Fusion Res.
Res. SERIES,SERIES, Vol.
Vol. 6 (2004) 000-000
318–321
J. Plasma Fusion

Theoretical Modeling and Application of Microwave
Reflectometry to Plasma Turbulence Study
BRUSKIN Leonid G., OYAMA Naoyuki1,
MASE Atsushi, SHINOHARA Kouji1 and MIURA Yukitoshi1
1ASTEC, Kyushu University, Kasuga 816-8580, Japan
Japan Atomic Energy Research Institute, Naka-machi 311-0193, Japan(Received: 9 December 2003 / Accepted: 13 March 2004)

Abstract
We discuss a possibility to reconstruct turbulence fluctuation spectra from single channel microwave
reflectometry data. The technique is based on the analytical solution of the full-wave equation of microwave
propagation in the axially symmetric plasma. As an illustration, we apply our technique to estimate the fluctuationlevel and poloidal wavenumber spectrum shape in the ELM-free H mode plasma of the JT-60U tokamak.
Keywords:
plasma diagnostics, microwave reflectometry, fluctuation spectrum

1. Introduction

Here ~ is the microwave frequency, fluctuations are
considered to be small (Born approximation), electric field
perturbations are integrated over the antenna surface v s , P is
the antenna pattern, H 2o and H 2n are the Hankel functions of
the second kind, J o is the Bessel function, v n = n 2 + A ,Ct = ~ 2 - h cr , p = ~r/c, t a and p a are the values of t andp at the edge of the plasma ra , E 0 is the unperturbed value of
microwave field, n0 and dn are the plasma density and its
fluctuation, dn = 0.5 for n = 0 and dn = 1 for other valuesof n . The unperturbed ~ 0 function is assumed axiallysymmetric, while the density fluctuations dn depend on bothspatial coordinates r, i and time t . Variations of dn are
assumed much slower than the microwave period 2r/~. The
analytical expression for the unperturbed electric field E 0 for
the same assumption (1) about the shape of the plasma profile
was found in [6].
In this paper we attempt to find density fluctuations dnsuch that the reflectometer signal simulated through the Eq.
(2) closely reproduces the one obtained experimentally.
The plasma turbulence is represented as a superpositionof the rotating modes, characterized by the poloidal andradial numbers (q, p) and yet unknown amplitudes C q, p , D q, p ,which are the aim of our study:

Microwave reflectometry has been studied theoreticallyand successfully applied to the measurements of plasma
profile and fluctuations in a number of magnetic fusion
devices [1, 2, 3, 4, 5]. The purpose of this paper is to develop
a reflectometry based method of measurements of turbulencewavenumber spectra in tokamak plasmas and to apply themethod to the JT-60U device. Our method does not require
new hardware design and installation in addition to theconventional O-mode reflectometer, which is already in placeon the JT-60U as well as on many other magnetic fusion
devices.

n2

n2. Poloidal spectrum reconstruction
For the plasma frequency radial profile representable as2~ 20 (r) = c A
r m +h(1)with the two constant parameters A and h , the analyticalsolution of the full-wave equation of microwave propagationand scattering in plasma was found in [6, 7] leading to thefollowing expression for the simulated reflectometer signal
fluctuations:

3i - u t/rd n (r, i , t)
= !e 2riq Dn0q =- 332H 2 (p )d e (t) = i r 2c
P d v ! 2n
#2 ~ -h v
n = 0 H n (p a )t# H o2 ( t a ) # J o ( tl) tl 1 20rcciisrr#b C q, p e 2r ip D + D q, p e - 2rip D l .

an2rn

(3)

r

(2)

Here u i is the poloidal speed, rc is the cut-off layer location,
D i is the angle u i T/rc covered by the rotating fluctuations
within the time period T , D r is the radial scale defined such

Corresponding author’s e-mail: bruskin@astec.kyushu-u.ac.jp©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Researchr# # [ d n cos (n i) cos (n il) + sin (n i) sin (n il)]-r

# E 0 ( tl, i l) ~ 20 ( tl)d n ( tl, i l, t)
d i l d tl .
n 0 ( tl)

1318

BRUSKIN Leonid G. et al., Theoretical Modeling and Application of Microwave Reflectometry to Plasma Turbulence Study

For an isotropic turbulence or for a given radialspectrum shape G r (p) we can determine C 0 G i (q) from theeq. (6) and thus finally obtain the average spectrum ofdensity fluctuations | C q, p | . It follows from the eq. (3) and therequirements of density dn to be a real function that the RMS
amplitude of density fluctuations c = v (dn/n0 ) is equal to

as to cover all the radial wavenumbers of practical interest(normally, the same range of k r with the poloidal
wavenumbers k i ).
Turbulent fluctuations are considered to be randomfunctions in a sense that relative phases of different modes
are random and (q, p) pairs are also random, different fordifferent realizations of dn (r, i, t) . It is thus reasonable tosearch not for the particular realizations of C q, p , D q, p , which
ideally reproduce the reflectometer signal on a given interval
0: T , but instead, to search for the quantities <| C q, p |> ,<| D q, p |> , averaged over many dn realizations of type (3),where the radial numbers p on each realization are randomly
generated for each poloidal number q.
Substituting eq. (3) into (2), representing de (t) as
Fourier series and taking into account the requirement of dn
to be a real function ( C - q, - p = C q*, p , D- q, - p = D q*, p ), weobtain:

Qc = 2 < G r2 (p)>A =R*- q, - p

C q, p + S*- q, - p

D q, p ,The O-mode reflectometer, installed on the JT-60U
tokamak, performs routine monitoring of plasma fluctuations
using the fixed frequency microwave channels [3]. The signal
amplitude and phase are registered using a quadraturedetector. In this section we apply our technique to reconstruct
the shape of the fluctuation spectrum from the quadraturecomponents of the reflectometer signal during the H-mode
phase of the plasma discharges #37433 and #37429 on theJT-60U tokamak. In the discharge #37433 the transition toELM-free H mode happened at 6.329 s, ELM activity started
at 6.355 s. The spatial distribution of the ambient plasma wascalculated from the magnetic equilibrium condition andapproximated by a cylindrical layer with A = 1.475 $ 1014 s-1
cm and h = - 7.9426 $ 10 23 s -2 . We first search for thestationary 2,010 points records within the reflectometersignal corresponding to the 35.9 GHz reflectometer channel,apply the Fourier transform to each 201 points interval andaverage the Fourier transform over 10 intervals to obtain thefunction <| A q | 2 > in eq. (6). Substituting it together with thesimulated functions R q, p , S q, p to the formulas (6) and (7), weobtain the fluctuation spectrum shown in Fig. 1. Soon after
the transition the poloidal spectrum has a monotonic shape, atlater stages a maximum grows at k i = 0.6 cm-1. The poloidal
velocity of fluctuations u i , necessary for the spectrum
reconstruction, was scanned from 5 to 20 km/s. The value of
17.5 and 20 km/s led to a fluctuation spectra such that thesimulated reflectometer signal (eq. (2)) closely reproduced
the experimental one. It is worth noting that charge exchange
recombination (CXR) spectroscopy indicates, although withpossibly large errors, that the poloidal speed of carbon ions
was +20 km/s. With the assumption of isotropic symmetric
spectrum, the amplitude of fluctuations is estimated as
c = 0.21 % at 6.336 s, c = 0.56 % at 6.343 s and c = 0.75 %at 6.347 s, increasing before the ELM activity starts. The
fluctuation amplitude of the H-mode turbulence was alreadyanalyzed in our previous work [7], and the above c values
are in agreement with the previous results. Since the radial
shape of the fluctuations spectrum is not known, the abovet# H o2 (t a ) # J o (t l) t l~ 20 (t l) 1 2rc0r# # [d n cos (n i) cos (n i l) + sin (n i) sin (n i l)]
an-r# E 0 (t l, i l) el2 r iq i
Dirle 2r ip D
$ * - 2r ip r l 4 d i l d t l .
eDr

(5)

r

In the eq. (4) the q index is considered to be positive,T
A q = 1 # de exp (2riq x ) dx are the Fourier coefficients ofT 0
T
the reflectometer signal.
For symmetric spectra | A- q | = | A q | and for| C q, p | = | D q, p | we obtain by averaging the first eq. (4):C 02 G i2 (q) =(7)3. H-mode turbulence in JT-60U(4)

32R q, p
H 2 (p )P d v ! 2n
4 = i2r 2c#S q, p~ -h v
n = 0 H n ( pa )n2iwhere Q is the amount of modes considered in the expression(3). If the radial spectrum shape function G r is unknown,
then of course, the precise value of c cannot be defined. A
rough estimation can be done by merely assuming a certain
radial spectrum type, for example, Gaussian spectrum or
isotropic spectrum. In a few test examples the c errors causedby wrong G r assumptions were in the range of 40-60 %.

where*20q=1A q = R q, p C q, p + S p, q D q, p*-q!C G (q) ,<| A q | 2 >< G (p)| R q, p | 2 > + < G r2 (p)| S q, p | 2 > . (6)2rHerewehaveintroduced
thenotation| C q, p | = | D q, p | = C 0 G i (q) G r (p) , explicitly separating thepoloidal G i and the radial G r wavenumber spectrum profiles.
The ensemble average in the denominator of the above
equation is taken over many realization of (q, p) pairs, andthe numerator is averaged over several realizations of thereflectometer signal. From the eq. (6) the shape of thepoloidal fluctuation spectrum G i (q) can be determined if theradial shape G r (p) is a known function. If the fluctuations are
isotropic G i (q) = G r (p = q) then the spectrum shapefunction G i can be derived through eq. (6) iteratively startingwith a uniform radial spectrum profile G r = 1 . We havenoted, however, that even the first iteration provided the G ishape quite accurately in all the examples we have looked at.
Moreover, our examples prove that even if the fluctuations
are not isotropic and G r (p) is unknown, the eq. (6) with theuniform radial spectrum G r (p) = 1 still defines the poloidal
shape function G i (q) with sufficient accuracy.
2319

BRUSKIN Leonid G. et al., Theoretical Modeling and Application of Microwave Reflectometry to Plasma Turbulence Study4(a)σ(δn/n0 ) = 0.21%

estimated spectrum
polynomial approximation

2

estimated spectrum
polynomial approximation
G [a.u.]

3

θ

2

100θ

G [a.u.]

10.4

0.8

1.2

σ(δn/n0 ) = 0.75%

21.6(b)012-1kθ [cm ]34Fig. 2 Poloidal spectrum reconstructed with the assumptionof u i = 7.5 km/s.

160(a)

500.4

0.8-1kθ [cm ]1.2σ(δn/n0 ) = 0.58%

1.64

estimated spectrum
polynomial approximation

3Fig. 1 Reconstructed wavenumber spectra of the JT-60U
edge plasma turbulence during the H-mode phase of#37433 discharge, t = 6.336 s (a) and 6.347 s (b).
The amplitude of fluctuations is estimated with theassumption of u i = 20 km/s.

2G [a.u.]10θ

values of c can be only treated as rough estimations of thefluctuation amplitude, with the accuracy mentioned in theprevious section. It is likely that the fluctuations in the edge
plasma area are not isotropic, since the longest possible radial
wavelength can't exceed the density pedestal width in the Hmode plasma (roughly 6 cm for the considered discharge)
while the poloidal spectrum is free of that limitation. The
shape of the poloidal spectrum, however, is defined withhigher precision than the amplitude, because it is not strongly
affected by the shape of radial fluctuation spectrum.
If the assumed u i is different from 20 km/s, thefluctuation spectrum changes as well. Those u i values lessthan 12.5 km/s result in unreasonable density spectra withsharp increase at large k i (Fig. 2), therefore we expect u i tosatisfy the relation u i > 12.5 km/s. In the discharge #37429
the ELM-free H mode plasma existed between 6.27 s and6.31 s. Similar to the shot #37433, the reconstructed poloidal
fluctuation spectrum gradually changed from a relatively flat
to a more bulgy shape (Fig. 3), and the estimated fluctuation
amplitude slightly increased before the ELMs. The value of
20 km/s is assumed for the rotation velocity of thefluctuations, assumptions less than 10 km/s resulted in thereconstructed spectrum shapes similar to Fig. 2, sharply
increasing at large k i .

400.4

0.8

1.2

σ(δn/n0 ) = 0.77%

1.6(b)

32

100

0.4

0.8
-1kθ [cm ]

1.2

1.6

Fig. 3 Reconstructed wavenumber spectra of the JT-60Uedge plasma turbulence, #37429 discharge, t = 6.272 s(a) and t=6.302 s (b).

4. Discussion and summary
The analytical technique discussed in this paper isintended to enrich the conventional reflectometry methodwith the measurements of fluctuation poloidal spectrum andamplitude. There are plasma conditions, however, when ourmethod is not applicable, such as the case of fast nonstationary events like ELMs and transitions. Multiple
scattering by strong plasma fluctuations reduce the accuracyof the method. Another situation when our technique fails to3320

BRUSKIN Leonid G. et al., Theoretical Modeling and Application of Microwave Reflectometry to Plasma Turbulence Study

Acknowledgement

correctly extract the fluctuation spectrum is the case when thefluctuations are not random, but instead, exhibit persistentpatterns. Our method relies on an assumption of random
density fluctuations when taking statistical averages, and thatis why its accuracy reduces if the plasma fluctuations are notrandom. For example, it is possible in our model to introduce
the fluctuations in a form of phase-locked structures such thatthe microwaves scattered from some possibly strong coherent
modes would always cancel each other near the receiver,while other weaker modes wouldn't, thus drawing amisleading conclusion about the relative strength of thosemodes. In our treatment we didn't consider such phase-locked
structures in plasma fluctuations. An interference of therandom modes averaged over many realizations does noterase the information about the average amplitude of thefluctuating modes such that stronger modes always produce
stronger scattering than the weaker modes of the same scale.
In summary, we have discussed a new extension of theconventional reflectometry, which allows us to reconstruct
the average poloidal spectrum and estimate the amplitude ofplasma turbulence. As an illustration, we have applied thespectrum reconstruction technique to the reflectometry of theJT-60U tokamak plasma.

This work was supported in part by Effective Promotionof Joint Research with Industry, Academia, and Government,
Special Coordination Funds for Promoting Science andTechnology, MEXT.

References[1] E. Mazzucato and R. Nazikian, Plasma Phys. Control.
Fusion 33, 261 (1991).
[2] M.E. Manso et al., Plasma Phys. Control. Fusion 43, A73(2001).
[3] N. Oyama and K. Shinohara, Rev. Sci. Instrum. 73, 1169(2002).
[4] R. Nazikian, G.J. Kramer and E. Valeo, Phys. Plasmas 8,1840 (2001).
[5] H. Hojo, Y. Kurosawa and A. Mase, Rev. Sci. Instrum.
70, 983 (1999).
[6] L.G. Bruskin, A. Mase, N. Oyama and Y. Miura, Plasma
Phys. Control. Fusion 44, 2305 (2002).
[7] L.G. Bruskin et al., Rev. Sci. Instrum. 74, 1473 (2003).

4321