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

44^th Annual Meeting

November 11 -November 15, 2002

Orlando, Florida

Objectives

Outline

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

• 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 volume-integrated 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 temperature-dependent resistivity.

Confinement time for PPCD with applied E_pol 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

• Decreasing parallel current in the edge is counter to expectation (and the initial intent of PPCD) of flattening the entire profile with edge-localized 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.

• The evolution of the parallel current profiles during the first 40 ms of the 18% PPCD simulation shows synchronized global evolution. In particular, the shallow current gradient in the core is quickly flattened in cases at S~2000 & S~6000

• The color green corresponds to the initial time when PPCD is applied.

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 one-tenth 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:

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 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

• 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:

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