“Profile consistency” features in strongly shaped Ohmic tokamak plasmas

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“Profile consistency” features in strongly shaped ohmic tokamak plasmas

H. Weisen, R. Behn, I. Furno, J.-M. Moret, O. Sauter and the TCV team

Centre de Recherches en Physique des Plasmas

Association EURATOM - Confédération Suisse

École Polytechnique Fédérale de Lausanne

CH-1015 Lausanne, Switzerland

Abstract. A large variety of shaped tokamak discharges have been created in the

Tokamak à Configuration Variable (TCV) with main parameters BT<1.5T, R0=0.88m,

a<0.25m, Ip≤1MA (Hofmann F., Lister J.B., Anton M. et al., Plasma Phys. Contr. Fusion 36

(1994) B277). They include limited and diverted discharges with elongations up to 2.58 and

triangularities between -0.7 and 1. Over the entire range of quasi-stationary sawtoothing

ohmic discharges investigated we observe that the widths of plasma pressure profiles,

and the inversion radii scale as the widths of the current profiles, , where

⟨ ⟩ refers to an average over the volume or, practically equivalently, the cross sectional plasma

area. These “profile consistency” features are in apparent agreement with theoretical consider-

ations and can be viewed as an extension of previous studies in circular discharges. The obser-

vations identify the parameter as the appropriate replacement for qa as scaling

parameter for arbitrary cross-sectional shapes. Using a simple current profile model with

0.8≤q0≤1 for the axial safety factor, we show that for sawtoothing ohmic discharges such

observations can be accounted for by the effects of sawtooth activity.

p⟨ ⟩ p 0( )⁄ j⟨ ⟩ j 0( )⁄

j⟨ ⟩ q0j 0( )⁄

2

Experimental observations and theoretical models supporting the idea that electron tem-

perature and/or the pressure and/or current density in tokamaks preferentially assume certain

privileged profile shapes have been reported for more than two decades [1-6]. For instance

“natural” current and pressure profiles from minimum energy models [1,2] are given in

circular geometry by j/j(0)=p/p(0)=(1+(qa/q0-1)ρ2)-2 and have profile widths equal to q0/qa.

Stationary entropy models [3] also lead to the observable prediction j/j(0)=p/p(0). Current

profiles as given by the above expression are also obtained from a model of filamented current

structure, although in that case there is no preferred pressure profile [4]. Arunasalam et al. [5]

have proposed a practical working definition of the profile consistency principle motivated by

tearing mode stability considerations and observations on the Tokamak Fusion Test Reactor

(TFTR) device [7]. It postulates the existence of unique, natural profile shapes for j(r) and

because of ohmic relaxation, for the shape of Te(r) or . The consequences of the postu-

late, as given in [5], are 1) that the normalized sawtooth inversion radius, ρinv= rinv/a, which is

related to the q=1 surface, and 2) the broadness of the temperature profile ⟨Te⟩/Te(0) are device

independent and are functions of 1/qa only. The same authors propose a third consequence

expressed as , which merely combines the postulation of

universal current profiles characterized by and the assumption of Spitzer

conductivity, .

Relations between profile widths, which appear to be consistent with the above theoret-

ical models, are indeed observed in all ohmic L-modes in TCV (Tokamak à Configuration

Variable, [8]), independently of plasma shape. This was borne out in a study of the effect of

plasma shaping on confinement of a set of some 300 discharges reported elsewhere [9]. A few

examples reflecting the variety of plasma shapes in the dataset is presented in Fig. 1. The

dataset comprises plasma currents in the range 0.1-1 MA and average densities ranging from

1.2.1019 to 1.2.1020m-3.

The observation, shown in Fig. 2, is expressed as ,

where the brackets indicate a volume average for the electron pressure and practically equiva-

lently a cross sectional area average for j. The proportionality factor cpe is typically in the

range 0.7-1. The electron temperature and density measurements were obtained using a

multipoint Thomson scattering system with 10-25 measurement locations in the plasma

depending on plasma size. Each data sample is an average over 3-5 individual measurements

obtained at 50 ms intervals (dictated by the laser pulse frequency) during the stationary phase

Te3 2⁄

r( )

Te03 2⁄

IpR0 cσa2Vloop(⁄ F 1 qa⁄( ) )∝

j⟨ ⟩ j0⁄ F 1 qa⁄( )=

σSpit cσTe3 2⁄

=

pe⟨ ⟩ pe 0( )⁄ cpe j⟨ ⟩ q0j0( )⁄=

3

(“flat top”) of the discharges. The times at which these measurements were made were not

synchronized to the sawtooth cycle.

The width of the current profile is known up to a factor q0 from the relation

(1),

where (2) and (3),

is the dimensionless average current density, j0 being the current density on axis, B0 the

toroidal field on axis, B0vac the corresponding vacuum field, R0 the major radius of the

magnetic axis and κ0 the central elongation. The expression on the right hand side of Eq.(1)

reduces to 1/qa in circular discharges when equilibrium corrections to the axial field (due to

paramagnetism and diamagnetism) are neglected and hence appears as a natural candidate

scaling parameter for “profile consistency” in arbitrarily shaped plasmas. We note that this

parameter can be evaluated without knowledge of the still somewhat controversial axial safety

factor. The core elongation and axial field used to evaluate this expression for Fig.2 were

obtained from the equilibrium reconstruction [10] and, for the former, confirmed by x-ray

tomography. For the purposes of predictive scaling they are found to be accurately given (at

least in TCV) by the following scaling expressions relating to global quantities:

κ0 ≈ 1+0.5.(κ95-1)+0.2.(κ95-1)2 (4) and (5),

where κ95 is the elongation at 95% of the poloidal flux.

Although Fig.2 only shows electron pressure, it is unlikely that the relation, which is

observed at all densities, is specific to electrons only. At average densities above about

5.1019m-3, representing 50% of the samples in the database, electron-ion energy equipartition

is such that central electron and ion temperatures cannot differ by more than 20% even if half

of the ohmic input power density is assumed to be transferred from the electrons to the ions.

TCV is equipped with a 200-channel X-ray tomography system which was used to

measure the sawtooth inversion radius in these discharges [11,12]. For shaped plasmas we

define the effective inversion radius as where Ap is the plasma cross

sectional area and Ainv is the area enclosed within the flux surface at which sawteeth invert. In

j⟨ ⟩q0j0---------

j⟨ ⟩ ∗ B0⋅ vac

κ0 1 κ⁄ 0+( ) B0⋅----------------------------------------=

j0

κ0 1 κ0⁄+( )B0

µ0R0q0------------------------------------= j⟨ ⟩ ∗

µ0R0 j⟨ ⟩B0vac

--------------------=

B0 B0vac⁄ 1 0.054 1 βp–( ) j⟨ ⟩ ∗ 2⋅+≈

ρinv Ainv Ap⁄=

4

Fig.3 we see that the inversion radius scales indeed as ρinv≈<j>/q0j0. This parameter provides

a better fit than inverse safety factor based expressions such as 1/q95. In particular the correc-

tion term for central elongation becomes important for κ0>1.6, whereas the equilibrium field

correction is important for the highest normalized current densities which are obtained at

extreme elongation. In the discharges with κa>2.2 of this dataset the paramagnetic increase of

the axial toroidal field is in the range 10-14%.

The increase of the inversion radius, which is linked to the q=1 surface, with average

current density can be understood from the limitation of the core current density by magneto-

hydrodynamic (MHD) stability. Although the exact value of the central safety factor is still

somewhat controversial, it is generally accepted that in ohmic sawtoothing discharges q0 is in

the range 0.8-1 [13,14]. We propose a model current profile given outside the sawtoothing

zone (ρ>ρ1) as jout=Eσneo+jboot where is the externally induced electric

field, σneo is the neoclassical conductivity and jboot is the bootstrap current, which in these

ohmic plasmas only contributes a few percent to the total current. Inside the q=1 surface a pre-

scribed safety factor profile of the form is assumed, where ρ1

is the normalized radius of the q=1 surface. For ρ≤ρ1 the corresponding current profile is

. (6)

Because of uncertainties in the measurement of Zeff we don’t use the absolute value of

jout. Instead Eσneo is renormalized such that jout(ρ1)=jin(ρ1), thereby matching the current

density profiles in the core and in the confinement zone. For any choice of the free parameter

q0, we then determine ρ1 from the condition . For these calculations the relevant

neoclassical expressions were taken from Sauter et al. [15] and evaluated using the experimen-

tal data. The trapped electron fraction was obtained following a formulation by Lin-Liu and

Miller [16]. Examples of the resulting profiles are shown in Fig.4 for κa=1.2 and 2.4. The

radial coordinate is defined as , where A is the cross sectional area of the flux

surface under consideration and Ap is the total cross sectional area of the plasma. The current

densities in Fig.4 are normalized to the axial current density for a circular discharge with q0=1

and B0=B0vac. The figure also illustrates the beneficial effects of elongation on the current

carrying capacity of the discharge. The q=1 radius from the model is consistent with the first

consequence [5] of “profile consistency” ρ1=F1(<j>⁄q0j0)≈<j>⁄q0j0. The scaling of ρ1, shown

in Fig.5 for q0=0.9, is seen to be identical to that of ρinv in Fig.3. The assumption q0=0.9

E Vloop 2πR0( )⁄=

q ρ( ) q0 1 q0–( ) ρ2 ρ12⁄( )+=

jin ρ( ) j0q0

q ρ( )-----------

B ρ( )B0

------------κ ρ( ) 1 κ ρ( )⁄+( )

κ0 1 κ0⁄+----------------------------------------- 1

ρ21 q0–( )⋅

ρ12

q ρ( )⋅-----------------------------–

⋅ ⋅ ⋅ ⋅=

j Ad∫ Ip=

ρ A Ap⁄=

5

provides the best overall match of ρ1 and ρinv, although good agreement is obtained for the

whole range 0.8≤q0≤1.

The effect of sawtoothing is to flatten, or at least limit the peaking of, the profiles of

temperature and density in the core. This suggests that the scalings of the widths of the plasma

profiles other than the current profile are to a large extend also a result of sawtoothing and

depend on the position of the q=1 surface. Fig. 6 shows the measured widths of ne, Te and pe

as function of ρ1. These widths are close to those expected if the profiles were merely trape-

zoidal (broken line) or trapezoidal squared (continuous line) with a flat portion up to ρ1. The

choice of simple flat core model pressure profiles, although not fully realistic, is motivated

both by the measurements and the fact that with ohmic heating alone the core power density is

too low for significant reheating and profile peaking during the sawtooth rise phase. The

profile shapes outside the inversion radius depend on transport. This however only has a

modest effect on the width <p>/p0 as seen when comparing the two model pressure profiles in

fig.6. It appears that profile widths, expressed as <x>/x0 are not very sensitive to details of the

profiles and hence to transport in the confinement zone (ρ>ρ1). We conclude that profile

relations such as <j>⁄j0≈<p>⁄p0 in ohmic plasmas are a consequence of sawtoothing and result

from the limitation of the central current density and from the flattening of the core profiles of

temperature and density.

In summary, the equation for the width of the current profile, Eq.(1), suggests the

scaling parameter <j>/(q0j0) for profile widths in arbitrarily shaped plasmas as a replacement

for 1/qa. Figs 2. and 3. show that this is indeed the appropriate choice for <pe>/pe0 and ρinv.

The proposed model current profile, for which q0 is in the generally accepted range for saw-

toothing discharges (0.8≤q0≤1) and which is determined by neoclassical conductivity and

bootstrap current in the confinement zone leads to a scaling for ρ1 which is identical to the one

observed for ρinv, as shown in Fig.5. Hence we conclude that the limitation of core current

density by the sawtooth instability is responsible for the scaling of ρ1 and ρinv. Moreover the

flattening of the core profiles of other quantities, such as pressure, inside ρ1 leads to profile

widths with values close to the one of ρ1 (and hence of <j>/j0) while being fairly insensitive to

transport and profile shapes in the confinement zone (Fig.6).

It appears as a coincidence that the effect of sawteeth is to mimic the profile consistency

relations in the sense of Aranusalam et al’s working definition [5] and predicted by the above

mentioned theories which are based on very different physics principles and do not include the

effects of internal MHD activity such as sawteeth. Therefore, despite the apparent agreement,

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the observations reported here should not be considered as supportive of these theories. They

do not however exclude that the processes implied by those theories may also be at work. The

existence of profile-shaping processes other than sawtoothing is suggested for instance by

observations of Te-profile resilience in the confinement zone of sawtoothing discharges [17].

These processes would be expected to determine profile relationships throughout the cross

section of sawtooth-free ohmic discharges, e.g. at sufficiently low <j>* or high q95. To be fair

the above mentioned theoretical predictions should be compared to observations on plasmas

which are not dominated by sawteeth or other forms of strong MHD activity. Unfortunately in

TCV, as in many other devices, ohmic operation with <j>* sufficiently low for sawteeth to

disappear often leads to strong tearing mode activity and an increased likelihood of disrup-

tions, which is why this part of the operational domain remains largely unexplored. Theories

aimed at explaining profile relationships and their scaling therefore should take the effects of

sawtoothing into account and/or make predictions on observable quantities which are not

directly affected by sawtooth activity.

Acknowledgement: This work was partly supported by the Swiss National Funds for Scien-

tific Research.

References:

[1] Biskamp D., Comments Plasma Phys. Control. Fusion 10 (1986) 165 [2] Kadomtsev B., Proc. Int. Conference on Plasma Physics, Kiev, 6-12.4 (1987) 1273[3] Minardi E., Physics Letters A 240 (1998) 70 [5] Arunasalam V. et al., Nuclear Fusion 30 (1990) 2111[6] Coppi B., Comments Plasma Phys. Controlled Fusion 5 (1980) 261[4] Taylor J.B. Phys. Fluids B 5 (1993) 4378[7] Young K.M.,Bell M., Blanchard W.R et al., Plasma Phys. Contr. Fusion 26 (1984) 11[8] Hofmann F., Lister J.B., Anton M. et al., Plasma Phys. Contr. Fusion 36 (1994) B277[9] Weisen H., Alberti S., Barry S. et al., Plasma Physics Contr. Fusion 39 (1997) B135[10] Hofmann F. and Tonetti G., Nuclear Fusion 28 (1988) 1871[11] Anton M., Weisen H., M.J. Dutch, Plasma Phys. Control. Fusion 38 (1996) 1849[12] Furno I., Weisen H., Moret J.M et al., 24th EPS on Controlled Fusion Plasma Physics,

ECA 21A (1997), part II, 545. European Physical Society, Petit-Lancy, Switzerland.[13] Levinton F.M., Zakharov L., Batha S.H. et al., Phys. Rev. Lett 72 (1994) 2895[14] Rice, et al., Fusion Engineering and Design, 34-35 (1997) 135[15] Sauter O., Lin-Liu Y.R., Hinton F.L. and Vaclavik J., Theory of Fusion Plasmas. Proceed-ings of the Joint Varenna-Lausanne Int. Workshop, Varenna, Italy, 1994. Edited by E. Sindoni,F. Troyon and J. Vaclavik. Published for the Società Italiana di Fisica by Editrice Compostori,Bologna, Italy (1994), page 337. ISBN 88-7794-066-2.[16] Lin-Liu Y.R and Miller R.L., Physics of Plasmas 2 (1995) 1666[17] Fredrickson E.D., McGuire K.M., Goldston R.J. et al., Nuclear Fusion 27, (1987) 1897

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Figures

12819, 0.55s

175 kA

12016, 0.7s

450 kA

12868, 0.72s

725 kA

9452, 0.85s

425 kA

Fig. 1. Examples of plasma cross

sections in the dataset.

0.1 0.2 0.3 0.4 0.5 0.6

0.1

0.2

0.3

0.4

0.5

0.6

<j>/(q0j0)

<p e

>/p

e(0)

1.0 <κa<1.25 1.25<κa<1.5 1.5 <κa<1.75 1.75<κa<2.0 2.0< κa<2.25 2.25<κa<2.6

Fig. 2. Widths of electron pressure

profiles versus widths of current

profiles (up to factor q0) in ohmic

discharges. Symbols refer to

discharge elongation at the last

closed flux surface.

0.1 0.2 0.3 0.4 0.5 0.6

0.1

0.2

0.3

0.4

0.5

0.6

ρin

v fr

om to

mog

raph

y

<j>/(q0j0)

1.0 <κa<1.25 1.25<κa<1.5 1.5 <κa<1.75 1.75<κa<2.0 2.0< κa<2.25 2.25<κa<2.6

Fig. 3. Scaling of inversion

radius from soft-x ray tomography

in ohmic discharges. Symbols refer

to discharge elongation.

8

0 0.5 10

0.5

1

1.5

j(ρ)

µ0

R0/

2B0

vac

effective minor radius ρ

#9888: <j>*=0.86 κa=1.21, κ0=1.16

#11877: <j>*=1.7, κa=2.42 κ0=2.14

q0=1 q0=0.9 q0=0.8

Fig. 4. Model current profiles for

two discharges with elongations

equal to 2.42 (top curves) and 1.21

(lower curves) and same qa=2.9 for

three choices of q0. The arrows

indicate the positions of the q=1

surfaces corresponding to q0.

0.1 0.2 0.3 0.4 0.5 0.6

0.1

0.2

0.3

0.4

0.5

0.6

<j>/(q0j0)

ρ(q

=1)

1.0 <κa<1.25 1.25<κa<1.5 1.5 <κa<1.75 1.75<κa<2.0 2.0< κa<2.25 2.25<κa<2.6

Fig. 5 Scaling of effective radii

of q=1 surface from current profile

model for q0=0.9 in ohmic

discharges. Symbols refer to

discharge elongation.

0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.1

0.2

0.3

0.4

0.5

0.6

0.7

ρ(q=1)

<p e

>/p

e0,

<T e

>/T

e0, <

n e>

/ne

0

<ne>/ne0 <Te>/Te0

<pe>/pe0

−− trapez. − trapez.2

Fig 6 Widths of electron density,

electron temperature and pressure

profiles versus effective q=1 radius

from model, assuming q0=0.9. The

broken and continuous lines refer to

trapezoidal and trapezoidal squared

model profiles with a flat part

extending to ρ(q=1).

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