Nonlinear Resistive MHD Simulations of PPCD in the Reversed Field Pinch

Jim Reynolds and Carl Sovinec

University of Wisconsin-Madison

Center for Plasma Theory and Computation

American Physical Society, Division of Plasma Physics

44th Annual Meeting

November 11 -November 15, 2002

Orlando, Florida


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

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

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


  1. Introduction & Background

  2. 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

We are using The NIMROD 3D Nonlinear MHD Code to solve the system of equations: []

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:


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)


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

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 temperature-dependent resistivity.

(Shown for S~2000 Case)

Confinement time for PPCD with applied Epol magnitudes varying from 1/8 to 1/3 of the applied toroidal electric field (labels 1-4). 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

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


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.

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.


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,, 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


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.


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:

  1. A benchmark case where there is no

    application of poloidal electric field and allow

    the alterred system to reach a state of


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

  3. 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-#
















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 time-varying.

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

More diagnostics are being constructed:


Finite Pressure Cylindrical RFP Simulations of PPCD

Modal Decompositional Study of PPCD