A REDUCED SET OF EQUATIONS FOR RESISTIVE …...Plasma Phlsics and Controlled Fusion, Vol 31. No. 9,...

Plasma Phlsics and Controlled Fusion, Vol 31. No. 9, pp. 1365 to 1379, 1989 Prinred in Great Britain

0741-3335'89 S3.00f .OO IOP Publishing Ltd. and Pergamon Press plc

A REDUCED SET OF EQUATIONS FOR RESISTIVE FLUID TURBULENCE IN TOROIDAL SYSTEMS

F. ROMANELLI and F. ZONCA Associazione EURATOM-ENEA sulla Fusione. Centro Ricerche Energia Frascati,

C.P. 65, 00044 Frascati, Rome, Italy (Received 10 June 1988 ; and in revised form 21 February 1989)

Abstract-A set of reduced equations is derived for the description of resistive fluid turbulence in the semicollisional and strongly collisional regimes, using the low-p Tokamak ordering. The equations obtained conserve energy in the collisionless limit and in the linear approximation reduce to models previously used to investigate the stability of resistive ballooning and microtearing modes.

1 . I N T R O D U C T I O N To UNDERSTAND the nonlinear evolution of the various classes of instabilities which are present in thermonuclear plasmas has been the goal of many theoretical inves- tigations. As the time and space scales involved are quite different, it is often useful to reduce the complete two-fluid system of equations in order to eliminate the instabilities and the oscillations which do not occur at the scale of interest. In order to accomplish this goal it is possible to make use of power expansions in terms of the inverse aspect ratio E = a/R, with a(R) the minor (major) radius of the device, which, for macroscopic modes, is proportional to the ratio between the characteristic perpendicular and parallel wavelengths and to the ratio between shear and compression AlfvCn frequencies.

Using such an expansion reduced equations have been derived first for the description of kink modes in a low-8 (8 = 87cp/B2) Tokamak (ROSENBLUTH et al., 1976 ; STRAUSS. 1976) and then extended to the high$ case (STRAUSS, 1977) as well as to include resistive MHD modes (Izzo et al., 1985; DRAKE and ANTONSEX, 1984; MASCHKE and MORROS TOSAS, 1989; MORROS TOSAS and MASCHKE, 1989). Such an approach has been shown to be successful both in reproducing the relevant physical effects associated with these instabilities and in reducing the computational require- ments for a complete nonlinear description, in particular by eliminating the compressional Alfven wave.

This approach has also been used in the description of small scale turbulence. Here the expansion parameter is the ratio between the ion Larmor radius (the characteristic perpendicular wavelength) and a macroscopic length (e.g. the minor radius) char- acterizing the variation of the equilibrium quantities. Model equations for low frequency semicollisional electrostatic turbulence have been derived as the Hasegawa- Mima equation (HASEGAWA and MIMA, 1977) and its generalizations which take into account the destabilizing effect of collisional momentum exchange (WAKATANI and HASEGAWA, 1984) and finite-/3 effects (HASEGAWA et al., 1980). Similar equations have been used in numerical simulations (FYFE and MONTGOMERY, 1979 ; WALTZ 1983,1985). Recently a four-field model of Tokamak plasma dynamics which contains

1365

1366 F. ROMANELLI and F. ZONCA

as particular limits the M H D reduced equations and the Hasegawa-Mima equation has also been proposed (HAZELTINE, 1983 ; HAZELTINE et al., 1985).

The aim of the present paper is to rigorously derive a set of equations for electromagnetic turbulence in low-p Tokamaks by using the reduced equations approach, i.e. an inverse aspect ratio expansion. The main result of the present analysis is the inclusion of the effect of parallel thermal conductivity which allows one to consider both the strongly collisional and the semicollisional regime. Therefore in the linear approximation the system of equations predicts the unstable collisional drift wave (CHEN et al., 1980), as well as the microtearing mode (DRAKE et al., 1980) and the resistive ballooning mode (CONNOR and HASTIE, 1985). In this respect we generalize the results of DRAKE and ANTONSEN (1984).

As the dynamics of these instabilities are almost independent of the ion physics, we restrict our analysis to the cold ion limit. This approximation is also commonly used in the analysis of the linear behaviour of these modes (CHEX and CHENG, 1980; CHEN et al., 1980; DRAKE and ANTOXSEN, 1984).

The equations obtained conserve an energy-like invariant in the collisionless limit and are suitable for high resolution simulation of Tokamak dynamics.

The plan of the paper is the following. In Section 2 the general formalism is presented. Then, after defining an ordering in terms of the ratio between the radial extension of the resistive layer and the macroscopic lengths, a set of reduced equations for equilibria with arbitrary /3 are derived. In Section 3 the equations are further reduced by using the low-p Tokamak ordering and the conservation properties are discussed. Concluding remarks are given in Section 4.

2 . G E N E R A L F O R M A L I S M A N D O R D E R I N G S The starting set of equations are the Braginskij two-fluid equations (BRAGINSKIJ,

1965) :

d -ne+n,V.v, = 0 dt

d e ‘dt

m n -v, = -nn,e(E+ve x B ) + R - V p

(3) d dt

$ n , - T + n T , V * v , + V . q = 0

d - n i + n i V . v i = 0 dt (4)

(5) d

m,n,-vv, = n,e(E+v, x B ) - R dt

where :

d a -- dt - +‘e ’

Resistive fluid turbulence in toroidal systems

R = neefie J - %,neb

.. f i = (f-bb)y, +bbylI

y i l = 0 . 5 ~ ~ = 0.5mev/n,e2

J l l 5 neT q = -rlT-b+fp2vJ~B-4.66n,p2vV,T- - ---bxVT-X bb*VT

e 2 meQe

x = %,neT/mev.

In equations (1)-( 10) “e” and “i” label respectively the electron and ion species, e is the electric charge of the electron, a, = 0.71, clj = 3.2, b is the unit vector in the direction of the magnetic field, n,(n,) is the electron (ion) density, T is the electron temperature, v, (VJ is the electron (ion) velocity, q is the electron heat flux, R is the electron-ion collisional momentum exchange, yI (yl ) is the perpendicular (parallel) resistivity, v is the electron-ion collision frequency, x, the parallel electron thermal conductivity, Qe = eBim, is the electron cyclotron frequency and pe = v,iR,, uTe = (T/me)’ * being the electron thermal speed. The term proportional to 2’ - 1 in equation (7) is the time-dependent thermal force (HASSAM, 1980). The electric field is written in terms of a vector and a scalar potential

with the magnetic field given by

Equations (1)-(12) are closed by Maxwell’s equations, i.e. by the quasineutrality condition (ne = n, = n)

V . J = O (1 3)

and by Ampere’s law

V x B = J . (14)

Before defining an appropriate ordering scheme it is better to write equations (1)-(5) in a more convenient form. As will be shown later, the perpendicular electron and ion velocities are dominated by the E x B and diamagnetic drifts ; therefore equations (2) and ( 5 ) can be solved formally for the perpendicular velocities.

1368 F. ROMAKELLI and F. ZONCA

v,, = YE +v, +vp,

where vE is the E x B drift equal for electrons and ions

E x B VE = ~

B2 '

Vde is the diamagnetic drift

V p x B vde = ~

neB2 '

v, is the drift associated with the resistive term R

and vpi (v,,) is the ion (electron) polarization drift

From the previous equations it is possible to determine the perpendicular current

B x V p B dv J, = n e ( q i -vJ = 7 + 9 x p 3

with p dvjdt = mene dv,/dt+m,n, dv,ldt. Equation (21) can be substituted into the quasineutrality equation (13), yielding

The parallel ion motion is determind by taking the scalar product with b of the sum of equations (2) and (5)

dvi minib*- dt = - b - v p

while the parallel component of equation (2) yields the generalized Ohm's law

Resistive fluid turbulence in toroidal systems 1369

(24) a

-neb*-A = neb .V@+b*R-b*Vp. at

Note that in equations (23) and (24) electron inertia has been neglected. It is now convenient to derive an expression for the divergence of the ion particle flux which will be needed in the continuity equation. Using some algebra the following equations are obtained

E x B B d B2 B2 at

Vsnv, = -* V n - n E - V x - -n-1nB

where use has been made of Faraday's law. Moreover,

and

Therefore equation (4) can be written as follows

an E x B B a B dv, at B2 B~ at d t -+-. V n - n E * V x - -n-lnB+V*- eB2 xnm,-

nv * B dv + B * V + - V - B [\;(Vp+pd;-$nVT)]=O. I (28)

Finally it is useful to note the following relation (TSAI et al., 1970) in the temperature equation

(29) 5 nT T B 2 meQ e

$nYde * V T + nTV *vde - v - __- b x V T = --(Vp+'n 2 V T ) * V X - B2 '

Equations (22), (23), (24) and (28) are exact and replace equations (l) , (2), (4), (5) and (13).

An appropriate ordering scheme has to be defined now. The ordering of perpendicular and parallel lengths is complicated by the multiple scale nature of the mode. Far from the mode rational surface equilibrium quantities and fluctuating quantities have perpendicular scale lengths of the order of the minor radius a and parallel scale lengths of the order of the major radius R. Close to the mode rational surfaces, fluctuating quantities have perpendicular and parallel scale lengths determined by balancing the rate of resistive diffusion with the characteristic frequency in Ohm's law


and inertia with line bending in the quasineutrality equation

with kll = k,x/Ls, k, being the component of the wavevector perpendicular to B on the flux surface, x being the distance from the mode rational surface, uA = B/(nnz,)' being the Alfven velocity and Ls = qR/s being the shear length, with q = 0(1) being the safety factor and s = q ' / q = 0(1) the shear. Equations (30) and (31) define the characteristic space and time scales

X, = 6/k,

U, = 6u,4/Ls

where, following general analyses, the quantity 6 = (f lLk;Ls/uA) 1 ' 3 has been introduced, which is related to the magnetic Reynolds number. The quantity 6 is the basic expansion parameter. Note that no ordering of 6 with E has been assumed up to now. This approach allows one to define a general set of equations for arbitrary equilibria and is similar to the linear analysis of GLASSER et al. (1975). In Section 3 the equations will be specialized to a low-fi Tokamak equilibrium with the ordering 6 = O ( E ' ) . In order to define the most general scheme, semicollisional ordering is assumed

which yields kip, = O(1) [or k,.p, = O(S)] with p ? = CJQ, C, = (Te/mi) ' I 2 . ai = eBtm,. The fact that finite Larmor radius corrections need to be retained restricts the present analysis to the cold ion limit Ti = 0. Note that this condition is equivalent to O / W , = 0(1) with U* = k,p,C,/L, and L, = I V n h - ' .

Finally an ordering for the fluctuating quantities can be obtained by equating the time derivative with the convective derivative associated with the E x B motion due to the electrostatic potential, yielding

and equating the electrostatic and electromagnetic part of the parallel electric field, yielding

which is consistent with the condition k l - BIB * V. Note that the operator b - V cannot be simply identified with k , also in the resistive region because, at the lowest order in


6, b * V is ordered as l iR. The parallel component of the fluctuating magnetic field is also ordered as

Note that the distinction between equilibrium and fluctuating scale lengths is phys- ically meaningful only in the resistive layer. In the ideal region it is only possible to distinguish between stationary and time-dependent quantities. Using this ordering scheme it is now possible to derive the reduced set of equations.

First it is possible to show that the perpendicular electric field is essentially electrostatic. To this aim the perpendicular component of the vector potential A is determined from equation (21) and Ampere's law, equation (14), yielding

Using equation (37) the E x B motion in equation (17) can be expressed as

The polarization drift can be determined from equation (20) using the lowest order velocity field. Note also that the v, drift turns out to be of order

Therefore it can be neglected compared to vE. Using equation (23), it can be shown that the parallel fluid velocity is ordered as

and the contribution of the parallel ion flow to the convective derivative is: in general, not negligible. Also, the term associated to the electron parallel flow has in principle to be retained in the electron convective derivative. However from equations (3), (23) and (24) it follows that

Therefore the operator z: b * V in equations (3), (22) and (28) yields higher-order terms than c"/i;t and it is possible to replace the d/dt operator with the perpendicular convective derivative everywhere


Note, however, that the d/dt operator in equation (23) has to be evaluated with the parallel convective derivative because b * Vg,, does not vanish at the lowest order.

It is now possible to evaluate the quasineutrality equation (22) term by term. The first term is the usual line bending term B.V(J.B/B2) -v J l / R which is partially cancelled by the curvature term.

The second term in equation (22) is the curvature term, which, taking into account equation (21), yields

B dv (j) B4 dt +Vp.V 7 X B =- X P - S V P B Vp-J vpqvx - - __ B 2 - B2

b . Jb-Vp +-- B2

The first two terms in the r.h.s. are negligible because they are ordered as 0(J2) with respect to the last term which is ordered as Jl, /R. Finally the last term in equation (22) is the inertia term. This term is O(6) and can be evaluated by using the equilibrium magnetic field and the E x B velocity given in equation (38), yielding

R (44)

The final expression of the quasineutrality condition is

(45) J * B 1 n m d

B * V T + B * V p x V , + V * " 9 - i - B B Be', dt V - @+0(J2)J , iR = 0

with d/dt given by equation (42). Equation (45) formally evolves the electrostatic potential (D. Therefore an equation for the pressure p is needed in order to close the system of equations.

It is first convenient to consider equation (28) for the density evolution. The leading term is the term related to the parallel ion motion B - V(nv, - BIB2) which is partially balanced by the curvature term. Following the derivation of equation (45), the latter term can also be written as

Referring to this term, the convective derivative and the term n(d/at) In B are of order 6. The term V * (B/B2 x nmi(dv/dt)) has already been evaluated in equation (44). Finally, the last term in equation (28) is of order

Resistive fluid turbulence in toroidal systems

nV*qiVp/B2 y,k: ----I = 0(1), dnidt o

1373

(47)

i.e. of the same order of the convective derivative. Therefore the final expression for the continuity equation is

dn +0(6) - d t = 0. (48)

Considering now the equation for the temperature, the terms in V * q different from the b x VT term can be written as

XI1 B -4.66V np,‘vVI T-B. V T B -VT. (49)

The leading term is the divergence of the parallel heat flux

Of the same order is flow,

the term coming from the divergence of the parallel electron

= o(6-I) B - VTJ , /eB Jl ,‘R n dTidt eon

-- and the curvature term similar to equation (46). Finally at the next order there is the convective derivative and the perpendicular transport term. Therefore the temperature equation can be written as follows

1374 F. ROMAKELLI and F. ZOSCA

where p: = m,Teq/(eBeJ2. Equations ( 4 9 , (48) and (52) can be combined yielding an equation for the pressure

( 5 3 ) dP + V * ( - 2.4 1 p:v~l,~VL T ) + O(6) - 0. dt

In deriving equation (53) use has been made of the fact that the parallel convective derivative can be neglected due to equation (41).

Let us now consider equations (23) and (24), which describe the parallel ion motion and the generalized Ohm's law, respectively. In both equations all the terms are ordered as nek,,@. Therefore we can rewrite them in the following way

(23 .bis)

B'Vp +O(6)eko@ = 0. (24.bis)

Equations ( 4 9 , (48), (52) , ( 5 3 ) , (23.bis) and (24.bis) are the reduced resistive equations valid for an arbitrary equilibrium. Note, however. that they suffer from a major limitation. Indeed, the reduced equations with the operator d/dt given by equation (42) do not conserve the total energy defined as

where the compressional magnetic field term associated with the aB/& term has been neglected, as will be shown in the next section. In order to conserve the total energy, we need to introduce the polarization drift in equation (48) and (23.bis) and the parallel velocity in the convective derivative of equation (53). This limitation is due


to the fact that while the polarization drift and the parallel velocity can be neglected in the convective derivative. the divergence of these terms is of the same order or larger than the divergence of the E x B motion. This problem has already been noted by DRAKE and ANTONSEN (1984) who indeed derived a two-fluid system of equations which conserve energy, by including the polarization drift and the parallel velocity in the didt operator.

(a) To retain the second-order quantities. In this case the equations of motion are very complicated because of the presence of the cubic (or higher-order) nonlinearities which make the numerical convolution very time consuming. On the other hand these terms do not appreciably modify the nonlinear evolution. The advantage is to conserve the global energy which makes the system of equations equivalent to the original system in the low frequency limit.

(b) To neglect the second-order quantities. In this case the equations of motion are much simpler but the total energy is not conserved. However, upon introducing additional approximations, an energy-like invariant can be derived. This system of equations is therefore an approximation to the original system.

So, in principle, two choices are possible, as follows.

We choose here the second approach which is the more realistic, taking into account that no simulations of resistive microturbulence actually exist. Indeed, as shown in the next section, we retain the minimum number of higher-order nonlinearities in order to conserve an energy-like invariant.

3 . L 0 W - b T O K A M A K In this section equations (45), (48), (52), (23.bis) and (24.bis) are specialized to a

low-p Tokamak equilibrium ( f l - 2). This further reduction is motivated by the observation that the reduced resistive equations, even though they do not contain the fastest time scales associated with the gyromotion, are not immediately suitable for numerical computation due to the large number of linear and nonlinear convolution terms and it is convenient to take advantage of the low-Tokamak ordering in order to eliminate many terms. These terms make a numerical computation very lengthy due to the third-order nonlinearities which appear in the inertia term. On the other hand many of those nonlinearities have no obvious physical meaning and it is therefore desirable to eliminate these terms.

To this aim it is convenient to consider an auxiliary ordering for the fluctuating quantities, obtained by balancing inertia, line bending and curvature for a low-p Tokamak equilibrium

xix, = O ( & ' 3 ) oi'o, - o,lo, = O(&4 3) ( 5 5 )

This ordering scheme is optimal in the sense that the resistive ballooning mode (which is the weakest resistive instability and, as a consequence, it requires one to retain the largest number of physical effects) is described in any collisionality regime (DRAKE and ANTONSEN, 1985; ROMANELLI, 1987). Moreover this ordering scheme can be shown to be consistent with the reduced equations of Drake and Antonsen. Upon assuming, for convenience, 6 = such an ordering yields


where wA = vAILs. Note that in equation (56) the ordering

= 0(&3) (56)

BJB = O ( F ~ ) gives the characteristic level of the mode-magnetic fluctuation level, while, in the resistive layer, the mode becomes essentially electrostatic and B"JB = O(e5) .

As the compressional component of the fluctuating magnetic field can be neglected and in the expression for the perturbed magnetic field only the lowest-order correction has to be retained it is possible to express B in the following way

B = roBoVq+V$ x Vq (57)

with q being the toroidal angle and = ieq + RA * Beq/Beq. Moreover, in the following, parallel and perpendicular directions will be considered with reference to the equilibrium magnetic field

From equations (56) it follows that

with JII = J * beq. Similarly, v, *BIB' = v, ,/BC,(1 + O(c4)), with q = v, *be, - J , /en . As the inertia term in equation (45) is O(E' ) , corrections of O(e3) in the line bending and curvature terms can be neglected. Following these directions the curvature term can be written as

1 1 B*Vp x V- - RoBoVq*Vp xV,[~+O(E~)].

B2 - Be,

In the inertia term the magnetic field can be replaced with the value on axis but the energy is still not conserved. On the other hand if the perturbed density is retained in such a term it is also necessary to retain the polarization drift in the convective derivative. In order to solve this problem in the simplest way we replace the equilibrium density with a constant value. Such a choice corresponds to considering a constant Alfven velocity and does not significantly alter the physics of the problem. With these approximations the final form of the quasineutrality equation is obtained

The operator dldt can also be simplified, yielding at the lowest order


Note that the perpendicular velocity in the convective derivative corresponds, in this approximation, to incompressible motion.

In equation (61) the line bending term and thus the parallel current are needed up to order E' . Therefore, in terms of $, Ampkre's law can be written as

As a consequence of these considerations we have that the flux $ as well has to be evaluated up to order c2 by means of Ohm's law, which reads

where we recall that yl l = 0.5 mev/ne2. In equation (64) density and temperature are needed up to order E ' , i.e. density and temperature fluctuations must be evaluated to the lowest order.

Let us now consider the continuity equation. In equation (48) the third and the fourth terms are negligible and the last term only needs to be evaluated at the lowest order, yielding

x (Vlp- :n,,V- T )

where again we have replaced the equilibrium density with a constant value in the inertia term. Similarly from equations (52) and (53) we obtain respectively the temperature equation

(66) 3T

+O(&)n- = 0 at

1378 F. ROMANELLI and F. ZOXCA

and the pressure equation

(67) 8P + v * ( - 2.41 p,2vneqvL T ) + O(&) - = 0 Bt

where p t = meTeq/e2Bi and x,, = a3p/mev. Looking at equations (65)-(67) we see that density and temperature fluctuations

are retained to the lowest order, which requires one to evaluate the ion parallel speed up to order E * . The term [vl-J,/en]be,B/Beq*Vp which is present in equation (67) is formally of order O ( E ) and has been introduced to conserve an energy-like invariant as noted in the following.

Let us come finally to the equation which describes the evolution of the ion parallel speed

Equations (61)-(68) constitute our final set of equations which conserve an energy- like invariant defined as

and the total number of particles in the nondissipative limit. Note that the invariant equation (69) differs from the total energy in the perpendicular kinetic energy term which is evaluated using an average value of the density.

4 . CONCLUSIONS In this paper the derivation of a reduced set of equations has been presented for

the description of the nonlinear evolution of resistive electromagnetic instabilities (microtearing modes, resistive ballooning modes and collisional drift waves) in a low- ,B Tokamak. This system, given by equations (61), (63), (64), (65), (66), (67) and (68), conserves the energy-like invariant of equation (69) in the nondissipative limit. The


equations retain the effect of the parallel electron thermal conduction so that the physics of the semicollisional regime is described correctly.

R E F E R E N C E S BRAGINSKIJ S. I. (1965) in Reviews of Plasma Physics (edited by M. A. LEOYTOVICH), Vol. I, p. 205.

CHEN L.; CHANCE M. S. and CHEXG C. Z. (1980) Nucl. Fusion 20,901. CHEN L. and CHEW C. Z. (1980) Physics Fluids 23, 2242. COYNOR J. W. and HASTIE R. J. (1985) Plasma Phys. contr. Fusion 27, 621. DRAKE J . F. and ANTONSEN T. M. (1984) Physics Fluids 27, 898. DRAKE J. F. and ANTONSEN T. M. (1985) Physics Fluids 28, 544. DRAKE J. F., GLADD N. T.: LIU C. S. and CHANG C. L. (1980) Phys. Rev. Lett. 44. 994. FYFE D. and MONTGOMERY D. (1979) Physics Fluids 22, 246. GLASSER A. H., GREEKE J. M. and JOHXSON J. L. (1975) Physics Fluids 18, 875. HASEGAWA A. and MIMA K. (1977) Pkys. Rev. Lett. 39, 205. HASEGAWA A., OKUDA H. and WAKATANI M. (1980) Phys. Rev. Lett. 44,248. HASSAM A. B. (1980) Physics Fluids 23, 38. HAZELTINE R. D. (1983) Physics Fluids 26, 3242. HAZELTINE R. D., KOTSCHEKREUTHER M. and MORRISON P. J. (1985) Physics Fluids 28, 2466. IZZO R., MONTICELLO D. A., DE LUCIA J., PARK W. and RYU C . M. (1985) Physics FIuids 28, 903. MASCHKE E. and MORROS TOSAS J. (1989) Plasma Phys. contr. Fusion 31, 563. MORROS TOSAS J. and MASCHKE E. (1989) Plasma Phys. contr. Fusion 31. 549. ROMAKELLI F. (1987) Second European Theory Meeting, Varenna. 1987. ROSEYBLLTH M. N., MONTICELLO D. A., STRAUSS H. R. and WHITE R. B. (1976) Physics Fluids 19, 198. STRAUSS H. R. (1976) Physics Fluids 19, 134. STRAUSS H. R. (1977) Physics Fluids 20, 1354. TSAI S. , PERKINS F. W. and STIX T. H. (1970) Physics Fluids 13, 2108. WAKATANI M. and HASEGAWA A. (1984) Physics Fluids 27, 61 1. WALTZ R. E. (1983) Physics Fluids 26, 169. WALTZ R. E. (1985) Physics Fluids 28, 577.

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