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# Axissymmetric, non-ideal MHD states with steady flow

. (page 1 of 2)
DOE/ER/03077-201
MF-102

Courant Institute of
Mathematical Sciences

Magneto-Fluid Dynamics Division

Axisym metric, Non-ideal MHD

W. Kerner and H. Weitzner

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U.S. Department of Energy Report

Plasma Physics
October 1983

NEW YORK UNIVERSITY

DISCLAIMER

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UC 20

UNCLASSIFIED

New York University
Courant Institute of Mathematical Sciences
Magneto-Fluid Dynamics Division

MF-102 DOE/ER/03077-201

AXISYMMETRIC, NON- IDEAL MHD STATES WITH STEADY FLOW
W- Kerner and H. Weitzner

U . S . Department of Energy
Contract No. DE-AC02-76ER03077

November 198 3

UNCLASSIFIED

L.^

Axisymmetric , Non-ideal MHD States with Steady Flow

W. Kerner* and H. Weitzner,

Courant Institute of Mathematical Sciences, New York University,

New York, N.Y, 10012, USA.

Abstract

Toroidal plasma configurations with steady flow are studied

in the framework of non-ideal MHD theory. The properties of

the resulting set of equations are examined.

The numerical solution of the two-dimensional, non-linear

system appears feasible, although the large variation in

the transport coefficients creates considerable numerical

problems .

c

'>

Max-Planck-Institut fur Plasmaphysik , EURATOM Association,
D-8046 Garching, Fed. Rep. of Germany.

- 2 -

O. Introduction

Tokamak discharges can be sustained for several seconds with
an energy confinement time approaching 100 milliseconds. These
experiments are, in general, not terminated by instabilities
in the form of a disruption. A suitable description of the
plasma behaviour is given by the macroscopic model (MDH) .
To reproduce the essential features of the experiment, the MHD
equations have to incorporate the non-linear dependence and
to include non-ideal effects. The full time-dependent problem
is, in our opinion, still far too complex for sufficiently
accurate numerical treatment. We therefore treat the long-time
evolution as an equilibrium problem, where the plasma passes
through a sequence of equilibrium states. By imposing axi-
symmetry, which is adequate for tokamak configurations, the
corresponding problem reduces to determining two-dimensional
equilibria. This is feasible with existing numerical techniques
and computing facilities.

The ideal MHD model with scalar pressure has been remarkably

successful for describing a plasma. Owing to the characteristic

Alfven time scale of the order of microseconds, the plasma cannot

be too far from an equilibrium. This feature is built into

H. Grad's 1 -1/2 D transport scheme /I/, where the plasma passes

through two-dimensional equilibria obyeing the equation

^ p = J X B and the profiles evolve as surface quantities.

This model is, however, only adequate if the plasma flow is

small and if the pressure and density are, basically, surface

quantities.

The large amount of additional heating in the form of neutral
beam injection, which is employed in all major tokamak experi-
ments, acts as a source for toroidal flow with a flow velocity
approaching the ion sound speed. The induced poloidal flow, on
the other hand, does not exceed a certain value, its damping
out leading to a poloidal dependence of the pressure and density
on magnetic surfaces. The ideal MHD model can easily be extended
to include flow. A code for computing such equilibria with
flow has been developed by Kerner and Jandl /2/. (Other

- 3 -

contributions and results are referenced in 111 .)
This model, however, neglects dissipative effects and is not
sufficiently general with respect to boundary conditions
and sources.

If resistivity is taken into account, toroidal equilibria
require a flow for their existence, as is pointed out in
Ref. /3/. Pf irsch-Schluter diffusion, Ref. /4/, sustains such
a pressure-driven flow. For ohmic discharges these flows are
usually small. But the additional heating drastically enlarges
the flow. The viscosity is just as important, and so is the
energy flux due to temperature gradients. Taking into account
the resistivity, viscosity and heat conductivity yield a set
of equations which allow a realistic simulation of an experiment.
The hyperbolic character of the continuity equation requires
sources for its solution. The remaining set forms an elliptic
set. The requirement of an elliptic characteristic is an important
point in our analysis since for mixed systems we expect tremen-
dous numerical difficulties. The essential role of the
continuity equation is then apparent. The choice of boundary
data and source terms should select certain equilibria and
forbid others. A close connection with experimental data is
possible. We thus can hope for results explaining the density
limit being observed in the experiments.

For our macroscopic model transport coefficients are required.
In the collision-dominated regime the coefficients are
explicitly known. More appropriate is the neoclassical regime,
where the trapping of particles is taken into account.
The coefficients have different values along and perpendicular
to the magnetic field. The consequences of this anisotropy
with respect to the numerical approximation are discussed.

The basic assumption in this paper is that the additional heating
deposited in the plasma in the form of neutral beam injection
and wave heating makes a two-dimensional treatment of the
equilibrium problem necessary - owing to the poloidal variation
of important quantities such as pressure and density.

- 4 -

The physical model, its mathematical properties and the
consequences for the numerical solution are analyzed.
The single-fluid model containing non-linear and non-ideal
features exhibits such a complexity that a detailed dis-
cussion of its properties - especially with respect to a
numerical approximation - is both useful and necessary.

We are aware of the fundamental role of the transport co-
efficients for our model. Experimental data show that some
coefficients differ from their classical or neoclassical
values and cause anomalous transport. A very interesting
aspect of our model is therefore its potential for determining
the transport coefficients.

The paper is organized as follows:

Section I presents the fluid equations used throughout the
analysis. In Sect. II the constraint of incompressibility is
incorporated into this model, and the equations are derived
and discussed. The non-ideal, compressible fluid is treated
in Sect. III. After the energy equation, the validity of the
incompressibility assumption is discussed. The general pressure
tensor is derived in Sect. IV, with only covariance and symmetry
properties being used. Such a derivation is useful for under-
standing the macroscopic features of this tensor, especially
if one aims to use a simpler form instead of the full tensor.
The transport coefficients are listed in Sect. V. The discussion
and the conclusions are then finally presented in Sect. VI.

- 5 -

I. Fluid Equations

To begin with, we list the MHD equations for a single-fluid
theory. This model relates the density 3 ' velocity u ,
scalar pressure p, pressure tensor P , internal energy e and
the magnetic field B:

(1) Continuity: ^ ^ ^ ^- l%Si I * 9s

(^ ^ ^-"^^ U = - Vp V T xS * \/-P

(2) Momentum: Yv\3 1^-^ * U.-V ) U = - Vp v ^

(3) Energy

S l^ * u-^") ^ - -^ " ^-[^ '^"-^^

where 2 denotes the heat flux and ^ and xi^ mass, reap, energy sources

(4) Ohm's law: ^\ ^ '^uA'^^ ^ ^*^''^'

where Oft is the resistivity tensor, tt\^ the Hall constant,
E the electric field and J the current.

(5) Maxwell: ^ â– - - V* E

By means of the thermodynamic relations the pressure is derived
from the internal energy and entropy S:

<6) 9"- ll^U ^" ^

- 6 -

II. Incompressible Fluid

The fluid equations for stationary, ax i symmetric configurations

are discussed in usual cylindrical coordinates r, 9, z,

where 6 is the ignorable coordinate. The equations are also given

in planar symmetry with z as the ignorable coordinate.

Introducing the condition of incompressibility

(7) V-U Â« o

replaces the energy equation (3). The pressure p can be eliminated
to avoid the thermodynamic relation, e.g. p =p (e, S) or p = p( 3 , T) ,
in the equilibrium calculation.

Equation (7) suggests the introduction of a stream function \
for the velocity:

(8) 41 = "VTv X "Ve V Oi "s)^

and in planar geometry

(8') OX - VX ^ ^1 â™¦ w '^I^

The equation of continuity ( 1 ) reads

implying that the density 3 is a function of \ only:

The condition that the magnetic field is divergence-free is
satisfied by introducing the poloidal flux function H' and
the poloidal current profile \ :

(10) ft = X/^x ^6 * 'X ^e

or

(10')

^ - '^V -^ ^2. â™¦ \^^

- 7 -

Ampere's law defines the current:
Using the relation
we obtain

(11) ; Â« - Ve X VX. - "^6ti\I.[^v^/t'-W -^6x^\ - ^6 ^0

or

(IT) ^ = - Ml X \JX - \)l ^^V = - ^l x"^-)^ - ^]4^^

Maxwell's equation ^xE Â» o is combined with Ohm's law, which

states a relation between the current and electric field,

eq. (4). Only a scalar resistivity is taken into account here;

(12) /V| \ - E * u x^

We have

(13) = -"^x ('v^\IBx\J'X^^ V Me^\]\^^\^o-'^^Â»'^V^6l

with Jq defined in eq. (11).

Evaluating the 9- component of this equation, we obtain

Next we operate with the vector product ^9 x on eq. (13), and
using the relation

^6x ^x ((*^^6x ^'\^| - ^6x|_^^K (^^JesxTjxY (^\3^^^\\x\Jv\] = o

we obtain

(15) ^ 1. 'V^^^ - ^^ r. XJX-^ft] =

- 8 -

and, furthermore,

(16) y"" ^- ^^"^A^"^ = ^oU ^ ^ ^ ^\-^^ ,

where r E is the constant toroidal electric field at r = r .
o o o

We now address the momentum equation (2)

To simplify the discussion, we begin with a scalar viscosity yx
and take the stress tensor from ordinary fluid theory:

(17) ^ - )A U.
with

(18) Vi,: - â€” ^ â€” - - -r 5u VU

In general coordinates the derivatives have to be replaced by
covariant derivatives. In cylindrical coordinates the stress
tensor for an incompressible fluid with axisymmetry is of the
form

\ - lax,,, - U-t'^-â€ž * ^.,h'\

a.

â– 5\

where ^ denotes the partial derivative â€” , etc.

- 9 -

The divergence of the pressure tensor is evaluated by
writing

(20) p = z: Jw R

where the ^y^ denote the unit vectors.
This gives

(21) \]-Â£ = z:. f ^^ i-\ 4 (?,-^^kvl

We get with
the result

Next we treat the term M- V u. and use the relation

(23) [^â– '^) ^ ^ i '^[^) - 4i X Vx ^

With the help of
and

- 10 -

we get the expression
(24)
The Lorentz force is given by

(251 ^|.^ - - -^, W - y, M'X * ^6 ^\Â«^H'-^6.

The momentum equation eventually assumes the form

The e- component of this equation reads:
The r-component is of the form

(26)

- ^ \7)A x\;x-ve 4 |A V,a/^i ^

- 11 -

and the z -component

<29) .V\.vU

The momentum equation contains the term Vp .

Usually the energy equation determines the internal energy e
(or the entropy S or the temperature T) of the system.
However, the incompressibility constraint (7) allows the
energy equation to be neglected. The pressure is eliminated
from the momentum equation and can be determined if u and B
are known. For this purpose we differentiate eq. (29) with
respect to r and eq. (28) with respect to z and subtract the
resulting equations from each other. Finally, we summarize
all the equations using the summation convention with respect
to the indices OC and B and the definition e^^ =1 if (X = r and 6 = z,
and -1 if (X = z and 6 = r, and end up with the system (SI) :

(32) T V (w\%ti'^ X ^'K- \ie + y^VH'v^XVe =

(33) '^^^[^%%,^/t) k vx-ve -
4 ^{Â±) K ^H^.\je.

- 12 -
In planar symmetry the system equations are of the form:

(30') V- [f^VX) = -V\x^X-^â‚¬ + VO'K^Jc^.gt,

(31-) - M ^HÂ» ^ ^o + ^^^ '^^^ ^^>

(32') V(\^suV V\- \1% + V^x\)^.\?i - \/-}X^Â£;i,

(33') \ \) (\^.5 0^,^"^ X \)X-^i = '^Â» ^â€¢j^^'^.o. - eocp. ^p.,>^^'X,j^-^

If there is no flow, then it holds that ^s and w * 0.
From eq. (32) it follows that Y = 'A (^^) and eq. (26) reduces

(34) -p,^ - YX^^h^ = V- (^^/y^^ .

The system (30' - 33') decouples into two pairs of equations.
The functions H* and 'K. are determined by eqs. (31') and (33').
Once ^ and % are known, the functions % and to are the
solutions of the linear system 30' and 32'. In the toroidal
system PO - 33) the functions % and Co also occur in eq . (33).
If % and Co are known , the functions ^ and 'X are determined
by eqs. (31) and (33). This defines an iteration scheme for the
numerical solution.

We examine the equations for the poloidal quantities X and ^
with given U> and 'X, for the toroidal case, eqs. (31) and (33).
For non-zero viscosity, ^ -^ O , this is a fourth-order
elliptic equation for \ , together with a second-order elliptic
equation for ^ . For this system only boundary data can be pre-
scribed and no profiles can be imposed, except the dependence
Q 5= 5^V( on the continuity equation. The other two equations
for the toroidal quantities to and % , eqs. (30) and (32) , with
given 'X and ^> also form an elliptic set for Oi and % .

- 13 -

The case with purely toroidal flow, i.e. \Â»0, is possible

if To'^o - ^c^o ^ ^r^V- iV^/r^^

which determines M^ . Ohm's law for % , eq. (30) , and eq. (32)

form a closed set for tc,^ and H* . Now ^ is no longer a function

of \ , but is arbitrary. The pressure balance, eq. (26) , where

(35) V(p . ^,V ^ W Â» V[^.)['^i^^^-X^)-0

is then highly degenerate. The two components of this equation
determine the remaining thermodynamic unknowns <^ and p .
If there is a solution with very small but non-zero toroidal
flow (o , its streamlines are, presumably, curves 5 ~ ^t.
of the above solution. It is by no means clear that with vis-
cosity such flows exist since the momentum imbalance may act
as a source of poloidal rotation even if no rotation is imposed
at the wall.

For zero viscosity, )> = , the system has quite different proper-
ties. The momentum equation (33) then reduces to

(36

The solvability condition for this equation is less clear. For
given H^ one has to solve a hyperbolic equation for ^\. This
implies that on closed field lines there is a solvability con-
straint. The free data on ^JX are needed for this constraint.

The discussion of the energy equation in the next section makes
it clear that the assumption of incompressibility is not satis-
fied for equilibria with flow. On the other hand, the system
(30 - 33) is quite complicated. We therefore conclude that the
considerable effort for its numerical solution may not be
worthwhile .

- 14 -

III. Compressible Fluid

In this section the energy equation (3) is included and the
assumption of incompressibility is given up. At first the general
Ohm's law, eq. (4) , is discussed. The resistivity tensor M is

defined as

(37)

^ = /V^^ X - i^i - ^^^^^ y

where 1Â£ is the unit matrix, b = B/\B| the unit vector in the
direction of the magnetic field and 'VV), and 'V^j^^ the values of
the resistivity along and perpendicular to the magnetic field.
This implies that

(38) V^.^= 'V^il^-^â€ž^ - %4, - /V\i^l ^ 'V^4.

where w = ( ^- b^ b

is the component of the current parallel to the magnetic field

and J^ the perpendicular one.

Using eq. (11) for the current yields

(39) ^, = ( w . w - \^o r) /^'^" ^ = ^u ^

With the introduction of the electric field

E = - Vc^ + toEo ^& ,

where C^ is the scalar potential (see eq. (16)), Ohm's law
takes the form

(40) f^^-^ ^ {y- '^x') l^ -^ ^H 4^ ^ ' -^ ' ^0^0^ ^ ^-^^
The projection onto ^^ yields

- ToEo - M. UH'

â– ^ '^^ Wx VX-ue.

- 15 -

Next we examine the \}^ - component of the Maxwell equation

which is equivalent to
and, furthermore,

We utilize eq. (41) to eliminate "\q and with

^"^ V|JVM'\^ V /v^â€ž %^

we eventually get the result

(42) \i. [f^ x)^") + wx ^Cr^V^e =

= M-^ ^ VX A (/V)â€ž-M^V^'"V^ ^\ , r i

'v^^ \^V\^ 4 <\^j,0c2.

L

- 16 -

Note that at most second derivates of V occur and that all
the % dependence is explicit. It is by no means obvious that
eq. (42) is elliptic. The characteristics are clearly those
of the equation

^4&x * m.-nx^ â™¦ <Â» %1(6^"''V ^^i^^ * '^. fen L

where B and B. denote the poloidal and toroidal components
of the magnetic field. If we set ''^f ~ -^ ^^^ \^t * ;^ >
the characteristics are

or

Ellipticity is equivalent to the positivity of the quantity

Hence for any v\i 2. and A'^^, > the system is elliptic,

Next we deal with the momentum equation (2). Writing the
velocity u

(44) ^i -= JLA. Vr + AT Ul + to ^6

yields the divergence in the form

(45) Vu = 7 lW., 4 (rv\^^.

- 17 -

The stress tensor is of the form according to eqs. (19 - 20):
(46) ^T>: - -^.i ^ ^.^

Making use of the expressions from the previous section, we
obtain for the r -component of V-\^

U XL

for the 9-component

and for the z- component

X
(49)

F^ =r V- V;^ V^> V ~ i}AVâ€¢AA^,^ - t VjAX^U-Ue

The term Jj.- ^ U involves the derivatives of the basis vectors
(50) ^.\Ju = Vr [UAJU - -^l -^ ^6 ii-^t) ^ ^1 AA.^>3-.

The momentum equation is then

51) M.S ^-^"ii - - Vp - ^, V^ - 7i vx ^ ve v-^xviv-ue ^ W

- 18 -

(52)

the Q- component

(53) iUA9u-^Â£^> ^ V^^'OX-VQ ^ x\Jy-^[t]- Y-^rT ~ }^T^>

and the z- component
(54)

Clearly this system is elliptic for non-zero viscosity.

Finally, we examine the energy equation. Introducing the
temperature instead of the internal energy, we obtain the
following energy equation:

(55)

The heat flux is related to the temperature

(56) -^ =r kVT - W, ^J * vcj^^J - v:^ bxT/T

where parallel VCy, and perpendicular K^ and V^^ are thermal
conductivities. The energy flux due to the viscosity is given
by

i? : vu = s ip.^ ^;

- 19 -

Inserting the terms of the pressure tensor yields

-f.u^a^ ii].

With these expressions the energy equation, which determines

(58) f^(^V)l + ^T V-li - VUVT^ - /V)i^i * /V|â€ž^â€ž + f -.Vu

The condition of incompressibility can be derived from the
energy equation. Equivalent to the energy relation (3) or (55)
is the entropy equation

With p = p ( 3 , ^ ) we can easily write an equation for the
pressure as

or

In the case of non-dissipate flow, where the right-hand side
vanishes, we recover the usual conditions for the validity of
incompressible flow, viz. the flow velocity must be small and
the pressure variations must be small compared with the mean
thermodynamic pressure. If dissipation is included, it follows
that, in addition to the above conditions, the dissipation must
not be so large that the right-hand side becomes comparable
with the left. Physically, the dissipation-induced pressure

20 -

variations must be small compared with the other pressure
variations in the system.

In a confiend plasma with substantial pressure variation, the
assumption of incompressibility is not justified but is rather
poor.

At the end of this section we summarize the set (S2) of com-
pressible equations for axisymmetry:

(1) ^ ^-U * ^. ^ Â« g5j

(41)

(42) "^-[ji "0^) V VH'^ V(7i).^ =

= ^. j ^f MX * -^'^^^ VH> -^

V^^V^ (toEo - juk-W ^ 'viH ^e)x^cvâ– u'xvâ– ^'â–

(52)

(58)

^^ 'v^il^H'l^ + 'v\â€ž X^

i ( - 1 . \ ^
(53) vw<^ vu . \J(o * ^'^J y ^%- ^e " "^& )

(54) M.^ ai.^r> . Cp. ^,\, ^^. ^^ =T,

^

- 21 -

With known velocity, the continuity equation is a hyperbolic
equation for the density. For its solution appropriate sources
have to be given. The remaining equations in the system (52)
form, as discussed, an elliptic system, where boundary data can
be prescribed.

For planar symmetry, the set of equations is of the form

(42-) Sj [i^^^X) vV^x 7(0.^1 =

â€¢* /^u y% % V*^ ; â€” =

(52-) M3ai.\?U * Cp* ^^\^ +H'.,^, = V>^M*l()A^-^\K-^x^v.Oe,

(53-) Wv^A^-Vv^ ((i - ^],^^ Y^^.-'^-h^-^ * ii^'^^\C^I^^'^-^^^
(54') AÂ»^3 u-^UJ * V^ X x/x-^i -V-^^w,

(58')

- 22 -

If there is no flow, i.e. M = O / it follows from eq. (53)
that X^XC^^ and eq. (51) reduces to the Grad-Shaf ranov
equation .

In the case with purely toroidal flow, i.e. ox â€¢* v ^ o ,
it follows that SX.-V - and ^-xa - 0. The continuity equation
then does not determine the density. Equations (41), (42) and
(53) form a closed set for ^' ,X and co . The two remaining
poloidal components of the momentum equation determine the
thermodynamic quantities Â«^ and p . As discussed for incom-
pressible flow, in the case with viscosity, the momentum imbalance
may act as a source for poloidal rotation even if no rotation is
imposed at the wall.

For zero viscosity, u. Â« O , the system has quite different
properties. The momentum equation reduces to

W\

^oA- Vu = - Vp - -^ \J^ - ^ ^'>^ + ve vxx^H^.^6

For given ^ one has to solve a system of mixed type with
sonic transition, which causes difficulties as discussed in
Sect. II.

- 23 -

IV. General Pressure Tensor

In the analysis so far a simplified pressure tensor has
been taken into account. A strong magnetic field, however,
requires a more general tensor than that from ordinary fluid
theory. This tensor is derived from microscopic theory;
see, for example, Braginskii / 5 /.

In our opinion it is useful to formulate the stress-strain
relations in a manner independent of the origin of the viscous
forces, but dependent only on covariance properties and simple
physical bypotheses. In particular, we assume that the stresses
are linear functionals of the strain matrix S :

(59) g.. = ^^ 4 â€” *

We can generalize this treatment to allow the coefficients
of the linear relation to depend on the invariants of that form,
namely on the traces Tr _S, Tr S_^ and Tr _S ^ . Additionally, we
assume that the stresses can depend on the polar vector B,
but on no other quantities. Again, the coefficients in the
linear stress-strain relation can be generalized to depend on
the invariants found from S and B. The stress matrix P must
be symmetric and invariant under inversion of coordinates,
since B is not. We shall construct the corresponding form and
show that it is essentially equivalent to that of Braginskii
hypotheses. We postulate that the viscous forces must increase
the entropy of the system and we assume that the resulting
viscous equations must generate an elliptic stress v â–  ^ â€¢
Finally, for convenience, but not essentially, we assume that
similar to an ordinary fluid there is no bulk viscosity, or
that the viscous stress only depends on

.60, ^J.. . ^,. - i 6,1; I =i:^|-:: - |=J.i(^-^

- 24 -

It is convenient to employ the vector B in two distinct forms,
We use B interchangeably as a row or column vector and we
introduce the standard matrix from electromagnetic theory.

(61)

B> '

e>u\

/

which is invariant under inversion of coordinates,
For any vector V it holds that

^ V = B^ '^ y

while

and

The stress tensor must be even in B and of any order in B .

Although we shall drop the bulk viscosity, for completeness
we give the most general stress tensor proportional to (V-x\ ) .
The most general form consistent with the covariance properties
is clearly

where uj* and u^ are the two bulk viscosity coefficients,
which we set equal to zero.

We construct the stress tensor from W^i , B and B.

We systematize the procedure by taking terms of increasing

degree in B and B. Clearly, the only term of degree zero is
1 2

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