PPPL--2731
DE91 004967
4_
¢
Microwave reflectometry for the study of
density fluctuations in tokamak plasmas
E. Mazzucato and R. Nazikian
Princeton Plasnm Physics Laboratoo"
Princeton University
Princeton, N. J. 08543
o
I
Abstract
The effect of small scale density fluctuations on the propagation of electro-
magnetic waves in an inhomogeneous magnetized plasma in the presence of a
cutoff is investigated. It is shown that, provided the fluctuation scale length is
greater than the free space wavelength of an incident plane wave, the scattered
field is strongly enhanced from fluctuations near the turning point. Numeric,ai
results for _ve propagation in a tokamak plasma demonstrate the feasibility of
refiectometry for the localized measurement of density fluctuations in the range
k±pi << 1.
, iER
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DIS'IRIBUTt_..,;,..,{.:.'-j:7;-: :: _....,.... ;,.,-..,, _. ,'<LIK,,,WED
1. Introduction
The causes of anomalous transport in tokamak plasmas are still unknown.
A popular conjecture, which finds limited support in theory and experiments, is
that the thermal and particle transport is enhanced by the existence of fine scale
turbulence. Indeed, theory predicts a large _riety of plasma _ves which are
driven unstable by density and temperature gradients, by dissipative effects and
by magnetically trapped particles (Kadomtsev and Pogutse. 1971; Tang, 1978:
Horton, 1984). Experimentally, observations with microwave (Mazzucato, 1976)
and laser (Surko and Slu_her, 1976) scattering have revealed the e.xistence of a
small scale turbulence in rough agreement with theoretical predictions.
Present obserx_tions show that the level of density fluctuation K increases
as the perpendicu!ar wave number k± decreases, as predicted by the mlzdng
length criterion Ft/n _ 1/k±Ln [where L_--(dlnn/dr) -_ is the densiD" scale
length]. This makes the interpretation of experimental data very difficult as most
m
scattering techniques detect long wavelength fluctuations with only poor spatial
resolution. On the other hand, very long wavelength modes, i.e., modes with
0
small poloidal and toroidal mode nmnbers, do not show correspondingly large
fluctuation levels. This implies that the turbulence spectrum must turn over at
some _ue of km where present obserx_tions lack spatial resolution. This is of
considerable importance since turbulent fluctuations with amplitudes below the
mixing length level and wavelengths much longer than the ion Larmor radius
pi could theoretically account for the obsern,ed plasma tramport in tokamak
(Liewer, 1985; Haas and Thyagaraja, 1986). Clearly the measurement of long
_velength fluctuations with improved spatial resolution over e.x.isting scattering
techmques is needed.
One tech_que _-ith the potentiM for prox4ding spatial])" localized measure-
'4
merits of long wavelength density fluctuations is micro_ve reflectomet_,. In
• fact, the first experimental evidence for the existence of a fine scale turbulence in '
tokamaks was obtained using microwave reflectomet_; on the adiabatic toroidal
' ' " J_" ".' ..... " ' ' INr' " ', '
compressor (ATC) tokamak (Mazzucato, 1975). This method, widely employed
in atmospheric studies, measures the reflection of electromagnetic _ves from
the plasma cutoff to obtain the electron density profile in inhomogeneous plas-
0
mas. The system ca2abe considered a special kind of interferometer where the
phase of the received _ve is determined by the refractive index along the _ve
trajectory, and also by changes in the position of the reflecting l_er.
Enhanced scattering from fluctuations at the cutoff in isotropic plasmas
_as first addressed by Pitte_y (1958) in reference to radio wave propagation
in the ionosphere. In this paper, we address the issue of wave propagation in
anisotropic plasmas and assess the relative capabilities of O-mode and X-mode
reflectometry for the local measurement of density fluctuations in tokamaks.
The paper is orga.n_izedas follows: the basic equations for the scattered field
in the Born approximation are derived in _ 2 for both the ordinary and the
ex-traordinax3 r mode of propagation. Some numericM examples are presented irt
,_ § 3 which simulate the case of a large tol_mak plasma. Finally, our conclusions
are presented in § 4.
2. Basic Equations
In this section, we derive the equations for the electromagnetic field scattered
by small density fluctuations in a magnetized plasma.
A plane electromagnetic wave E = Noex_pi(a20t- k0. r) is launched into
a plasma with a magnetic field B which, in Cartesian orthogonal coordinates
(x', y', z'), is aligned in the y' direction and is only a function of z'. The plasma
occupies the region z' > 0 with the electron density distribution given by
= + :'.t), (1)
i.e., where a plane stratified plasma equilibrium is perturbed by weak (1_ [ <<
° fi_) irregularities which are uniform along magnetic field lines. While we asstmm
that the free space _avelength A0 is comparable to the spatial scale length A of
i_ lT I ' , 71 I II ' ' | I] ' ' I,' I
plasma, fluctuations, we assume the _ve frequent3 ' aJ0 to be much greater thaxl
the bandwidth of density fluctuations so that the time dependence of Eq. (1) can
be ignored.. Finally, we assume k0.B = 0. This geometry reproduces the typical
0
reflectometer comq.guration used in the investigation of density fluctuations in
tokmzlaks (Mazzucato, 1975; Cripwell et al., 1989), the major simplification being
the omission of the magnetic shear in the description of the wave propagation
which is justified by its smallness in this type of magnetic configuration.
, Under these assumptions, the electromagnetic field may be separated into
two independent modes of propagation, each described by a scalar differential
i
l
[ equation in the two _riables x / and z' (Budden, 1961). The dependent _riable
is the y' component of the electric field E for the ordinary, wave, and the 5/
component of the magnetic field H for the extraordinax3 __ve. Using standard
notations and introducing the change of coordinates (x, y, z) = (kox', _y', 'ko:'),
the equation for E is
V2E + eoE = 0 , (2) ,
4
" while the equation for H is
1
61
V2H - -- VH. Vel + i-- iy. VH x Ve2 + elH = 0 (3)
61 62
where i_ is the unit vector along the y-axe, and
= x x (u - x) - (u- -
--_, 61 : U(U-X)_],-2' E2 -- ._" ' (4)
and v << a., a small effective collision _equenQ" which takes into accolmt weak
wave damping. W_hen t_ = 0, eo = 0 for X = 1 and el = e2 = 0 for X = 1 + ]'.
Using Eq. (1)we may put Eq. (4) in the form
I,
4
where le'il<< I. We shall assume that dgi(z)/dz _ 0 at z - zc, and that, the
thickness of the evanescent region behind the cutoff is iIlally free space wave-
lengths such that tunnelling effects may be ignored. Then, we may proceed by,
$
making the Ansatz, to be verified a posteriori, that
" E = Eo + Z En,
n>O
=
where IEol_ O(I) axldIEn>ol_ O(%n). Prom thetwo lowestordertermsofEq.
(2)we obtainfortheordinarymode
I
I V_Eo+_o(=)Eo=o, (6)
i
mad
V2E_ + _o(z)E_ = -_o(X, z)Eo(z) • (7)
Similarly, by assuming for the extraordinary mode
• H=Ho+EH_ ,
n>O
" with II-Iol_ O(1)and IHn>o _ O(71=), we obtain
•g]
V2H° e1-1rHo. Vgl + z-_iy, rHo x Vg2 + g_Ho - 0 , (8)
and
1 _7H1. Vgl+ igl
,i V2H1 gl g-_iy. VH1 x V_2 + glH1 -
]
• -- -- i--ly • _'7'H 0 X _"-- "Jr- -- _'7292 -- _lHo .
gl ez e2 e2 ro gl
Let us first consider the case of the ordinao' mode with a_ incident plane
,,_tve which in ,._cuum takes the form ex'p[i(wot- aoX- 2oz)l, with/3o = (1-
" a2o)1/2 so as to satis_, the w_ve equation for free space propagation. Taking
• solutions of Eq. (6) in the form
ii Eo=Eo(z)exp(--ic_ox),5