Nonlinear Resistive MHD Simulations of PPCD in the
Reversed Field Pinch
Jim Reynolds and Carl Sovinec
University of WisconsinMadison
Center for Plasma Theory and Computation
American Physical Society, Division of Plasma
Physics
44^{th} Annual Meeting
November 11 November 15, 2002
Orlando, Florida
Objectives

To study the magnetohydrodynamic activity
present during Pulsed Poloidal Current Drive (PPCD) in the
Reversed Field Pinch .

To present key results observed in PPCD
simulations with particular emphasis on fluctuation
reduction, improved energy confinement and correlation with
experimental observations.

To
explore the influence of PPCD on reduction of the dynamo
fluctuations associated with the dominant core
resonant tearing mode.
Outline

Introduction & Background

Finite Pressure Simulations of PPCD in
Cylindrical Geometry
A. Observed Reduction in Magnetic
Fluctuations
1. Magnetic energy spectra
2. Contributions to Ohm's Law &
Powerflow
3. Reduction of magnetic
stochasticity
B. Improved energy confinement time
C. Parallel
current profile evolution
III. Modal Decompositional Study of PPCD
A. Methodology
B. Fluctuation reduction due to applied
C. Parallel current profile evolution
D.
Contributions to Ohms Law and Power Flow
IV. Conclusions

is the primary drive
mechanism for several m=1, n~2R/a instabilities that generate
magnetic fluctuations communicated through the entire global
system through nonlinear coupling aided by small radial
spacing between core resonance surfaces and the large
amplitude of driven tearing
perturbations.

The fluctuations lead to a stochastic
magnetic field topology that fuels energy transport.
Particles can follow stochastic field lines from the core
region out to the wall.

[G. Fiksel ,et al, PRL 72, 1994 1028]

PPCD has been experimentally observed to
dramatically decrease the amplitudes of core resonant tearing
modes that are responsible for more than 90% of the
magnetic fluctuations in the RFP. [J. Sarff,et al, PRL 72,
1993 3670]

Recent PPCD experiments have produced an
order of magnitude increase in energy confinement time.

[B.E.Chapman,et al, POP 9,2002,2061]

Confinementimprovements is maximized in experiments where direction is reversed after an application
of a series of edge
pulses at the edge in order
to maintain [B.E. Chapman,et al, POP 9, 2002 2061]

The underlying mechanisms governing core mode
suppression during PPCD are not currently well
understood.
We are using The NIMROD 3D Nonlinear MHD Code to solve the system of equations:
[http://nimrodteam.org]
_{}
_{}
_{}
_{}
_{}
_{}
anistropic heat
flux
ohmic heating
Finite Pressure Simulations of PPCD in
Cylindrical Geometry
Simulations are started from standard RFPs. At
this point, we are applying just poloidal electric field with a
magnitude that is constant in time while maintaining constant
loop voltage. We use perfectly conducting boundary conditions at
the wall.
Parameters in the cases reported here are:
S~[2000, 6000, 10000]
Pm=1 _{}
_{} Axial modes used:
_{} R/a=3
16x8 mesh (S~2000) 16x16 mesh (S~6000)
Fluctuations Show Rapid
Response to Applied
Application of as 18 % of the applied shows a marked
reduction in fluctuations. The magnetic fluctuation response is
almost instantaneous, even for core resonant modes
at S~2000.
(The magnetic energy spectra plots show the
natural logarithm of magnetic fluctuation energy decomposed by
toroidal mode number (n).)
Here for an aspect ratio of R/a~3 the m=1, n =
[6,7] modes
are the dominant core resonant modes.
A simulation at S~6000 with the same parameters
as the case shown above shows n=7 decline, while the drop in n=6
is less dramatic (not shown). The modes recover
faster.
Adjusting the value of
to 36% of at S~6000 shows amplitude reduction resembling the S~2000
case. Note that in this case n=7 displays dominant
behavior. (shown above)
Background
In a typical PPCD experiment,
an applied electric field largely parallel to the edge magnetic
field is applied at the plasma edge in attempt to
"flatten" out the profile.
In the
Madison Symmetric Torus (MST) toroidal magnetic
flux is removed by application of a voltage pulse train to the
toroidal gap in the conducting shell during PPCD. (Data
Courtesy of J. Sarff)
Current Simulations Apply a Poloidal Electric
Around the Boundary that is not time variant. Here at
t=0.1 the poloidal electric field used in current simulations is
applied for demonstration.
Future simulations will use a time dependant
waveform.
Reduction in n=6 Dynamo Power Density With
PPCD
Plots of power density,
_{}
before and during PPCD show the insensitivity
to edge current n = 6 (at S~2000).
Strong Evidence for n=6 Fluctuation
Reduction

Changes in the difference between the
volumeintegrated nonlinear power and
direct Ohmic loss for each toroidal component indicate
overall growth or decay for a mode. The n=6 response
is strongest, despite being resonant in the core.
(S~2000)
Application of Poloidal
Electric Field Shows Dramatic Reduction
in <> Fluctuations
Globally
Shown below are contours
of before application of poloidal
electric field and after a few tearing times at
S~2000.
Increase In Instantaneous Confinement Times
Thermal transport responds promptly to the
reduced magnetic fluctuation level. More dramatic responses are
anticipated with temperaturedependent resistivity.
(Shown for S~2000 Case)
Confinement time for PPCD with applied
E_{pol} magnitudes varying from 1/8 to 1/3 of the applied
toroidal electric field (labels 14). A comparable simulation
without PPCD is also shown (0).
The S~6000 displays a simular growth for cases
where a poloidal electric field is applied. An increase is seen
in instantaneous energy confinement time commeasurate with
applied field strength.
Simulated Parallel Current Profile Behavior is
Consistent With Experimental Measurements of Edge Current

Decreasing parallel current in the edge is
counter to expectation (and the initial intent of PPCD) of
flattening the entire profile with edgelocalized current
drive.
The result is consistent with edge probe
measurements [B. E. Chapman, et al., Phys. Plasmas 7, 3491
(2000)] of parallel current in the outer 1/10 of the MST plasma
during PPCD.
Illustration
S~6000
Pinch Flow Evolution

Prompt changes in both the parallel current
at r=0.4a
and core dynamo mode amplitudes indicate
electric field
propagation faster than resistive current
diffusion.
For this system, such changes must be ideal
MHD in character. The increased inward Poynting flux
resulting from PPCD would tend to further pinch the plasma
column.
The global change in the pinch flow profile
supports this idea.
The global changes are
central for understanding the behavior of the magnetic
fluctuations. For example, the initially large m=1,
n=6 mode resonant in the core is
relatively insensitive to edge parallel current at
S~2000.

Applying a poloidal electric field that is
18% of the applied toroidal electric field leads to
significant profile changes in less than onetenth of a
resistive diffusion time.
Reversal parameter evolution from the start of
the induced transient at S~2000.
The pinch parameter evolution is mostly due to
the flux change; toroidal current remains approximately
constant.

Magnetic Fluctuation Reduction Shows
Reduction in Field Stochasticity
At the largest PPCD level, core fluctuations
are reduced enough to produce an island in the magnetic
topology.
This correlates with recently reported PPCD
experimental HXR emission evidence showing runaway electron
energies up to 100 Kev. For electrons to accellerate to this
energy would require the particles to transit the torus in MST
10000 times. This points to evidence of partial flux restoration
present in the core region during PPCD. [B.E. Chapman, et.al.,
POP 9, 2002 2064]
The temperature response is
significant at S~2000 and S~6000. The results reflect the
reduction in due to applied poloidal electric field in a system with
anisotropic heat conduction: .
Illustration
s~6000
Modal Decompositional Study of PPCD
Motivation
Simulations both with zero Ò & finite pressure effects have
demonstrated rapid response of core resonant modes to
A simplified system with reduced aspect ratio and
n=[1,0] axial modes examines effect of applied poloidal electric
field in a system
where the dominant resonant core mode is
isolated. System with adds nonlinear coupling.
Methodology
We start with a nonlinear saturated RFP case
with 43 axial modes at S~2000. R/a~3 Ò=0
The RFP has been formed from perturbations to a
paramagnetic pinch equilibrium.
The RFP case exhibits n=6 dominance in the
fluctuation spectra.
Adjust the aspect ratio to R/a ~ 0.5 by
modifying the length of the cylinder to a length corresponding to
one full spatial period of the n=6 mode. This makes the n=6 mode
the n=1 mode in the new geometry, n =12 becomes n=2, i.e.
This isolates the dominant n=1 mode in the new
geometry. All other modes except the mean field can now be
removed if we wish.
We reset the simulations by reading in the
fluctuations from the 43 axial mode RFP case. We keep selected
fourier components
Three Simulations have been run for each
truncated spectrum.
For each choice of modes to include a set of
three simulations is ran:

A benchmark case where there is no
application of poloidal electric field and
allow
the alterred system to reach a state
of
saturation

A case where is applied
immediately upon restart. (transient)

A case where the alterred system has
been
allowed to evolve and reach a state
of
saturation before application of
(saturated)
Our choices of modes to include are:

# of included modes

New Geometry n#

43 mode RFP case n#

one

zero

zero

two

0,1

6,12

three

0,1,2

6,12,18

six

0,1,2,3,4,5

6,12,18,24,27

eleven

0,1,2,3,4,5,6,7,8,9,10,11

6,12,18,24,27,30,33,...

_{Application of Epoloidal Along
the Edge Results in Dramatic Growth Suppression of the Dominant
Core Resonant Tearing Mode}
The restart case from the 43
mode RFP simulation keeping only n=0,1(formerly n=0,6)
demonstrates pronounced growth of n=1 in the absence of nonlinear
couplings (no applied )
The same case with applied immediately
after restart shows dramatic suppression of the transient growth.
Here
is not timevarying.
In the case where the system has been allowed
to reach a state of saturation before subsequent application of
, the
n=1 mode is also suppressed. The suppression is not as dramatic
as in the transient case. The rapid decline in n=1 energy shown
here after nearly 12 ms after application is due to a loss of
resonance.
The introduction of additional modes
demonstrates the same transient behavior in the presence of
for
the dominant mode. The presence of additional higher n modes has
little effect upon
the mode suppression. (here the dominant mode is
depicted in green).
Parallel Current Profile Evolution For n=0
Only
In the case where all modes except n=0 are
removed upon restart the parallel current profile relaxes into a
paramagnetic pinch profile after about a system tearing time in
the absence of an applied poloidal electric field. ( follow
traces from green to red in time and shown here is the evolution of the
profile over 57ms) . Here the <VxB> fluctuations have been
effectively turned off.
Immediate application of shows a marked
increase in parallel current in the region r<0.8 over the same
time scale.
Parallel Current Profile Evolution For Cases
with Additional Fourier Modes
The addition of modes has a minor effect on the
parallel current profile evolution in comparision to the case
where poloidal E has been applied to a system with just n=0
above. (shown here for a simulation with 11 axial modes)
The behavior depicted in reduced mode cases
above can be observed to make a substantial contribution the
evolution in the original 43 mode RFP case. There is more growth
in the core region observed here.
Discussion of Modal Decomposition Study of
PPCD

Removing fluctuations (n=0 only case ) in
the simplified system shows striking simularity in parallel
current profile evolution to the case where poloidal electric
field is applied to suppress the dominant fluctuation mode.
The removal of fluctuations causes edge current to migrate
inward.

Application of poloidal electric field
suppresses growth of the dominant tearing mode in the system
even in cases of rapid transient growth where nonlinear
coupling has been have been removed from the system (see n =
0,1 case). In the simulations shown there is a
pronounced direct effect on the n=1 mode.

Simulations show at Low Lundquist Number the
n=0 parallel profile evolution is a key component in
evolution of the parallel current in systems with many axial
modes.
More diagnostics are being constructed:

A diagnostic that measures the radial
velocity
of the mean poloidal and toroidal flux
contours to
determine the role of pinching in the
parallel current profile evolution during application of

Diagnosis and analysis of fourier
component contribution evolution in the
power density to uncover the underlying
mechanisms responsible for the dramatic modification of the
dominant tearing mode.
Summary
Finite Pressure Cylindrical RFP Simulations of
PPCD

Simulations demonstrate the reduction of
dynamo
fluctuations with the application of a
poloidal electric field around the conductive boundary

The simulations show the reduction of field
stochasticity with the application of applied poloidal
electric field

A marked increase in energy confinement time
is
observed. We expect more dramatic increases
with
the addition of more physics into the
simulations

The simulated parallel current evolution is
consistent
with experimental probe measurements
performed on
MST
Modal Decompositional Study of PPCD

Application of poloidal electric field can
suppress rapid transient growth of the dominant tearing
mode.

Parallel current profile evolution during
application of poloidal electric field has reliance upon the
mean profile evolution after the reduction of
fluctuations.

Edge behavior of the profile evolution may
be attributed to reduced fluctuations.

Future Work

We want to find the MHD processes that
govern the experimentally observed phenomena during PPCD
experiments

Study of PPCD in systems with increased
physics such as toroidal geometry, temperature dependant
resistivety, and time dependant poloidal electric field
waveforms.

Study of the MHD mechanism behind the m=0
bursting behavior reported during PPCD experiments.

Exploring elliptical cross section geometry
effects nonlinear fluctuation amplitudes and coupling in the
absence of PPCD