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Paper No. PI:19
THE 13th INTERNATIONAL STELLARATOR WORKSHOP
TRANSPORT IN HSX ELECTRON CYCLOTRON HEATED PLASMAS AT 0.5 T*
Talmadge, J.N., Anderson, D.T., Anderson, F.S.B., Almagri, A.F., Lechte, C. (1),
Gerhardt, S.P., Radder, J.
The HSX Plasma Laboratory, University of Wisconsin-Madison, USA
(1) TJ-K Laboratory, Christian-Albrechts-Universität Kiel, Germany
E-mail: [email protected]
A 1-D code, ASTRA, is used to model transport in HSX when heated with a 28 GHz gyrotron
at a magnetic field of 0.5 T. The absorbed power is based on a ray-tracing calculation. Results are
compared when HSX is run in the usual quasihelically symmetric configuration and one in which the
symmetry is broken (mirror mode). When the radial electric field is included in the transport
calculation, the two configurations should be fairly similar at the low power and low magnetic field
for the present set of experiments. Instead, it is observed that the stored energy and confinement time
for the QHS mode of operation at low density is significantly better than for the mirror mode. The
good confinement of high energy particles in the QHS mode is the possible cause for the observation.
A 28 GHz gyrotron, with maximum output power of 200 kW, is used in the Helically
Symmetric Experiment (HSX) to heat electrons to the low collisionality regime. The normal
configuration in HSX is quasihelically symmetric (QHS) and has a dominant n = 4, m = 1
component in the magnetic field spectrum. With a set of auxiliary coils, the quasihelical
symmetry can be broken with the addition of a toroidal mirror term to the spectrum, n = 4, m
= 0. Here n is the toroidal mode number and m is the poloidal mode number. With 50 kW
input to the plasma, we want to explore experimentally whether it is possible to observe
differences in the confinement time, electron temperature, and electric field between the QHS
and mirror modes of operation. To do this, we compare the results of a 1-D transport code to
the experiment.
The ASTRA code was originally
developed at the Kurchatov Institute to model
tokamak transport and later modified by
0.8
Karulin1 to simulate stellarators. It has since
0.6
been updated to model low power, second
harmonic electron cyclotron heating (ECH) at
0.4
0.5 T. A transmission line takes the hollow
power density profile of the TE02 output of the
0.2
gyrotron and transforms it to a Gaussian-like
0
profile in the HE11 mode with a series of mode
0
0.1
0.2
0.3
0.4
r/a
converters. An ellipsoidal mirror focuses the
microwave power onto the magnetic axis with a
Fig. 1: Power deposition profile for 2nd
beam waist of 2 cm. The output spot size has
harmonic extraordinary mode absorption
been measured experimentally and compares
from ray-tracing calculation (red) and
well with the design parameters. Using the
broader profile used in ASTRA code.
appropriate beam width and curvature radius at
the plasma boundary, a ray-tracing code was used to compute the power deposition profile in
the plasma.2 For the ASTRA code, we assume a deposition profile somewhat larger than the
Power density
1
calculation, P = P0 [1-(r/a)2]40. Figure 1 shows a comparison of the two profiles. The
integrated total absorbed power is determined by the decay of the stored energy as measured
by a diamagnetic loop.3
The expressions for the particle and heat fluxes in ASTRA, including the off-diagonal
terms, are dependent on moments of the monoenergetic diffusion coefficient. The diffusion
coefficient is calculated for a broad range of test particle energies, densities and electric fields
using a Monte Carlo code. The magnetic field spectrum is determined from a finite-size
model for the magnets and includes the modular ripple due to the discrete coils. The data
from the Monte Carlo code is then fit to a six-parameter analytic expression for the diffusion
coefficient originally developed by Shaing4 and later modified by Painter and Gardner5. A
simple form of the monoenergetic diffusion coefficient can then be obtained that fits the
numerical data over a broad range of electric field, magnetic field, collisionality, particle
energy and particle mass. The expression is given by the following:
D

2
 t C6Vd2

2
 2  C1~ 2  C2 ( E   B ) 2  C3 B2  C4  B ~
 ,
E
 B  C5Vd , Vd  K , ~ 
  , t  r / R
rB
eBr
C6
(1)
The radial electric field is calculated at each spatial region during the iteration of the
1-D code by setting the ion and electron flux equal to each other, i(r,Er)=e(r,Er). Figure 2
shows the solution of the ambipolarity condition for the mirror configuration. At a density of
about 1  1012 cm-3 the calculated electric field is on the order of 90 V/cm, which is in rough
agreement with experimental measurements. This method of calculating the electric field is
valid for the mirror configuration, but for the QHS mode of operation the parallel flow
velocity has to be obtained from the momentum balance equations on a flux surface. Further
modifications to the code will be done to accommodate this feature in the future.
In general, electron transport is not
neoclassical. Therefore, the total electron thermal
conductivity is given by
0
2
/m s)
10
Flux (x 10
19
   e, nc   a
-1
10
-2
10
-100
-50
0
50
Electric Field (V/cm)
100
Fig. 2: Solving the ambipolarity constraint
for the mirror configuration. The electron
flux (red) is equal to the ion flux (black) at
~ 90 V/cm. This is for Pabs ~ 15 kW.
(2)
where a is an anomalous thermal conductivity
given by ASDEX L-mode scaling,
Te3 / 2
1
a ~

RB [1.1  (r / a) 2 ]4
(3)
Even with the anomalous electron thermal conductivity and the self-consistent radial
electric field, at full magnetic field strength of 1.0 T and an ECH power of 200 kW, the
central electron temperature is predicted to be several hundred eV higher for the QHS mode
than for the mirror configuration. However, at the reduced power and field strength of the
present experiments, the effect of the radial electric field for the mirror configuration is to
lower the neoclassical transport for the mirror configuration so that the anomalous
contribution dominates for both the QHS and mirror configurations. Figure 3 shows a plot of
the calculated central electron temperature as a function of density. With the radial electric
field included in the neoclassical transport equations, the mirror and the QHS modes have
roughly the same temperature. Only if we arbitrarily set the electric field to zero is a 100 eV
temperature difference expected for the two configurations. Measurements of the electron
temperature profile will occur later in the year when the Thomson scattering and Electron
Cyclotron Emission diagnostics come on-line.
A comparison of the confinement time measured
experimentally with the ASTRA simulation is
shown in Figure 4 for the QHS and mirror
1
configurations. At the higher densities, the
0.8
experimental confinement times for the two
0.6
configurations are roughly the same. At the lower
densities, the confinement time for the QHS mode
0.4
is actually increasing while it is flat for the mirror
0.2
mode. The ASTRA calculation, for constant
0
absorbed power, shows a confinement time that
0
1
2
3
increases with density, as expected for an
Density ( x 1012 cm-3)
anomalous thermal conductivity that scales as
Fig. 3: Calculated Teo for the QHS mode
3/2
0.6 -0.6
T
, which is
e . This gives a dependence ~n P
and for the mirror configuration with
close to most of the common scaling laws used for
finite Er (red).Also shown is the case for
stellarators. At these low power levels, the ASTRA
the mirror with Er = 0 (blue).
results show that there should be little difference in
the stored energy or confinement times between the QHS and mirror configurations, with or
without the electric field included in the neoclassical transport. This is borne out by the
experimental results, except at low density.
e
T (keV)
1.2
3
Confinement Time (msec)
3
Confinement Time (msec)
2.5
2
1.5
1
0.5
0
0
0.5
1
1.5
12
-3
Density (x 10 cm )
2
2.5
2
1.5
1
0.5
0
0
1
2
Density ( x 1012 cm-3)
3
Fig. 4: Experimental confinement times (left) as a function of density, comparing QHS mode (red)
to mirror mode (blue). This is compared to the ASTRA calculation (right) showing the QHS and
mirror mode with finite Er (red), as well as the mirror mode with Er set to zero.
To determine whether changes in the turbulent driven transport might account for the
differences between the experimental results and the numerical calculations, measurements
were made at the plasma edge of the density and potential fluctuations for the low-density
QHS and mirror configurations. The particle fluxes due to fluctuations are shown in Figure 5.
For the mirror mode of operation, the flux is directed inward. From this data, there appears to
be no correlation at low density between the improved confinement times for the QHS mode
compared to the mirror.
2000
-100
-2000
-80
mm from wall
-60
-40
-20
0
1000
0
).u.a( 
 (a.u.)
-1000
0
-1000
1000
-2000
-100
-80
-60
-40
mm from wall
-20
0
2000
Fig. 5: Turbulent driven fluxes for the QHS mode (left) and mirror mode (right) as a
function of distance from the vacuum vessel wall.
We have, however, observed that for the QHS mode at low-density, the hard X-ray
flux is much more intense than for the mirror configuration at the same density. With 2 nd
harmonic ECH, the wave is energy is preferentially absorbed by particles with large
perpendicular velocities. Guiding center calculations indicate that such particles are very well
confined in the QHS configuration. It appears then most likely that such particles are
responsible for the larger stored energy and confinement time for the QHS mode at low
densities.
* Research supported by USDOE under Grant No. DE-FG02-93ER54222.
[1] N. Karulin, “Transport Modeling of Stellarators with ASTRA”, IPP 2/328, Dec., 1994.
[2] K.M. Likin and B.D. Ochirov, Sov. J. Plasma Phys. 18 (1992) 42.
[3] A. Almagri et al., paper OII.8, this conference.
[4] K.C. Shaing, Phys. Fluids 27 (1984) 1567.
[5] S.L. Painter and H.J. Gardner, Nucl. Fusion 33 (1993) 1107.