Fusion Plasma Physics || Power Balance

17 Power Balance

In order for any plasma to maintain thermonuclear temperatures, once heated, there mustbe a balance between heat sources and power losses. This balance may be written

P� C P˛ C Paux D PTR C PR (17.1)

Here, P� is the ohmic heating by currents flowing in the plasma, P˛ is the 3:5MeV perD–T fusion event that is released in the form of ˛-particle kinetic energy and hence remainsin the plasma because the charged alpha particle is confined, and Paux is auxiliary powerinput to the plasma by neutral beams, electromagnetic waves, etc. The power losses areby transport, PTR, and by radiation, PR. The radiation loss includes impurity line andrecombination radiation, bremsstrahlung and cyclotron radiation.

Neglecting P�, which becomes negligible at high temperatures, Eq. (17.1) may bewritten

14n2 h��if U˛

�1CQ�1p

�D

3nT

�EC n2LR (17.2)

for the purpose of discussion. We have assumed nD D nT D12n, used LR to represent the

total radiation emissivity by all processes, used U˛ D 3:5MeV, and defined the plasmaenergy amplification factor.

Qp � P˛=Paux (17.3)

All quantities in Eq. (17.2) are appropriate averages over the volume of the plasma, as willbe discussed in this chapter.

17.1 Energy Confinement Time

17.1.1 Definition

The balance of the total (rotational-kinetic plus thermal) energy in a plasma is

PWrot C PWth CHQ D PHEAT � PR (17.4)

where

Wrot �

Z a

o

D12nm�2

EV 0. / d (17.5)

is the rotational energy,

Fusion Plasma Physics. Weston M. StaceyCopyright © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimISBN: 3-527-40586-0

Weston M. StaceyFusion Plasma Physics

Copyright © 2005 WILEY-VCH Verlag GmbH & Co.

464 17 Power Balance

Wth �

Z a

o

D32p CEf

EV 0. / d (17.6)

is the thermal energy, including both “thermal” ions and electrons (32p each) and “fast”

ions in the process of thermalizing .Ef/, PHEAT D Paux C P� is the total heating powerintegrated over the plasma volume, and

HQ �

Z a

o

hr �QiV 0 . / d (17.7)

is the total energy flowing out of the plasma. The integrals are from the center of theplasma . o/ to the LCFS . a/, and h i indicates a flux surface average.

The total energy flux

Q Dn�

12nm�2 C 5

2p��oC fqg C f� � �g � Qconv CQcond CQvis (17.8)

consists of convective, conductive and viscous components. There are corresponding com-ponents of the energy loss defined by Eq. (17.7).

The energy confinement time is intended to represent the mean time energy is con-fined, so it must be defined in terms of the ratio of the plasma energy content to the plasmaenergy loss rate. However, there are many possible definitions. The plasma energy contentmay be considered to be just the energy content of the “thermalized” ions and electrons(Wth without the Ef term), the thermal energy content of the thermalized and fast parti-cles .Wth/, or the thermal plus rotational energy .WthCWrot/. Similarly, the plasma energyloss rate may be due to convection, conduction, viscosity and radiation or to any one orcombination of these processes.

For our purposes here, we will define “thermal” energy confinement times by usingWth to represent the plasma “thermal” energy content. We may define separate energyconfinement times for the separate energy loss mechanisms indicated in Eq. (17.8)

� convE �

Wth

H convQ

; � condE �

Wth

H condQ

; �visE �

Wth

H visQ

(17.9)

Radiation is usually excluded from the definition of energy confinement time. Thus, a total“thermal” energy confinement time is

1

�ED

H convQ CH cond

Q CH visQ

WthD

1

� convE

C1

� condE

C1

�visE

(17.10)

17.1.2 Experimental Energy Confinement Times

If we use Eq. (17.4) to replace HQ in Eq. (17.10), we can express the thermal energyconfinement time in terms of experimentally measured or otherwise known quantities

1

�expE

DPHEAT � PR � PWth � PWrot

Wth(17.11)

17.1 Energy Confinement Time 465

In practice, PWrot is usually ignored, but this can introduce error for strongly rotating plas-mas. Ef is sometimes omitted from measured Wth, in which case a “thermalized” thermalenergy confinement time with no fast particle contribution is obtained.

One of the purposes of measuring energy confinement times is to check theoreticaltransport models, which can be used to evaluate the Hx

Qthat are used in Eq. (17.10)

to obtain a theoretical thermal energy confinement time. Most commonly, the theoreti-cal transport model is for the thermal conductivity, �, which enters the conductive (q D�n�rT ) energy confinement time. Equating Eq. (17.10) to the measured energy confine-ment time and solving for

1

� condE

D1

�expE

�1

� convE

�1

�visE

(17.12)

provides a correction to the measured energy confinement time that is needed for suchcomparisons.

17.1.3 Empirical Correlations

When experimental and theoretical energy confinement times are compared (albeit usuallywithout the corrections indicated above), it is usually found that neoclassical transporttheory overpredicts the confinement time by an order of magnitude or more. This has ledto correlation of measured energy confinement times in order to obtain expressions thatcould be used in predicting the power balance in future tokamaks.

The confinement behavior can be conveniently put into four categories. The first coversohmically heated plasmas and the other three relate to plasmas with additional heating.The two basic modes of auxiliary heated plasma confinement are the so-called L (for low)and H (for high) confinement regimes. The fourth category covers a variety of plasmas forwhich operational procedures have been found which produce enhanced confinement.

17.1.3.1 Ohmically Heated Plasmas

At low density it was found that the energy confinement time scaled as

�E.s/ D 0:07�n=1020

�aR2q (17.13)

where n is the average electron density, a and R the minor and major plasma radii, and q

is the cylindrical equivalent edge safety factor. (Unless otherwise indicated, MKS units areused in this chapter.) The improved confinement with increased density is in conflict withthe decrease in confinement predicted by neoclassical theory.

As the density is increased the linear improvement of �E with n is lost and �E saturatesat a density

nsat.m�3/ D 0:06 � 1020IRM 0:5��1a�2:5 (17.14)

where I is the plasma current, M is the atomic mass of the ions in amu and � is the plasmaelongation, b=a.


Figure 17.1. Improved ohmic confinement mode in ASDEX. The broken curve, with the open circlesand crosses, corresponds to the saturation mode; the upper lines are for two different scans in IOCmodes

By controlling the density in such a way as to maintain a peaked density profile it hasproved possible to extend the linear scaling into the so-called Improved Ohmic Confine-ment (IOC) regime. Results from the ASDEX tokamak are given in Fig. 17.1.

17.1.3.2 L-Mode Confinement

In order to increase the plasma energy content above that achieved with ohmic heatingalone, additional heating was applied using beams of energetic neutral particles or RFwaves. The results were initially disappointing in that, for given operational conditions,the confinement was found to degrade with increasing power. By analyzing the resultsfrom several tokamaks Goldston obtained the confinement scaling law that bears his name

�G.s/ D 0:037IR1:75�0:5

P0:5a0:37(17.15)

17.1 Energy Confinement Time 467

Figure 17.2. Comparison of the experimental values of confinement time from a number of tokamakswith the L-mode scaling � ITER89-P

E

(I in MA, P in MW), where P is the applied power. The degradation is apparent throughthe factor P0:5 in the denominator.

Although the scaling was obtained before the large tokamaks such as JET were oper-ational, it was also found to describe the results on these machines. In order to improvethe predictive capability for the proposed tokamak reactor ITER, an extended data base,including the larger tokamaks, was used to generate a more precise form of Goldston’sscaling. The resulting confinement time, which was given the name ITER89-P, is given by

� ITER89-PE .s/ D 0:048

I0:85R1:2a0:3�0:5�n=1020

�0:1B0:2M 0:5

P0:5(17.16)

(I in MA, P in MW) where B is the toroidal magnetic field. A comparison between thisformula and data from a number of tokamaks is shown in Fig. 17.2.

17.1.3.3 H -Mode Confinement

It was found that when sufficient power was applied to an L-mode discharge the dischargemade an abrupt transition in which the edge transport was apparently reduced, leading toedge pedestals in the temperature and density, as discussed in chapter 15. The effect of thiswas to produce roughly a doubling of the confinement time.

This behavior was subsequently obtained in many tokamaks, and analogues of theITER89-P L-mode scaling law have been derived for H -mode discharges. The results arerepresented by

� ITERH93-PE .s/ D 0:053

I1:06R1:9a�0:11�0:66�n=1020

�0:17B0:32M 0:41

P0:67(17.17)


(I in MA, P in MW). It is seen that the dependences are generally similar to those ofthe L-mode.

A more recent H -mode correlation is

�IPB98.y;2/E

.s/ D0:144I0:93B0:15

�n=1020

�0:41M�0:58R1:97�0:78

P0:69(17.18)

where P .MW/ is the heating power, I.MA/ is the plasma current, and � is the elongation.

17.2 RadiationRadiation is a major power loss mechanism in plasmas, as well as being a useful diagnos-tic tool. Acceleration of plasma particles due to their mutual interactions and due to theirinteraction with the electric and magnetic fields results in the emission of electromag-netic radiation. We will develop a general formalism for radiation fields resulting from theacceleration of charged particles and will apply this formalism to two important radiationprocesses – bremsstrahlung and cyclotron emission – in this section. The third, and mostimportant, radiation process in plasmas arises from orbital electron transitions in partiallyionized impurity atoms that are present in the plasma and were discussed in chapter 13.

17.2.1 Radiation Fields

First, we consider the radiation field from a charged particle moving with velocity �.In Fig. 17.3 we denote by A the location of the particle at time t when it is accelerated andradiation is emitted. Let C be the point at which the radiation is “observed,” or computed,at a later time t C jrj=c, and let D denote the actual location of the accelerated particle atthis time. Let B denote the “virtual” location of the particle at t C jrj=c if it had not beenaccelerated. In computing the scalar and vector potentials observed at C at time t C jrj=c,the position and motion of the particle at time t must be used. These delayed, Lienard–Wiechart potentials are

�.r/ D1

4�0

e

jrj � .r � �/=c(17.19)

and

A.r/ D14��0e�

jrj � .r � �/=c(17.20)

The magnetic and electric fields at C due to the charged particle can be computed fromthese potentials,

B D r �A (17.21)

and

E D �r� �@A

@t(17.22)

17.2 Radiation 469

Figure 17.3. Radiation field nomenclature

Using these equations, and identifying the 1=r -component of the resulting fields with theradiation field, we obtain

Erad 'e

4�0c2r3r � .r � P�/ (17.23)

and

Brad '�0e

4�cr2. P� � r/ D �0Hrad (17.24)

where the nonrelativistic limit, j�j=c � 1, has been assumed. The acceleration of thecharged particle is P�. The Poynting vector for the radiation field is

Prad � Erad �Hrad De2

.4�/20c3r5Œr � .r � P�/� � . P� � r/ (17.25)

The outward radiation flux per unit solid angle (d˝ D sin � d� d� in spherical coor-dinates) is found by computing the power flux through an incremental surface area, dA Dr2 d˝, on a sphere of radius r centered on A:

dWrad

d˝D r2 .Erad �Hrad/ � Onr D

e2j P�j2 sin2 �

.4�/2 0c3(17.26)

where Onr is the unit vector along r in Fig. 17.3, and � is the angle between r and P�. Thetotal power radiated by the accelerated charge is

Wrad D

Z 2�

0

d�Z �

0

dWrad

d˝sin � d� D

1

6�

e2j P�j2

0c3(17.27)

A generalization of this development to remove the nonrelativistic approximation ( j�jc�1)

leads to

dWrad

d˝D

e2j P�j2 sin2 �.4�/20c3

�1 �

�

ccos �

��6(17.28)

instead of Eq. (17.26).


The rate at which radiated power originating at point A at time t passes through anincremental surface area at point C at time t C jrj=c � t 0 is

d2Wrad

dt d˝

!d˝ D jErad �Hradj

�dt 0

dt

�r2 d˝ (17.29)

Using Eq. (17.28) and

dt 0

dtD 1 �

r � P�

jrjcD 1 �

�

ccos � (17.30)

we can write Eq. (17.29)

d2Wrad

dt d˝

!d˝ D

e2j P�j

.4�/20c3sin2 �

�1 �

�

ccos �

��5d˝ (17.31)

The general Poynting vector, which is valid for relativistic as well as nonrelativisticparticles, can be constructed by the same procedure used to obtain Eq. (17.25), but withoutmaking the nonrelativistic approximation. The result is

jPradj � jErad �Hradj De2jr � .r� � P�/j

2

.4�/2�020c5jrj6

�1 � �

ccos �

�6 (17.32)

where

r� � r �jrj�

c(17.33)

(refer to Fig. 17.3). Finally, the general expression for the outward radiation power fluxper unit solid angle is

d2Wrad

d˝ dt

!D

e2

.4�/20c3s5jr � .r� � P�/j

2jrj (17.34)

where

s D

�1 �

r � �

jrjc

�jrj (17.35)

This formalism is generally applicable to the radiation produced by the acceleration ofa charged particle. Specific types of radiation fields are obtained for specific accelerationmechanisms in the following sections.

17.2.2 Bremsstrahlung

Bremsstrahlung – the German term for “braking radiation” – results from the collisionaldeceleration of charged particles. A photon is produced with energy equal to the energyloss in the collision process. Bremsstrahlung arises from any two-body collision betweencharged particles. However, in the nonrelativistic limit (j�j=c � 1), the radiation fields

17.2 Radiation 471

produced by the two particles in a like-particle collision exactly cancel, so that it is onlythe electron–ion collisions that produce bremsstrahlung.

Quantum mechanics must be used to treat electron bremsstrahlung correctly. However,the classical theory of the previous section provides a reasonable approximate result andsome insight into the physics of the process.

The Coulomb scattering law gives

j P�j 'ze2

4�0mex2(17.36)

For the transverse acceleration of an electron incident upon an ion, with impact para-meter x. Using this expression for acceleration in Eq. (17.27) leads to an expression forthe rate of energy loss in a single electron–ion collision

Wrad Dz2e6

96�330c3m2

ex4

(17.37)

This energy loss lasts for a time of about x=j�ej, and the number of such collisions experi-enced per unit time by an electron in passing through a plasma with ion density ni is ni�e .Thus the total bremsstrahlung power radiated by ne electrons is

Pbrem D ninej�ej

Z xmax

xmin

Wrad

�x

j�ej

�2�x dx (17.38)

The result obtained from Eq. (17.38) differs only by an order unity numerical factorfrom the correct result, which is (T in keV, other in mks)

Pbrem D 1:7 � 10�38z2nineT1=2

e

�W =m3

�(17.39)

When more than one ion species is present, z2 in Eq. (17.39) is replaced by

zeff �Xj¤e

njz2j

ne(17.40)

A small impurity concentration significantly enhances the bremsstrahlung power radiated.

17.2.3 Cyclotron Radiation

Charged particles in a magnetically confined plasma spiral about the field lines with gyro-frequency

˝ D �eB

m(17.41)

and gyroradius

rL D�?

j˝j(17.42)

The centrifugal acceleration of these charged particles is a significant radiation processin sufficiently hot plasmas.


Figure 17.4. Cyclotron radiation nomenclature

Using the nomenclature of Fig. 17.4, the magnitudes of the particle velocity and accel-eration are

j�j D rL˝ (17.43)

and

j P�j D rL˝2 (17.44)

The direction of P� is always radially inward, and the direction of � is always tangentialto the gyro-orbit (motion along the field line is ignored). For a fixed observation point, theangle ˛ between the axis of rotation (field line) and the direction to the observation pointis approximately constant. We use the relations

� � r D j�jjrj cos �

P� � r D j P�jjrj cos � tan�

r� D r ��jrj

c

(17.45)

to evaluate the term

Œr � .r� � P�/�2 D j P�j2r4

��1 �

�

ccos �

�2�

�1 �

�2

c2

�tan2 � cos2 �

�(17.46)

that appears in the general radiation formula, Eq. (17.34), allowing that expression to bewritten

d2Wrad

dt d˝

!D

e2j P�j2

.4�/20c3

h�1 � �

ccos �

�2��1 � �2

c2

�tan2 � cos2 �

i�1 � �

ccos �

�5 (17.47)

17.3 Impurities 473

Making use of the trigonometric relation, cos � D sin˛ cos�, and writing the unitsolid angle as d˝ D sin˛ d˛ d�, we can integrate Eq. (17.47) over a solid angle to obtainthe rate of energy loss from a single particle because of the acceleration associated withits gyromotion,

�dWrad

dt

�D

e2

6�0c3

r2L˝4

�1 �

�rL˝c

�2�2 (17.48)

where we have made use of Eqs. (17.43) and (17.44). Because .rL˝2/e � .rL˝2/i, the

cyclotron radiation is generally much larger for electrons than for ions.The computation of the total power emitted from a plasma by cyclotron radiation of

electrons is quite complex, involving as it does relativistic electron distribution functions,the correlation of radiation from different electrons, the absorption and reemission of thecyclotron radiation by the plasma, and the reflection of the radiated power by the surround-ing wall. An approximate formula for the total cyclotron radiation power density emittedfrom a plasma with electrons in a Maxwellian distribution is (Te in keV, otherwise mks)

Pcyl D 6:2 � 10�17B2neTe.1C Te=204C � � � /�

w=m3�

(17.49)

17.3 ImpuritiesIdeally, D–T fusion would involve the presence of only hydrogenic species in the plasma.However, plasma impurities are unavoidable. At a minimum, He ash from the D–T fusionreactions will be present. Plasma–surface interactions are also inevitable, resulting in therelease into the plasma of atoms and molecules from the solid structural components byevaporation, sputtering, etc.

Plasma impurities have both harmful and helpful consequences. The principal harmfulconsequence is cooling of the main plasma. The radiation power function varies greatlyamong elements, (see Fig. 17.5). That fractional impurity level which results in radiationpower equal to 50% of the alpha-heating power is shown in Fig. 17.6. Clearly, it is muchmore deleterious to have high-Z elements present in the core than low-Z ones. Figure 17.5brings out a second critical difference between low-Z and high-Z impurities, as regardstheir radiative properties.

Low-Z elements become completely stripped of their orbital electrons at relativelylow temperatures, Te � 1 keV. This results in a sharp decrease in the radiation powerfunction to the low residual level given by bremsstrahlung radiation. The high-Z elements,however, retain some orbital electrons, even at the high temperatures of the core plasma,and their line radiation continues to make them efficient radiators. Fortunately, the intrinsicimpurity helium can be tolerated more than other impurities. Nevertheless, a maximumlevel of about 10% is acceptable for a reactor-like device.

A fit for impurity radiation from a plasma is

Pimp

�Mw=m3

�D .1C 0:3Te/ � 10�43nenzz

.3:7�0:33 lnT / (17.50)


Figure 17.5. (a) The radiation loss or power function R (defined as Lz elsewhere) and (b) the meancharge NZ as functions of Te for C, O, Fe and W

Figure 17.6. Fractional impurity level which produces a radiation power equal to half of the alpha-heating power

where T is in keV, z is the atomic number of the impurity ions, and the other quantitiesare in mks units.

Impurities present in the fusion core of the plasma are also harmful because of fueldilution. The total plasma pressure, for a given field strength, B, is limited by MHD insta-bilities. The plasma ˇ (�

Pi niTi=.B

2=2�0/) summed over electrons and all ion speciescannot exceed the MHD beta limit. The electrons released by the impurities can “use up”a lot of this permitted maximum plasma pressure, even for impurity fractions which arerather low, thus diluting the beta-fraction of fuel ions.

Low-Z impurities radiate near the edge of the main plasma, where Te is low enoughthat the ions are not fully stripped. This is potentially very advantageous, as was discussedin chapter 15 regarding the radiative mantle. It is also potentially harmful. Sufficiently

17.4 Burning Plasma Dynamics 475

strong radiation in the periphery of the confined plasma can reduce the electrical conduc-tivity so much that the tokamak current profile contracts, leading to an increased destabi-lizing current gradient inside the q D 2 surface. This then results in a disruption of theplasma, as will be discussed in chapter 18.

Unless impurity levels are kept below a critical level, there is the risk of a sputteringrun-away catastrophe. While the normal incidence ion yields of hydrogenic species on allsubstrates are less than unity, self-sputtering yields often exceed unity. Since most sput-tered neutrals become ionized within the plasma they can return to the solid surfaces asmultiple charged ions and be accelerated by the sheath potential drop to very large impactenergies.

Plasma impurity effects are not all deleterious, and we consider also their beneficialeffects. As already discussed, the greatest benefit of impurities is volumetric power loss,so long as it occurs either in the SOL or near the periphery of the main plasma, and doesnot compromise the energy confinement. Such disposal of the exhaust power is greatlypreferable to the intense, highly localized power deposition by particle impact on the verysmall divertor target areas, � 1m2, that characterize magnetically confined devices. Thephotons can distribute the exhaust power over most of the large wall area – hundreds ofsquare meters for a reactor-like device. Furthermore, photons do not cause sputtering. It isalmost certainly essential that reactor-like devices dispose of a substantial portion of theexhaust power through radiation.

17.4 Burning Plasma DynamicsA set of global equations may be written to describe the power and particle balances in aD–T plasma operating under fusion “burning” plasma conditions.

dni

dtD Si �

12n2i h��if �

ni

�i(17.51)

dn˛dtD S˛ C

14n2i h��if �

n˛

�˛(17.52)

dnzdtD Sz �

nz

�z(17.53)

3

2

ddt.neTe/ D P� C P e

aux C14n2i h��if U˛e �Qie � PR �

3

2

neTe

�eE

(17.54)

and3

2

ddt.niTi/ D P i

aux CQie C14n2i h��if U˛i �

3

2

niTi

� iE

(17.55)

The subscripts i , ˛, z and e refer to ions, alpha particles, impurity ions and electrons,respectively. The quantities in Eq. (17.51) to Eq. (17.55) are appropriate spatial averages ofthe previously defined quantities with the same symbol. The particle confinement times, � ,are defined in a similar manner as the energy confinement times, �E , but from the particlebalance.


The sources (Si, S˛ , and Sz) depend on the particle and heat fluxes leaving the coreand upon the specifics of the boundary region, plasma chamber wall and any impuritycontrol mechanism present. Any external fuel source also contributes to Si. A simple,order-of-magnitude estimate of these sources can be made when there is no impurityremoval mechanism and the boundary region is sufficiently tenuous that charge exchangeis unimportant,

Si D RWi

ni

�i(17.56)

S˛ D RW˛

n˛

�˛(17.57)

and

Sz D�RW

z C Y Wz

� nz

�zC Y W

ini

�iC Y W

˛

n˛

�˛(17.58)

where RW and Y W are the recycling coefficient and sputtering yield, respectively.We now examine some of the problems that will be encountered in maintaining a

steady state power balance in a thermonuclear plasma and consider some control mech-anisms that can be used to achieve this objective. The problems fall into two categories,composition changes and dynamic instabilities. We can use an overall power balance onthe plasma to examine the problems,

dUdtD .P˛ C Paux/ � .PR C PTR/ (17.59)

where U D 3nT is the plasma internal energy, the terms in the first bracket on the rightside are the alpha and supplemental power sources, and the terms in the second bracketare the radiation and transport power losses. A power balance is achieved when the RHSof Eq. (17.59) vanishes.

Assume that a power balance has been achieved at the beginning of a burn cycle, andconsider the effects of the accumulation of wall eroded and fusion alpha impurities uponthe power balance. The most obvious effect is the increase in PR, which is usually theprincipal effect. The accumulation of impurities may also change PTR � .�E/

�1. If themaximum value of ˇ is limited by MHD stability requirements, then the accumulation ofimpurity ions must be compensated by a reduction in the principal plasma ion concentra-tion, so that N / .niCn˛CnzCne/T remains constant. This reduces P˛ D n2i h��ifU˛ .The net effect of impurity accumulation is usually to decrease the plasma temperature.

Again, assume that a power balance has been achieved, and now consider the effectof a small temperature perturbation. A linear expansion of Eq. (17.59) about the powerbalance condition yields,

dUdtD

��@P˛

@T

��T C

�@Paux

@T

��T

��

��@PR

@T

��T C

�@PTR

@T

��T

�(17.60)

If the RHS of Eq. (17.60) is positive the response to the perturbation is such as toincrease the temperature perturbation. Thus the power balance is unstable if��

@P˛

@T

�C

�@Paux

@T

��

��@PR

@T

�C

�@PTR

@T

��> 0 (17.61)

17.4 Burning Plasma Dynamics 477

Because the fusion cross section increases sharply with the temperature (up to aboutT D 80 keV for D–T fusion), the alpha heating term is destabilizing. The sign of @Paux=@T

depends on the type of auxiliary heating. The radiative loss term is generally negative, thusdestabilizing, for impurities which are partially ionized (see Fig. 13.9). The transport lossterm can be stabilizing or destabilizing depending on whether �E varies inversely (e.g.most “anomalous” transport theories) or directly (e.g., neoclassical with �E � T 1=2) withtemperature.

This power balance instability can be analyzed quantitatively by using Eq. (17.51) toEq. (17.55), which are of the generic form

dxi

dtD fi.x1; : : : ;x5/ ; i D 1; : : : ; 5 (17.62)

The equilibrium solutions that obtain when the power balance is achieved satisfy

fi.x10; : : : ;x50/ D 0 ; i D 1; : : : ; 5 (17.63)

To examine the stability of the power balance condition to perturbations in density andtemperature, expansions about the equilibrium solutions,

xi.t/ D xi0 C�xi.t/ ; i D 1; : : : ; 5 (17.64)

are substituted into Eq. (17.62) and the resulting equations are Laplace-transformed andlinearized to obtain

5XjD1

��@fi

@xj

�0

� sıij

��xj D 0 ; i D 1; : : : ; 5 (17.65)

Eq. (17.65) can be solved for the frequency, s. When Re.s/ < 0, the equilibrium is stable.A qualitative picture of the thermal stability properties of a fusion reactor can be

obtained from the contours of constant auxiliary power in the hnei–hT i space, commonlyreferred to as POPCONs. The POPCON plots are calculated here from the energy balanceequations for the electrons and ions, obtained from the equilibrium versions of Eq. (17.54)and Eq. (17.55).

For each point in the hnei–hT i space, we fix ne and the electron (or ion) temperatureand we solve the resulting nonlinear algebraic system of power balance equations for theion (or electron) temperature and the required auxilary power Paux.

The thermal stability properties of a point are determined by the slope of the resultingPaux curve. Points are unstable to temperature excursions if @Paux=@T < 0, since, if thetemperature increases from its equilibrium value due to a positive temperature perturba-tion, the plasma moves to a region where the required auxiliary power for equilibrium islower. Therefore, a net energy gain heats the plasma and removes it further from the orig-inal equilibrium point. This temperature excursion lasts until a stable operating point isfound in the hnei–hT i space (in reality the beta limit will likely be exceeded and a disrup-tion would terminate the discharge).

The shape of a POPCON, and hence the stability properties of the plasma, depends onthe energy confinement scaling, but even for the same confinement scaling it is sensitiveto several parameters such as the Zeff, the thermal alpha-particle concentration, and the


Figure 17.7. POPCON for ITER, assuming Goldston scaling with enhancement factor H D 2:0 andalpha-particle concentration of 5%

L-mode enhancement factor H . In Fig. 17.7, the POPCON for Goldston scaling, assumingthermal alpha-particle concentration n˛=ne D 5% and H D 2, is shown. Also shown onthe same plot are curves of constant fusion power of 500MW and 1000MW. There aretwo possible ignited operating points at each of these power levels, one unstable in thelow hT i, high hni region, and one stable in the high hT i, low hni region.

Problems for Chapter 171. Calculate the experimental energy confinement time of a plasma of volume 24m3 that

is maintained at a constant density n D 3 � 1019 m�3 and temperature T D 5 keVby neutral beam injection power of Pnb D 5:5MW. The measured radiation from theplasma core is 1:5MW.

2. Calculate the predicted energy confinement times of an H-mode deuterium plasma withI D 3MA, R D 1:7m, a D 0:6m, � D 1:75, n D 3 � 1019 m�3 and B D 3T usingthe ITER93-P and the IPB98(y,2) empirical scaling laws.

3. Calculate the bremsstrahlung and cyclotron radiation power from the plasma of prob-lem 2 operating at 5 keV. Calculate the impurity radiation that would result from a 5%carbon concentration and from a 1% iron concentration.

4. Solve Eq. (17.51) to Eq. (17.53) and Eq. (17.56) to Eq. (17.58) for the equilibriumdensity levels for deuterium–tritium plasma ions, alpha particles and carbon impuritiesfor a 10 keV plasma.

5. Derive an explicit power balance stability criterion by substituting explicit expressionsfor the various power terms in Eq. (17.61).

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Fusion Plasma Physics || Power Balance