-

field uansport remains one of the most challenging areas in current fusion effort. Considering the richness

of the underlying physics and the cornplexity of modeIIing turbulent processes of any kind (cf.

Aerodynamics), the situation - namely the lack of sound theoretical b a i s for cross-field transport - is Iikely to remain unchanged into the near, if not the distant, future. Fortunately, exhaustive knowledge of the

underlying physics is not a necessary condition for the development of a technology, as the aerospace

industry will sureiy attest; rather it motivates empirically based (interpretive) models. Finally. target

quantities are very sensitive to upstream quantities (cf. the two-point model), whicb introduces numerical

stiffness into the paramctric search-.

The above considerations were the primary incentives for developing m alternative solution

method for the edgc plasma transport equations, broadly referred to as the Onion Skin Method (OSM) on

account of the flux surface nature of the solution (plasma transport equations arc solved on each flux

surface in tuni). The method consisu of replacing the cross field terms in (2.4.8) by source terms QL whose

func tional dependencc on U is specifred (for now WC assume that QL depend on U but not on VIU),

Ideally, plasma transport dong the flux surface as well as the volumemc sources in OSM and 2D codes

would contain the samc physics, the only difference being the treaunent of cross-&id plasma transport

(diamagnetic drifts are neglected in kceping with 92.1, but could be incorporated witbout much difficulty in

the future; since diamagnetic drifts lead to transport dong a flux-surface, they do not affect conciusions

regarding the validity of the onion skin approach). OSM, as defined above, rests on the pivotal assumption,

referred to from now on as the QACI] ansart, that the functional dependence of cross-field sources. &ml,

has only a small effect on the plasma solution U(a, given the flux surface average value of <a>. It is

somewhat surprising that with so much redundancy in 2D code development (EDGE2D. B2 and UEDGE,

despite cenain differtnces, are solving the same set of equations, (2.4.8)) this cruciaiiy important postdate

has never been pruperly examineci. We may rewrite the QJU] amatz as a series of variational

conditionswu741,

to be satisfied for each component of U and QL. At this point, the full weight of functional analysis~Sh'7J1

could be brought to bear on (2.4.11); it is. however, more instniciive, and physically revealing. to adapt a

simpler approach by exptoiting the hyperbolic/elliptic (convective/diffusive) smture of (2.4.10) and

including from the outset the cross-field sources appropriate for tokamak edge plasmas. We begin by

noting that if al1 elements of U are non-zero than U may be reconsuucted given FII(U). (2.1.22); this

translates into a series of variational conditions on 6Fll / 6QL similar to (2.4.1 1 ). Rewriting (2.4. IO- 1 1 ) in

steady-state form,

we ree thu F*' depends explicilly on QO' and ati' and irnplicitiy (via U) on Qk) and Q'" . k t j. We now

investigate (2.4.1 1) for the mass, momentum and energy equations (2.4.10) in the limits of svong and weak

cross-field sources QI@ as compared with îhe neutrai sources Q ~ ' . Intioducing the relative cross-field

saen@ fL(i)= <QL%/<Q% and the source-mget ratio. P) = C Q ~ ~ W ( F V ) + G % ~ . We note that the Q A U ~

onsarz (2.4.1 1) is satisfied when fi(j' << 1, for d l j. provided that parallel fluxcs dominate El >> F,, G, >>

GL; in other words, the validity of the OSM approach should improve with stronger neutral sources. Let us

consider the cross-field sources in limiter and divertor cdge pfasmas:

a(') for limiters arise due to the intirnate contact of the limiter with the core plasma, such that most of

the particies ionize in the core and diffuse lrross the LCFS into the SOL, (fin > 1 ); although we expect

the distribution of partides dong the LCFS to bt fairly uniform. due to the closed topology of core

flux-surfaces, the cross-field fluxes may peak in regions of strong radial density gradients. For

divertors, Qc") may be similar to the limiter scenario described above (low-recycling) or localized in

the near target region (high-recycling). We will consider the high-recycling case, as representative of

divertor configurations in general (fLn < 1).

&(" arise due to core plasma rotation or cross-field viscosity; since plasma momentum flow to the

targets is determined by the Bohm critenon, these effects are smdl in cornparison to charge exchange

losses (typically the cross-field flow of momentum across the LCFS is asswned to be zero in 2D codes

for both the limiter and divenor configurations) :. f,'*' 1. The Qfl] amtz for momentum

transport is strengthened by the fact that cross-field momentum fluxes transport convective momentum

alone; it is easily shown that the maximum amount of momentum which could be transported in this

way is smaller than the parallel momentum flux in the plasma

and w4' arise due to power generation in the core plasma which then diffuses conductively

across the LCFS entering the SOL (the powers dcposited into electron and ion channels appear to be

roughly equalm); since the power rernoved by plasma-neutral interactions is limited to the input

power. then > 1, fi('' > 1 (this situation applies for both limiter and divertor plasmas); as with

particlc flux. we expect the distribution of radial powcr dong the LCFS to be fairly uniform. increasing

with radial temperature gradients; for divenor SOL. there is no power inflow downstream of the X-

point.

The above considerations reduce the problem to examining the mass and energy transport equations in two

lirniting scenarios of core ionization (fln > 1. fiE > 1) and SOL ionization (fp c 1. fLE > 1). The QJUI

ansarz (2.4.1 I ) is satisfied for the momentum uansport equation. j = 2. in both of these cases.

Core ionization: the QJU] ansarz may be violatcd for particle transport Cj = 1, Flnu = an) so that

the functional dcpendence of QLn[U(sfi)l would effect the parallel plasma flow pattern, nu(sI); fortunately,

in case of dominant core ionization we expect &"(a to Vary only gradually with sa due to the closed

topology of the core flux surfaces, see (a) above. The QJU] ansarz is satisfied for the electron energy

transport due to the IÇ = T : ~ scaling. as demonsinted by (2.4.3-5). so that the flux-surface profile of

a E ( s 3 has only a small effcct on T,(Q. The ion enefgy transport contains both convective and conductive

parallel fluxes; the former depend on an(qJ since F:') = (5/2)T,nu. whüe the latter are inwnsitive to

e ( s , ) due to the 9 = ~i~ scaling. We expeçt the energy transport by ion convection I?i3) to dominate

over electron conduction G"' oniy in regions of super-sonic flow, which (by the Bohm criterion) we

anticipate only in the vicinity of the targew (ie. downsncam of the X-point). Elsewhere in the SOL, electron

conduction dominates and the QAU] ansarz is satisfied. We may conclude that with core ionization

dominant, parallel plasma flow and ion convection are indeed sensitive to radial particle diffusion, but due

to the location and distribution of the particle and cnergy sources, the sensitivity to the energy source is

weak. The QJU] ansarz is only violated for the mass transfer equation, affecting nu(sa).

SOL ionization: the QJU] unsaîz depends in this case on the effect of strong neutral particle

sources in the SOL (S=, S,) on the flux tube mass balance. If target panicle fiuxes are specified as

boundary conditions (as is traditionally the case in OSM), then che flux tube integral of Si, - S, determines

t!e amount of over/under-ionization with respect to the spccified target flux; the cross-field sources are

adjusted such that the flux tube inte@ of QLa offsets this particle irnbalance. The QJU] a w t z is satisfied

for the mass uanspon equation as long as the ratio J Q L ' ~ ~ 1 - 61% remains small (since ion

convection is only weakly dependent on Q;(+) while the eaagy source aE(sa) is largest u p s m of the

X-point where convection is small, it would be satisfied for energy transport as well). If however <fi"'>

approaches unity. the QJUJ ansarz may be vioIated and the solution may become sensitive to the specified

functional dependence of WnM. The conditions for the validity of the OSM approach are sumsnarized below:

a) suong magnetic field (mt »1) implying that Fr >> FL C;I >> GL,

b) strong temperature dependence of parallel conduc tive energy transport, a a T'", C) if dominant core ionization, then nearly unifonn influx of particles and power across the LCFS

d) if dominant SOL ionization. then only a moderate degree of over-under ionization, 0.5 < f " < 2.

e) availability of radial information in the divertor (near target) region, ud"" udiag(r).

Condition (a) is necessary in order to inuoduce the dominant transport directions and impose the flux-

surface georneuy. Condition (b) satisfies the QJU] anxar: when near target energy flux is available, cf. the

two-point model. which shows that Tu is weakly dependent on the sa profile of cross-fieid energy sources.

~ ~ ~ ( s , ) . Condition (e ) assures chat sufficient information is available ro solve the transport equations on

each flux surface. It is worth notiog chat since this radial information cornes from diagnostic measurements.

it contains the effccts of cross-field fluxes between the diffcrent flux surfaces within the SOL.

OSM was initidly developed in quadrature f o m in which the cross-field sources were

renormalized based on the imposed target values[su971. One of che contributions contained in this thesis is

the formulation of a more flexible framework for OSM based on variational techniques (applicable for

problems in which the integral of a function is known but not its functional dependence; the latter is then

specified and the solution satisfying the desired integral value is obtained). In the OSM context. the

variational technique developed will be referred to as Diagnostic Variance Minirnization (DVM). It

consists of adjusting the cross-field sources in such a way as to minimise the statistical variance I I 3

between calculated quantities U(r,s) and selected diagnosticly obtained quantities e ( r . s ) . where Z = (U-u*)-, defincd tenn by tem, is a normalized error between U and e. The variational form of

OSM is more general than is needed for the present analysis, which is based only on target Langmuir probe

measurements of r and Te. for which e ( r . s ) = e ( r ) = {& ïL, TcO . T~}*- This form is introduced

in order to provide a fiamework for future extension of OSM to other types and larger sets of diagnostic

data. At this point, let us digress a little and briefly review the field of edge plasma diagnostics1sm- -. AUc891. In g e n e d . the measurement of edge plasma conditions requires better radial localization in

cornparison to core measurements; aside fiom this fact, the same diagnostic methods can be employed.

Etectron density and temperature are routinely measured by a variety of techniques:

a) microwave interferometry [&)]

b) microwave reflectometry [ ~ ( r ) ]

c) Thomson scattering (of laser photons by plasma electrons) [ ~ ( r ) , Te(r)J

d) LIDAR (iight detection and ranging: timc of flight technique) [n&), T,(r)]

e) ECE (electron cyclotron emission) [Te(r)]

f) Hydrogen emission spectroscopy &, H, lines) [n&), T,(r)]

g) Impurïty emission spectroscopy (eg. He, Li, C lines) [n&, Te@)]

h) Langmuir probes (target or reciprocating; single or mple) [n&). T,(r)]

When multiple lines-of-sight are available. tomographie reconstruction is possible, eg. Thomson scattering

array on the D-IFID tokamak [n&,s), T,(r,s) in the divenor]; this is potentially a very powerful diagnostic

tooI providing invaluable information about divertor edge plasma phenornena, such as detachment, and

offering an ideal opportunity for code-expriment comparisons. The ion temperature, aithough more

diffrcult to measure, can be obtained via chargeexchange neutral spectroscopy or retarding field ion energy

analysis. Plasma flow velocity. or the local Mach nurnber, can be estirnated via directional Langmuir

probes. Doppler shift spectroscopy, or impurity puff measurements, eg. the carbon plume experiments on

Alcator C-mod; unfonunately, due to interpretation errors and scarcity of diagnostic data, the flow velocity

is generally unknown. In addition to plasma variables [&, Te, Ti. MJ. a wide range of spectroscopie data.

ranging from visible light to soft X-rays, is usuaily available in the cdge region, providing information on

neutral hydrogen distribution and various plasma-neutral (atomic/molecular) interactions, eg. two-body

recombination rate can be inferred from the associated photon spectrurn. Al1 of thesc diagnostic

measuremcnts could potentially be used in conjunction with the OSM-DVM approach; the cross-tield

sources could be adjusted based on a diagnostic vector UdY<(r.s) which would include several diagnostic

data sets with specified weight factors. Therein lies part of the appeai of the OSM-DVM approach, which

could combine great quantities of disparate diagnostic data into a unified, consistent picture of edge plasma

behaviour. Since the OSM-DVM analysis for larger diagnostic sets would follow the same route s for the

simpler target Langmuir probe case (provided radial information was available) the present analysis may be

viewed as a preliminary sketch towards a much broader range of OSM-DVM applicability. For the time

king, the only claim king made is that the cross-field sources are adjusted according to some diagnostic

error E, ie. a = El.

The replacement of the radial derivative by cross-field sources allows the transport equations

(2.1.22) with an added source term QLCU(r.s), S(r,s)] to be solved in 1-D sweeps dong each flux surface,

from which a 2-D plasma solution U(r,s) rnay be constnicted (neuual sources @(r,s) are obtained from

neutrai-transport codes. in the prcsent work fkom N I M B U S ' ~ ~ ~ ) . It is imporîant to note thai although each

flux surface is solved independently of the others at each time stage. neunal sources Q(r,s) depend on the

transport of neutral particles and hence on the entire 2-D plasma profile. = Q@(r,s)] and Qm =

Q[U(r,s)]. When U[Q1 and Qm are iterated in this way, the 2-D nature of the solution becornes apparent.

The sources on the right hand side of (2.4.10) consist of three parts:

a) Plasma sources, Q,[U(r,s)] given in (2.1.22). contain the interaction between ions and electrons. With

the assurnption of equal ion and elecaon parallel flow velocities, the only contribution comes from ion-

electron energy exchange via Coulomb collisions and the interaction with the plasma electric field;

b) Neutrat sources, Q[U(r,s)]. include plasma-neuaal (atomic/molecular) interactions discussed in 92.3,

and have to be caiculated based on neutrai nanspon equations. In the work presented here, the

volumeu-ic sources of ionization SU, recombination S,, ion-neuual charge exchange Scx, electron

energy loss due to ionization, dissociation and excitation Q,, and ion energy loss due to ionization,

recombination and charge exchange Qd are cdculated using a Monte-Carlo neutral transport code,

M M B U S ' ~ " ~ ~ . The effect of impurity cooling is estimateci by Q1.. f=Q, and Q . fzQqi- In the

default model, the neutral birth velocity dong B is set to zero, v~ = O, which conesponds to an upper

limit on the amount of momentum loss due to CX collisions. The effect of ion cooling/heating due to

neutral interactions is neglected in the default model in keeping with the EDGE2D code, Qqi = O, as is

the effect of impurities on the ion energy balance, fii = O; an impurity enhancement factor of unity is

assumed for the electrons. fi, = 1. The default form of Q3 is thus (2.3.5) with Qqi = O. fii = 0. and fi, =

1 ; the sensitivity of the solutions on Sm-, v ~ . Qqi . and fi, will be explored in 54.2.2.

C) Flux expansion sources, QB[U(r,s)], include the effects of magnetic geometry. ïhey are calculatcd

based on the equilibrium B field found by solving the Grad-Shafranov equation (A.5.7).

Cross-field sources Q [ U , E] simulate the effect of cross-field plasma transport. Their functional

dependence on U must be specified, and may be considered as the key modelling assumption. As in al1

variational approaches. the choice of the test function, here QJU, Ej should reflect a compromise between

content and sirnplicity, that is betwcen known features of the desired function (in our case, the diffusivc

nature of cross-field fluxes) and srnallest number of free-parameters (a minimum of four per ring are

required to impose the four external constraints per ring, two at each target, provided by Langmuir probe

data, {To . r ~ , TeVo . T ~ J J " ~ ). Based on these criteria the default form of QJU. E] was constnicted as

follows: the particle source QI"' was raken proportional to density p = U<') (equivalent io Vl(DLV,n) with

the sarne radial length at ail Q locations), the momentum source a"' to plasma fiow pu = u"'

(cquivalent to VL(qLVLnu) wiîh the same d i a 1 length A,,'OLat ail locations). while the power sources of

ions QI'3) and electrons QL(1) to double-step functions fiom inner to outer X-point locations with their

value in the inner and outer regions adjusted independently (an approximation of uniform power influx

across the LCFS with no influx downstream of the X-points; the division into inner and outer halves is the

lowest order refinement necessary to match possible power asymrnetries between the two targets); power

deposited in«> the iow and elecuow was assumed to k equal (as suggested by expenment)-. In the

future, a more complicated, difisive f o m of QI involving radial gradients could be in t rodud: a'" a DLVLn. a(*' = qLVLnui c) a n&VLTi . a nbVITe; in whkh case adjusting Q, would be

equivalent to varying the cross-field coefficients. The default cross-field sources can be swnmarised in

vector form:

where sux O and sllx. are the s, locations of the X-point and c(r) are parameters adjusted in order to

minimise the siatistical variance between U(r.s) and ~ ~ ( r ) . Considering only target Langmuir data (ro . r ~ , Tco , T * ) ~ ~ , we can constnrt two cornparison vectors çap(r) and ~ ( r ) ,

for which the statistical variance IEII may be defined as some vector n o m I1.H of the relative error Z(r), eg.

as the sum over al1 computationa! rings of the square of the elements of E(r).

ç(r ) -çd;u* ( r ) .Z(r)= IIzI=CIE(~~ '

ç W ( r ) r

Finally, we can state the DVM condition for the minimisation of II31 with respect to &).

NumMcal strategies for solving this multivariable optimisation problem are discussed in $3.5. It is

worth noting that the DVM technique is not particular to the onion-skin mehod, and could be included

within existing 2D fiuid codes to increase theu diagnostic content and interpretive ability (in that case the

specified cross-field transpon coefficients would be adjusted in order to m i n i e the diagnostic variance).

The case of strong volurnetric recombination poses a difficulty for the DVM approach based

entirely on target measurements (naditional OSM); the target is no longer at the heart of the active region

(of particIe, momentum and energy sources) which determines the upstream conditions. In the limit, the

recombination fiont becomes a mobile 'gas targct* in which nearly d l plasma particles are neuualizedml.

Let us consider this 'gas target' scenario, in which the plasma is entirely extinguished before striking the

soiid target, sucb that no = To = O. The standard OSM approach would clearly fail in this case, because no

plasma information is available at the target. For finite, but weak, plasma-target contact we expect a wide

range of upsaam conditions to correspond to a small range of target conditions, an unfavourable situation

frorn the point of view of accuracy. It then becomes usefd to seek additional upstream information in order

to further consirain the solution- In the prescnt study the upstrcam (mid-point) separatrix density nuq will

serve, when ntcessary, as this additional constraint; its numerical implementation will be discussed in 93.5.

Possible target boundary conditions have been presented in 92.2. In the default model. they consist

of: a) the velocity being limited by the target sound speed according to the Bohm critenon, MO = ~g / C, 2 1,

and b) the poum fluxes k ing related to the target temperature by the sheath heat transmission coefficients

for ions and elecuons. qi = %Tbo. se = yeTJO1 ye = 5.0, = 2 5 + &2(~LO+~L~)/(2TiD)- The Mach

dependent term of was added in order to insure that the ion conductive heat flux vanish at the target. G~(~)

= O (we can see this by wri ting F&)' + GP' = xTiOro1 insening ib& = uo / ( ( T ~ + T ~ ~ ) / ~ ) ' ~ and isolating for

~ 2 ~ ' ) . The effect of boundary conditions on the final soIution is examined in 64.23.

Taken together equations (2.1.22). (2.3.5). (2.4.10-16) constitute a self-consistent interpretive

model of the edge plasma, which will be referred to from now on as the OSM-DVM default rnodel. The

following sections will discuss the numerical implementation of the OSM-DVM model (the 0SM2 code)

and the extensive application of the OSM2NMBUS code package, including a study of the sensitivity of

the default mode1 solutions io changes in modelling assumptions.

3. Numerid Solution of Modd Equatkas: the OSM2 code

The OSM-DVM edge plasma mode1 equations (plasma transport (2-1.22). neutral transport (2.1.1).

field equilibrium (A.5.6), cross-field sources (2.4.10- 15)) form a self-consistent . mathematicai system; the

present chapter will be devoted to numencal sîrategies employed in the solution of this s y s t e a

The Grad-Shafianov quation (AS.6) must f m t be solved in order to obtain the equilibrium

magnetic field B(R,Z) and constnict a concsponding numerical grid X. Thc nmaining equations are solved

on this grid, that is in magnetic coordinates (r,s), using a four step iterative cycle:

a) plasma transport equations are solved using the methods of computational fluid dynamics (CFD)

for given neutral and cross-field sources: U [ Q , QI]

b) neutral transport cquations arc solved using Monte-Carlo (MC) techniques: Q-

C) the new MC sources are relaxed with prcvious sources Q O ~ ~ to assure numerical stability of the

MC-CR) itenave cycle: Q3 IU] = ( 1 - % w l d (U1+ %QL- [ZJ], q « 1

d) the cross-field sources are caiculatcd according to the DVM technique: Q, FU.

The above iterative cycle is continued until steady state is reached, that is until U no longer changes

significantly with further iterations; It may be represented schematically as,

CFD relaurtion with Q,&[u]

3.1 MHD Equations: Magnetic Gnd

As mentioned in gA.5.1. magnetic equilibrium is calculated by numerically solving the elliptic Grad-

Shafranov cquation in the poloidal plane of the tokamak (a@ = O). using boundary conditions which

include mcasurements of the edge magnetic field; iwo typical solven used for this purpose are EFIT(~'"'~'

and SONNET[^'^'. The calculated magnetic field B(R,Z) is used to generate a poloidal grid X which

conforms to the magneiic geomeuy (lateral faces of grid cells form magnetic flux surfaces). Fig.3.1.1. At

cach ceIl center both poloidal and toroidal components of the magnetic field are storcd. from which the

field line pitch angle and the connection length between targets La may be calculated,

The magnetic grid X is assumed to contain the following information, Fig. 3.1.2.

I Si. j 5.j 1

l sIiSj rli., ~ 2 ; ~ ~ r2i. t i~ (1, .... Nr } x = ~ 3 ; ~ ~ djj ~ 4 ; ~ ~ '4, j E {O ..... N, + I ]

R;., 2,. j Ai. j

where sü and rd are the poloidai and radial cmrdinates of the ce11 center with indices i and j, sli j to Sqy

and rl, to r4, are similar co-ordinates of the four al1 corners, R,., and &., are cartesian toroidal co-

ordinates of the ce11 center, A, is the poloidal ceIl ami (ce11 volume - 2xR,.,Au) and and Ba3 are the

toroidal and poloidal components of the magnetic field at the ce11 center (Br, n O, by the de finition of the

magnetic surface). WC will denote a single grid cc11 by X,; ail cells with the same i index lie along a single

flux surface forming a wmputational ring Xi. Target Iodons correspond to the j = O and j = Ns+l indices.

We begin the iterative cycle by specifying neutral and cross-field sources, @(X) and a@), and

keeping these sources constanr. solving the plasma transport equations (2.1.22) to obtain U(X), (since the

final solution is insensitive to initiai conditions. the exact form of the initial sources is not important; they

may be taken fiom a prcvïously converged solution or pcescribed analytically). A solution may be obtained

using the standard nunerical methods of CFD-! based on replacing temporal and spatial

derivatives by corresponding discrete operators,

If the relatively coarse, poloidal grid X is chosen as a substrate for discrete spatial operations, the grid point

locations could not be adjusted to regions of steeper gradients without re-solving the Grad-Shafranov

problern. Therefore, for reasons of irnproved spatial accuracy, wc define a series of finc. almg-B grids x', (one for cach computational ring Xi) of NI, interna1 points at locations xk, k E ( 1 ,...,No}, N,, > N,. and two

boundary poinrs xo = O and X ~ b l = Ln, where x is a distance along the fieid line from the outer target, x = s". The two types of grids are linked via (3.1.1). the anrilytic transformation x ch s bcing isomorphic: x =

x(s). s = s(x); the discrete transformation Ti: vx(Xi) + yr,(~:) is however not isomorphic for arbitrary X,

and x:, that is T~''T; # 1. since for NU > Ns no general suucture exists which would permit complete

reuieval of information (a situation analogous to entropy growth and irreversibility in

t h e r r n ~ d ~ n a m i c s ' ~ ~ ' ) . We can however construct Ti and T," ruch chat hy.(~i)dx = lys(x(Xi')dx over any

region on [O&]. and such transforms will be used to map Q onto X: and U ont0 Xi; U and Q will be

stored on both type of grids to prevent loss of information via repeated application of T~-'T~.

W e choosc 6, to be a three-point, cenual difference operator. the simplest operator of 2d order

accuracy: ( 6 , ~ ) ~ = [vk+in - ycktld 1 [xk+ln - xk+,,ml, where half-indices indicate mid-point locations, X ~ + I R =

(xk + xkil) 1 2 ; similarly. ( & ~ h + , ~ = [v~+I - ~ t l / [XM - xt]. Plasma variabtes and sources are defined on

the grid points, Ut and Q , and are lincariy interpolatcd from these values to any other location. R u e s Fa

and G,, , which WC write simply as F and G. are caiculated at the mid-points xk+ln in order to give a

gradient at the g i d poinü xk. Spatial discretisation on X' may be npresented schematically as.

or emphasizing the non-uniformity of &' ,

After discretisation. the system of four coupled partial differentid equations is transformed into a system of

4N algebraic equatioas, which we write in vector form as

where we have introduced the residuat sl, as a measure of local convergence of the solution (in steady state

Il93ll = XkI5Rkl E O). Square brackets in the last two expressions indicate the dependence of the !luxes on U;

convective fluxes are calculated from interpolated U vaiues, while conductive fluxes are obtained from the

gradient of U. both at the mid-points xt+ln.

The temporal derivative may be discretized using a two-point (Euler) discrete operator (2" order

accurate): (&\y)" = [FI - y"] / At, where index n denotes the time stage. so that uW1 = Un + (&U)" At.

Since we are interested only in steady state solutions, time accuracy is not required and time t may be

viewed as a rel.znation variable. T i marching methods are customarily distinguished into explicit (&U)" =

3tn and implicit (&U)" = %*', the former based on already known Un, the latter on next-stage, as yet

unknown uW'. Explicit methods are generally far less stable, although the degree of stability depends on

the details of the disretization and equation system k i n g solved. cg. the explicit Euler method with central

differencing is unconditionally unstable for purely convective flues requuing artificial dissipation terrns.

The stability limitations of explicit methods m y be understood as tfie requirement that information must

uavel between adjacent g i d points during the update time step d tml ; based on characteristic (eigen-)

velocities of convective and conductive fluxes. CF = (u, u i ç), CG= (qJpe, 6pi/pu) = (XJLm , qilpLvJ, two

approxirnate limits are imposed on the update time step A4 < m i n ( ~ 4 ~ . A&?,

In a typical tokamak edge plasma, conductive fluxes would be much more constraining on the explicit

updates (~4' << A ~ L ) due to high values of electron heat diff'usivity X, and the square dependence on grid

spacing (very small near the target). In short, the system exhibits a large degree cf numerical stiffness and

is il1 suited to explicit time marching (requiring prohibitively long convergence rimes)- In contrast, the

implicit formulation offers greater numeticai stability, allows longer time steps, and leads to faster

convergencem1; for these reasons, implicit Euler time marching was selected. With implicit updates.

infinite tirnes steps are potentiaily allowed, but are limited in practice by implementation of boundary

conditions (often non-implicit) or strong non-linearities in the equations; time steps on the convective time

sale AC: are more appropriate (still a great improvement over A&?. We may write the update to U. as

where H represent artificial dissipation fluxes, to which we return shortly. In implicit f o m , the update to

Un is expressed in tenns of the next stage un". The new values rnay be obtained by expanding Q. F. G and

H to first order in AUn = U-' - Un and defining Jacobian matrices for each quantity.

Inscrting the abovc into (3.2.6) we obtain,

The above is a iinear system which may be solved for AUn. The expression in the square brackets is a

sparse 4N x 4N matrix composed of 4x4 blocks related to the Jacobian matrices. For the purpose of

evaluating these matrices, it is convenient to introduce a shorthand notâtion for Ci,

and perform the differentiaiion in this notation:

A number of comments are in order:

i ) in differentiating G we treated qi. iri = nXi and K, = nXe, as constants, which greatly simplified calculation

of JG (denoted below as JG@); this omission docs not adversely affect stability. provided that JGO is

evaluated at each tirne stage using qi(U''), q(Un) and ~(ü"). Had we assumed that dl three coefficients

exhibit the classicd (non-flux limited) dependence. .r, a T:~, chah rule differentiation would yield.

I f the flux-limited dependence of q, and 8pi . both of the form 11yt = III$' + llp, is also considercd. then

the fincil JG would consist of clrissical and flux-limited contributions,

O 0 O [p) Qc

Since the above Jacobians do not enter the residual%, which alone acts as the forcing function for AU. they

affect stability and rate of convergence. but not the steady state, convcrged solution U. We find that. the

removal of Jacobian tenns corresponding to calmer source terms, eg. JB or the first rnacrix in J,, has little

effect on either convergence or stability,

ii) in Je. the B-field graâient VnB/B may be caiculated and n o r d as ~~k = Bk (%+, - xCL) I (Bk+, - Bk.l). 3R SR iii) in Jp we treated Q' as a hinction of U, G'~' IlT] = T ~ R ' = d B ,

iv) since Q3 and QL do not change in the course of CFD convergence. theu Jacobians vanish,

v) it is important to temcmber that G, occurring inside the Jacobian matrices are operators which act on AU

in (3.2.9). and should not be applied only to their argument in the rnarrix. This becornes clear is we examine

the structure of 6JG in the AU basis. A term-operator in JG, (a&@). gives rise to a term-operator in &JG ,

6,(a6$), which when acting on AU produces terms tri-diagonal with respect to the AU basis,

Artificiai dissipation fluxes R were introduced for reasons of numerical stability. Let us consider

the situation without H: as a consequence of spatial discretisation, grid related harmonics are introduced

into the solution; they are either attenuated or augmenteci at every time stage, the latter leading to numerical

instability (this behaviour depends on the operators 6, and 6,. the grid spacing Q, the temporal step size At.

and the nature of the equations; it has been studied extensively and stability limits for al1 practical schemes

are readily availablelHal). The case of central two-point differencing G, and explicit Euler tirnc rnarching 6,

applicd to locally hyperbolic (convectively dominated. c ~ > c ~ or lul+c,> A x l ~ , ) equations is destabilized by

a decoupling of altemate grid points[H'"'!. A number of methods of dealing with this problern may be found

in the 1iteraturerRœBZHm1: a) splitting the convective fluxes according to direction and discretizing using an

upwind 6,. thcn limiting the local flux to prevent oscillating overshoots, the so-called TVD (total variance

diminishing) methods. and b) introducing artificial dissipation fluxes which dampcn numerical osciIlations

during convergence, while adding only a small error to the final solution (it has been be shown that flux-

Iimiting methods Iead to adficial dissipation terms). Accordingly. artificial dissipation fluxes suggested by

Jarneson were adap tediJmsZ1,

where pk is the pressure, q= [(5/3)(pi + p3/p]'n is îhe adiabatic sound speed, and d2' and fi' are constants.

In this form. &H contains both 2* and 4" order dissipation te-: the 2* order tenn is largest in regions of

strong pressure gradients, (cg. shock waves or deuched fkonts), while the 4h order term. which effectively

darnpens higher harrnonics, is clipped in these regions. The facobian matrix is particuiarly easy to evaiuate,

leaving the same spatial operators as in H.

We now have al1 the components necessy to find the update AUn. The Iinear system could be

solved using standard LU d e ~ o m ~ o s i t i o n ~ ~ at a cmmputationai cost - ( 4 ~ a ) ~ . We notice however that the

matrix expression in the square brackets of (3.2.9) has a highly sparse structure (pentadiagonal, if al1 terms

are retained; tri-diagond, if 4" order terms are ~eglected from &JH ). We choose the second varÏant, a

decision justi fied by later analysis, and write the linear system

where each of the AI . Bk and Ck represent 4x4 matrix blocks. Such a linear system is most efficiently

solved using a block uidiagonal Thomas a~~or i th rn [~~" . R"21. developed initially for scalar tri-diagonal

sysrems. but easily extended ro block ui-diagonal systcms by replacing scalar algebra with rnauix

operations, App.3 (it requues a series of 4x4 rnatrix inversions on the returning sweep for which LU

decornposition was used). Computational cost is reduced from - ( 4 ~ l l ) ~ to -43~u2, that is by a factor of -NI.

The updaies AUn and the new state variables un" = Un + AUn are thus obiained. The entire process is

repeated until the relative changes ro Un are small, (AUn/Un ) « 1.

Boundary conditions discussed in $2.2 and 82.4.3 (the Bohm criterion and energy transmission

coefficients for ions and elcctrons) may be numencaily impIemented in a number of ways. The key feature

is the transition from a hyperboiic to an elliptic system at the target; since Mo > 1 by the Bohm criterion,

the hyperbolic nature of equations at the target is guaranteed. The simplest implementation method is to

cxplicitly update the grid boundary variables. Uo and after each sweep of the solution Uk on the inner

grid points, k~ ( 1 ....,Na) ; the boundary updates AU* and AUNICII should rrflect both physical and numerical

consuaints (known as floating or numerical boundary conditions, which involve extrapolations of U h m

the inner grid points in order to assure continuity of Uo and UWl with tJw inncr Ud and may be calculated

by a rnethod similar to the inner point updates AUk. Let us define the consûa.int vector as 'l'du); then the

desired updates AUo may be obtauied by Newton's m e i h ~ d ~ ' ~ .

where Extro(AI,Az) denotes a Iinear extrapolation of A(x) from the values at A(xl) and A(x2) to the target

m. The evaluation and inversion of the Jacobian matrix proceed m exact analogy to (3.2.9-1 1). As

already mentioned, explicit updates may destabilize near target cells. A more stable method, short of the

fully-implicit treatrnent (which for reasons of fiexibility we would likc to avoid), is to make the updatcs

semi-i rnplicit by solving for Tio + UO" fiom = O and for TCo 4 u:" from Y:" = O. where Fin and

Gia were expressed in terms of Uo and UI according to (3.2.4) and (3.29); the updates to U"' and U<*' am made explicitly according to (3.2.16). A further improvement involves a srnall iterative loop in which tht

updated values of Uo and UNII are diffuscd into the grid by a series of explkit updates of the few near target

cells, Uo and UN+1 are then updated semi-implicitly, and the cycle repearcd until local convergence. In our

case, it was found that the one-tirne semi-implicit method is more efficient for T, c 10 eV, while the

cyclical semi-implicit method is superior for Tc> 10 eV.

The main motivation for inuoducing the fine grid xn was to eventually adjusi grid point locations

in order to rnaximize spatial accuracy. There are two main approaches for adapting the prid to the

s o l u t i ~ n ~ ~ " ~ ~ ~ : a) introducing more grid points. b) redistributing the grid points according to spring-likc.

variational, or conuol function methods. Selecting control function redistribution rnethod of ~ i s e r n a n ~ ~ ' ~ ' ,

which effectively adapts X" to both interna1 and boundary errors in U. we inuoduce a uniform grid h, lis { O ..... Nll+I J. Ct+l - tk = 1. and treat X" as an isornorphic mapping fmm 5. xk = xr(6r>. WC wili inducc

diffusion of x i in 5-space in response [O a conuol function il(U(xi(&))), which will determine the grid-

point density in 5-space (grid points are clustcred in regions where ll is large). This process may bc ~ n t t e n

as an ellipdc differential equationwdwl. which we discretire using a 2* order. central-difference 6:

where b's are parametric constants and il was subdivided into geometric and adaptive contributions. The

former consuain the near target ce11 size (we select I l g such that IUrUIWo < 10%); this consuaint is

relaxed exponentially away from the targets,

The latter impose changes to inner grid point density in 5-space in response to some measure of solution

variation fi, , which we choose as the II- I I1 n o m of the difference Uk+1 - Ut-,. These are n o d i z e d to the

intervai [O, 11 and in the form wk defineû below are subjected to clipping and smoothing ~ ~ e r a t i o n s [ ~ ~ ~ ' ,

Adaptive conuol function Il' measures the relative gradient of wk in 5-space,

Combining (3.2.17-20) and solving the tri-diagonal system (3.2.23) using the Thomas algorithm, we obtain

new values of xk of the grid xU; typical parameter choices are 13 = ( 10.1.113.10.2). The process is repeated

wirh U interpolated to the new xk values from the input U(x& until the relative changes to xk are srnaIl.

AX,"/X," cc 1. The final (new) grÏd is then relaxed with the input (old) grid to assure gentler evolution: xfl = E,X',,~,,. + ( 1 -G)x ' '~~~ . G - 0-03.

We can now combine al1 the above modules, and present an algorithm of the cdge plasma solver.

Given sources Q(X) = Q3(X) + QL(X). edge plasma variables U(X) are obtained by mapping the sources

onto the along-B grids. Q(X) + Q(x~). converging the CFD solvcr on each X: grid. and mapping thc

solution back onto the poloidal grid. u(x~") + U(Xi). The final algorithm is given below:

Salve JW~V LX, Q(m Sofve ring [i, xi"', Q(x,-~')] Do i = itcn. id, R e d udd(x;')

Q(Xd -. Q ( X ~ b o p

Solve Ring [i. x:, Q ( x ~ ) ] SWCCP RUig [i, x:, Q&)]

A h p t Grid [i. X: u(x:)] Erit when mair(9Pk) < €9 << I

Solve Ring [i. x:. Q ( x ~ ) ] End loop

Q&') + QG&) Return u(x:) End do

Rerurn U(X)

sweep Ring @, Q(X'II Find Fluxes:

Find Qp QB:

F M Jacobians:

Find Residuol:

Find Mairù:

Inverî M&:

Update U:

Up&fe BC:

(3.2.21)

A&pt Grid [i, xi"', u(x/')]

{xO. u(x31 + lna. ngI {na, ne) +xi" Exit when &:/Xi" c< 1

End loop

&" = G!' , MY + (1 -axil'dd Return x,"

3.3 Neutra1 Equatiorrs: Monte-Cario simulzition

Having solved the plasma transport equations on the potoidal grid X, we proçeed to (re)calcuIate

the neutral sources a. The kinetic equation describing the transport of neutral species. (2.1.1) with Z = 0,

rnay be solved using Monte-Carlo s i m ~ l a t i o n ~ ' " ~ ' ~ . This numerical technique consists. in the prcsent

conrext, of stochastically generating a large number (NMC) of particle trajectories based on a specified

plasma background U(X) and atornic/molecular cross-sections. As a consequence of the statisticat law of

large nurnbers, for Nwc >> 1 the probability distribution obtained by averaging over the simulated

trajectories wili approach the exact solution of the kinetic equation. with an error proportional to 1 /dIUSfc.

Let us consider one such neutral particle trajectory. A neutral Hz molecuie is created al somc

location x = (r.s.@) and velocity v = (v,, v,, v,) with a probability prH'(x.v) determined by: a) the ion flux to

the surface ro(r), b) the rate of volumeuic recombination S,(x), c) the wall. target and plasma

temperatures T W d ( s ) , Tdr), Ti(x), T,(x) (we will work on a per particle basis, so that jdxdv prH'(x, v) = 1; to

obtain actual magnitudes we would multiply the results by a given source strength), If an atom is launched

from (born at) the target, Ivl is chosen according to To and its direction according to some fonvard peaked

angular distributionMq, eg. cos28; if it is launckd from the plasma, Ivl is detennined by the Franck-

Condon energy specmim and iis direction has a nearly isotropie angular distrib~tion"~'. me trajectocy is

constructed by explicitly updating x and v through small t im intervals At, x"" = xm + vaAt, where n

denotes the time stage. For ncutral particles. v is only changed by a collision or reacuon with another

particle (charged or neutral); locally calculated mean free paths determine the probability that a particular

interaction takes place in the small interval &. At each time stage, this is simulated by a uniforrn random

variable Ç E [O, 11 with corresponding branching ratios. If a scattering event does occur, the new velocity v'

is obtained by simulating the kinematics of the collision with appropriate random variables. If the molecule

is dissociated into two H atoms, each of these is then followed in tum. A trajectory is terminated when a

neutrai particle is either ionized or absorbed by the wall.

The final probability Pf, that a neutral particle of species 'a' is found in ce11 Xij (sum over grid

equals unity, I, Pfü = 1) and the cell average <WJ, of any trajectory related quantity $, may be

obtained in terms of integrals of a ccll based delta function 6[q,Xq] over WMc trajectories of species 'a*

which we denote by (xk(t),vk(t)l k = 1,. . ..NaMC,) , a = (Hz. Hl,

Multiplying PFü by a given source strength and dividing by cell volume 2rtRijAu we obtain the number

density n,(X), velocity u,(X)= <v> and temperature T,(X), 3TJ2 = rn,<v,'>/2, for both molecular, a = Hz,

and atomic, a = H, hydrogen. This approach is equivalent to calculating macroscopic quantities <w> by

integrating over phase-space density p(xl, vk I k = 1 ,. . .,ET), (A.2.12), since the set of WMC trajectories

{ xk(t).vk(t)l k = 1 ,. . .,NaMC, } is an estimacc of p(xk, vk I k = 1,. ..,Na), which bccornes exact as NiMC + Ni - 10'~. Line integrals over the stepwise linear trajectories may be replaced by Riemann surns (IAxk/Ax,,l > 1

is required) at the cost of a reduction in accuracy but with savings in mernory storage and quadrature time.

For the purpose of calculating discrete events. such as excitation or ionization reactions. it is

simpler to cumulatively store al1 the instances of a given event in a grid-based array N~(x), which

divided by NMC gives the average number of events in a given cell per particle launched. Combining event

densities na(X) with calculated n,(X), ua(X) and T a s ) , the atomic-molecular sources (&a), S,,(X).

Q,(X). Q+(X)} + Q3W) may be estimated. Again. as N M ~ + -, al1 the estimates become exact; in

practice N, - lo4 is usually sufficient to achicve - 1 8 error in ralar quantitics (vector quaniities. such as

u,, aiways contain largcr errors for the same NMC). S e v e d neuual vansport codes based on Monte-CarIo

techniques are commonly used in the fusion community <EIRENE['~'. DEGAS-*'. N I M B U S ~ ~ ~ ' ) .

Al1 the results presented here were obtained with the NIMBUS code, aithough EiRENE has also been run

as validation.

The Monte-Carlo technique rnay also be used to simulate impurity (neuual and ion) transport in

the edge plasma, with the a h of calculating the fiaction of target impurity atorns reaching the core, their

redeposition rate and the role they play in plasma detachment and M A R E formation. The trajectones of

neutral impurity species (entering the plasma fiom the target as a result of physical or chernicd sputtering

with a probabili ty determined by experimentall y obtained yield expressions, and a speci fied veloci ty

distribution) rnay be followed in the same way as the hydrogenic neutrals. M e r ionization, the equation of

motion (A.2.3) of an imprtrity ion (whether atomic or molecular) becomes.

where the forces on an impuriry atom due to collisions with hydrogenic (field) ions. ~f and F~'. rnay be

obtained from plasma kinetic theorywl, cf. (A.2.30-37)- Assuming a shifted Maxwellian impurity ion

velocity distribution fz(v), îhe parallel forces rnay be erpressed in a fluid-like for~n'~"''.

The fust tenn of this equation is the impurity pressure gradient force; the second term is the friction force

on the impurity ions moving with dong-B fluid vetocity vu exerted by the background ions moving with

parallel fluid velocity u (t, is the stopping time); the final two terms are the electron and ion temperature

gradient forces, which arise because Coulomb collisions are more effective for colder particles, (the

elecuons and H' ions striking the impurity ion from the colder side. give it more momenturn than do the

hotter particles striking from the opposite side). In principle each of the above forces can act in either

direction - toward or away from the solid target. However, i n the sirnplest case the plasma flows toward the

targct at al1 points and thus the frictional force tends io supprcss divcrtor leakagc of impuritics. The

tcmperature usually rises away from the target, VllT > O. so that the T-gradient forces push the impwity ions

in the upstream direction, tending to increase divertor leakage. The impuri ty pressure-gradient force

generally acts in both directions, away from the location where the ions are created by neutral ionization.

The pressure gradient force can be interpreted on an individual ion basis as parallel diffusion and may be

ueated as a random-walk problem (the parallei velocity o f the ion is changed at each time step At by +,

( T J ~ , ) ' " ( ~ A V ~ ) ' ~ , where T,, is the Coulomb parallel collisional diffusion time, and the sign is chosen

randornly). The cross-field force FP rnay be expressed with the help of an anornalous diffusion coefficient

D;, and the update to the ion velocity is aiso constructed in a Monte-Carlo rnanner ,

The transport of irnpurity atoms, cither sputtered fiom the waIl or injected into the plasma, may be followed

in this way through successive ionization stages: eg. for Carbon, CO ++ C+ ++ C" t.. . The irnpurity

rranspon results presented in this thesis use the DIVIMP based on the pnnciples diwussed in b i s

section.

In $3.2, we saw that numencal stability of the CFD method requires knowledge of the neutral source

Jacobian, (a& /au. During CFD convergence the sources Q3 fernain constant, (%/au) = O, but change

abruptly afier a cal1 to the neutral M-C code from which new are retumed. if the change in Q is too

large, instability may result Since the sources are calculated using a Monte-Carlo method, (a@/aU) is not

known a priori. A numericd estimation of this Sacobian requires a differential increment of each spatial

variable. and becornes prohibitively expensive. A more efficient method of assuring Cm stability consists

in rclaxing the sources in such a way chat the plasma quantities change sufficiently slowly. In general,

having found the new MC sources, we update the next stage sources as,

The exact form of e[U1 may be optimized to give the quickest convergence without loss of stability.

Clearly. the final solution does not depend on the form of %[W. since Q3- - Q~~~~ in steady state. Due to

the stochastic nature of Monte-Carlo simulation, we expect some variation (- I / ~ N ~ , ~ ) between successive

QZnCU even if plasma background U remains constant. In addition. the M-C pan of the iterative Ioop is more

cornputationally expensive than the CFD part. The tirne efficiency of the iterative loop rnay be optimized

(ie. the ratio t M C / b > I reduced), by performing MC runs with fewer particles. and compensating for the

largcr oscillation in QSmw by reducing the relaxation factor E~[U]- The effective number of MC particles

uscd to obtain the final source Q3 may be estimated ris Nsfck2[U]. Typically. €,[U] was found by rcquiring

chat QSrtCw change by at most - 10% per iteration.

- VariamXMinimizri . . . 3.5 Cross-field Sources: tion

The final step in the iterative cycle consists of adjusting the cross-field sources according to the DVM

method, in order to k t match givcn diagnostic data. In 52.5.3. ihc DVM technique was presented as a

multi-variable optimization problem. The radial dependence of U and 5 is discretized when this problem is

projected ont0 the poloidal M d X: 50) + 51 , U(r.0) + Uio , U(r- + ULNM. where i refers to the

computational ring Xi. Depending on the number of nondegenerate ûce parameters N, and the number of

diagnostic consiraints N-, this pmblem may be under (Na < N d , well (Na = Ndu$ or over (Na > N d

c o n ~ t r a i n e d ~ ~ ~ ~ . In the fmt case. thcre exists a family of (ei} for which llEll = O, in the second case. only

one such b. and in the thud case, none at al1 (in al1 three cases a minimum of llZll may be found, which

does not vanish only if the problem is il1 posed). We will assume heft that the problem is in fact weli posed

(although the analysis which follows applies regardles of this assumption) as is the defadt choice of si and

çiw for target Langmuir probe data. The particular path taken by the control variables ci should not effect

the final state, provided that al1 components of gi and are involved in the optimization. We can construct

an incrernent A4i by expanding &[Ed for s d l Si and applying the Newton methodW4l.

with E~ acting as a smali damping constant. From (2.4.13-16) we note thah

From which we can construct an approximation to the inverse of J and write down the final form

in which departures from ro* and are used to incrernent cross-field particle and momenturn sources

(if rtiag = r/iaC. then a(2' = O is expected and observed) and departures from ~,,d"' and r,.Liag to

increment cross-field energy sources (same for electrons and ions). This simple method has proved efficient

for the default choicc of iw for target Langmuir probe data and any nom II4 as long as llLll vanishes at

its minimum. The general Newton method may prove superior for lafger diagnostic cornparison sets.

As mentioned in 92.4.3, for strong volumcûic recombination, it is useful to impose an additional

upsueam constraint, which in this study has k e n chosen as the upstream density on the separatrix, n,q. To

impose this constraint, the growth of the radiative density peak must be controlled. which amounts to

limiting the growth of recombination. Due to an cxponential dependence of <sv>, on Te and the added

effect of n = lm due to pressure conservation, the growth of S, = &ni<-, may be effectively opposed

by srnall changes (-0.1 eV) in the target temperature. This method has been implernented within a feedback

loop similar to the one disfussed above, with "' measuring the depamue from n? and Gf5) containing an

update to target Te. The constraint is one sided, with upstream density not allowed to exceeds nucep, but

allowed to fa11 below this limiting value.

In the previous two chapters, a model of the edge plasma was developed, and a numencal method of solving

the model equations was outlincd In the present chapter, ihese equations are solved in a variety of scenarios: a) the

transition from attached to detached conditions is studied via a scries of solutions ranging in target Te from 32 eV to 0.5

eV, and To = loU to 1 p m-2s'1, b) the sensitivity of the solutions to several modeling assumptions is examined for

conductively lirnited (8 eV) and detached ( 0 5 eV) conditions, c) the OSML/NIMBUS solution is compared with a 2-D

edge code EMjEZD/NIMBUS by using the target values obtained with the latter code as the diagnostic target data for

OSM2. and d) an attempt to model cxperimental measurcments is made by comparing the 0SM2 simulation with

reciprocating probe data on the JET tokamak As part of the validation procedure ttie results of (a) above are examined

in detail: the sources are broken d o m into the neutral, cross-field, plasma and flux-expansion contributions, the flux

balances of mas, momentum and energy are divided into their hyperbolic (convective) and elliptic (viscous,

conductive) components, and the OSM2 solutions are compared with the two-point model predictions (2.4.6); the

cornparison with the EDGEZD code, point (cf) above, may also be viewed as a validation step. Solutions are displayed

on two types of figures: along-B plots and poloidal contour plots. The former provide target-to-target flux-tube profiles

at four radial locations in the SOL and contain etectron temperature Te [eV), ion temperature Ti [eV], plasma flow nu

[loU rn-2s-'], Mach number M 1-1, electron density n, [ l p m"], and total pressure p, = F "' = Pe + Pi + [Pa]; the

target-to-target (connection) lengths for the four chosen flux tubes, denoted by their ring numbers (ir=8 is the fmt SOL

ring beyond the separatrix), and the locations of the inncr and outer X-points are given below,

LI [ml <saX/4>, (saX&)in

i r=8 119 0.1345 0.843

ir= 10 98 0.0874 0.886

ir= 12 92 0.0706 0.900

ù= 14 88 0.0730 0.893

The latter plots offer a poloidal view of the divenor region showing Te [eV, Ti [eV]. n, ~rn-~] , M[-1, atomic hydrogen

density n~ [m"], molecular hydrogen density nm [m-3]. ionization rate Si, [m-3s-']. recombination rate S , [m-3s-'].

charge exchange momentum loss rate Q, [Palm]. elecuon power loss rate Q, [ ~ m - ' 1 . average H atorn energy EH [eV].

rate of H, line emission from neutral hydrogcn H, [m-3s"], densities of neutral carbon and its first four ionization statcs

CO. C'. ... .C<+ [m.'] and the total energy radiated by al1 carbon atomdions Qc [ ~ r n - ~ ] (carbon transport was calculaicd

using the DNIMP Monte-Carlo code using the hydrogenic plasrna/neutral background obtained from the

OSM2NiMBUS calculation; only carbon originating at the Large& was considered in order to better isolate the effect

of dcirichmcnt on irnpurity transport; both physical and chemical sputtering were included). As mcntioned in 93.2. iwo

grids were used to generate the solution: the neuual sources are generated on the fixed poloidaf (coarse) srid, whiie ihe

CFD solver is convcrged on the adjustable, dong-B (fine) grid; since the rnapping of sources and plasma variables

between the two grids results in partial loss of information, al1 variables are stored on both grids. In the figures below,

the poloidal contour plots refer to the coarse grid while the dong-B plots refer to the fine grid. The largest difference

between these two solutions usually occurs within the fust cc11 of the coarse grid, which is too cmde to properly resolve

the ncar target gradients (see $4.1 below).

In the fmt set of solutions, the gradua1 onset of divenor plasma detachment (discussed in 92.4 in terms of the

four experimentai regirnes) is simulated by f ~ n g the target particle flux To = t ï ~ u o = l d ) rn-L1 and progressively

lowering the target temperature Ta = (32, 16, 8, 4, 2, 1, 0.5) eV; the magnetic geometry corresponds to an L-mode

discharge (shot 34859) on the JET tokamak with a MARK-1 divertor configuration. Both To and Tco are specified as

radially uniform and qua1 at both targets (including the private flux region); the uniform radiai profiIes, obviously not

physical for power entering the SOL fiom the core across the separatrix, were chosen to emphasize that radial

variations in the final solution are caused by magnetic geometry and neutrai transport rather than by radial variations of

carget conditions. ie. that the iteratcd solution is inherentiy 2-D, rather than a collection of 1-D along-B solutions. The

choice of syrnmeîrïc boundary coaditions was made for similar reasons, in particular to draw attention to asymmemes

generated by the magnetic field geomeuy. The dcfault OSM2 mode1 was used for al1 cases in 84.1, with two

modifications: a) the private flux-ngion was trcated in a simplified manner: plasma variables were Iinearly interpolated

on each flux tube, based on the specified values at the targets, b) the flux-tube integrated input power was not allowed

to increase radially away fiom the separatrix; for some of the cases in this section, this physically motivated constraint,

produced radially decaying target temperature profiles.

The f m t case = 32 eV, Fig.4.1.1-5) demonsuates ail the feanrres of a sheath limited SOL. The

temperature is nearly uniform dong each flux tube. with Te and Ti decoupled due to weak collisionaiity (r, = TF is

too srnall at these conditions CO relax Te and Ti to a common value; Ti > Te due to srnaller ion conductivity K, cc c); the

fact that Tiso - 2 reflects the difkrence in the sheath boundary conditions for ions and electrons (same input power

for both species). Plasma flow nu and Mach number M decay exponentially away fiom the targets with a charactenstic

Iength for ionization. & « L,, (the flow tums sonic at the sheath-edge, = 1; since the s c d e of the sheath is so much

smaller than the length of a flux tube, this is not clear on the full scale figures, but can be seen on the close-ups neu the

inner target, eg. Fig.4.1.10), leveling off upstrearn to - 2056 of To in the direction of the outer target; due the short

penetration length. only - 10 % of the neutrals reach the core, f,,'"" = 0.1 1. The stagnation point occurs near the inner

targct (not far from the inner X-point, sr% - 0.8) due to poloidal asymmetry in the rnagnetic field. although the

boundary conditions are symmetric (the asymmctq is related to the dominant toroidal B-field, which varies with the

major radius as !IR); this cffect can be offset by raising the plasma flow at the inner target (or by inclusion of drift

effects) which shifrs the stagnation point towards the rnid-point. Both n, and p,, reach their maximum values very close

to the target (sPiL& - 0.993. n+l& - 2.15; al1 values refer to the inner peak on the separatrix flux tube) and rernain

fairly constant dong each flux tube; in solutions with different trirget pressures at the inner and outer targets (say. Te,o =

32 eV and Tc.L = 8 eV. not shown) p, changes continuously between the two target values. Along-B variations in n,

and p,, rnost noticeable beyond the outer X-point. are caused by an ionization peak in this region, which modifies the

nu profile and gives nse to a local viscous flux gradient, G'~' = -T(i Vau; the drop of p, away from the targers is due

entirely to parallel viscosity. Flux expansion Q~" ' plays a relatively small role in the momentum balance; its relative

contribution is studied in 84.2. Thc weak radial gradients seen in Fig.4.1.1, indicate radial unifonnity of the neutral

sources; thus the presence of a finite VLTe = dTJdr < O arises due to a radially decaying input (deposited) power

profile, dPJdr < O. The unphysical increase in ups t~am &nsity away from the separaerix is a consequence of the

unphysical choice of uniform radial target profiles (a more physical scenario with radially detaying boundary

conditions will k considered in 84.24). The relatively low upsucam density (40'~ m") as compared with typicai

experimental values (1-3x10'' md) is a consequence of the relatively low To (the choice of low ro was made to

illustrate a particular type of detached transition, in which the detached ionization front reaches the X-point and

develops into a MARFE-I*c radiative region). Higher density cases (ro = 3xld) and 1 0 ~ m'2s'1) for attached (8 eV)

and detached (0.5 eV) conditions are prcsented near the end of 94.1

Source profiles, Fig. 4.1.2, rcveal exponentid decay away fiom the targets, characteristic of a short neutrai

penetration length (the double step hinction profiles of a''' and a('), which are clearer for lower Te cases, cg.

Fig.4.1.9, represent the input power and are divided into the inboard and outboard parts; their asymmetry reflects the

greater volumetric energy losses Q, next to the inner target). The power balance shows the dominance of conduction

over convection, especidly in the electron channcl. while the momenturn balance indicates a presence of charge-

exchange losses even at - 70 eV (although their net effect is no larger than that of the viscous flux). It should be noted

that conservation of mas, momenturn and energy, is equivalent to convergence of the solution set (2.1.22); typically

1%-At/UI < 104. Poloidal contours, Fig.4.1.34, provide a 2-D rtprcsentation of the dong-B profiles, Fig.4.1.1-2. The

core density and temperature were specified by assumuig a constant radial increment of 250 eV and 5x10" mJ per core

ring (flux surface). The ionization front Si, responsible for ihe majority of the electron cooIing Q, and the H, line

ernission is clearly 'attached' to the target (recombination is negligible); the density of neutral species also peaks in the

vicinity of the target (due to short penetration lengths) with atoms peneuziting further than molecules. As a result of

thermalizing collisions, the average neutrai energy EH appmaches the average ion energy 3TJ2. In Fig.4.1.5, the

Mon te-Car10 simulation of carbon transport using the DIVlMP code is displayed. The carbon neutrals originating at the

target are quickly ionized such that only the C? peneaates significantly beyond the X-point, and only the C? and C.?

States (not shown) reach the inner core. Most of the carbon radiation coincides with the c2+ ionization state, peaking

next to the target (the default mode1 of Q = f ,Q, with f, = 1 gives a rough estimate for the impurity radiation,

although Qc is less extended than Q,).

The next two cases (16 eV, Fig.4.1.6-7; 8 eV Fig.4.1.8-12) demonsuate a transition to the conductively limited

regime. characterized by the presence of suong T, gradients in the cool, near target region (Te is conductively

dorninated, Te = s , , ~ : . on the entire spatial domain); Tc and T, rernain dccoupicd with TjT, - 2. Thc plasma flow nu and

Mach number M decay less rapidly (with a ionger half-length) due to lower ionization rate and stronger neuual

penetration (the position of the stagnation point is unaffected, since changes in boundary conditions are symmetric).

The density pcak separates from the target. sW& - 0.982. with a decrease in the relative pc&-to-target density ratio.

n,&, - 1.4. In contras[ to the sheath-limited case, pressure riscs away from the target as a results of charge-exchange

collisions overcoming the viscous flux, Fig.4.1.9. Along-B variations in p,, near the outer X-point are also suppressed

by the larger Qcx. An increase in the radial Te, Ti and p,, gradients close to the separatrix indicates suonger plasma-

neutral interactions Q3 on the nez-separatrix flux tubes requiring larger input power (this may be explained by a

weaker poloidal field in the Mcinity of the X-point (Be + O, B, + B). which decreases of the field line pitch angle, tan'

'(B$B); consequently. field lines musi vaverse a longer distance IO cover the same poloidal intcrval. su = (B/Bo)s. and

an ion travelling dong such a flux tube has a longer distance in which to inuract with the neutral species contained in

the region enhancing the plasma-neutrai rractivity and producing larger ncutral sources for the plasma, Q). The

asymrneuy in the input power is more pronounced due to stronger electron cooling near the inner target. Examining,

Fig.4.1.9, we see that the plasma-neutrai interaction {S, Qcx, Qqc) region is more extended, although most neutral

species rernain confined to the divertor region with a maximum near the targci; neutral species penetrate furthcr due to

lower Skallowing a larger fraction of neucrals to reach the core, 6-- 0.22. The same is crue of carbon atoms and ions,

FigA.1. i 1, although the penetration of carbon ions is governed primarily by VITe forces; the radiative power departs

somewhat from the Q, distribution, but has a comparable integrated value. In FigA.1.10-11, which are close-ups of

Fig.4.I.8-9 in the vicinity of the inner target, the effect of grid coarseness is exarnined; it is seen that for attached

solutions, the relatively coarse poloidal grid X is suffkient to resolve the near target gradients (the same will not be m e

for detached conditions as we will see in Fig.22-23 ahead).

Since the 8 eV case will be the object of a parametric sensitivity study in 94.2, it is worth commenting on the

relative contributions of consatutive sources, Fig.4.1.9. In the mass and momentum equations, both Qt and Qg are

small in comparison with @, that is &and Qcx. In the energy equations. net power enters the flux tube by cross-field

transport with equal arnounts deposited to each species; in addition, the ions gain energy from the along-B electnc field

E,,, while k ing cooled by equipartition collisions with the electrons (the latter effect dominates the plasma source,

Q y'3) c O. while Q, = O is specified in the default model); electrons gain energy by equipartition and loose energy to 4 (with a net heating effect Q,"' > 0) and by atomic interactions, Q, c O. Flux expansion sources redismbute ihc power

along-B according to the variation of the total B-field (QB > O beyond the outer X-point, & < O between the outer and

inner mid-planes, and QB « Qqc near the targcts). As a consequence of the Mach dependent , the ion conductive flux

G'~' vanishes at the targets, unlike the electron conductive flux G'"' which is comparabIe to the convective flux p4' ; the

dominance of conduction is most pronounced around the stagnation point at which convective fluxes approach zero.

The next three cases (4 eV Fig.4.1.13-14; 2 eV, Fig.4.1.15- 17; 1 eV, Fig.4.1.18- 19) demonsuate a gradua1

transition to weakly detached conditions. The separation of the density peak from the target (& - 0.974.96,

n,,l/n, - 1.4+2.6. fi" - 0.29+0.44) and total pressure loss become more pronounced (p,o,Jp,, - 1.5+3.3: wZ), Q ~ ( ' ) << Q~"'). especially on the separatrix flux tube. The region of neuual penetration. plasrna-neutral interaction and

high recycling progressively expands towards the X-point (that is, the ionization front moves away frorn the target with

the consequent extension of the nu profile). The increase of peak density is reflected by a growine degree of over-

ionization on the separauix flux tube, as indicated by ~ l ( " c O (Srcc increases significantly, but rernains small in

cornparison with Sk). This has two consequences: a) since Q,"' = n, in the default rnodel. a"' c O Ieads to a net

partick sink near the target and a local maximum in the plasma flow nu, b) combined with flux expansion, i t also gives

rise to a net particle sink in the upstream region, which tends to reduce the plasma flow nu (under more symmeuic

conditions these effects gives rise to flow reversal, in which two additional stagnation points appear, such that plasma

flows towards the targets in the innermost and outermost regions, and away frorn the targets in the upstream regions, cf.

the higher density case below; flow reversal has also k e n predicted by 2-D codes and has been confirmed

e ~ ~ e r h e n t a l l ~ ~ " ~ ; the potential of flow reversai to enhance impurity lealcage into the core makes it irnporîant for

fusion cxperiments). The dominant role of convection in the divertor region produces an inflection i n thc curvaturc of

the Te profile and a transition to supersonic flow at the sheath edge. in the same region, Ti and Te approach a common

value due to higher collisionality and more effective energy equipartition al iower temperatures. The vansport of

carbon also incr-s due to colder divenor conditions; higher ionization States (ch. Clt) concentrate dong the

divenor legs (extending fiom the target to the X-point), as does the radiated power density fiom carbon Qc (the ratio of

the maximum Qc/Q, rises fiom 0.61 for the 8 eV case to 1.9 for the 1 eV case). The increase of impurity cooling is

very important in deuchrnent, eventually leading to the formation of a MARFE-like radiative region in the vicinity of

the X-point (& is treated irnproperly in the default mode1 which uses & = f&. since Q, is spatially rnuch broader

than Qz = Qc).

The 0.5 eV case, Fig.4.1.20-26, is an example of a detached plasma (p,Jp,,, - 6) without significant

recombination. The trends outlined in the preceding paragraph have progressed funher (a - 0.93. n&ni - 8,

fi,,,- 0.651, with the ionization front, the density peak and the recycling region reaching the X-point- The most

striking new feature is a plateau region in the Te profile, in which conduction is negligible and nearly al1 the energy is

carried by convection. The plateau electron temperature is more than twice the target Te because of two physical

mechanisms: a) the electrostatic sheath acts as a low energy filter for the electrons, allowing only the more energetic

electrons to reach the target, and effectively cooling the near target plasma; this process is represented by the relatively

high value of the sheath heat transmission coefficient for the electrons, y, - 5, in cornparison to the purely convective

value of - 2.5; the corresponding kineiic boundary condition, F?'+ GO')= 2.5 T,Jo - &VaTe = y,T,J0 gives rise to a

temperature gradient of (ViTJT',)o = (y, - 2,5)T& at the target, which requires Te to increase away from the target as

long as y, > 2.5 (note that y, = 5 results in Fo'J' = in the T-plateau region V,TJT= cc 2.5TdK, and convection

dominates); assuming negligible sources in the target-plateau transition region. we have TI = To and 2.5TeITi = yeTe.oTo

where the subscript 1 denotes the T-plateau value; this leads to an estimate of Tcl/Tco = y, / 25, which with y, = 5

predicts a plateau temperature of twice the target value, b) the self-consistent plasma (pre-sheath) eiecmc field &,

largest in the near target region. transfers energy from the electrons to the ions, enhancing VIT,. The above near target

gradient occurs on a scale much shorter than the ionization length, and comparable to the collisional mean-free-path

(for both 8 eV and 0.5 eV); the latter point is reassuring, since in any fluid-like description, the collisional length scale

offers the lower limit to the resolution of any plasma quantity. The above arguments also justify the introduction of the

adjustablc fine grid. since the inadcquacy of the corne grid in resolving the near target gradients becomes evident. cf.

Fie.4.1.22. In the T-plateau region, the plasma flow nu exceeds the mget value with a noticeable peak in the nu

profile; an effective particle sink due to over-ionization and cross-field sources is responsible for this inversion in the

flow profile (recombination 1s present but comparatively small, <Sd/<Siz> - 0.02. where <Q> denotes a flux tube

integral of Q; in many early, and some more rcceni. explanations of detachment. rccombination was proposcd as the

dominant process; bis view would be discredited by the above solution (sec also $4.2.2 where momentum Ioss pcrsists

even with S, = O). Thc creation of a cold, dense, convective, collisional and highly radiative region between the

ionization front and the target is characteristic of advanced stages in plasma detachment. foloidal contours show the

extent of this region vis-à-vis the divenor geomeuy; the ionization front is seen to reach and bend around the X-point

(the front can not extend into the core plasma since the neutrals are ionized shortiy after entering the core). The

refraction of the ionization front at the X-point may be viewed as an asymptote of plasma detachment, with the core

playing an increasingly dominant role henceforth; this interaction cdminates in the formation of a dense, radiacive

(MARE-like) region at the X-point (It should be emphasizcd that this thesis contains only steady-state resuits, with no

information about temporal c hatacteristics of the obtained solutions; the tenn 'evolution' , appearing frequentl y below,

refers to numencal convergence, rather than iemporal tvoiution). The carbon atoms and ions are concentrated dong the

legs of the divenor and the X-point, radialy peaking on the separauix (max. QdQ, = 1.35); the expansion of the

radiative region into the core would cause dong-B transport on the core flux surfaces, as well as radial transport across

the separatrix, neither of which are handled properly by the OSM approach.

A surnmary of the plasma values at the location of the inner density peak of the separatrix ring, ir=8, (s&),,,

is given beiow. Progressive separation from the target is accompanied by an extension of the convective region,

increased over-ionization and penetration of hydrogen neutrals and carbon ions into the core.

The next four cases examine attached and detached solutions in the higher density range; the target particle

flux is increased threefold and tenfold with respect to the 8 eV and 0.5 eV cases. As Fig-4.1.27-28 (& = 3x10" rn-'S.',

Tc* = 8 eV) and Fig.4.1.29-30 (ro = 10'~ ni-%-' , Td = 8 eV) indicate, a warm plasma becomes progressively more

attached as the density increases (steeper gradients of Te Ti, nu, M, and n, near the target) due to a linear reduction of

the neutral penetration tength with density. The higher input power raises the upstrearn temperature and pressure, but

the relative pressure drop remains roughly the same, that is viscous, at al1 densities. The ionization Front, the elecuon

cooling region and the carbon cloud are also pushed closer to the target (Qc/Q, = 0.35).

Higher dcnsity detachcd solutions are shown in FigA.1.3 1-32 (To = 3x10" m-'s". = 0.5 eV: nuxP - 1 0 ' ~

m.') and Fig.4.1.33-34 (To= 10 '~ m"s". = 0.5 eV. nuxp - 3x10'~ rn-)): in both cases. the upstream separatrix density

was consuained by the method discussed in $2.4.3 and 943.1, although for the 3 x 1 0 ~ case this consuaint is only

neccssary during the evolution transient. We note that detachment appears mainly near the separritrix (often seen in expc r imenr s~~f~ f l~ - plfi71 ). The increase of near target density has two consequences: a) reduction of neutrril penetration,

which leads to a translation of the reactive region towards the target, a shortening of the convective Te plateau and a

suppression of the supersonic flow into the sheath ( e l ) , b) the appearance of significant volumetric recombination

with S, >> Si, at the density peak (the region of maximum S, is known as the recombination fiont) with the flux tube

average5 approaching unity, <S&/<SQ - 0.1 for To = 3x10~ and <S,&<Sp - 0.5 for ro = loU. The density peak.

sometimes referred to as a plasma buffer, evolves due to a synergism of three physical processes: a) hydrogcnic and

impurity radiation, which cool the divertor plasma, b) charge-exchange collisions, which remove dong-B momentum

from the plasma, and c) volumetric recombination, which increascs the recycling rate between the ionitation and

recombination fronts. Recombination. though not responsible for pressure Ioss (since Scx » S, nearly al1 momentutil

is removed by charge exchange collisions, even when <S,&/<S* - l), plays a major role in this 'condensation'

process by altering the dong-B flow profile and ttie plasma energy balance ( S , has a net cooling effect at lower

density and a net heating effect at higher density). The above synergism has been studied in some detail, both

theoretically and numericdly, and is believed to bt rcspoasible for the experimentally observed unstable translation of

the active region towards the point-'. It is immnive to note that the relative pressure drop pwJpm, and peak-to-

target density ratio nP.l/n, increase only moderately with higher density. an increase which can be atmbuteci more to

S, than to To. If we consider that plasma-neutral interactions involve a product of neutral and plasma densities. and

scale linearly with n, we would expect o d y lhov process involvhg d. such as recombination. to lead to non-

linearities in the degree of plasma detachement The above two cases conf ' i i this prediction. The appearance of flow

reversal on the separauix flux tube is related to a higher degree of over-ionization. QL") c O. Although the reversed

flow is weak in absolute terms, the dong-B variations of nu would be sufkient to cause strong flow-reversal if the

inner particle flux was allowed to increase or if QL"' was assumed to have a weaker density scaling (hence a more

uniform dong-B profile; as mentioned in $2.4.3. strong over-ionization increases the sensitivity to QI'L' which could

lead to inaccwacies in the OSM2 solution; this point will be explored in 84.2.1 ahead). Finally, we note chat the

appearance of the peak in the separaûix p, profiles is a numericd rather than a physical effect, It is caused by loss of

information while transfemng the solution U from the fine, dong-B grid & on which it is converged to the coarse.

poloidal grid X used for NIMBUS calculations. This is confmed by cornparing p, of Fig.4.1.3 1 which is based on

grid X with F"' of Fig.4.1.32 wbich is based on &; the absence of the total pressure peak in the latter case, shows that

grid coarseness introduces inaccuracy to the coupled OSMUMMBUS cycle. In the future, it would be desirable to

refine the coarse Monte-Carlo grid adaptively in order to prevent such behavior (such adaptation is k ing developed as

part of a separate Ph.D. project for the EIRENE poloidal grids generated by the SONNET magnetic code, and may be

available for the OSMUEIRENE code combination within the near future).

The final case in $4.1. Fig.4.1.35, shows an example of non-uniforrn radial iarget profiles (Tc and To

decreasing roughly linearly across the target from 8 eV to 1 eV and 10") m-'s-' to 10" m"s-'); these profiles are

somcwhat more rcprcsentativc of expcrimenr. Radial variations of Tc, T,, nu, ne and p,, clcarly reflcct the imposed

boundary profiles; i t is intercsting to note the increase of radial nu gradients with temperature, caused by a shortcr

neuual penetration length. Carbon impurity transport is enhanced dong the hotter separalnx flux tube, with a larger net

pcnctration to the core.

In concIuding this section, let us compare the above solutions with the predictions of the two-point model,

discussed in 92.4.3. This study may be considered as part of the validation process, which along with procedural

transparency and code documentation, is essentiai for providing the readerluser with a sufficicnt degree of confidence

in code results (the alternative, a black box approach, is rarely conducive to good science, while having the potential to

crcate disastrous rnistakes). The modified two-point model equations (2.4.6) with the target and upstream plasma

conditions obtained by the 0SM2 solution, will be used to calculate the convective. radiative and pressure corrections

(4, 4, 6p)2PM. The, corrections will then be utimated independcntiy by examining the sources and fluxes of the

0SM2 solution. (4, 4, %)OsM and the two ses of numbers will be cornparecl for the attached-detached transition of

94.1 . Based on (2.4.6) and the implied defini tions, we may write the above corrections explicitly as,

Table 4.1.1 contains a summary of inner target and upstream (mid-point) conditions for the solutions ranging

from the sheath limited (32 eV) to detached (05) regimes, at four radial locations (the same flux tubes as in the figures,

ir = 8, 10. 12, 14). The trends already observed in the course of the section are well brought out by this sumrnary:

upstream conditions remain rather insensitive to even large changes in target conditions. in agreement with the

predictions of the two-point model. Table 4.1.2 shows the input power PL, the inner target heat flux ~ L L and the

comparison between the correction factors as detined in (4.1. L); inverse delta's are shown in order to increase accuracy.

The observed trends also agree with the observations made earlier: radiative, charge exchange and convectivc

contributions increase with lower target temperature and the onset of detachment. The agreement between the 2PM and

OSM factors is very good for 6, (which indicates energy conservation in the CFD equations), reasonably good for 8,

(slight différences can be attributed to viscous fluxes). and least good for 6, (convectional contribution is to be assumed

uniform in the modified two-point model. while in fact it is confined to the recycling (divertor) region. so chat $"" <

6,0s''; as expected. the discrepancy grows with detachment). The systematic error may also be attributed to the factor

71-1 in the denominator of 4"" which corresponds to uniform power deposition: for the default X-point to X-point

profile this factor cakes on some value between 714 (uniform) and 712 (upstream). A slightly more direct comparison is

shown in Table 4.1.3, whcre the upstream h, Te and (T,+Ti)/2 values of the 0SM2 solution are cornpared with the

simple (uncorrected. 4''" =6PM =1) and corrected (40SM. %OsL( fiom Table 4.1.2) two-point model. We can

conclude that the 0SM2 solver broadly agrees witb the conected two-point treatment across a wide range of SOL

conditions.

4.2 Sensitivity to Moddiag Assumptions

The validation of any mode1 requues an examination of the sensitivity of the solution to pivotai modeling

assumptions. In the present context, this pivotal assumption is precisely the QJU] ansatz. discussed in 94.2.1. In

addition, by changing a single parameter in a multi-variable problem we are able to isolate the effect of any physical

process represented by the modei equations. thereby examining its role in the overall behaviour of the system. The

division of this section reflects the morphology of sources introduced in 53.4.5: the sensitivity to cross-field sources

Ql, neutral sources Q, plasma sources Q,, flux-expansion sources Qe and boundary conditions (y,, yi) is studied in

turn. However, it should be stressed that while the sensitivity to QI lies at the very heart of the maner, rhe sensitivity to

the other sources does not (ie. it will not affect our conclusions regarding the validity of the OSM approach). Due to

space limitations, only dong-% plots are shown for each case considered, provided that noticeable departure from the

default mode1 has occurred; changes of key quantities are also summarized in Table 4.2.1. The study will focus on one

attached (case A: 8 eV, ~d-' m2s-l) and two detachcd cases (case B: 0.5 eV, los m"s-'; case C: 0.5 eV, 3 x l P m-*s-').

4.2.1 Sensitivity to cross-field sources, QL: the QAW] ansan

In the context of the OSM approach. as defined by (2.4.10). the crucial assumption is that plasma solution U is

only weakly sensitive to the formal (user specified) dependence of the cross-field sources on U, QL[U]. This

assumption which due to its pivotal importance has been named the QJU] ansatz, has already been examined in

$2.4.3. where it was concluded that spatial distribution of the particle cross-field source a'" raises the largest question

about the validity of the OSM approach. It is therefore appropriate that we begin our analysis at this point. In the

default modet, QL") = VI) = II, was assumed, corresponding to radial diffusion with a constant radial lengtfi. In

Fig.4.2.1-2 rhe effect of assuming QI'" a constant is examined. As expected. case A is virtually unaffected. with only

slight changes to upsvearn flow (nuu&5%) and density (nU?5%); the arrows indicate increaseldecrease and are followed

by per-cent chanps. Case B is only mildly affected. wiih srnall changes in peak (nm.L208. noF'~lO%) and

upstrcam (n,-120%, T,Tzo%) values, suonger along-B gradients in p,, and nu. and the appearance of weak flow

reversa1 on the separatrix ring (as expected for a more unifomn cross-field particle sink due to over-ionization. C",, >

1 , QI"' < O). This relative insensitivity. despite strong difference between a(') a ne and = constant (note that

npeak/n,, - 15). is consistent with the conclusion drawn in 82.4.3, that the QJU] ansarz is satisfied for the mass transport

equation provided that overlunder-ionization is no[ excessive. f<" < 2. For case C. a weaker variation was adapted. QI"' = n,lE, which changes the max-min along-B variation of n, from - 20 to - 4.5 (this choice was preferred at higher

dcnsities over the highly unphysical QL") = constant). nie results. Fig.4.2.3. show a pronounced increase in flow

reversal and upsueam Ti. accompanied by a reduction in ~hc degree of detachment (p,,Jpm,&25%) for al1 SOL rings.

Peak values of density and plasma flow remain roughly the sarne. We must conclude that as the degree of ionization

increases beyond f'' > 2, the OSM2 solutions becorne sensitive to the assumed &')[u] scaling, a conclusion which

has been anticipated by the earlier anaiysis of the QJU] ansatz. However, we should qualify the above staiement with

two comments: a) the changes for case C, which are at most a factor of 2. are stili within the m a n of accuracy of

most edge diagnostics. b) the adjunment of the spatial form of Q L " ' ~ hu iis quivalent in varying the assumed DI

values in 2D fluid codes (in both cases. the cross-field uansport is the biggest unknown). In keeping with the DVM

approach, additional upstream information should be used to adjust this scaling in such a way as to best match al1 the

diagnostic data (both target and upstream); it is clcar that for detachmeni, it is important to have input information both

below and above the region of intense atornic interactions; the task of including this extended upstream data will be left

for future refinemenis.

The sensitivity to the W2'm dependence was investigated in a similar manner, Fig.4.2.4-6. As the results

indicate, the effect of the cross-field momentum source is comparable to that of the particle source: weak for case A

and B. and pronounced for case C (since the momentum source is introduced in the default OSM-DVM mode1 in order

to match the specified target flow patterns, we may expect its role to increase with the appearance of recombination

which directly alters these patterns). We should however note, with selfconsistent ï, data available at both targets, the

cross-field momentum sources should remain relatively s d 1 .

The effect of changing the cross-field (deposited) power profile from the default. uniform double-step function

(2.4.13) to a pressure dependent a pi, QL") = pI) double-step function is negligible, aside from a slight ( ~ 1 0 % )

decrease in %W. Fig.4.2.7-8. The same was found for a field dependent = B. &(4) a B) double-step function

(not shown). In al1 three cases, the asymmetty remains heavily shifted towards the inner side, with Iittle alteration

(<IO%,. Despite the highly artificial division into inner and outer regions (which could be removed by a linear

variation), the plasma solution is remarkably continuous across the upstream rnid-point- We can safely conclude that

due to the r. = T;V': waling, the crossfield power scaling ~ r ' " " [ ~ 1 does noi significantly affect the final solution

(irrespective of the degree of detachment).

4.2.2 Sensitivity to neueral sources, Qg

The effect of neutral sources, unlike that of cross-field sources, is not an issue touching on the question of the

validity of the OSM approach (neutral and along-B physics, including , Q, . Qg and (y,. y,), may be pro_gressively

rcfincd as supcrior atomic modcls becorne availablc). As a consequcncc. the followinp sections may bc vicwcd as an

exploration of the physics rcpresented by the mode1 equations, rather than as validation of the OSM approach as such.

The effect of turning off recombination is shown in Fig.4.2.10-11 (case A is not shown. since the change is

imperceptible). Case B is vinually unaffected. with a slipht drop of the densiry and plasma flow peaks (np"'L5%:

nuw""5%). In contrat, case C shows strong changes on thc separauix ring (nPcakI3~%; ~ u ~ ' ~ T s o % ; pLT15%;

~,.,T20%) and weaker changes clsewhere in the SOL. Notably, p,,Jp,,, is unchanged on the separauix ring, while

being reduced from -4 to -3 on outer SOL rings. This behaviour may be explained by noting that even in the presencc

of significant recombination (S, - Sa, nearly al1 momentum is rernoved by CX collisions (Scx >> S,, a result

dependent solely on the ratio ~av>,J<av>~~ ce 1, see Fig.2.3.1). We may deduce that recombination alters the degree

of detachment indirectly, via the particle and energy balances (by coolingheating the plasma and altering the plasma

flow in the active region). The next step is to consider the degree of ova/under-ionization f " and the default QL") =

scaling (in both the detacheci cases, the separatrix ring is over-ionid, f!') > 1. QL'" < 0, with outer SOL rings under-

ionized. fl)< 1 , QI"' > O). Consequently. by setting S, = 0, the separatrix ring becomes more over-ionized (et)? 30%.

from 2.7 to 3.7) and QL(') c O becornes an evcn stronger sink for plasma particles (this is the reason for the initially

surprising increase of n,w for S, = 0); lilrewise, the other SOL rings become less under-ionized which decreases the

positive QL('). Unfortunately, the effect of S, on the energy balance could not be ueated properly since only the

+13.6eVxS, term is controlled directly, ihe other t e m (radiative cooIing due to recombination into excited atomic

states) are returned en-mass from NIMBUS. We may conclude that although recombination contributes to detachment,

it is not necessary for its appearance.

The effect of neutrai velocity on momentum l o s is show in Fig.4.2.12-14. where vIH = O of the default

model, which represents the maximum amount of momentum rernoval from the ions, was replaced by V: = 0 . 5 ~ . ie. the

neutrals were assumed to move everywhere at half the plasma velocity. Both of these assumptions are crude, and more

advanced estirnates of vpH should be made in the future. Case A is only mildly affected by this reduction in the amount

of momentum loss (p,,~p,& 10%. fR' 150%) wUle the impact on cases B and C is more pronounced (n*&50%;

P m L J p m L t & ~ ~ % , f(2)-h0%). The results confirm that momenturn loss is the dominant physical processes responsible for

plasma detachment (ie. the upstream-to-target pressure ratio is directly conelated to the mornentum rernoval rate; the

presence of over-ionization and recombination introduce non-linearity into the otherwise linear dependence).

In t+e default model. ion-neutral energy term Qqi was set to zero in agreement with the default senings of the

EDGE2D code; the effect of non-zero & (calculated according to the 1 s t paragrisph in 02.3) is examined below.

Fig.4.2.15-16. For case A, ion-neunal eaagy exchange is much smaller than elecuon cooling, Qqi « Q, and the

default solution remains vinually unchangui. With Q, turned on. there is no solution for case B which would satisfy

Te, = 0.5 eV (the reason is simple: in the Te plateau region, Te < 2 eV, the ions are heated by charge-exchange collisions

with Franck-Condon atoms more effectively thm they are cooled by the ekctrons, which in turn loose energy due to

Q, < 0; the target electron temperatwe can not be reduced to 0 5 eV, even if the input power approaches zero). This

observation points to the potential importance of Qqi for detachment, and we musc conclude that the case B (0.5 eV.

lo3 m.'s") is deficient in physical content with the default setting of Qqi = 0. and must be treated with caution. A

solution may be found for case C because at highcr densities electron cooling comes to dominate over CX ion heating

{Q,, > Q,,) ailowing the plateau temperature to drop below the F-C value of - 2 eV. Input power and upstream

temperature increase to cornpensate for the Qqi energy losses in the T, > 2 eV region (pLT30%; T,-,7306). The degree

of detachment is reduced (p,,~p,,&20%) fm al1 SOL rings, while on the separatrix ring the desee of over-ionization

increases (t1'T25%). leading to flow revend and panially compensating for the reduction in p,Jp,,, due io Q,,

heating). A still higher density case (0.5 eV. 10" m"s-'). Fip.4.2.17. shows that Q , providcs a stabiliting rnechanism

rnissing from the default model: Te, adjustmcnts are no longer necessary to stabilize the solution (ion heating opposes

further recombination and density peak growth). The finite Qqi model is clearly the preferred ahernative.

In Fig 4.2.18-20, the effect of neglecting elecuon cooling due to impurities is shown (this is done by setting fi

= O in Qz = fzQw < O. cf. the default value of fi= 1). Since this reduces the power removed from the plasma, the power

deposited into the SOL musc also decrease in order to achieve the samc target Te. This is observed in varying degrees in

the ihree caser (A: ~ ~ 4 3 0 % . B: ~ ~ 4 5 0 % . C: P,Iu)<~); due to the r. a T:~ raling. the decrease of upstream Te is

smaller for hotter plasmas. The degree of detachment p,,#p,, is only marginally altered by sening f i = O, which is to

be expected since the near target conditions, which detennine the arnount of momentum loss, remain largely

unchanged; however in each case the same pressure ratio correspnds to smaller input power. We may therefore

conclude that impurïty radiation enhances detachment by cooling the plasma Since the treatment of impurïty radiation

in the default model is relatively crude, future refinements are strongly advisable (an iterative coupling of the 0SM2

and DlVïMP codes would pmvide the optimum, albcit computationally costly, algorithm).

4.2.3 Sensitivity to plasma sources, Qp

The effect of plasma sources is confined to transferring energy between the ion and electron channeis;

however, since energy is transpocted within and removed fkom these channeb by different physical processes, the

energy coupling provided by electron-ion collisions has a strong effect on the overall plasma behaviour. Plasma sources

Q, (2.4.14). are composed of a coIIisional term Q, E (p,-pi)/% , which tends to equaiize Te and Ti, and an elecmc field

term, Q = uVllp,, largest in the vicinity of the target, which transfers energy fiom the electrons to the ions.

The role of collisiond energy equipartition may be exarnined by setting Q, = 0, Fig.4.2.21-22. The ion

temperature is seen to rise drastically when the strong ion energy sink is removed (the cross-field input power into the

ion and electron channels is assumed to k equal in the default model, Q": = Q(~'J This rise of near target Ti to values

of wel1 over 5 eV, sharply d u c e s the rate of charge-exchange collisions and the associated momentum loss from the

plasma (p,,Jp,,, < 2); this is not caused by the energy dependence of the charge-exchange cross-section, but rather by

the drop of ion and neutral dcnsities (pressure balance with higher Ti and TH ). In short, collisional energy equipartition

is necessary for the appearance of detachment, due to its role of cooling the ions in the divertor volume.

4.2.4 Sensitivity to flux expansion sources,

Flux expansion sources arise in the plasma transport equations due to dong-8 variations of the magnetic field

(2.1.22). Their effect on the default solutions is best explored by setting QB = 0, Fig.4.2.23-25. In a11 cases. the

scparauix flux tube is the most affected, the totai pressure profile becoming more uniform and symrneuica1. For crisr:

A. the separatrix density and flow profiles approach the profiles elsewhere in the SOL, suggesting that radiai

differences present in the default case A are caused by B-field variations near the X-point (where the scparatrix and

outcr SOL magnetic geomeuy differs the most) rather than by neutral source variations. The effect on case B is sirnilar.

with only rninor changes in the overall plasma solution (nF"k.L20%; p,,Jp,,, - samc). For case C. the cfkc t of QB = O

becomcs more pronounced, increasing the degree of over-ionization and upstream temperature on the separatrix ring

(f"T20s; T ~ . . ~ O % ; n*f20%: n u P Y L h 0 ~ ) , while decreasing the degree of detachment (p,~p,,,L~~%.) on the

outer SOL rings. It is reassuring that, although a measurable effect, flux expansion does not greatiy alter the overall

plasma behaviour (QB is often ignored in simpler treaunents of the SOL, such as tbe two-point model or some of the

earlicr versions of onion-skia modeling).

3.2.5 Sensitivity to kinetic boundary conditions, {y, 11

The kinetic origin of the ion and electron sheath heat transmission coefficients was discussed in some derail in

52.2, and their effect on the near target plasma profile has already been alluded to in 64.1. In this section. this effect is

exarnined by reducing first y, and then yi by unity.

In Fig.4.2.26-28. y, has been reduced b m the default value of y, = 5 to y, = 4. Case A 1s only mildly affccted

by the smaller power required to satisQ the bouadary conditions, q, = yeTeTg (PLL~w; Te.u,.-same). The most direct

effect on Case B is the lowering of the Te plateau (since lower y, requires a smaller VaTe at the target, see 02.2). which

increases rLp"L by - 30%; the degree of d e t a c h n t p d p , , remains unchanged. The effect becomes more pronounced

for case C. when the plateau temperature govms the density peak evolution; wiL lower Te. n,* increases and the

solution is stabilized with the additional constraint II,,- < 10'' m').

Sirnilarly , in Fig.4.2.29-3 1. yi has ban rcduccd from the default value of y, = 2 5 + M ~ ~ ( T ~ . ~ + T ~ ~ ) ~ ( ~ T ~ ~ ) to r, =

1.5 + ~ ( T c o + ~ L o ) / ( 2 ~ i o ) . For case A, the near target Ti increases since a higher ion temperature is necessary to carry

the same amount of ion power to the target. qi = %Tiro (note that, ion convection is much sbonger than ion conduction

at the target); this cames over into slight changes to n, and p,, but since Tc, and not Ti, determines the amount of input

power, PL remains vinually unchanged. For case B. the effect of lowering yi by uni9 is nearly the same as chat of

lowering y, by the same amount. (the plateau Te decreases and nw increases as before). with one notable difference:

namely a suppression of the supersonic flow at the target The similarity between the sensitivity to y, and yi for lower Te

is consistent with strong electron-ion energy coupling (via collisional equipartition, Q,) for colder plateau conditions.

4.3 Cornparison with the EDGEZD/NIMBUS code

As was mentioned in 02.4.3, the onion-skin methocl is not intended to replace but to complernent the existing

2D fluid codes; it is an interpretive tool designed primarily for experimental use. By the same token existing 2D fluid

codes are primarily predicrive tools. although they are also frequently used to interpret experirnents. A cornparison of

thcse two approachcs would not only provide an assessrnent of the onion-skin method, but could tell us a lot about thc

relative importance of cross-field transport in the edge plasma. ln this section. such a cornparison is presented based on

the O S M W B U S and EDGEZDNiMBUS codes (the latter code was selected because of its close association with

the JET tokamak). The radial target profiles of Te and To extracted from the EDGE2D solution are passed into OSM2 as

thc diagnostic target consuaints. The two solutions are then compared everywhere in the SOL.

The EDGE2DMIMBUS solution. shown in Fig.4.3.1-4. was obtained with the following input: power flux

across the separatrix PL = 1 MW, separatrix density n, = O . S * I O ' ~ m". cross-field transport coefficients DL = 0.15

m2/s. XiL = 1 rn2/s. xCL = 1 m2/s. and kinetic boundary conditions y. = 5.0. y, = 2.5 + &2(~e.o+~i8)/(2~i.o). The SOL

plasma appears to be well attached and conductivcly limited, with some asymmeuy between the inner and outer urges

(increasing away from the seperatrix) and with significant radial gradients across the SOL (b, Te, Ti, and p, decay

away from the separatrix).

The OSM2/NIMBUS solution, shown in Fig.4.3.5-8, wos obtained by speci@ing the EDGE2D target values of

plasma flow and electron temperature ( To, Tt, Tc& Tc') (r) as the diagnostic constraints for the OSM-DVM approach.

Special care was taken to insure that OSM2 and EDGEZD rnodtling cquations differed only in the treatment of radial

fiuxes: a) neutral sources, including Qm and Q,, were calculated dücclly by NIMBUS, b) impurity contributions were

neglected, fi = O, c) the same kinetic boundary conditions {y, , x) were used, d) flux limited heat conductivities were

replaced by classical values, e) the cross-field source scaling QJUl was specified as for the default OSM-DVM mode1

of in 92.4.3. Comparison reveals close agreement between the OSM2 and EDGE2D solutions, both in the along-B and

cross-field directions, Fig.4.3.9-IO, with ody minor differences: OSM2 predicts slightly larger dong-B gradients near

the targets, more extended and asymmecricid flow patterns, and more uniform total pressure profiles. The near target

drop of separatrix p, in the EDGE2D solution is accompanied by spatial oscillation (perhaps due to the coarseness of

the near target cells). The pressure drop on the separatrix in the EDGE2D solution has been Iinked to cross-field

viscous effects; when cross-field viscosity is se; to zero in EDGEZD, total pressure becomes more uniform. improving

agreement between the two codes . Aside fiom these relatively srnail differences, the overd1 2D plasma distribution is

very closely replicated. It is worth considering the implications of this result, namely that the claim sometimes made

against the OSM method as king devoid of cross-field content is misinformed, The radial information enters 0SM2

through the radial profiles of the diagnostic target quaniities, r&), Te.o (r), ïL(r), Tc.' (r). The cross-field sources QI are

then adjusted in response to these radial profiles. Replacing FL + GL (calculatcd with specified DL . riL . xeL ) by the

much sirnpler QL has apparently only a minor effect on the final solution (at least for the conductively-limited regime

considered here). The only explanation consistent with this result is that the precise ffux-tube location of cross-field

transport is far less important chan i ts flux-tube integrated value (rapid dong-8 uansport effec tivel y redistri butes both

particle and power entenng the flux tube. the redistribution of particles king slower due to ion inertia).

4.4 Comparison with expriment: reciprocating probe (RCP) measurements on JET

The final application of the OSM2/NIMBUS code examined in this thesis is perhaps the most important. Since

the onion-shn mcthod was repeatedly identified as an interpetive tool, its uitirnate test must involve a comparison with

experiment. For this purpose. the upstream reciprocating Langmuir probe (RCP) data on the JET tokamak was

cornpared with the results of thc OSM2/NiMBUS simulation (target Langmuir probe data was uscd as the diagnostic

constraint); Li-beam and LIDAR measurernents are also available for some shots, but the RCP data and the threc

methods are in rough agreement as to the radial decay lengths of n, and Tc. Five shots were selected from more than

twcnty cxamined: an Ohmic shot (48310. t = 54.3 sec). three L-mode shois (15702. t = 6 1 sec; 45791. t = 63 sec:

45798. t = 63.5 sec), and one H-mode shots (47734, 61 sec). Dcfault modciing assumptions of g2.4.3 werc adaptcd.

with the exception of Qcx and Q,,, which were obtained directly h m MMBUS.

For the Ohmic shots, the 0SM2 solution, Fig.4.4.1-2, cxhibits the farniliar conductively limited behaviour

with strong radial gradients and relatively low upsueam densitics. The RC probe is inserted and withdrawn from the

SOL vertically near the top of the toms (R = 3.25 m). Unfonwately. there is no reliable method of determining the

location of the separatrix. Because of the large size of the plasma in comparison io the small SOL thickness. the errors

rnay be significant. The procedure often employed to relate target and upstream flux tubes relies on total pressure

conservation in the absence of detachment In other words the upstream radial profiles are shifted such that p,u/p,t - 1

(in our case this required a shifi by 18 mm); flux tubes are then identified by the distance away from the separatrix at

the outer rnid-plane, rd . The measurtd and caiculated values are plotted as a function of this ordinate in Fig.4.4.3-4

(the RCP data is highly scattered and in the absence of experimental error analysis, 20% error bars were assumed for n,

and Te, and 50% error bars for the Mach number). The measured (RCP) and predicted (OSM) values agrce to within the

assurned experimental enor, slight deparcures on the separatrix ring are iikely caused by loss of accuracy as the probe

enters the core plasma. Plasma flow differs substantially for a number of reasons: a) Mach probe interprctation depends

on the anomalous cross-field transport coefficients, and may contain a large systematic error. b) drifts in the

diamagnetic direction, which are at present not included in the OSM2 code, have been shown to significantly alter

along-B flow patterns in the SOL, c ) the assumed cross-field particle source may need to be varied in order to mtch

the observed flow. Siniilar level of agreement is obtained for the L-mode shots, Fig.4.4.5-10, and the H-mode shot for

which RCP data was available, Fig.4.4.11-12 (it is difficult to say whether the agreement with the H-mode Mach

number is significant or simply fortuitous). It is hoped that the above results illustrate the utility of the OSM-DVM

approach (each OSMUMMBUS solution required only - 6 hrs of CPU time on a typical IBM RS6ûûû work station):

with a modest irnprovement in microprocessor speed, 2-D SOL plasma distributions may be reconstructed based on

target diagnostic data within one hour (this is the ciosest to real-timehter-shot analysis which is possible at present).

5.0 Conclusions

The improved OSM approach, as defined by (2.4.10). has k n successful at modeling al1 the regimes of

tokamak edge plasma behaviour, including plasma detachment. The regions of validity of the above approach have

been investigated both theoretically, by investigating the validity of the QJU] ansatz, and numerically, by examining

the sensitivity of the solution on the assumed form of cross-field sources, leading to similar conclusions: a) the solution

is highly insensitive to cross-field power sources for ail SOL regimes. b) it is only wealdy insensitive to cross-field

momentum and particle sources for attached SOL conditions, and c) it may become sensitive to cross-field momentum

and particle sources for certain detached conditions. Since the arnount of cross-field momentum flow plays a relatively

smaiier role than cross-field particle flow in actual tokamak edge pIasmas, the iatter effect may be singled out as the

key sensitivity; the requiretnent that overlunder-ionization with respect to the targets does not exceed a factor of two on

any flux tube may be used as a rough validity criterion of the above OSM approach. As discussed in 54.2.1, this

criterion can be mitigated by iniroducing additional diagnostic data upstrtam of the ionization front; even in the present

form, the sensitivities observeci for strong detachment remain within the bounds of accuracy for most edge diagnostics.

Code-code comparison with a standard 2D edge plasma modeling approach (EDGE2D/NIMBUS vs.

OSM2NiMBUS) has s h o w good agreement between the two codes. This positive result is the best confirmation so Far

of the predicted correspondence between the OSM and the standard modeling approach, and one of the key results

presented in this work- It establishes, at the very least, the practicd usefulness of the OSM approach, which produces a

nearly idenucal solution in a much short time (-6 hours) than that required for the EDûE2D parameter scan (- 1 week).

Code-experiment comparison between the upstrearn profiles of Te and n, obtained by OSM2/NLMBUS and

those measured by reciprocating probe data on the JET tokamak for Ohmic, L-mode and H-mode discharges, further

reinforce the above conclusions; the profiles agree to within the experimental error of the probe, estirnated at 20%.

The improved OSM methocl could greatly enhance our interpretive capability of edge plasma behaviour. It has

the potential for incorporating unlimited number of diagnostic channeis, for being used in 'real time* for inter-shot

analysis, for exuaction of flux-surface averaged anomalous cross-field uanspon coefficients, and for inclusion within

cxisting edge plasma codes such as B2, EDGE2D or UEDGE. It is presently being installcd as a routine analysis tool at

JET. and funher improvements to the OSM2 code (drifts. currents. cross-field source dependence on gradients.

upstrearn diagnostic data) are anticipated.

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Fig.2.1.1: Charged particle orbits in the presence of a magnetic field B

Fig.2.2.1: Plasma-swface vs. neutral-silrface interactions

Fig.2.2.2: Ion and elecuon velocity dislributions Fig.2.3.1: A xhematic of hydrogen recycling in the collisional plasma. the collisionless pre- inchding bath voIumetric and wall interactions. sheath and rhc elcctrostritic shcath.

Fig.2.3.2: Energy spectnim of a hydrogen atom, H Fig.2.3.3: Energy spectnim of a hydrogen molecule. Hz as a function of the nuclear separacion

Fig.2.3.4: Radiative potential for several clements Fig.2.3.5: Hydrogenic cross-sections for dominant as a function o f electron temperature. T,[Do182) plasrnaheutral interactions as a function of Te[Jari9S]

Fig.2.4.1: Schematic rcprcstntation of Limiter SOL Fig.2.4.2: Schematic qrtsentation of divenor SOL

Fig.3.1.1: A typical poloidal grid used for divertor Fig:3.1-2: A close up of the same grid in the vicinity edge plasma modeling o n the JET tokamak This of the X-point. including the divenor volume. grid was usa i for majonty of the results in 84.

Fig.4.1.1: Along-B plots of plasma variables for the 32 eV. 1 0 ~ m.'s-' case at four radial locations in the SOL (solid Iine, separarrix ring, k=8; dashed line, ir=lO; dotdashed line, ir=12; dotted line. ir=14).

Fig.4.I .2: Along-B plots of sources and fluxes for the 32 eV. IO* mas-' case on the sepaaaix ring. k=8. Sources: light grey line, cross-field; dark grey line, neutrai; black line. flux expansion; dotted line, plasma; thick line, total Q. Huxes: grey line. F; black line. G; dotted line, Q intepted from O to s,,: thick line. sum.

Figure 4.1.3: Poloidal contour plots of plasma variables and neutrai densities for the 32 eV case (absolute value of the Mach number is shown).

Figure 4.1.4: Poloidal contour plots of neutrd sources for the 32 eV case: ionization So [m*3s"], cecombination ~,[m"s-'1, , momenturn removal Qa = a u - vkH)Sa [Paim] and elecrron cooling Q, ~ r n - ' ) (absolute vdue of Q is shom). Also i ~ l u d c d is the average acon enngy EH 5 I [eV] and intensity of H, Iine emission [m' s- 1.

Figure 4.1.5: Poloidal contour plots of the carbon cloud originating at the target for the 32 eV case. Show am particle densitics of the carbon neutrals and the fmt four ionization s t v u [m-'1. and the total amount of radiation from al1 the carbon species [ ~ m - ' 1 (compare Qc with Q, of the previous plot).

Fig.4.1.6: Along-B plots of plasma variables for the 16 eV. 1 0 ~ m'*s" case at four radial locations in the SOL (solid line, separatrix ring, i r 4 ; dashed liiie, ir=lO; dot-dashed line. k 1 2 ; dotted line, ir=14).

Fig.4.1.7: Along-B plots of sources and fluxes for the 16 eV. 1 0 ~ ~ r n - 5 - I case on the separatrix ring, i r 8 . Sources: Iight grey line, cross-field; dark grey the, neutral; black line, flux expansion; dotted line, plasma; thick line, total Q. Huxes: grey line, F; black line, G; don& line, Q integrated from O to %; ihick iine, sum.

FigA.I.8: Along-B plots of plasma variables for the 8 eV. 10" rn'*s-' case ai four radial locations in the SOL (solid line, scparatrix ring. i d ; dashed line, ir=lO; dot-dashed line, ir=12; dotted line, ir=14).

Fig.4.1.9: Along-B plots of sources and fluxes for the 8 eV, IO= m'*il case on the separauix ring. ir=8. Sources: light grey line, cross-field; dark grey Iine, neuaal; black line, flux expansion; dotted line. plasma; thick line. total Q. Fiuxes: grey line, F; black Iine, G; dotted line, Q integrated h m O to ss; thick line, surn.

Fig.4.1.10: Along-B plots of plasma variables for the 8 eV. IO= m'*s-'case in the vicinity of the inner target on the scparavix ring, ir=8. Circles indicaie fine grid point values, while squares indicate coarse grid points.

Fig.4.1.11: Along-B plots of sources and fluxes for the 8 eV, loU rn-2s-' case in the vicinity of the inner target on the separatrix ring, ir=8. Sources: light grey line, cross-field; dark grey line, neutral; black Iine, flux expansion; dotted line, plasma; thick line. total Q. Fluxes: grey line. F; black line. G; dotted line, Q integrated from O to s,,; thick line, sum.

Figure 4-1-12 Poloidal contour plots of the carbon cloud originatuig at the targct for the 8 eV case. Shown are particle densities of the carbon ncuuais and the first four ionization states ~rn-~] . and the totai amount of ndiatîon h m al1 the cPrbon rpcies [ ~ rn " ] (cornparc Qc with Q, of the previous plot).

Fig.4.1.13: Along-B plots of plasma variables for the 4 eV. IO= m=Ls-' case at four radial locations in the SOL (solid line, separauix ring, -; dashed linc, ir= t 0; dotdashed line, ir= 12; dotted line, ir= 14).

Fig.4.1.14: Along-B plots of sources and flunes for the 4 eV, loU m"s" case on the separatrix ring. ir=8. Sources: light grey line, cross-field; dark grey Iine, neutral; black line, flux expansion; dotted line, plasma; thick line, total Q. Fluxes: grey line, F; black line, G; dotted line, Q integrated fiom O to SI; thick line, sum.

Fig.4.1.15: Along-B plots of plasma variables for the 2 eV. loU m'*il case at four radial locations in the SOL (solid line, separauix ring. k 8 ; dashed line. ir=lO; dot-dashed line. Ll-12; dotted line. ir=14).

FigA. 1.16: Along8 plots of sources and fluxes for the 2 eV. 1oU m-2s-' case on the wparatrix ring, i-8. Sources: light grey line, cross-field; dark grey line, neutral; black line, flux expansion; dotted Iine, plasma; ihick line, total Q. Fluxes: grey Iine, F; black line, G; dotted line, Q integrated From O to su; thick Iine, sum.

Figure 4-1-17 Poloidal coatour plots of the carbon doud originating at the target for the 2 eV case. Shown areparticte densities of the carbon neutrais and the first four ionization States [m"]. and the total amunt of radiation h m al1 the carbon species [~m-'1 (compare Qc with Q, of the previous plot).

Fig.4.l. 18: Along-B plots of plasma variables for the 1 eV. 1oU m-2s" case at four radial locations in the SOL (solid line, separauix ring, ir=8; dashed line, k l O ; dot-dashed line, ir=12; dotted line, ir=14).

Fip.4.l. 19: Along-B plois of sources and fluxes for the 1 eV. los m%-' case on the sscparatrix ring. ir=8. Sources: light grey line, cross-field; dark grey line. neutral; black line, flux expansion; dotted line, plasma; thick line. total Q. Fluxes: grey Iine, F; black line. G; dotted line, Q integrated h m O to SR; thick line. sum.

Fig.4.1.20: Along-B plots of plasma variables for the 0.5 eV. 10" rn'l case ai four radiai locations in the SOL (solid Iine. separatnx ring, id; dashed line, ir=lO; dot-dashed line. ir=12; dotted line, ir=14).

FigA.I.2 1 : Along-B plots of sources and fluxes for the 0-5 eV, 1oU rn-'ll case on the separatrix ring. k 8 . Sources: light grey line, cross-field; dark grey line, neutral; black line, flux expansion; dotted line, plasma; thick iine, total Q. Fluxes: grey line, F; black line, C; dotted line, Q integrated from O to SU; thick line, sum.

Fig.4.1.22: Along-B plots of plasma variables for the 0.5 eV. IO= mm"i'case in the vicinity of the inner target on the separatrix ring, i d . Circles indicate fine grid point values. while squares indicate coarse Md points.

Fig.4.1.23: Along-B plots of sources and fluxes for the O S eV. 1oU rn-*s-' case in the vicinity of the inner iarget on the separatrix ring, ir=8. Sources: light grey line, cross-field; dark grey linc, neutra]; black Iine, flux expansion; dotted line. plasma; thick line, total Q. Fluxes: grey Iine, F; black line, G; dotted line, Q integrated from O to sl; thick line, surn. The energy sources arc dominated bv the clcctric field term of the plasma source.

Figure 4.1.24 Poloidal contour plots of plasma variables and neutrai densitics for the 0.5 eV case (absolute value of the Mach number is shown).

Figure 4-1-25 Poloidai contour plots of neutrai sources for the 0.5 eV case: ionization SU, recombination S, , momcntum removal Qa = m(u - vUL)Sa pdm] and elactron cooling Q, [~ rn - ' ] (absolute vaiue of Q, is shown). Also inchidcd is the average atom emrgy [eV] and intensity of H, line emission [m"s"].

Figure 4-1-26 Poloidai contour plots of the carbon cloud originating at the target for the 0.5 eV case. Shown arc particle dtnsities of the carbon neutrals and the fvst four ionization states and the total amount of raâiation h m al1 the carbon species ~ m - ' ] (compare Qc with Q, of the previous plot).

Fig.4.1.27: Along-B plots of plasma variables for the 8 eV. 3* 10~rn-'s-' case at four radial locations in the SOL (solid line, separatrix ring, i d ; dashed line. i-10; dot-dashed line, ir=12; dotted line, ir=14).

Fig.4.1.28: Along-B plots of sources and fluxes for the 8 eV. 3* lot) rn-'s-' case on the wparacrix ring. ir=8. Sources: light grey line, cross-field; dark grey line, neutrai; black line, flux expansion; dotted Iine, plasma; thick line, total Q. Fluxes: grey Iine. F; bIack line. G; dotted line. Q integrated from O to s,,; thick line. sum.

Fig.4.1.29: Along-B plots of plasma variables for the 8 eV,

0.0 0 2 0.8 1 .O O.' sl,/L,,

102' m%-' case at four radial locations in the SOL (solid line. separatrix ring, ir=8; dashed line, ir=lO; dot-dashed Iine, ir=12; dotted Iine, i-14).

Fig.4.1.30: Along-B plots of soums and Ruxes for the 8 eV, loZ4 m'2s-' case on the separavix ring, ir=8. Sources: Iight grey line, cross-field; dark grey line. neutrai; black line, flux expansion; dotted line, plasma; thick line, total Q. Fiuxes: grey line, F; black fine, G; dotted line. Q integraied from O to s,,; thick Iine. sum.

Fig.4.1.3 1 : Along-B plots of plasma variables for the 0.5 eV. 3* 1 0 ~ ma2s'' case at four radial locations in the SOL (solid line, separatrix ring, ir=8; dashed line, k 1 O ; dot-dashed line. ir=12; doned line, ir= 14).

Fig.4.1.32: Along-B plots of sources and fluxes for the 0.5 eV. 3* loZ3 m-2~-' case on h e separatrix ring, ir=8. Sources: light grey Iine, cross-field; dark grey line. neutral; biack line, flux expansion; dotted line, plasma; thick line, total Q. Fiuxes: grey line. F; black line. G; dotted line, Q integrated from O to su; thick liw, sum.

Fig.4.1.33: Along-B plots of plasma variables for the 0.5 eV, 1oZ4 m-2s" case at four radial locations in the SOL (solid line, separauix ring, ir=8; dashed linc, ir-10; dot-dashed line, ir=12; dotted line, k 1 4 ) . The evolution was consuained by the additional upsueam constra.int, nuYP - 3* 1 0 ' ~ m".

Fig.4.1.34: Along-B plots of sources and fluxes for the 0.5 eV. 10" m"s" case on the separatrix ring. ir=8. Sources: light grey line, cross-field; dark grey line, neutral; black line, flux expansion; dotted line, phsrna; thick line, total Q. Fluxes: grey line. F; black line, G; dotted line, Q integrated from O to sl; thick line, sum.

Fig.4.1.35: Along-B plots of plasma variables for the radiaily decaying target profile case at four radiai locations in the SOL (solid line, separatrix ring, ir=8; dashed line, ir= IO; dotdashed line, ir= 12; dotted line, ir= 14).

nu-t 1 0 A 2 3 m-2s-

1 .O 2 .O 1 .O I .O

1 .O 1 .O 1 .O 1 .O

1 .O 1 .O 1 .O 1 .O

1 .O 1 .O 1 .O i .O

I .O I .O 1 .O I .O

1 .O 1 .O I .O f .O

1 .O 1 .O 1 .O 1 .O

3 .O 3 .O 3 .O 3 .O

10.0 10 .0 10.0 10 .0

Ti-u ne-u eV l(r19m-3

Table 4.1.1 : A listing of inner target and upstream (mid-point) conditions at four radial locations in the SOL (the top row is the separatrix ring, ir =8, followed by ir = 10, 12 and 14) for the attacheddetached transition discussed in section 4.1.

Table 4.1.2: A cornparison of the 0SM2 results and 2PM predictions in terms of the correction factors defined in the text by equation (4.1.1); the rows correspond cxactly to Table 4.1.1. The fmt two columns give the power deposited in the ring and the power reaching the inner target (same flux units).

p-113 - - Te-u n-u n-u ne-u Tave-u Tave-u Tave-u

Table 4.1.3: A similar cornparison as in Table 4.1 .Z but in terms of the upsueam densiiy, electron temperature and average temperature predicted by the simple 2PM. modified 2PM and OSM2.

Fig.4.2.1: Along-B plots of plasma variables for case A with unifonn cross-field particle source at four radial locations in the SOL (solid line, separatrix ring, ir=8; dashed line, ir=10; dot-dashed line, ir=12; dotted line, ir=14). Compare with Fig.4.1.8.

Fig.4.2.2: Along-B plots of plasma variables for case B with uniform cross-field particle source at four radial locations in the SOL (soiid line, separatrix ring, k 8 ; dashed line, ir=IO; dot-dashed line, ir=12; dotted line. ir=14). Compare with Fig.4.1.20.

--

1 MI-]

Fig.4.2.3: Along-B plots of plasma variables for case C with a square root density dependent cross-field particle source at four radial locations in the SOL (solid line, separatrix ring, k 8 ; dashed line. ir=lO; dot-dashed line, ir-12; dotted line, ir=14). Compare with Fig.4.1.3 1 .

Fig.4.2.4: Along-B plots of plasma variables for case A with uniform cross-field momentum source at four radial locations in the SOL (solid line, sepamtrix ring, ir=8; dashed line, ir=lO; dot-dashed iine, ir-12; dotted line, ir= 14). Compare with Fig.4.1.8.

Fig.4.2.5: Along-B plots of plasma variables for case B with uniform cross-field momentum source at four radial locations in the SOL (solid iine, separauix ring, id; dashed line. k I O ; dotdashed line, ir=12; dotted line, ir=14). Compare with Fig.4.1.20.

Fig.4.2.6: Along-B plots of plasma variables for case C wiih a square root density dependent cross-field mometum source at four radial locations in the SOL (solid line, separatrix ring, i d ; dashed Iine, ir=lO; dot-dashed line, i~ 12; dotted line, ir=14). Compare with FigA. 1.3 1.

Fig.4.2.7: Along-B plots of plasma variables for case A with pressure depedent cross-field energy sources at four radial locations in the SOL (solid line, separatrix ring, *; dashed line, ir=lO; dot-dashed line, ir=12; dotted line, ir= 14). Compare with Fig.4.1.8.

Fig.4.2.8: Along4 plots of plasma variables for case B with pressure depedent cross-field energy source at four radial locations in the SOL (solid line, separaûix ring. ir=8; dashed line, ir=lO; dotdashed line, ir=12; dotted line, ir=14). Compare with Fig.4.1.20.

Fig.4.2.9: Along-B plots o f plasma variables for case C with pressure depedent cross-field energy source at four radial locations in the SOL (solid Iine, separatrix ring, ir=8; dashed line, ir=IO; dot-dashed line, ir=12; dotted line, ir=14). Compare with Fig.4.1.3 1 .

Fig.4.2.10: Along-B plots of plasma variables for case B without volumetric recombination, ai four radial locations in the SOL (solid line, separamx ring, ir=8; dashed line, ir=lO; dotdashed line, ir=12; dotted line, ir=14). Compare with Fig.4.1.20.

Fig.4.2.11: Along-B plots of plasma variables for case C without volumeuic recombination, at four radial locations in the SOL (solid line, separauix ring, ir=8; dashed line. ir=lO; dot-dashed line, 1 ~ 1 2 ; dotted line. ir=14). Compare with Fig.4.1.3 1 .

Fig.4.2.12: Along-B plots of plasma variables for case A with neutrai velocity equal to half the plasma velocity, at four radial locations in the SOL (solid line, separatrk ring. il=%; dashed line, ir=lO; dot-dashed line, ir-12; dotted line, ir=14). Compare with Fig.4.1.8.

Fig.4.2.13: Along-B pIots of plasma variables for case B with neutral velocity equal to half the plasma velocity, at four radial locations in the SOL (solid line, separatrix ring, ir=& dashed line, ir=lO; dot-dashed line, ir=12; dotted line, ir= 14). Compare with Fig.4.1.20.

Fig.4.2.14: Along-B plots of plasma variables for case C with neuual velocity qua1 to half the plasma velocity, at four radial locations in the SOL (solid Iine, separatrix ring, -; dashed line, ir=lO; dot-dashed line, ir=12; dotted line, ir= 14). Compare with Figd.l.3 1 .

Fig.4.2.15: Along-B plots of plasma variables for case A including the ion energy term Q+ ar four radial locations in the SOL (solid line, separauix ring, ir=8; dashed line, irz10; dotdashed line, ir=12; dotted line, ir-14). Compare with Fig.4.1.8.

Fig.4.2.16: Along-B plots of plasma variables for case C including the ion encrgy terrn &. at four radial locations in the SOL (solid line, separatrix ring, k=8; dashed line, ir=lO; dotdashed tinc, k 1 2 ; dotted line, ir-14). Compare with Fig.4.I .3 1.

Fip.4.2.17: Along-B plots of plasma variables for the 0.5 eV, 10" m'2s'' case including the ion energy term Q,. at four radial locations in the SOL (solid line, separatrix ring, b8; dashed line, ir=lO; dotdashed line. ir=12; dotted line, irz14). The evolution was stable, even without the hEP constra.int. Compare with Fig. 4.1.33.

Fig.4.2.18: Along-B plots of plasma variables for case A without impurity cooling (& = O) at four radial locations in the SOL (sotid line, separatrix ring, ir=8; dashed line. it=lO; dotdashed line, ir=12; dotted line, ir=14). Compare with Fig.4.1.8.

Fig.4.2.19: Along-B plots of plasma variables for case B without impurity cooling (Q = 0) ût four radial locations in the SOL (solid line, separairix ring, k 8 ; dashed line, ir=lO; dot-dashed line. k l 2 ; dotted line, ir=14). Compare with Fig.4.1.20.

Fig.4.2.20: Along-B p[ots of plasma variables for case C without impurity cooling (& = O ) at four radial locations in the SOL (solid Iine. separatrix ring, ir=8; dashed line, ir=lO; dot-dashed line, ir=12; dotted line. ir=14). Compare with Fig.4.1.3 1 .

Fig.4.2.21: Along-B plots of plasma variables for case A without electron-ion energy equipartition (Q, = O) at four radial locations in the SOL (solid line. separatrix ring, ir=8; dashed line, ir-IO; dot-dashed line, ir=12; dotted line, ir= 14). Compare with FigA. 1.8.

Fig.4.2.22: Along-B plots of plasma variables for case B without elecuon-ion energy equipartition (Q, = O) at four radial locations in the SOL (solid line, separairix ring, dashed line, ir=lO; dot-dashed line, ir= 12; dotted line, ir= 14). Compare with Fig.4.1.20.

Fig.4.2.23: Along-B plots of plasma variables for case C without elecuon-ion energy equipartition (Qi = O) at four radial locations in the SOL (solid line, separatrix ring, ir=8; dashed line, ir=lO; dot-dashed line, ir-12; dotted line, ir= 14). Compare with Fig.4.1.3 1.

Fig.4.2.24: Along-B plots of plasma variables for case A without flux expansion t e m s (QB = O) at four radial locations in the SOL (solid Iine, separatrix ring, k 8 ; dashed line, k l O ; dot-dashed Iine, ir=12; dotted line, ir=14). Compare with Fig.4.1.8.

Fig.4.2.25: Along-B plots of plasma variables for case B without flux expansion tenns (QB = O) at four radial locations in the SOL (solid line, separauix ring, ir=8; dashed line, ir=lO; dot-dashed line, ir=L2; doned line, irzl4). Compare with Fig.4.1.20.

Fig.4.2.26: Along-B plots of plasma variabIes for case C without flux expansion t e m (QB = O) at four radial locations in the SOL (solid line, separatrix ring, i d ; dashed Iine, ir= 10; dot-dashed fine, ir= 12; dotted line, ir= 14). Compare with Fig.4.1.3 1.

Fig.4.2.27: Along-B plots of plasma variables for case A with the electron heat transmission coefficient reduced by unity, at four radiai locations in the SOL (solid line, separauix ring, ir=8; dashed line, ir=iO; dot-dashed line, ir=12; dottcd line, ir= 14). Compare with Fig.4.1.8.

Fig.4.2.28: Along-B plots of plasma variabks for case B with the electron heat transmission coefficient reduced by unity. at four radial locations in the SOL (solid line, separatrix ring. ir=8; dashed line, ir=lO; dot-dashed line. ir-12; dotted line, ir= 14). Compare with FigA.I.2û.

Fig.4.2.29: Along-B plots of plasma variables for case C with the electron heat transmission coefficient reduced by unity, at four radial locations in the SOL (solid line. sepratrix ring, ir=8; dashed line, k l O ; dot-dashed line, ir=12; dotted line. i-14). 'lhe evolution was constrained by the additionai requirement that nuYP - 1 0 ' ~ m". Compare with Fig.4.1.3 1 .

Fig.4.2.30: Along-B plots of plasma variables for case A with the ion k a t transmission coefficient reduced by unity, at four radial locations in the SOL (solid Iine, separatrix ring. *; dashed line, ir=lO; dot-dashed Iine, k 1 2 ; dotted line, ir= 14). Compare with Fig.4.1.8.

Fig.4.2.3 1: Along-B plots of plasma variables for case B with the ion heat transmission coefficient reduced by unity, at four radial locations in rhe SOL (solid line, separatrix ring, id; dashed line, ir=lO; dotdashed line, ir=12; dotted line, ir= 14). Compare with Fig.4.1.20.

Fig.4.2.32: Along-B plots of plasma variables for case C with the ion heat transmission coefficient reduced by unity, at four radial locations in the SOL (soiid line, separatrix ring, ir=8; dashed line. ir=lO; dot-dashed line, ir= 12; dotted line, ir=14). The evolution was constrained by the additional requiremeni chat nuYP - 1 0 ' ~ m-3. Compare with Fig.4.1.3 1.

Case list

default A

Table 4.2.1 : A summary of some key results of the parametnc sensitivity study for the separamx flux tuk. ir=8. Target conditions: Case A: 8 eV, 1 0 ~ ~ rn-'s-'; Case B: 0.5 eV. loU rn%-'; Case C: 0.5 eV. 3 x 1 0 ~ ~ m- 's-l. The variations are defined in the text.

Fig.4.3.1: Along-B plots of plasma variables as calculated by EDGEZDINIMBUS at four radial locations in the SOL (solid line, separatnx ring, ir=8; dashed line, ir=ll; dot-dashed line, ir=14; dotted line, ir=17).

Fig.4.3.2: Along-B plots of plasma variables in the inner divertor volume as calculated by EDGE2D/NIMBUS, at four radial locations in the SOL (solid Iine, k 8 ; dasfied Iine, ir=l 1; dot-dashed line, ir=14; dotted line, ir=17).

Figure 4.3.3: Poloidal contour piots of plasma variables and neutrai densities as calculatcd by EDGE2DMIMBUS (absolute value of the Mach nurnber is shown).

Figure 4.3.4: Poloidai contau plots of neutrai sources hom EDGEZD/MMBUS: ionkation S,, recombinaiion S, , momentum rernoval Qcx = rn(u - vur)Scx [Palml and electron cooling Q, ~ m " ] (absolute value of W; also the average atom energy EH [eV] and intcnsity of H, line emission [m"s"].

Fig.4.3.5: Along-B plots of plasma variables as caiculated by OSM2/NIMBUS at four radial locations in the SOL (solid line, separatrix ring, ir=8; dashed line, ir=ll; dot-dashed line, ir=l4; dotted line, ir=17).

Fig.4.3.6: Along-B plots of plasma variables in the inner divertor volume as calculated by OSM2/MMBUS, at four radial Iocations in the SOL (solid line. i d ; dashed line. ir=ll; dot-dashed line. i r 4 4 ; dotted line, ir=17).

Figure 4.3.7: Poloidal contour plots of plasma variables and neutrai densitics as calculated by OSM2MIMBUS using the wget TCo and TO of tbe EDGE~DIMMBUS result as Langmuir probe data.

Figure 4.3.8: Poloidal contour plots of neutrai sources from OSM2/NIMBUS: ionization SU, recombination S, . momnturn removai Qa< = m(u - vu<)% [Pdm] and clcctron cooling Q, [ ~ r n - ~ ] (absolute vaiue of Q,): also the average aiom energy EH [eV and intensity of H, line emission [m-3s"].

Fig.4.3.9: Along-B plots of a cornparison between EDGE2D and 0SM2 solutions, Fig.4.3. I and Fig.4.3.5 at two radiai locations in the SOL (solid line. 0SM2 at k 8 ; dashed line. EDûE2D at ir=8; dotted line, 0SM2 at i-14; dot-dashed line, EDGE2D at ir=14).

Fig.4.3.10: Along-B plots of a cornparison between EDGE2D and 0SM2 solutions. Fig.4.3.2 and Fig.4.3.6, in the inner divenor volume and two radial locations in the SOL (solid line, 0SM2 at id; dashed line. EDGE2D at ir=8: dotted line, 0SM2 at ir=14; dot-dashed line, EDGEZD at ir=14).

Fig.4.4.1: Along-B plots of plasma variables obtained with OSM2/NIMBUS for the ohmic JET shot 483 10, t = 54.4 sec at four radial locations in the SOL (solid line, separatrix ring, ir=6; dashed Iine, ir=9; dot-dashed line, ir= 12; dotted line, i-15).

Fig.4.4.2: Along-B plots o f plasma variables obtained with OSM2/NLMBUS for the ohmic JET shor 483 10, t = 54.4 sec in the vicinity o f the inner q e t and at four radial locations in the SOL (solid line, separatrix ring. i d ; dashed line, ir=9; dot-dashed line, ir= t 2; dotted line, ir= 15).

Fig.4.4.3: JET shot 48310, t = 54.4 sec (Ohmic). Radial profiles of plasma variables measured by the upstrearn reciprocating probe (RCP) and calculated using OSMZ/NIMBUS (squares) ploned as a function of the radial distance away from the separatrix at the outer midplane; error bars of 20% reflect oniy the RCP data scatter. The RCP data was radially shifted in order to satisfy flux-tube pressure balance, r m = 18 mm (results are shown only over the extent of

the cornputational rcgion). Since TF was not available. T:~ = 7YM was assumed.

Fig.4.4.4: ET shot 483 10. t = 54.4 sec (Ohmic). Same cornparison as in Fig.4.4.3. except with T~~~~ estirnatcd based

on the ion-to-elecuon temperatuture ratio predicted by the 0SM2 code. T~" = T:*

Fig.4.4.5: JET shot 45702, t = 61 sec IL-mode). Radial profiles of plasma variables measured by the upsueam reciprocating probe (RCP) and caiculated using OSM2fNIMBUS (squares) plotted as a function of the radial distance away from the separatrïx at the outer midplane. Radiai shifi, 20 mm. T~~~~ = T~~~~ assumed.

1001-';!-" ..;- h d [mm fi Il

O

Fig.4.4.6: JET shot 45702. t = 61 sec (L-mode). Same cornparison as in Fig.4.4.5. except T~~~~ = T~~~~ CTiOsY~COs")*

Fig.4.4.7: JET shot 4579 1, t = 63 sec (L-mode). Radial profdes of plasma variables measured by the upstream reciprocating probe (RCP) and calculated using OSM2/NIMBUS (squares) plotted as a function of the radial distance away from the separatrix at the outer rnidplane. Radial shift, 20 mm. T ~ ~ ~ = TY* assumai.

Fig.4.4.8: JET shot 45791. t = 63 sec (L-mode). Somc cornparison as in Fig.4.4.7, except TiR8= TioSM * ( T i O S M , y " ) .

Fig.4.4.9: ET shot 45798, t = 63 sec (L-mode). Radial profiles of plasma variables measured by the upstream reciprocating probe (RCP) and calculated using OSM2lNIMBUS (squares) plotted as a function of the radial distance away from the separauix ai the outer midplane. Radial shift, 20 mm. T~~~ = TY* wumed.

Fig.4.4.10: JET shot 45798, t = 63 sec (L-mode). Same cornparison as in Fig.4.4.9. except T~~~~ = T~~~~ +.

f l i 0 S " ~ * S " ) .

35 ' f i T

Fig.4.4.11: JET shot 47734, t = 61 sec (H-mode). Radial profiles of plasma variables measured by the upstream reciprocating probe (RCP) and calculated using OSMWNIMBUS (squares) plotted as a function of the radial distance away from the separatrix at the outer midplane. Radial shift. 8 mm. xRCP = T~~~~ assumed.

Fig.4.4.12: ET shot 47734. t = 61 sec (H-mode). Same cornparison as in Fig.4.4. I I , except TFcP = TYSM * ( ~ ~ ~ ~ ~ r r ~ ~ ~ ~ ) .

A.l Energy ia îhe XXI Century and N u d m Fusion

The millenium may be viewed as a tuming point in humankind's energy usage. Up to now technology

of energy production evolved in complexity; with the retreat of the nuclear fission industry and gradua1 r e m to

fossil fuels, this trend is about to end-

The earth's lithosphere is rich in organic fuels created over millions of years from animal and plant

remains. Hurnans began exploring these non-reaewable fossil deposits only in the comparatively recent past

(currently roughly two fifths of the world's energy are produced from crude oil, another two fifths each from Gib89J coal and natural gas. and the rcmaining fifrh fiom n u c k v and hydroelectric p w e r ' ). and bave aot yet had

to face the consequences of their eventuai and inevitable cxhaustion. Based on current estimates of known

resources and present world energy consurnption. we may expect a fossil energy crisis within the next

However, the moment of crisis may corne much soaner, if the hypoihesis linkiag the observed

gIobal warming to Ca releasc from fossil fuel combustion was substantiated, and if carbon emission limits

were imposed to prevent the anticipated consequences of global warrning (cnatic weather patterns, drastic

c lirnatic changes. rising sec Icvel, descrtifi~ation[-~~ Seh891). Although validation of the causal link between Ca

ernission and global warming will require at least a couple of decades, present evidence seems to support a

correlation between the two.

In the second half of the XX century, nuclear fission was seen by rnost as the logical antidote to this

future energy crisis, and indeed, fkom a technological perspective it still appears as a viable option for many

centuries via the cycle[lm831. Noaetheless. public apphension about the health hazards of nucleat

exposure in case of an accident has led to the gradua1 decline of support for the fission alternative. It is the

author's opinion that although the dangers of fission reactor opcration, as portrayed by the media, are typically

exaggerated by facnial ignorance or delibcrate partisanship. t k still serious and long-term hazards posed by

spent fuel disposal have not been met with a satisfactory strategy to contemplate a world e n e r a economy based

on nuclear fission alone.

The remaining alternative of sufficient magnitude to become the main energy source of our planet

(soon to contain nearly five bilIion inhabitants in rapidly expanding industrial economics). which rcgrcttably

cxcludcs such 'rcnewable' encrgy sources as solar. wind. geo-thermal or tidc power, is the process of nuclear

fusion. The universe is full of fusion reactors in the form of stars (fossil fuels themselves are remnants of solar

fusion energy, indirectly captured by floral photosynthesis). Fusion energy reserves Vary from large to vinually

incxhaustible depending on the fuel cycle utilized (the deposits in units of 1985 world energy use[Fun9'': crustal

6Li - 10'. oceanic bLi - IO', oceanic D - 10'~. lunar - ' ~ e - i d , jovian 'He -10"). Unlike fission, the reaction

produc ts are not radioactive, althoug h structural irradiation by neutrons would produce a radioactive inventory ;

with a proper choice of low activation materials or neutron-scarce fusion fuel cycles (eg. D 'H~) , one could

expect total onsite radioactivity many orders of magnitude s d l e r than in a fission reacto?]. The caveat a

propos fusion is twofold: first, both physics and technology required for designing a fusion reacror are not yet ai

our disposa1 and require further research (read - $10"); s e c d l y , the abundance of fossil fiiels on today's

markets ( c d e oil at $10 a baml) undermines the incentive to develop this sophisticated and probably more

expensive alternative.

Nuclear fusion refers to a catcgory of nuclear reactions in which two nuclei produce a nucleus of

greater mass. such as[MiY781,

D + T + 4 ~ e (3.52) + n (14.07)

D + ' ~ e + ' ~ e (3.66) + p (14.6)

D + D + ' ~ e (0.82) + n (2.45)

D + D + T (1.01) + p (3.03)

p + 6Li + ' ~ e (1.72) + -'He (2.29)

p + "B + 3 ' ~ e (8.68 total)

(A. 1.1)

where p, D. T are short for 'H, 'H. 'H. and the vaiues in brackets are average energy yields in MeV. Most of the

energy releascd in a fusion feaction is carricd away by the less massive product, usually a proton or a neutron,

according to the ratio of product masses. Since fusion reactions arc governed by the short-ranged nuclear force,

nuclei must approach each other to withn a few fm (10 'IS m) for significant wave function overlap. Although

suc h encounters are frequent between neutral particles (eg. neutron scattering), the elecuostatic repulsion

between like charger reduces the interaction probability by a factor exponential in relative v e l ~ c i t ~ [ - ~ ~ .

(A. 1.2)

where ahfu denotes a cross section for a fusion reaction between species 'a' and 'b*, c = ma $/2 the energy in

the center of m a s frarne, c e and &,e the nuclear charges, and inh the reduced mas. This barrier is smallest

when the charge-mass product &&, rnabl" in the numerator of the exponential is minimized. that is for hydrogen

isotopes (for which it still amounts to hundreds of keV). As can be seen from Fig. A. 1.1. D-T is by far the the

rnost reactive fusion mixture. Averaging the product of the fusion cross section $"'&(it) and the relative velocity

v = v, - vb , over the velocity distributions of both products f,(v), fb(v), assumed Maxwellian with a single ion

temperature Ti (in energy units), we obtain the thermal fusion reactivity,

The integrand peaks at E - 5Ti, in the tail of the Maxwellian f(~)'~''~'. The fusion power density becomes

(A. 1.4)

where 6a is the Kronecker delta function, na is the nurnbn density of specics 'a*, and bfm is the energy

released per fusion reaction.

The goal of a fusion reactor is to sufficiently confi# the reacting species such that the energy released

by fusion reactions is sufficient to sustain the reaction (ignition criterion). A milestone on the road towards

ignition, is the so called energetic breakeven, for which the heating power is equal to the fusion power

produced. Introducing a net reactor efficiency q, which includes both the heating and thermal-to-electric

eficiencies, energetic breakeven may be stated as an ineqdity involving fusion, radiative and convective-

conducuve (10s) power densities(m'm',

where we have also defined the global tnergy confinement time TE, and assurned that radiative losses are

dominated by brernmstrahlung radiation. Inserting the expression for fusion power density and assuming the

same Ti for al1 ions sspecies, we arrive at the energetic break-even (Lawson) ~ n t e r i o n * ~ ~ ,

(A. 1.6)

Ideally, the reactor would become ignited, that is the fusion power absorbed by the plasma would be sufficient

to compensate for al1 power losses, and extemal heating could be turneci off. Since only a fraction CL, of the

fusion power is deposited in the plasma (for a D-T mixture, 8096 escapes in the form of neutrons), the ignition

criterion rnay be expressed as

(A. 1.7)

In rems of the energy muitiplication factor Q = pfU / phCa', the point of energetic break-even is defined as Q= 1,

and that of ignition as Q + -.

Many different fusion reactor schemes have ken conceived over the years'GLXa.

a) dual ion beam collision scheme proved ineffective due to space charge losses,

b) multidirectional laser heating of frozen hydrogen pellets followed by shock wave implosion, so called

inertiaI confinement fusion (ICF), has demonsuated proof of principle in a series of classified tests, but

remains plagued by spherical Rayleigh-Taylor instabilities and technological problems linked to the

pulsed nature of pellet implosions,

c ) catalyzation of fusion reactions by short-lived muons. whose larger mas decreases the nuclear separation,

proved ineffective due to parasitic muon capture by alpha particles,

d) cold fusion has at best revealed a new multi-body nuclear ptocess, but of insuffkient magnitude to prove

usefui for energy production,

e) magnetic confinement of ionized fusion fuel is to date the most widely developed method. A number of

magnetic configurations have been investigatediGlgOl (arnong them the 2-pinch in the 5OVs, magnetic

mirror in the 60's, stellarator and tokamak starting in the 60's). and although the verdict is far fiom

complete, the toroidai tokamak concept has emerged as the clear leader among the experirnental designs,

and the most promising candidate for a fmt generation fusion reactor. The nearly two decades of

experiments on a nwnbcr of large tokamaks CITTR, D-ID, JT-60, JET) and many smaller ones, has

greatly extended o u understanding of the basic physics of magnetized plasmas, and provided us with the

database for extrapolating to reactor relevant regirncslWm- At the moment the best results (ET '97)

hover about the energetic bfeakcven point expresseci by the Lawson criterion. The next generation

experimen t (International Thermonuclear Experimental Reactor, or ITER), whic h has been designed

jointly over the last d e u d e by the USA, EU, Russia and Japan. is cxpected to reach ignitionm1, a point

at which no external heathg would be required to sustain the hision reactions- Recentiy doubts have

arisen concerning the construction of this experiment, due to its multi-billion dollar price tag and a low

price of crude oil.

The eventual development of commercial fusion energy seems inevitable: it offers a virtually unlimited,

efficient, safe, non-polluting energy source, based on a technology which is alteady within our grasp. However,

this development may be driven either by econornic incentives or by political impetus; consequenlly, the first

fusion reactor could see the light of day as early as 2020, if international political cornmitment to fusion

persists, or as late as the end of the coming century, if its future is abandoned to economic factors alone.

Let the nature of a fluid be assumed to be such that. of its parts which lie evenly and are continuous, thaz which is under the fesser pressure is drïven along by that under the greatcr pressure.

ARCHIMEDES

By the term plasma ( X A a q i a [anc-gr.]: shape, creamre. phantom. fiction) we shdl rnean a large

ensemble of charged particles undergoing strong collective interactions'-l. We thus begin the discussion of

plasma physics by considering how a single charged particle behaves within this ensemble.

A.2.1 Charged Panide Dynamics

The classical motion of a particle with kinetic energy €3 and potential energy can be descnbed by

Hamil ton's eqirationsFD7q,

where q and p are its canonical co-ordinates (position and momentum). For a non-relativistic particle of mass m

and charge Zc in the prcsence of an elcctrornagnetic field. the Hamihonian H(q.p.r) b e ~ o r n e d ~ ' ~ ~

which leads to the equation of motion,

In the above, + and A are the potentials of the net fields (E = VQ - JNat, B = VxA), which we will divide into

contributions due to extemal sources (E,,,, B,,) and due to microscopie interactions between charged panicles

(E,,,. Bi,,). The fields evolve according to Maxwell's equaiions[J"7s1 (for a plasma E = 1. p = I ).

where p+ and j denote net charge and current densities (CGS units throughout SA.2). In the case of a unifonn

magnetic field (V BI = 0) and vanishing eiectric field (E = O). we find from (A.2.3) that the B-field only alters

the velocity perpendicular to B, vl = vL(cosp i, + sinrg ig), cp = ~ t , causing the particle to gyrate about the B-

axis with frequency o, and radius r ~ , normally called ttie Larmor radius,

Note that the electrons gyrate much faster (o, >> o, ), in tighter orbits (rk « r ~ i ), and in the opposite

direction to the ions. The finai trajectory is a helix of radius r~ and pitch angle tan" (v, / vJ, with the helicai

axis of the orbit, or its guiding centre, following the magnetic field line, Fig. A.2- 1 - In the presence of additional

non-uniform or varying 3 and E fields, suffkicntly weak to be t r d as a perturbation to the helical solution,

that is satisSing the so-cailed adiabatic invariantsm7".

the guiding center of the gyrating particle, found by averaging (A.2.3) over a gyration period &~~dtpC?n

acquires a transverse cirifi velocity due to a non-unifotmity of the orbit caused by: a) a transverse E-field, b) a

transverse gradient of the B-field, c) a curvature of the B-fieId, d) a time varying E-field. Its dong-B velocity, va

, changes with the paralle1 B gradient to preserve the mapetic moment of the ~ r b i t ~ ~ ~ ~ ' ~ .

(A. 2.7)

In the presence of strong B and E gradients, the trajectory is no longer helical but follows a complicated path

dependent on the local field strengths. and the guiding center description ceases to be useful.

Since charged particles in a plasma interact via the long range Coulomb force, we expect the ensemble

to exhibir strong collective behaviour. Solving the Poisson equation for an ion test charge in a single ion-species

p~asrna[bsO1.

with the assomption of Boltzmann distribution for the electrons and unpenurbed ions. we find ihat the Coulomb

potential is reduced exponentially by the mobile electron population with a scale length AD. called the Debye

length,

(A. 2.9)

This means that electric fields can only pcnetrate into tfic plasma a distance on the order of the Debye iength

before hey are shielded; as a consequence, average charge &nsity over dimensions L >> & is vanishingl~

small, ie. the plasma is quasi-neutral, i n, - ni I << G, and n, r f i ; the characteristic response urne of this process

may be estimated as l/+ ,

where cc is the thermal electron velocity and o, the electron plasma frequency. Collective behaviour dominates

when the number of parîicleo in a sphere of radius & is largew'.

Inserting the expression for into the above, WC find that A, often called the discreteness parameter, is also the

ratio of potential to kinetic energy of particles in the plasma. This leads us to believe that the standard methods

employed in kinetic theory of gases, namely an expansion in the discreteness parameter, shouid remain

applicable for plasmas as well, with the collective Coulomb interactions replacing discrete scattering events.

A.2.2 Statisticai Mechanics

An ensemble of N particles, N >> 1, can be described by a phase-space density p( qj, pj, t I j = 1,. . . ,N),

suc h that any microscopie quantity y gives rise to a maaoscopic quantity Y = c ~ > . defined as an average over

phase-space with the weight hinction pba '-,

The time evoiution of p is governed by the Liouville equationbg"

where { .., } is the Poisson bracket of ciassicaI mechanics, and H the Hamiltonian of the ensemble. In quantum

mechanics, H and p become operators and ( . . . ) a commutator. We are justified in neglecting quantum effects

as long as the thermal elecvon deBroglie wavelength is stnailer than the average inter-particle ~ ~ a c i n ~ ' l . " ' ~ ~ .

Likewise. we can neglect relativistic effects if the thermal electron velocity ce = (TJ~)'". is sufficiently

smaller than the velocity of light cW9'. We may therefore consider the region of validity of classical

mechanics to be h / w e < 14-l'~ and cJc < 0.1.

For a single ion-species plasma the phase-space density and the non-relativistic Hamiltonian (for which

the interaction potential is purely electrostauc. V- ) cm be written-'

where we removed fields generated by externai sources from the Hamiltonian of the ensemble. so that p=mv in

(A.2.2); the external field Hamiltonian will be added to the finai LiouviIIe operator at a later stage. The

extension to multi-ion species is straightforward, once we note that I;(Z,NJ = O in a neutral mixture.

In themial equilibrium aplat = (H. p } = O, and p relaxes to the canonical d i s m b ~ t i o n ~ ~ ~ ' .

where p = 1 / T, and 6 is the partition function, The link to thermodynarnics is provided by the expression for

the free energy, F = - (i/B)-lnc, fiom which al1 other thermodynamic quantities rnay be obtained using the well

estabfished formalism. In evaluating 5, we may perforrn a cluster expansion in terms of the Mayer functions gjk

= exp(-cpjd - 1. which transfomis integrals of the form /dNq.exp(-$xj.@ into IdNq-&(l+gjd. The problem

then reduces to evaluating cluster integrais of gjk, a task Iess daunting chen the original. We will not dwell on the

results of these calcuiation~~"~~'. except to note that the depanure from the ideai gas law is of the order of the

discreteness parameter. p, = n,T, + O(AJ, a€ {i,e}.

A.2.3 Plasma Kinetic nieory

We would like CO predict the probability of finding a particle of a given species in a single ion-specics plrisrna.

ae {i,e), at position q, with mornentum p,; this probability is given by the one-particle distribution function f

"(q.p.t) defined by integrating the phase-space density p over al1 but one pair of c o - o r d i n a t e ~ ~ ~ ~ ~ ' , which we

dcnote by a common j index, j= 1.. . ..N with N=N,+N,,

Here we have aiso defined the two and t h e psmfle distributions. f * and f *. with each species index

spanning over both ions and elecuons, a,b.c E (i.e); for tike-particle distributions. it is usehl to consider the

first particle. with indexes 'a' and '1'. as a test partick, and the following particies, 'b' . '2'. and 'c', '3'. as field

or background particles. Using the mentioncd cluster expansion-', these can k recast in tenns of the two

and lhree particle correlatioa hiactions g* and g" .

where we expect gd/rp - A and gaC/M~ - A', with the discretenes parameter A c< 1. Nexr we expand the

Poisson bracket in the Liouville cquation and express the result in terms of the propagation and interaction

operators- ""l. Li and L*.

(A. 2.18)

The cffect of extenial fields is included in the propagation operator La. Integrating (A.2.18) over d"''qds-'p and

inserting the cluster expaodcd f *. we ~btain['@~

A sirni1a.r integration of (A2.18) over ds-zqds-'p yiclds

p = ( ~ $ + L % ) f ~ f ~ + 1 (A. 2-20)

where we have expanded the urne derivative, used the previous result for aP/& and a&&. and neglected terms

of order b2 to close the BBGKY hierarchy; taken together (A.2.16-20) represent the kinetic equation we set out

to find. Since the choice of îhe test particle is arbitrary, we can recast this equation into a more familiar form by

switching to cartesian coudinates (ql = x. pl = mv). bringing the microscopic field term (L*+Lbi)f f ont0

the left side and adding the extemal and the microscopic contributions to form the net fields,

The remaining terms on the right hand side, Ca, are norrnally referred to as the collision terms. Explicit time

dependence appearing in g* refers to the physical assumption that pair correlation hinctions relax much faster

than the single-particle disoibulion functions; it allows us on the one hand, to neglect the cime-history of the

correlation in cdculating Cd , and on the other, to mat f' as constant over the time needed for correlation g* to

relax. These arguments are refemd to as Bogoliubov's characteristic time hierarchy.

The rnorphology of Lrinetic equations corresponds to diffcrent assumptions used in evaluating C*. The

simplest, narnely ga = O, leads to the weii known Vlasov e q u a t i ~ n ~ ~ ~ for which Ca = O (it is a slight

misnomer to cal1 this approximation coiiisionless, since Coulomb interactions are present in the microscopic

field Eh,; the usage refers to the fact that &, does not increase piasma entropy nor generate plasma transport; it

can however dissipate field-energy as demonstrated by the 'collisionless* (Landau) dacnping of E-M

wavesrM6'; bowing to common usage. we use the term 'collisionkss' to mean Ca = O). The standard approach

of gas dynamics is to retain terms on the order of the discretencss parameter A; the calculations have been

canied out independently by ~ a l e s c u ~ ~ ~ , ~ u e r n s e ~ [ ~ ~ , and Lenard[Lma1 (1960). There are a number of

ways to proceed of which we will adopt the Green's fiinction approach. First, we notice that only the first term

on the right hand side of (A.2.20) is not-homogenous with respect to g*, and so only this term acts as a forcing

function to the evolution of g*. W e c m thus express g'(t) as an integraI of the forcing function and an

unknown propagator W over al1 previous tirnesucm B*ul.

where for conciseness we abandoned the parricle index. which wiH be linked to a and b: thus. a = (ql. pl). b =

(q?, pz) with either particle being an ion or electron. Direct substitution of the above into (A.2.20). with no

external fields and f constant in time, yields a time-evolution equation for the propagator W.

(A. 2.23)

The propagator W may easily be obtained in (ka) space by effecbag a Fourier transform with respect to x,-x,'

and a Laplace transform with respect to t > 0,

Integrating over va givcs an expression for the interaction integralfihgz',

1 j d v a w k ( V a , v ;

(o-k -va M k , W )

Inserting the above into (A.2.24). we have the final form of the propagator,

7

(A. 2-24]

(A. 2-25)

The fmt term is simply a frce-particle propagator, Mile the second is a propagator due to the Coulomb

interaction, includmg the tffect of exponcntial shiclding contained in the Vlasov dielactric function E(k,o).

Armed with Wk we are ready for the collision term Ca. We carry out the same Fourier-Laplace

uansfonn on g* as we have on W and integrate over the time variable,

The collision tenu Cd may now be constructed from the Fourier components ghk by an inteption over k that

is over the scattering collision parameters,

The final steps after combining the last three equations require extensive use of contour integrals and the (Ich92j analytic properties of the tiansformed propagator; we present the final result without m e r delay .

In the process of intcgration over k the presence of the dielectric function removes the singularity encountered

at long range. sincc €(U- v) = 1 I kl for k « 1. A short range limit on the order of the DeBroglie length or

the 90" scattering distance must still be assumed. k- - max(mavJh, 3~./2&,e~). since cluster expansion breaks

down ai such small distancesrhm1: this lirnit usually appears as the so called Coulomb logarithrn. In Aa = In

(LA,,). Symmeuy consideration reveal a number of general properties of the BGL collision term which we

can summarize as follows: it is symmecric with respect to v; it conserves the average density, velocity and

kinetic energy of f(v); it relaxes the f(v) towards an equilibrium Maxwellian distribution, for which it is

identically zero.

The same form of the collision term but without electron shielding, &(k,k- v) = 1, was first obtained by

anda au'-'] (1937). who introduced a long-range limit, & -1 &, in integration over k The BGL and

Landau te- clearly agree for ~ ( k & v) - 1 + 1 / Pb2. It is noteworthy that Landau initially arrived at this

expression not by a cluster expansion, but by looking for a collision term of the Fokker-Planck type, appropnate

for grazing ~oilisions~~-~,

2 c, = --. I a a v Adfa(v)+--:Bdfa(v)

2 avav (A. 2-30)

and by including the kinematics of binary, unshielded Coulomb scattenng; it is reassuring that both methods

give identicai results. Intcgrating by parts we can transfonn (A.2.29) into the Fokker-Planck form. which we

will find useful in evaluating plasma relaxation times. Intrducing the Rosenbluth (Trubnikov) potentials H*

and Gh we obtainm

The physical meaning of Aa and Bb, which we denote by dn* and dvAv>* to avoid confision with

magnetic field quantities, becomes apparent if we consider a beam of mono-energetic

Maxwellian field particles with a thermal velocity cb = fi 1 rnb)ln.

test particles and

(A.2.32)

Insening these into (A.2.30). we find that d v > * and d v A v > * determine. respectively, the rates of slowing

down and isotropization of the initially mono-energetic test particles by collisions with the background

particles: they are often referred to as the coefficients of friction and diffusion in velocity space. Effecting the

integrals in (A.2.3 1) over fb(v) of (A.2.32). the Rosenbluth potentials b e c ~ r n e [ ~ ~ ~ . " " ~ ~ ~

(A. 2-33)

Using these resul ts, we can evaluate the Fokker-Planck coeficients paralie1 and perpendicular to the test

particle velocity TO by evaluating the gradients in (A.2.3 1) at v=v& as requifed by f,(v) of (A.2.32),

(A. 2-34)

Plots of 0. a' and Y are shown in Fig. A.2.1, and theu asymptotic limits are given below.

2v' 2Y - ) - G<<I I Wv')+l. Y(v')+-T. C » l (A. 2.35)

t 3 6 ' v -

Ir is useful to ncmnalize the above by defining characteristic tir ne^^.^'^^ for slowing down ?. deflection

and energy excbange f. which measure the effectïveness of collisions in relaxing the initial test particle

distri bution, f,(v.kû) = RV-vO). towards the equilibrium background distribution fb(v),

(A. 2.36)

Examining the characteristic tirnes with respect to vo. nb and Tb, we c m make several observations:

for fixed vb al1 relaxation processes increase linearly with density nb, and decrease with the background

temperature as Tb -3R; :. hot plasmas approach the collisionless limit.

slowing down is independent of vo for vo « cb. falling as vo -' for vo >> cb ,

broadening of the velocity spectrum (isotropization) proceeds slower at higher energies; l/? decreases as

v,-' for vo « cb. and as vo " for v~ >> cb.

energy exchange decreases as v i ' for vo << cb, and as vo -* for vo >> cb

The effect of particle mass on relaxation becornes evident if we define z,t, as a relaxation time of test parucles of

type a E (ive) by collisions with field particles of type b E (i.e). and relate these to ion and elecuon

characteristic times ri and t, which we write down explicitiy for future reference [GiIBZ BoogOl 1

(A. 2-37)

ClearIy, like particle collisions are more effective at transferring momentum and energy, and electrons tend to

relax faster than ions. The evolution towards equiiibriurn proceeds as follows: fmt, the etectrons relax by self-

collisions to a Maxwellian fCM(v; Te) on a timescale &; next, the ions relax by self-collisions to a Maxwellian

fiM(v; Ti) on a timescale q oc (m&)lRr, ; l ady , ioneleciron collisions relax the two distributions to a

common Maxwellian fM(v; Ti = Te) on a timescale G? a (r~/m&.

A.3 Braginskii and CGL cquations

A cornplete set of Bragiashi transport equations may be found in the NRL Plasma ~ o r t n u l a r ~ ~ ~ , p.36.

A derivation of the CGL. or double adiabatic, equations may be found in 3 text by ~ i ~ a m o t o ~ " ~ ' ' ~ ~ . p. 14 1.

Historically. the fmt solution to the problem of plasma-solid contact appears in a now classic paper by

Tonks and ~ a n ~ m u i f ~ ' (1929). The authors assumed cold, collisionless ions. for which they obtained values

cp, = - 0.84 and Ms = 1.14.

The case of warm, collisionless ions has betn solved analytically by Emmert et al.-"] in the limit

X,,«L; we will summarize their results as usehl benchmark for further refinemnts. Their method consists in

casting the ion Vlasov cquation into kernel form by rewriting the equation in terms of the toial energy, mvx2/2 + eZq(x), which is an integral of motion, and choosing the source kernel so as to produce a Maxwellian f(x,v,) in

the absence of potentid gradients. The Vlasov equation is integrated in space and energy to yield ni(x), which

together with ~ ( x ) given by (2.2.3). form an integraldifferentiai equation for the potential Mx). The authors

solved this equation anaiytically, with the rcsult cxpftssed as a uanscendental equation for y, E ecp, 1 Te and the

parameter ( = me 1 Ti > 0,

The normalized sheath voltage w,(<) was found to increase monotonically with the ratio from O for cold

electrons, to 0.854 for cold ions. The normaiized surface potential , the ion particle flux To and the power

flux for ions qd werc cxpfessed in terms of the sheath potentiai,

The asymptotic limits of the dimensionless functions Bi and pi are given below,

(A. 4.2)

(A. 4.3)

Comparing the above resuits with the ion particle and energy fluxes cdculated for a haif-space Maxwellian, we

see that and p represent enhancement factors due to the presence of the sheath. They are not sensitive to the

elcctron tcmperature as long as T, > ZT,; the ion flux is enhanced even when Ti » ZT,.

The above formulation was extended-l for wcakly collisional ions by including a BGK collision

operator in the ion kinetic equation,

Like-particle collisions do not affect the O., la and 2d velocity moments of &(va; by irnposing thex

constraints on the BGK operator, the shifted Maxwellian parameters &x), uo(x), To(x) were related to the actual

ion dismbution fi(x,va. The ion density ni (x) was once again expressed in terms of integrals over space and

energy, and consequent quadratures were evaluated numerically (stable solutions were found onIy for the case

of weak collisionality, & - L)- The effect of collisioos on tbt potentials and fluxes into the sheath. was minimal.

Whether this insensitivity to self-collisions would pcrsist for & L is not clear. However, since self-collisions

only effect the shape and not the averages over the distribution, we would expect the lower moments to be less

altered than the higher ones; in fact, the only signifiant difference between the two modeIs was a 10% increase

of the ion heat flux qi - ~<v*v,>. The same authors investigated the effect of strong ion-ion c ~ l l i s i o n a l i ~ ~ ~ ~ ~ ' , using a set of fluid-like

moment equations equivalent to a simplified form of the drift-kinetic moment equations (2.1.22), with V = v - u = v - cvX>iL, < v p = <vz> = O,

where <...> denotes an average over f(v). The above equations are not vaiid in the sheath itseif, since quasi-

neutrality was assurned in their derivation. but are relatively accurate up to the sheath edge. Two sets of closure

relations were pmposed: a) in the collisionless limit, perpendicular energy was taken as constant. CV; > = Ty /

m, and the flow was assumed adiabatic. <v,) > = 0; the latter condition was supponed by a cornparison with

previous kinetic results from which the contribution of < ~ . f > ro the total ion heat flux was found to be small.

b) in the fuily collisional limit, the velocity distribution was assumed isorropic in the frame moving with thc

velocity u. hence cv,' > = <v.' > and once again cvX3 > = O. Using the above relations two sets of moment

equations were obtained. Rather than stating these explicitly. we compare their point of sonic transition, that is

the singularity in duldx. The collisionless and collisional sound speeds are thus found as.

indicating that ion collisions reduce the flow velocity at the sheath edge (assumed to be sonic). The moment

equations were solved for Ty = Te and the two source distributions: Sl(x,v;T,), corresponding to an isotropie

Maxwellian ionization source, and S2 = Slv, which provides no particles with zero velocity. Since stagnant ions

must be accelerated to u - c, by the Efield, the sheath potential for SI is larger than for S2.

A similar analysis was performed for ion-neutral collisions, by allowing the BGK operator to aiter

energy and momentum densitie~'*~'. In this case. wdl and sheuh potentials increaseri, with the potencial drap

across the sheath remaining fairly constant; particle and energy flues into the sheath were reduced. Very strong

ion-neutrai collisionality was investigated kinctically by ~ i e r n a n n ~ ' , with the assumption of cold ions, Ti = 0.

Similar effect on potentiais and fluxes was obscrved; in addition, the ion velocity distribution at the sheath edge

was significantly broadened by neutral collisions, with a width of the order of Tm.

The various ueatments of non-magnetized ion flow into the electrostatic sheath are reviewed in a

recent article(-'. Comparing the normalized fluxes and porentiais, Tab1e.A-4.1. and the shapes of fi (v) at the

sheath edge, Fig.A.4.1, we note a surpnsing robusuiess of the estimateci values over a wide range of modelinp

assumptions; this insensitivity justifies the use of simplified expressions for the heat transmission coefficients

(at l e s t in the context of edge plasma modeling).

AS Tokamak Coir Pluma

This appcndix is intended as a short preface to tokamak physics. in which the basic physical processes,

experimental findings, and technicai problems are outlined. The idea of a 'shon preface' to such an extensive

field as tokamak research may strike the reader as both audacious and ludicrous. TO appease his anger, we make

our aim explicit: certain physical processes play an essential role in our region of interest - the tokamak edge

plasma - and these are most simply discussed in the context of tokamak physics at large. Those areas which

have little bearing on our anaiysis, such as plasma beating or wave propagation, receive only a passing mention.

AS. 1 MHD equiIiùrïum

The term TokamoL; is a Russian acronym for toroidal-chamber-magnetic-cuil; it refers to a family of

toroidal plasma devices in which the toroidal magnetic field, generated by external current carrying coils, is

stronger than magnetic fields generated by plasma currents, Fig. A.5.1. A schematic of the dominant tokamak

fields and currents may be stated as foiiows:

poloidal coil current. je'&' * roroidal magrietic field, Be

ro roidal plasma CU ment. j, - poloidal magnetic field. Bo

where the jeco' is held constant, sometimes in superconducting coils, while j, is induced in the plasma either by

an external transformer circuit, the traditional method, or by current drive systems based on non-linear wave

absorption, neutral bcuri injection, or self-indueed (booütrap) plasma curentwm. Calculation of the

equiIibnum B-field (or rather the field-plasma system) is simplified by treating the ions and electrons as a single

fluid. Starting from the Braginskii transport equations (A.3.1). we add the electron and ion equations for mass

and momentum transport assuming quasi-neutrality, and replace the energy transport equations by an adiabatic

closurc relation. p a pyl y =5/3. Cornbining the ûbove with Maxwell's equations (A.2.4). we arrive at the

cquations of rnagneto-hydro-dynamics (MHD)[' ' '~~~.

If collisions are neglected entirely, the resistivity q, becomes zero, and the rnodel is known as ideal MHD. The

MHD approximation is valid for global plasma behaviour, that is plasma evolution on length and time scaies

much longer than those characterizing microscopie processes, >> (b . rd. and o cc (op ,a, ). ln time

dependent form, it may be used to study a variety of plasma waves with k = 2 d and 61 in the above range:

broadly speaking, electromagnetic waves (highiy non-isotropie with respect to B, and exhibiting cut-offs and

resonances due to particle gyration), pseudo-acoustic waves (differing for ions and electrons due to mJm, >> 1)

and field-inertial or drift waves (such as the Ahen wave II B coupling ion inertia and B-field tension)isM21. In

steady state form, the MHD approximation may be used to find the equilibrium state of the field-plasma

systemmg7. "m.

So far we have worked in generalized co-odinates; to proceed M e r we must select a CO-ordinate

system appropriate to the tokamak geometry. The toroicîai-Cartesian co-ordinates (R 2. 9) may serve as one

such reference frame; here R is the distance fkom the major axis of the toms, Z is the vertical distance from the

mid-plane, and is the toroidal angle- Assuming toroidal symmeq at the outset. a-, the V- B = O condition

allows us to write the poloidal component of B as a gradient of some function y,

Magnetic field lines form surfaces of constant y, which is related to the poloidal magnetic flux cutting across an

equatoral ring R = (= Ri < R c R2) by the integral'My7g1.

For this reason it is known as the flux function, and the surfaces of constant y as flux surfaces. Examining the

components of the pressure equation, jxB = Vp, dong B. j and Vy, we conclude that pressure is constant dong

a flux surface, B-Vp = 0, and current flow is CO-planar with the flux surface, j-Vp = O. We separate the current

density into toroidal and poloidal components,

The poloidal current passing the samc ring 3t is related to RB, = 4).

Finally, taking a scaiar product of j and Vy, and using j-Vp = j-Vy = O we obtain a non-linear, elIiptic

differeniial (Grad-Shafranov) equation'-9.Ycm, relating y and the two flux functions p(y) and @(w).

In practice it is useful to uansform this equation into flux CO-ordinates (r, 8, 6). Here we choose the fust

ordinate as a flux-surface minor radius (radial disiancc away fiom the poloidal center), r = r(yr), and denote iu

value at the last closcd flux surface (LCFS) by r- . The second ordinate 8 is the poloidal angle; alternativeiy

we could have chosen the poloidal distance along a flux surface. s, or the distance along B, %. Using the

generalized variable relations of SA.3, we obtain expressions for the Jacobian J, axmc tensor m. line element

dx and the gradient operator V in the new fnww".

which allow us to writc the Grad-Shafkanov equation in temu of flux CO-ordinates,

(A. 5.9)

To solve the Grad-Shabanov equation in either form. additional constraints are required. In practice, a number

of altcrnatives are presentwm: a) specifying y on some flux surface, or the poloidal shape of any flux surface.

b) taking the coil current distribution as given, and iterating the plasma current distribution with successive

solutions, c) iaking account of direct measurements of Be near the edge of the plasma, and sirniliarly iterating

the solution. Specialized numerical solvers are routinely emptoyed to generate such magnetic equilibna; the

calculateci magnctic field B(r,8,4)) provides the substrate for plasma transport calculations; typicai poloidai

contours of B and radial profiles of B, . j, and p are shown in Fig. A.5.2.

As a final comment, we note that on a given flux surface field lines may not cross each other due to the

vanishing divergence of B; the ratio of toroidd to poloidal revolutions of a field line is therefore constant for a

flux surface, and we denote this quantity, known as the safety factor due to its importance for global plasma

Flux surfaces with rational values of q. that is with m = q-n. where m and n are both in

(AS. IO)

itegers. are known a

resonant surfaces; on these surfaces field lines r e t m to their original poloidal locati~n after m toroidal and n

poloidaI revolutims (we shall retain this convention when discussing plasma instabilities with wave number k =

m& + nie). Fiux surfaces with inational values of q, contain field lines ergodically accessing the entire flux

surface. retwning to their initial location only after an infinite number of revolutions.

A 5 2 MHD Instabilities

In deriving the Grad-Shafkanov equation, steady state was assurned. a/& = O. Temporal evolution of

the resulting equilibria would reveal their stability properties. To this effect, the tirnedependent MHD equations

are subjected to two types of perturbation analysismul:

a) variational energy analysis, based on the fact that dynamical systems evolve in such a way as to

minimize their potential energy; if a perturbation is found to lower the w potential energy of both field and

plasma, as calculated ftom the MHD equations, the equilibriurn is unstable;

b) eigen-mode anaiysis. in which the linearized MHD equations are treated as an eigen-system. usudly

separable in the toroidal and poloidai directions with k = rni, + ni& which yields eigen-functions (modes) and

their eigen-frequencies; if the imaginary part of the frequency is positive, the mode is expected to grow,

destabilizing the equilibrium.

We restrict ourselves to merely summarizing the results of such analysis, which for many years formed

the very heart of the fusion effort, Most geaerally, MHD instabilities are dnven either by radiai gradients in the

current distribution, dj(r)/dr, or by radial pressure gradients in regions of convex field curvature, dp(r)/dr. they

are stabilized by the tension and compressibility of the magnetic field liaes. in other words by the B-field

energy, ~ ~ / 8 1 t . The dominant destabilizing modes and the constraints they place on the equilibrium plasma are

listed below[wm:

a) ideai kink modes q(0) > 1.0. q(rrnd/q<o) > 2.0

b) resistive kink (tearing) modes a q(0) > 1.0, q(rm)/q(0) > 2.0

C) ballooning (pressure) modes a S = (r/q)(dq/dr) > S,

fj = p/(B%x) < B, - O. 1 5r&(Rq(rLcFs))

d) verticd mode BZ adjusted via active feedback

Kink modeswa781 lead to twisting of the plasma column, as defined by the flux surface; they are strongest at

long wavelengths (at low mode nurnbers n,m), originate on flux surfaces resonant with the mode, and are driven

prirnarily by dj(r)ldr. Tearing modeslfm"l are the resistive equivalent of the kink modes. and are driven by the

same mcchanisrn; both of these modes grow when B, is weaker than Be aaywhere in the plasma, or less than

twice Be at the edge (we shall examine the reasons for rhis shortly). Ballooning modes[c0n781. driven by gradients

in the plasma pressure dp(r)/dr, occur localiy on the convex side of the field; they are stabilized when the

change of magnetic field pitch angle, often termed the magnetic shear. S = (rlq)(dq/dr), is sufficient to oppose

the interchange instability; rhey also place a limit on the ratio of plasma to magnetic pressure. pumSrl. Finally.

thc vertical or axi-symmeuic mode is important for suongiy non-circular poloidal cross-sections. but it may be

stabiIized by a vertical field from external current carrying control coils.

It is worthwhile to explore the consequences of finite plasma resistivity in more detail. In complete

analogy to the analysis of invariant tori manifolds in classical mechanics. the resonant surfaces are more

susceptible to the onset of chaosrM9'. Fig. A.5.3; according to the Kolmogorov-Arnold-Moser (KAM)

t h e ~ r e r n ' ~ ~ ~ ~ , when non-linear perturbations are introduced into a dynamidly conservative sysrern, invariant

ton (representing the integrals of motion) initially break up into islands on the rwonant surfaces (the number of

islands per surface determined by the periodicity of the tom, each island following a helicai mjectory around

the torus), thcn into bounded crgodic volumes, which grow with the strength of the pe-ation, eventuaily

filling the entire island region as well as linking al1 the islands into a continuous ergodic volume. Therefore, a

region initiaily cornposcd of a resonant (noncrgodic) surface, bounded on eiîher side by ergodic surfaces, is

transfonned in10 a completely ergodic volume; the initial 2-1-2 topology is destroyed in favour of 3-3-3, where

spatial degrecs of freedom are indicated. The regions in which the volumetric ergodicity first appears are the X-

points of the Wands and the separatrix which defines hemrw. ALI of the above features apply directly to the

magnetic flux surfaces in a tokamak; magnetic islands have indeed beea observed in tokamaks using

tomographie The non-lineu perturbations responsible for topology breaking may be caused by

a) finite resisitivity, leading to E fields dong current fiows, hence ExB drifts, b) oscillations due to MHD

modedwaves, c) any kinetic process not included in the ideal MHD picture, eg. plasma interaction with neutral

beams, pellet injection, fast particles, etc. The rcsulting changes to the magnetic topology have consequences

for h t h stability and transport, and conspire to make the tokamak plasma one of the most complicated

dynamical systems ever studied.

Quasi-Iinear analysis of MHD quations reveals chat islands grow in response to dj(r)/dr and

saturate due to magnetic field line d i f i s i ~ n ~ ~ . A penurbation in the form of a radial magnetic field Br with

wave number k = mi, + nie Ieads to an island of width Ar,

with the expression evaluated at the location of the fiactured magnetic surface; note the inverse scaling with

mode number (m=2 is usually dominant). It follows from the MHD equations (A.5.2)- that the radial B-field

diffuses due to finite resisitivity q,.

which allows us to estimate the saturated island width as the solution of the following expression~sYk"J

where crq is constant, and the characteristic growth time as

where T,, is the characteristic resistive time. At larger perturbations, chaotic behaviour begins, that is field Iines

are no longer consoained to panicular surfaces but access the bordering volume region ergodically; these

regions appear first near resonant surfaccs and grow as the streagth of the perturbation increases.

The change in magnetic topology is believed to play a key role in two tokamak instabilities, known

colloquially as sawteeth and disruPtioieWm. Sawteeth are oscillations of n. and Te near the center of the

plasma; they have been linked to an rn-1 mode occurring whenever a q=I surface exists in the plasma In one

sawtooth period, j,(r), Te(r), and n&) evolve until a q=l surface is formed and the instability commences,

quickly destroying the surface; the e f f d v e constraint on the q(r) profile is that q(0) > 1.0. The exact process of

the fast phase of this instability is still uncertain, but magnetic reconneaction leading to the transfer of e n e r a

between a hot, inner region and a cold, w te r region is the preferred explanationw@'.

Sawteeth oscillation are for the most part benign modulatioas of core density and temperature;

disruptions on the other hand are spcctacular insîabilities leading to compkte destruction of eq~ilibriumw*.wl.

The plasma current is entirely quenched, the plasma pushed ont0 the vcssel wall, causing large heat fluxes, eddy

currents and mechanical stresses (via jxB forces). Our understanding of disruptive phenomena is still far from

complete, but the sequence of events leading to a dismption and the mechanisms involved at each stage are now

fairly clear. Briefly, disruptive instabilities bcgin with a gradua1 growth of the m=2 resistive kink mode, and

terminate with a critical multi-mode island r eco~ec t ion causing strong ergodization of the core plasma, rapid

flattening of j(r) and complete loss of confinernenr. The onset of the m=2 resistive kink mode coincides with the

overlap between a q=2 surface and a region of steep negative current gradient. dj(r)/dr < O, usually near the

edge of the plasma. This is brought about by,

a) an increase in the current which causes the q=2 surface to move outward (the central q value can not exceed

unity due to sawteeth oscillations via the m=l resistive kink mode),

b) radiative cooling of the outer core, which inmeases local resistivity q, contracts the current profile j(r), and

moves the steep current gradient region inward towards the q=2 surface.

The stability lirnits imposed by the two cases am q(rLCFS) > 2 and ~ O - ' ~ ( ~ R / B ) > q(rmS). respectively; the latter

criterion. known as the Greenwaid density limit, places a lower lirnit on edge density n(rLms) (despite the fact

that ii is usually expressed in terms of the flux-average density <n> = Inrdr I kdr) and constitutes the most

important operational incentive for modeling the tokamak edge plasma.

The evolution of the m=2 resistive kink mode involves growth and ergodization of magnetic islands on

the q=2 surface: the measured radial magnetic k l d Br allows us to estimatc the width of the m=2 islands at the

time of current collapse using the expression for the island width Ar (A.5.I 1). This width is found to reach a

value of - 0.2rms. Why the fast phase sets in ai this relatively low value is not clear, but an interaction between

ergodized islands of different resonant surfaces which grow until the ergodized regions begin to overlap and

merge (that is when the island widths are comprable in size to the separaiion between surfaces), offers a Iikely

explanation; the process may be described as a critical topologicai phase transition. In the fully ergodized

plasma, radial gradients are removed on the timescale of parallel a s p o r t and rapid loss of codinernent

follows.

AS .3 Kinetic Instabilities and Transport i B

in the 1 s t section. we saw that even a relatively simple (single fluid or MHD) description of the tokamak

plasma predicts complicated global behaviour. In reality, local variations of f d ~ v ) and f,(x,v) do not conforrn to

the constraints of the MHD model, but evolve according to the kinetic equations, (2.1.1). In this section we will

consider phenomena arising Erom the kinetic equations, in particular modes with k and 61 on the order of the

rnicroscopic scale and time lengths, A - niin& , r d and o - max(w, ,&).

Kinetic modes, dso called micro-instabilities, may be categorized according to the fields involved

(e lectros tatic vs. electromgne tic), the oscillating particles (ions vs. electrons), the particle trajectory in a

tokarnak geometry (trappcd vs. fiee), the degrce of collisionality (collisionless vs. coliisional), in short by the

type of kinc tic wave fiom which the mode originatesVa7". Micro-instabilities are typicall y inves tigated by

subjecting a quasi-limarizcd kinetic equation to small perturbations; in order to determine the saturation level of

a given mode, fully non-linear calculations are required. The reader may find it usehl to consult a recent review

of tokamak micro-instabilities in order to appreciate the full complexity of the probkm~cm~l . As an example,

we can consider the micro-tearing mode, which is a kinetic analogue of tbe MHD-tearing (resistive) mode

discussed in the previous section. Quasi-linear anaiysis reveals that high-m resonant surfaces dissociate into

srnail scale magnetic islands in regions of suong Te gradients and i n t e d i a t e collisionality; the island

saturation width for Ars > r~ is determined by the influence of the island on local transport. The process is a

rnicroscopic equivalent to large-scale ergodization, and has been proposed as one explanation for the dismptive

collapse.

Micro-instabilities have a dramatic effect upon cross-field plasma transport, which we shall consider

briefly. According to classical theory, 5A.3, cross-field fluxes aise due to un-like particle collisions Cab and

radial gradients of f,(x,v), that is Vl(n,u,T); classical cross-field transport coefficients of 5A.3 correspond to a

diffusive process with radial step r' = 1/B, (A.2.5). and collision frequency ta -', (A.2.37).

Since collision frequencies are much srnalier than gyration frequencies ( s r , >> l j. we expect cross-field

transport to be much slowcr than transport parallel to B (Le&). In toroidal geomeuy, for which Be a UR,

and low collisionality, the CGL equations predict that sorne of the particles become uapped in elongated,

banana shaped ~ r b i t d ~ ' ~ ~ ~ ~ . The radiai widrh of such an orbit may be considered as the radial diffusive step,

leading to the so called neo-classical e ~ t i m a t e d ~ ~ ~ ~ ,

(A. 5-26)

It should be noted that aside from a mornetricd enhancement factor, the dependence on B is unaffecteci. For

snonger collisionaiity. the neo-classiai enhancement is less profound since a particle is likely to suffer a

collision before completing the banam orbit. Another mechanism leading to an increase in radiai transport is

particle trapping due to toroidal asymmetries or ripples in the magnetic field, caused by finite spacing between

field coils. It leads to an e ~ ~ r e s s i o n ~ ~ ,

( cal kppk [V? - 1 (AB) ( C O L )cl B npplr

Needless to Say. exprimcntally exa r*d values of SL. that is paniclc (D il . D L) and energy ( X . x CL )

diffusivities disagree quite sharply widi the above esthates; the inferrcd cross-field transport appears srronger

by as much as two orders of rnagnitide, with the discrepancy being larger for electrons than for ions. The

observed dependence on the magnetic field is much weaker, roughly linear rather than quadratic, a 1/B; this

is known as the Bobm s c a ~ i n ~ [ ~ " ' .

The search for the underIying transport mechanism is complicated by the difficulty of obtaining an expression

which would successfùl ceproduce eltpcrimental data under ail operating conditions. In fact, the accepta!

working procedure has been to exuact global particle and energy confinement times rather chan local transport

coefficients. It is now generaily believed that plasma turbulence, driven by locally saturated micro-instabilities,

is rcsponsible for the greater part of the observed cross-field t r a n ~ ~ o r t ~ ~ o ~ ~ ~ ~ ~ " ~ . Th e suongest evidence for this

clriim are measured correlations between and plasma-field fluctuations. Sn, 6T, 8E, SB.

Fluctuations in the local elecmc field give rise to a fluctuating ExB drift velocity 6v,. If corrclated

with (6n . 6T). that is if (n ,T) fluctua& with the same frequency as 6vL, their relative phase determines the

direction and magnitude of the cross-field fluxes (TL . q 3 ; we note that (V,n .VLT) producc fluctuations (6n.

8T) corrctated in-phase with 6vL, and fluxes (L , qJ in the direction opposite to the gradients[""ssi. WC dcfine

thcse fluxes as averages over some tirne t' longer than the fluctuation p e n d w-' but shorter than the

macroscopic time scde Tau,

The diffusion coefficient may be estimateci in terms of Fourier cornponents of the fluctuation^^^^-^^,

where .ri, i s the correlation time of the k-th component of the fluctuation. For srnall amplitudes, the fluctuations

are likeiy to be deconelated due to parallel chermal motion Tt, - l m I . drifi motion g - lfaml. or collisions rt

- l/al ; for large amplitudes, decorrelation occurs as a result of cross-field motion itself, with fk - l/k16vt. In

other words, at high amplitude DL exhibits the Bohm-Iike. l A ,

Fluctuations in local 6-field, such as those present around

parallel fluxes in the cross-field d i r e c t i ~ n ~ ~ ' ~ ~ ,

behaviour,

(A.5.21)

islands, redireci

(AS. 22)

where we replaced ST, by &(6BL)VLT with & ( S B j some fluctuation dependent heat diffusivity. We can

estimate the k-th component of the diffusion coefficient of the magneticfield lines as

where we assumed a weak turbulence approximation for the rnagnetic correlation time tkB. Considering paralfel

motion of species 'a' dong the diffusing field lines. the radial step size depends on the degree of collisionality:

if ik » T ~ ~ . then 6rk - ( D ~ ~ v ~ T ~ ' " and D~ - (6r&tk - ~ ~ ~ v ~ l ; i f ri, < 7kB. then 6rk - (GBI/B)v,~, and D~ - (6rd2/rk

- ( ~ B I / B ) ~ V ~ ' T ~ . With these assumptions. we find chat cross-field transport due to magnetic fluctuations exhibits

a weak scding with B, which depends on the 6BJB relation.

Lastly, large sale magnetic islands may be considered as providing a virtual short circuit for particlc

and heat flow, and we expect plasma variables, especially T&). to be nearly constant across the island region.

Fig.A. 1.1 : Fusion reactivities as a function of plasma temperature @Ai174].

O ' G 2 3 4

Fig.A.2.2: Rosenbluth (Trubnikov) potentials[Kra80]

Coll'less coll'al coll'less coll'al fluid fluid kinetic kinetic

Fig. AA. 1 : Ion velocity distributions at the sheath edge Table A.4.1: Values of sheath quanti ties as predicted by various kinetic models [Pos84] predicted by various kinetic models [Sta90].

Fig.A.5. I : Schematic of the tokamak concept FigA.5.2: Tokamak flux-surface geomeuy

Fig.A.5.3: Typical tokamak fields and cunents Fig.A.5.4: Magnetic island structure (m=3) in as a function of the distance fiom the major axis. poloidal cross-section of a tokamak. showing regions

of stochastic behaviour around the X-points.