A onedimensional model for radiative thermal equilibrium and stability of thetokamak edge plasmaShishir Deshpande Citation: Physics of Plasmas (1994-present) 1, 127 (1994); doi: 10.1063/1.870920 View online: http://dx.doi.org/10.1063/1.870920 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/1/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Multifaceted asymmetric radiation from the edge (MARFE’s): A general magnetohydrodynamic study in aonedimensional tokamak model Phys. Plasmas 1, 2623 (1994); 10.1063/1.870589 Stability of a onedimensional plasma in slab geometry Phys. Fluids 30, 3502 (1987); 10.1063/1.866431 OneDimensional Impenetrable Bosons in Thermal Equilibrium J. Math. Phys. 7, 1268 (1966); 10.1063/1.1705029 OneDimensional Plasma Model at Thermodynamic Equilibrium Phys. Fluids 5, 1076 (1962); 10.1063/1.1724476 OneDimensional Plasma Model Phys. Fluids 5, 445 (1962); 10.1063/1.1706638

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A one .. dimensional model for radiative thermal equilibrium and stability of the tokamak edge plasma

Shishir Deshpande Institute For Plasma Research, Bhat, Gandhinagar-382424, India

(Received 26 May 1993; accepted 27 September 1993)

The multifaceted asymmetric radiation from the edge (MARFE) instability observed in the peripheral radiative layer in the high-density tokamaks is considered for a spatially smooth profile of radiative-loss function in equilibrium. The instability which is driven by the impurity radiation and opposed by the finite thermal conductivity, was considered previously by Drake [J. F. Drake, Phys. Fluids 30, 2429 (1987)] as a surface wave on a sharp radiative layer (step-function-like) at the edge. A novel exponential model is presented for the radiative loss function and it is shown that the two crucial features of MARFE: (i) stability to poloidally symmetric perturbations and (ii) a threshold value of radiative loss for the growth of asymmetric perturbations, explained earlier by Drake's theory, are contained in this model of radiative equilibrium, also. Since, from the earlier theory, these features appear to be equilibrium specific and related to the form of radiative-loss function, the present work broadens the class of radiation profiles which can explain the MARFE, by recovering all the old results by an entirely different mathematical treatment.

I. INTRODUCTION

The multifaceted asymmetric radiation from the edge (MARFE) phenomenon) typically arises when the plasma density in tokamaks is increased towards the density limit.2

The density limit primarily arises due to the loss of power balance at the high densities. The radiative power loss due to impurities increases with density and at the limit, equals the power input. Upon further increase of density a ther­mal collapse is expected2 to trigger a disruption of the tokamak plasma discharge.

The MARFE, in a sense, is a predisruptive, quasi­steady state of the tokamak discharge, generally achieved by increasing density while keeping the plasma current constant. During the MARFE an intense toroidal belt of radiation forms at the small major radius side of the torus. Cool and dense plasma characterizes the region of MARFE. The onset is observed to be just below the den­sity limit,3 n-O.8nc and upon further increase of the den­sity, the radiation can poloidally symmetrize to form a detached plasma (DP) or lead to a disruption. The impor­tance of the MARFE arises from the fact that it affects the heat load and its distribution on the wall, so a greater understanding of MARFE may allow us to control edge conditions in a desirable way.4 The crucial features of MARFE are (i) onset somewhat below the density limit, (ii) poloidal asymmetry, and (iii) simultaneous density enhancement with temperature decrease. In a recent paper, DrakeS has shown that a simple model for radiative ther­mal equilibrium of a cylindrical plasma column explains the above features of the observed MARFE. In this model a constant heating rate and a constant perpendicular ther­mal conduction coefficient is assumed with a step function form for the radiative power-loss function [L=L(T)=Lo(n)8(TL -T)], as a crude model for the impurity radiation in coronal equilibrium.6 The character­istic temperature, T L, is much smaller than the core tem-

perature, so the radiation is essentially localized to the edge.

The crucial success of Drake's theory lies in qualitative explanation of the following observed features of the MARFE: (i) stability of poloidally symmetric perturba­tions and (ii) growth of poloidally asymmetric perturba­tions when the radiative losses cross a certain threshold. Furthermore, it was demonstrated that these features can­not be explained unless the information of the underlying equilibrium is used in the perturbation analysis. It is im­portant to realize that even if the basic driving mechanism for the instability-negative derivative of the radiative power-loss function-is operative, the stabilizing effect of the perpendicular heat flux from the core cannot be ig­nored. It is this effect which is related to the MARFE features above. In Drake's theory, this relationship comes from the equilibrium radiation layer width (and its depen­dence on Lo) and decides whether perturbations will grow or not. If the model for the temperature dependence of radiative loss is changed then the spatial profile of equilib­rium radiation also changes and so it is of interest to know the stability features of the MARFE above.

In this paper, we probe a radiative equilibrium derived by taking a novel exponential model for coronal radiation, in order to examine the apparently equilibrium-specific na­ture of the MARFE features mentioned above. We take L(T) = Loe-TITL to be the form of radiative-loss func­tion with temperature. Exact eqUilibrium and stability analysis are then carried out. Although the resulting math­ematical treatment is entirely different, all the results ob­tained previously by Drake are recovered. It is thus shown that the crucial features of MARFE are also contained in this model for L(T) so that a broader class of L(T) may be able to explain the MARFE phenomenon.

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II. THERMAL EQUILIBRIUM WITH RADIATION

In this section we set up the radiative thermal equilib­rium for the edge region of tokamak plasma. In this region the heat received from the core plasma by perpendicular conduction is radiatively lost by the impurities in the edge plasma.

To study the equilibrium and stability we use a simple set of equations5 describing, density n, parallel fluid veloc­ity vII and the temperature T,

an h

at+b.V(nvll )=0, (1)

mn aVIl = -h·VP I at ' (2)

3 aT A h

2nat-b,VXII b·VT-Vl 'Xl Vl T

= -nTh· VVII +H-L. (3)

The unit vector h=BI B is along the direction of the mag­netic field and XII and Xl are the parallel and perpendic­ular thermal conductivities. The heating rate H is generally smaller in magnitude compared to the radiative loss L in the edge region, so we shall ignore it in the present calcu­lation. We shall show later that its inclusion does not change results qualitatively. However, the physics of His retained while setting up the vital boundary conditions which decide the heat flux escaping from the core to the edge region.

The one-dimensional equilibrium which describes the balance of perpendicular heat conduction and radiation is given by setting alat=h·V=O in Eqs. (1)-(3),

d2T TIT

Xl dX2=Loe- L-H. (4)

Here, X is the local slab variable, X=a-r, and VI has been replaced by dldX because the edge localized radiative layer is thin, and hence curvature terms can be justifiably ignored. We now introduce Le to represent that value of Lo at which all the input power is radiatively lost. Physically, increasing Lo beyond Le can trigger a thermal collapse of the existing equilibrium. Whether the plasma attains a new detached state} by radial contraction, or disrupts by even­tually becoming unstable magnetohydrodynamically, is not contained within the scope of present calculation, therefore we shall simply say that there is no equilibrium beyond Lo= Le' With dimensionless parameters p2=. Lol Le ,

1/1"== TIT L' and a 2=.2XI T LI Lc' we can rewrite Eq. (4) as

(5)

Here, the prime (') indicates differentiation with respect to the dimensionless variable x=.Xla and h=2HI Le' The parameter a is the characteristic spatial scale of variation of equilibrium and is estimated as follows. Let T c be the core temperature, so that the core heat balance is given approximately by Xl T /a2-Hthen a2-a2HTLI(LeTc). By definition, Le is the radiation rate at which the total input and radiated powers are equal, therefore,

128 Phys. Plasmas, Vol. 1, No.1, January 1994

11'a2H-211'aL~, which along with the relation for 1::,.2, shows that HI Le is of order T LITe' Thus a-aT LITc<t.a as T L <t. Te' This shows that the radiative layer thickness a is much smaller than the radius of the plasma column a. Also, the local heating term h in Eq. (5) is small compared to unity. We therefore treat this effect perturbatively, later in this section. As in Ref. 5, we take the relationship be­tween Xl' Te, and H as Xl Te=Ha2/4, so that Lc=HT /2T L using a=aT LITe. Equation (5) is now solved for the boundary conditions

(6)

The latter condition in (6) follows from the requirement that as one moves closer to the core region (increasing tf;) the radiation diminishes and that the total heat flux should correspond to the total input power. As the radiative layer is thin, this can be realized even when r is close to a,

dT 2 211'aXI dr 'Z11'a H.

The solution of (5), subject to (6), is given by

tf;=2In[p cosh(x+o)] or tf;=21n(cosh x+€ sinh x), (7)

where o=cosh- 1(lIp) and e2=1_p2. Note that, when p2 = 1 (Lo = Lc) all the input power is radiative1y lost [tf;' (0) = 0] and there is no heat outflux from the boundary at x=O (r=a). Note also, that € represents the fraction of the outgoing heat flux, tf;' (0) =2€, so that €=tf;'(O)/tf;'(x>l). Finally, note that the neglect of h in (5) is justified when e -", > h (p being typically of order unity), so that tf;max<ln(lIh). At large values of x, tf;-2x, this gives us the range of validity of (7): O<x;Sln(l1 Jh).

The radiation profile is given by

L=Loe-"'= Lc sech2[x+cosh- 1(llp)]. (8)

Defining the width (w) of the radiation layer as that dis­tance (in units of a) from the edge where L falls to 1/ e times its peak value, we get

(

eI/2+ (e2_ p2) 112) w=ln 1+(1_p2)1l2 ,

where e=2.718.

(9)

When p2<1, W'Z 1/2_p2(l + 1/e)/4, and when p2=1,

This shows that w= 1 or that the layer thickness is of order a even when Lo is increased up to the limit Le' We thus get the result that radiation is still localized in the edge when the equilibrium is lost. These results are consistent with those of Drake.s

We now obtain the correction to the solution (7), due to the local heating constant h in Eq. (5). Imposing the boundary conditions tf;(0) =0 and tf;'(0) =2€, the solution with corrections of order h is given by

Shishir Deshpande

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t/l=21n{p cosh [x+l>-h1o(x) n, (10)

with Io(x) =I[t,b=t,bo(x)]. Here, =2In[p cosh(x+8)], and

I r'" u du I(t,b)=g Jo (l-p2e u)3/2·

The radiation profile with corrections of order h is given by

(11 )

Since Io(x) > 0, the local radiation loss increases by a small amount (~h) compared to that given by (8). Note that t,b' (x> 1) -.2, so that the boundary condition t,b' (0) =2£ keeps the same fraction of heat outflux as before. The local radiation, therefore, must increase to compensate the (small) local heating of the plasma.

III. STABILITY ANALYSIS

Linearizing Eqs. (1)-( 3), substituting al at by y, b· V by ikU and defining SII =kTI C;/(Y+kTI C;), we obtain the following equatioE for the spatial structure of the per­turbed temperature T:

(3 ) _ _ a2f 2+S11 nYT+kTI XII T-Xl ax2

( aL -aL)

=- n an +T aT . (12)

The density and temperature perturbations are coupled by the sound wave

n T ;;=-SII T· (13)

The right-hand side of Eq. (12) is the driving mechanism for the radiative-condensation instability. As mentioned earlier, aLlaT is generally negative and hence leads to the cooling-enhanced-cooling mechanism. Equation (13) shows that a negative temperature fluctuation enhances density locally, as the sound wave effectively flattens the pressure along the field lines. This "condensation" effect (accumulation of density) drives the instability further. Using (13), aLlan=2Lln, and aLlaT= -LIT L> Eq. (12) can be written as

d2f ~+(blsech2z-a1)f=0. (14)

Here,

z=x/a+l>, b1=2(1+2SII ), a1=9(3/5+SII ).

The profile of the equilibrium radiation appears through the "sech2

" term in Eq. (14). The normalized growth rate r is defined as r=5yna2Ti/(2Xl T~) =20YTHTi/3T~, with the plasma heating time TH defined as TH=3nT /2H. In deriving (14) we have ignored the logarithmically slower variation of equilibrium temperature [see Eq. (7)], compared to the sharp variation of equilibrium L [Eq. (8)]. Equation (14) has a well-known solution,7 which is

Phys. Plasmas, Vol. 1, No.1, January 1994

written in terms of hyper[eometric functions, S obtained with the requirement that T -.0 for large values of z,

T=TN(sechz)PF(a,b;c;u), (15)

where, p=aj!2, a=aj!2+1I2-(b1+1I4)1I2, b=al12

+1I2+(b1+1I4)1I2, c=aj!:'+1 and u=(1-tanhz)/2. The boundary condition T = 0 at x = 0 (or z = <5) yields

the eigenvalue condition

F(a,b;c;1I2-£/2) =0, (16)

where £ has already been defined as £= (1- p2) 112 = ( 1- Lol Lc) 112 and the coefficients a, b, and c depend on the eigenvalue y.

Let us discuss the poloidally symmetric (k ll =0) per­turbations first. Such perturbations do not produce the condensation effect [see Eq. (13 )], however, the stabilizing influence of parallel thermal conduction is also absent. Had this been the only stabilizing factor, one should have ob­served a poloidally symmetric instability first. The role of perpendicular heat conduction was clarified first by Drake,5 who showed that a model thermal equilibrium of a plasma column is stable to such perturbations. It was also shown that perpendicular thermal conduction alone can give rise to a threshold value of Lo, beyond which only, the perturbations with kll *0 would grow. We demonstrate that our results are consistent with those of Drake in this regard. For kll =0 perturbations, we obtain

3 a= (359) 1/2 -1, a1=5 9, b1=2,

(39) 112 (39) 112

b = 5 + 2, c= 5 + 1.

We expect the radiative instability to be stronger when the radiative loss L is higher, Le., when p2;::;; 1 or £<1. In this limit, the eigenvalue condition (16) can be solved analyt­ically. Taylor expansion of F near £=0 yields

ab r(b/2+ 112) £2 r(a/2+ 1) r(b/2 + 1) 0,

(17) r(a/2+ 112)

where r(z) is the standard gamma functionS of argument z. To obtain (17) we have used the relation

F(a,b;c;1I2) = 1T1I2r(c)/r(a/2 + 1I2)r(b/2+ 112).

Since the second term in (17) is of order £, the gamma function in the first term must be near its pole, Le., a;::;; - 1. For cases of our interest, r;;:': 0 so that a;;:': - 1. It is easily seen that both terms in (17) are positive and there is no solution for r in the range of interest. Note that for £=0, one obtains r=O, so the poloidally symmetric perturba­tions are marginally unstable when Lo= Lc.

We now discuss the poloidally asymmetric perturba­tions. As mentioned earlier, such perturbations give rise to a condensation effect that is destabilizing. This effect is most pronounced when y<k ll Cs , so that the sound wave has enough time to flatten the pressure perturbations. To obtain the maximum possible growth rate for a given £ we ignore the stabilizing influence of XII ' so that

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al=Y, bl=6, a=yI!2-2,

b=1'1l2+3, and c=1'1l2+ l.

Using these in the Taylor expansion of F near E=O [Eq. (17)], we note that, a::::: -1, so that terms balance; how­ever, the deviations of a around - 1 are significant only in the first term of (17). We therefore put a = - 1, and b= -4 in the second term above and use the form of r(z) near z=O, r(z)::::: 1/z, to write l' as

1'=(1_32E)=[ I-~ (1- ~:r/21. (18)

This is the growth rate of radiative condensation instability in the absence of parallel thermal conduction. The thresh­old value of Lo/ Lc is found by setting 1'=0 in (16), and solving for the threshold value of E defined as Et • In this limit, a= -2, b=3 and c= 1, so that

F( -2,3;1;1/2-E/2) =0, (19)

yields a polynomial8 of degree 2 as a= -2, given by

I-6z-<iz2=0, (20)

where z=t( I-Et ). This gives

Et= 1/Y3 or Lml Lc=2/3=0.66. (21)

Here we have introduced Lm to represent the threshold value of radiative power loss below which the poloidaUy asymmetric perturbations are stable. For Drake's model this value was found to be 0.75L{;. A point regarding the relation JLIJn=2Lln should be mentioned. It is custom­ary to express coronal radiative loss L as n2g( T). The expression for L is actually6 L ( T) = nn II z( T), where n I is the total impurity density and I z( T) is the cooling rate for an impurity with nuclear charge given by Z. By writing, nIl n = E and g ( t) = E! z< T) one gets the usual expression for L(T). In coronal equilibrium there are no other pro­cesses except ionization and recombination for impurities. One therefore cannot expect a change in n I' for, it is only redistributed over its charge states as temperature is changed. This leads to JLIJn= Lin, which is without the factor of 2 taken earlier.s In the present calculation for r and L m , the changes are only quantitative, e.g., the expres­sion of l' is changed to

A [{0-3 ({0-1 r( {0/2) ) r= 2 J; r(1/2+ J17/2

( Lo) 112] X I-I: '

c (22)

compared to (18). Similarly the threshold value Et gets altered as

(1- {0 1+ {0 .. l-Et)_

F 2 ' 2 ,1, 2 -0,

which is closely approximated by

P3/2(Et ) =0,

130 Phys. Plasmas, Vol. 1, No.1, January 1994

(23)

(24)

where Pv(z) is a Legendre polynomial8 of index v. The exact relation (23) gives a 4% correction to the index in (24). Using the elliptic integral representation of P3/ 2 the threshold value of Et and hence, Lm can be obtained. We find that

(25)

It is possible to qualitatively estimate the change in the growth rate and the alterations of the threshold value of P due to the presence of small heating in the edge region. The profile of L [Eq. (11)] when used in the stability analysis generates a small deviation from the "potential" which arises in (14). Correspondingly, there are small changes in the eigenvalue. These changes can be found from the orig­inal "unperturbed" solutions as follows. Let y" + V(x;A)Y=O, represent the problem at hand. Letyo(x) represent the solution for the eigenvalue ..1.0 and V(x;Ao) = Vo(x,Ao). A small perturbation VI (x) riding on Vo will generate a deviation Al from the original eigenvalue ..1,0' It can be shown that

f Yo(x) VI (x)Yo(x)dx

f Yo(x)(J Vo/JAo)Yo(x) dx '

where JVo/JAo=JVo/JA at ..1,=..1.0, For our case,

Vo=b l sech2[x+cosh- l( 1/ p)] -al

and

VI =hbtIo(x)sech2[x+cosh- l ( 1/ p)]

Xtanh[x+cosh-l(1/p) J.

(26)

We have found that for a fixed p, the increment of l' is positive. Next, taking 1' ..... 0, and treating Pt (threshold) as an eigenvalue, the increment of Pt is found to be negative.

The slight enhancement of the growth rate and con­comittant decrease of the threshold P or (Lm) arises due to the enhanced radiative loss in equilibrium. We have al­ready pointed out earlier, that for the equilibrium bound­ary conditions considered, the radiative loss increases so as to balance the local heating.

IV. CONCLUSION

Structure and stability of radiative equilibrium of the tokamak edge plasma is presented for a model radiative power loss which is an exponentially decreasing function of temperature. The merit of such a model is that, while it shows JLIJT<O, as the coronal model does,6 it also per­mits exact solutions for equilibrium and stability analysis. It is shown, by an alternative mathematical analysis, that poloidally symmetric perturbations are stable while the asymmetric perturbations grow when radiation loss ex­ceeds a certain threshold value. This value is just below the density (or radiation) limit Lc. and in qualitative agree­ment with experiments.3 These crucial and apparently equilibrium-specific features of the MARFE, explained by Drake earlier, are also contained in our model equilibrium which is qualitatively different. The overall results are con-

Shishir Deshpande

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sistent with those obtained by Drake suggesting that a broader class of radiation functions possesses the stability features of the MARFE mentioned above.

ACKNOWLEDGMENT

The author is grateful to Dr. P. K. Kaw for helpful discussions.

lB. Lipschultz, B. laBombard, E. S. Marmar, M. M. Pickrell, J. L. Terry, R. Watterson, and S. M. Wolfe, Nucl. Fusion 24, 966 (1984), and references therein.

2V. A. Vershkov and S. V. Mirnov, Nucl. Fusion 14, 383 (1974); M. Murakami, J. D. Callen, and L. A. Berry, ibid. 16, 347 (1976); A. Gibson, ibid. 16, 546 (1976); P. H. Rebut and B. J. Green, in Plasma

Phys. Plasmas, Vol. 1, No.1, January 1994

Physics and Controlled Nuclear Fusion Research, 1976, Berchtesgaden (International Atomic Energy Agency, Vienna, 1977), Vol. II, p. 3; N. Ohyabu, Nucl. Fusion 9, 1491 (1979).

3B. Lipschultz, J. Nucl. Mater. 145-147, 15 (1987). 4p. C. Stangeby and G. M. McCracken, Nucl. Fusion 30, 1225 (1990); B. LaBombard and B. Lipschultz, ibid. 27, 81 (1987); M. Shimada, H. Kubo, K. ltami, S. Tsuji, T. Nishitani, and The JT-60 Team, J. Nucl. Mater. 176-177, 122 (1990).

5J. F. Drake, Phys. Fluids 30, 2429 (1987). 6D. E. Post, R. V. Jenson, and C. B. Tartar, At. Data Nucl. Data Tables 20, 397 (1977).

7p. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Part II, Chap. 12.

8 Handbook of Mathematical Functions with Formulas, Graphs and Math­ematical Tables, edited by M. Abramowitz and I. A. Stegun (Dover, New York, 1970).

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