J. A Tataronis.

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COO-3077-46
MF-68



Courant Institute of
Mathematical Sciences

Magneto-Fluid Dynamics Division



RF Energy Absorption Due
to the Continuous Spectrum
of Ideal Magnetohydrodynamics



|. A. Tataronis



AEC Research and Development Report

Plasma Physics
March 1974



New York University



UNCLASSIFIED



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



MF-68 C00-3 077-^6

RF ENERGY ABSORPTION DUE TO THE CONTINUOUS
SPECTRUM OF IDEAL MAGNETOHYDRODYNAMICS

J. A. Tataronis

March 197^



U. S. Atomic Energy Commission
Contract No. AT (11-1) -3 077



UNCLASSIFIED



Abstract

The relationship between energy absorption and the continu-
ous frequency spectrum of the linearized equations of ideal mag-
netohydrodynamicg is investigated. We limit our considerations
to incompressible fluid perturbations for which the continuum
stems from resonant surfaces where the oscillation frequency
equals the local frequency of an AlfvSn wave. Details of this

absorDtion nrocess arp mustr-ai-oH >w nhfo^^n. +-u~ — ~

ERRATA

COO- 3077-46 MP- 6 8

RF ENERGY ABSORPTION DUE TO THE CONTINUOUS
SPECTRUM OF IDEAL MA GNETOHYDRODYNAMCS

J. A. Tataronis



32: Lines 12 - 14 should be replaced by the following sentence
"It was pointed out in the previous section that when the
applied force varies sinusoidally in time, the fields in
the plasma also vary sinusoidally in time with constant
amplitude except at the point x = x (w ) where the fields
grow in time . ,T

36: In line 13, the number ''0.07'' should be replaced by "7."



After submission of the report for printing, the following referei
concerning Alfven wave heating was referred to the author:

Hasegawa, A., Chen, L. 1974. Phys . Rev. Letters, 32, 454.



Abstract

The relationship between energy absorption and the continu-
ous frequency spectrum of the linearized equations of ideal mag-
netohydrodynamics Is investigated. We limit our considerations
to incompressible fluid perturbations for which the continuum
stems from resonant surfaces where the oscillation frequency
equals the local frequency of an Alfve"n wave. Details of this
absorption process are illustrated by obtaining the response
of the diffuse sheet pinch to an external current source.
Expressions are derived which give the rate at which energy is
transferred to the plasma and show the spatial distribution
of the absorbed energy. The results indicate that the absorp-
tion is enhanced if the plasma profile has regions of large
spatial gradients. This enhanced absorption is a consequence
of the presence of surface waves.



1^ Introduction

Conservative, dynamical systems under the linear approxi-
mation have the property that they can display dissipative
effects if there exists a sufficiently large number of degrees
of freedom (Sturrock 1961; Grad 1969). In simplest terms
apparent dissipation arises when the system has a continuous
frequency spectrum. The normal modes which make up the con-
tinuum can phase-mix in time, leading to decay of macroscopic
variables. Plasma models, free of any apparent loss mechanism,
are often encountered which are characterized by a continuum
and are consequently dissipative. Among these models we
mention a thermal plasma described by the linearized Vlasov
equation (Landau 1952; Weitzner 1963), a cold non-uniform
plasma (Barston 1964) and certain configurations, such as the
screw pinch (Grad 1973; Goedbloed and Sakanaka 197^), the
theta pinch (Weitzner 1973; Grossmann and Tataronis 1973) and
the sheet pinch (Boris 1968; Tataronis and Grossmann 1972;
Uberoi 1972; Tataronis and Grossmann 1973), described by the
linearized equations of ideal magnetohydrodynamics . (In what
follows the initials MHD will designate the term magnetohydro-
dynamics) .

A second implication of a continuum is that the system
can irreversibly abosrb energy when under the influence of an
externally applied force. The absorption is a consequence of
a resonance which occurs when the spatial distribution and the
time behavior of the force matches a mode in the continuum.
A characteristic of this absorption is that it occurs locally



in the space in which the continuum is defined. This relation-
ship between a continuous spectrum and energy absorption has
been recognized in the past (Grad 1969; Baldwin and Ignat 1969;
Tataronis and Grossmann 1972; Hasegawa 1973), and calculations
have been made to determine the effectiveness of this method
of energy transfer as a means of heating plasmas (Baldwin and
Ignat 1969; Brownell 1972; Tataronis 1973; Grossmann, Kaufmann
and Neuhauser 1973)-

In what follows we shall be concerned with this absorption
mechanism under the approximation of ideal MHD. We shall
limit our considerations to a particular plasma configuration,
namely the diffuse sheet pinch, for which it has already been
shown that the linearized MHD equations have a continuous spec-
trum (Boris 1968; Tataronis and Grossmann 1972; Uberoi 1972;
Tataronis and Grossmann 1973)- Under the approximation that
the fluid motion is incompressible, we obtain the response of
the plasma to an external sheet current located in a surroundinf
vacuum region and determine from the response the energy
transferred to the plasma. The force which drives the plasma
is derived from the pressure exerted on the plasma surface
by the magnetic field induced in the vacuum by the sheet cur-
rent and the plasma current. As we shall demonstrate, this
scheme leads to efficient coupling of energy to the interior
of the plasma region even though the plasma motion is



The results of this paper were first reported at the October
1973 meeting of the Plasma Physics Division of the American
Physical Society, Philadelphia, Pa. (Abstract in Bull. Am. Phys .
Soc. 18, 1291 C1973)).



incompressible. The absorption rate is measured in terms of
the effective impedance. that the plasma imposes on the sheet
current. It will be shown that as a consequence of the
continuum, the impedance has a resistive component which, for
parameters and dimensions typical of high beta, high density
plasmas, can have values exceeding one Ohm at frequencies on
the order of 2 MHz. This resistance is comparable to what
has been found for excitation of the lower hybrid resonance
by a method similar to the one assumed here (Puri 1973) > with
the advantage that the absorption described below occurs at a
relatively low frequency.



2. The Plasma Model and Basic Equations

The specific plasma model we study is the planar sheet
pinch with the equilibrium magnetic field J3 C= B z) parallel
to the z axis of an (x,y,z) Cartesian coordinate system as
illustrated in Fig. 1. The equilibrium pressure p(x) and
mass density p(x), as well as B^Cx), depend only on x and are
symmetric with respect to x = 0. The plasma is confined to
the region -b < x < b . For |x| > b , there is a vacuum where
p =0, ^ is independent of x and an externally supported sheet
current sheet £ is present which is directed parallel to the
y axis and depends sinusoidally on z,

£ s (x,y,z,t) = J s Ct)[SCx-c) - 5tx+c)]cos kz y CD

In this expression k is a real wave number, 6Cx) Is the Dirac
delta function £ has the dimensions of current per unit area
and J (t) is expressed in terms of current per unit length,
length being measured in the z direction. The sheet current
is the source of the external force which acts on the plasma
surface.

We descrihe the plasma dynamics by means of the linearized
equations of ideal MHD with resistivity and viscosity neglected
and incompressibility assumed. In Maxwell's equations, the
displacement current is neglected. Since the assumed equili-
brium magnetic field is free of curvature, we have (Jg • V)£ = 0,
implying that the MHD equations can be reduced to the following
form:



vi£ (2)

at'



P^-^->V-1f



.g-0 (3)



CO



^ - -V x ; C5)



*1 = £ V x 8i C6)



where y is the vacuum permeabily, ^, jgp and £ are perturbations

of the velocity, magnetic and electric fields, and p represents

* —1
a perturbation of the total pressure, i.e. p =p 1 + y £*$]_•

In the vacuum the fluctuating electric, 6£ v , and magnetic, 6£ y ,

fields are determined by,



6B v = Vc|>
3(



rr - " v x 6 Sv



C7)



(8)



where the magnetic potential satisfies,

V 2 (j> = 0. (9)

At x = b, the plasma and vacuum fields are connected by the
usual continuity conditions of ideal MHD, namely 6B y is parallel
to the fluctuating plasma-vacuum interface and the total pres-
sure p + B 2 /u is continuous across the interface. After



linearization, these two conditions imply the following equali-
ties at x = b,

8 x 6 Sv = C Q "«^v Cl0)



jl £ v ' 6 £v = P* CXI]



where B ( = B 2) is the equilibrium magnetic field in the
vacuum region and (i is a unit vector normal to the plasma-
vacuum interface of the equilibrium and directed into the
vacuum. Finally, the following jump conditions across the
sheet current are to be satisfied,

n, x C6^ v ] = y^ s C12]

e - isj^j = o ci3i

where the brackets designate the jump in the enclosed quantity.
The symmetry of the sheet current implies that the excited
mode is characterized as follows,



v, = (v x , 0, v z ), (14)



1 = CO, E y , 0), (15)

El = C B x' °- 3 z } > (16)

h = (o, V 0)j Cl7)



6£ v = (6B x , 0, 6B z ), (18)

6£ v = (0, 6E y , 0) (19)

where in each expression the parenthesis on the right enclose
the x, y, and z Cartesian components of the vector on the left
Furthermore, it can be readily established that the Induced
fields possess a spatial symmetry with respect to the plane
x = 0, namely,



v x (-x,z,t) = -v x Cx,z,t), v z (-x,z,t) = v z (x,z,t), (20)

E y (-x,z,t) = -E y (x,z,t), (21)

B x (-x,z,t) = -B x (x,z,t), B z (-x,z,t) = B z (x,z,t), (22)

J y (-x,z,t) = -J y (x,z,t), (23)

p*(-x,z,t) = p*(x,z,t). (2H)



SB and 6E have the same symmetry as B, and E respectively.

Finally, since the imposed surface current, eq. (1), depends

on z in the form cos kz, it follows that v , B , E , J , p*,

SB and 6E also have the form f(x,t) cos kz, while v , B and
z y z x

SB depend on z as g (x,t) sin kz. In what follows the depen-
dence of the fields on z as either coz kz or sin kz will not be
explicitly shown except in cases where ambiguity would result
by leaving the z dependence implicit.



In the sheet pinch geometry, the plasma fields and energy
can be expressed explicitly in terms of v . In particular,
by combining eqs . (2) and (3) 5 one finds,

1 3v x
V z = ~t IT (Xjt) sln kz > (25)



^ = -■-l£ fi CxJ.4 T + £ k 2 B 2 (x)]^Cx,t)cos kz, (26)



where v satisfies,

gk( -\ + i- k 2 B 2 )^] -k 2 (p-^2 + h t 2 B 2 )Y- = 0. (27)
3x 3t 2 y 3x 9t 2 y X



The remaining fields are obtained from v and v with the aide
of eqs. (4)-(6). As a consequence of the spatial symmetry in-
dicated by eqs. (20)-(2 i l), it suffices to solve eq. (27) in
the region x > subject to boundary conditions at x = ,



v x (0,z,t) = 0, (28)

and at x = b ,

v (b,z,t) = jL 6E(b,z,t), (291

x v y

p*(b,z,t) = ^ B v 6B z (b,z,t), (30)



where and vanish as x ■* od .

The equation which expresses the rate of change of the
plasma energy in terms of v is derived by introducing ^ (x,z,t),
the displacement of a fluid element from its equilibrium position



3



t

£ = f «•* (3D





By integrating eq. (2) with respect to t, with the initial
conditions set equal to zero, scalar multiplying by y_ and
the integrating over some plasma volume V bounded by a sur-
face E , one finds the following statement of energy conserva-
tion in terms of ^ and %,
2

l d &¥ TT " ? X, ' ^ • v > 2 ^ - - f «* ■ *P - (32)

V £

where use was made of the incompressibility condition V • y/ =
For the volume V, we choose the region bounded on the x axis
by x = ±b , on the y axis by y = and y = L , where L is an



arbitrarily chosen dimension, and on the z axis by z = and
z = 2ir/k . By taking into consideration that v and p* are
antisymmetric in x and depend on z in the form cos kz while



integration and the integration over y and z in the left hand
side of eq. (32) can be readily accomplished, yielding,

dT = -"IT 2 " v x (b,t)p*(b,t), (33)

where W, the plasma energy , is given by,

W = -^Z- f dx[|p(x)^ 2 (x,t) + Jp k 2 B 2 (x)£ 2 (x,t)] . (3 c (45)



, » J a M .-kx _



where D(u) is a function of u determined by the boundary
conditions at x = b . The electric field has been written as
the sum of two terms: the first term represents the electric
field induced in the vacuum directly by the sheet current, while
the second term Is the component of the electric field resulting
from the plasma motion. If the plasma is removed, then the
second term vanishes. The boundary conditions at x = b , eqs.
(10) and (11), can be expressed in terms of v and E . After
a Laplace transformation is performed, they read,



v(b,w) = =±- SE(b,w) , (46)

x B y y



9v k 2 B 36E



(47)



At x = we have



v C0,w) = . (48)



12



Equations (40 ) — C^8 ) form a self-consistent system which deter-
mines v and E .
x y

To solve the system we introduce a function \\> (x,cd) which
satisfies eq. (40), is free of singularities on the real x
axis in the range _< x _< b for w on the Laplace contour, and
satisfies the following conditions at x = ,

K0,w) = , (49)



§f(0,u>) = k . (50)



v (x,w) must then be proportional to i> :



v x Cx,w) = C(a>)Kx,uO (51)



The constants C(w) and D(w) are obtained by substituting eqs .
C44) and (51) into eqs. (46) and (47) yielding for lm(w)


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Online LibraryJ. A TataronisRF energy absorption due to the continuous spectrum of ideal magnetohydrodynamics → online text (page 1 of 2)