Thomas G. Jenkins, University of Wisconsin-Madison

in collaboration with

Dalton Schnack, Carl Sovinec, Chris Hegna, Jim Callen, Fatima Ebrahimi

University of Wisconsin-Madison

Scott Kruger, Johan Carlsson Eric Held, Jeong-Young Ji

Tech-X Corporation Utah State University

Bob HarveyCompX

SWIM Project Meeting 23 September 2008 Princeton, NJ

Goal of Slow MHD campaign – numerically simulate ECCD stabilization of NTM’s

Experimental efforts to stabilize neoclassical tearing modes via electron cyclotron current drive have been very successful [La Haye, Phys. Plasmas 13, 055501 (2006)].

For ECCD, the RF-induced current is relatively small (of the same order as the electric field) – small expansion parameter Add an RF term to the kinetic equation:

Quasilinear diffusion tensor from RF source

Gyrophase-averaged Fokker-Planck Coulomb collision operator

RF terms appear in the fluid equations

Taking fluid moments in the conventional manner yields

Can approximate fα as alocal MaxwellianMaxwellian, here –RF perturbations are small

Closure calculations for qα, πα are also affected by RF.

Ultimately, the closure scheme and the small parameter expansion yield a self-consistent set of fluid equations for ECCD-influenced MHD.

While work proceeds on that front, we can study simpler models (e.g. resistive tearingmodes, rather than NTM’s) to gain physical insight. (Neoclassical effects enter the Rutherford equation additively.)

Consider the electron momentum equation, which gives rise to the MHD Ohm’s law.RF-induced momentum yields an additional term,

A physically reasonable form for the lowest-order effect of the RF is

λrf = RF amplitude

f(x,t) = space/time dependence

For these simulations, model the RF as a Gaussian function in the poloidalplane – neglect toroidal variation for now.

Ramp up on some intermediate timescale (slow compared to Alfvén timescale, fast compared to resistive timescale), neglect closure problem.

What physics results arise from the inclusion of this term in the MHD model?

Parameters:(Rrf,Zrf) = position of RF peakwrf = half-widthλrf = amplitude

Island width data obtained from field line traces – expensive.

Marked reduction in the size of saturated islands can be achieved – proofof principle.

δEmagnetic = volume integral of δB2/2μ0 Island width ~ √δψ

δEmagnetic ~ w4

NIMROD can now make continuous plots of island width (recent diagnosticdevelopment by S. Kruger). (Functionality not used here, though.)

Tearing mode can be influenced by the RF in two major ways:

Modification of tearing parameter Δ’ -will affect linear growth rate and saturated island width -easy to diagnose – proportional to linear growth rate

Modification of helical current profile μ = μ0 (JB)/(BB) -becomes important as mode saturates nonlinearly -not so easy to diagnose. Conceptual picture:

B0

J0J0

δJ, δB

B0 JB

Pack grid around (2,1) and (3,1) rational surfaces

Begin with equilibrium where the dominant instability is the (2,1) resistive tearing mode

Allow only toroidally symmetric (n=0) modes in simulation, ramp RF up to steady state – yields new RF-modified ‘equilibrium’

Then allow n=1 (and higher) modes into simulation, check linear growth

For fixed RF current(centered on (2,1) rationalsurface), the growth rate (and thus Δ’) is reduced as the poloidal cross-section of the RF spot is reduced.

In agreement with resultsof Pletzer/Perkins [PoP 6,1589 (1999)].

For fixed RF current (centered on (2,1) rational surface), the saturatedisland width is also reduced for smaller RF poloidal cross-sections. Here,IRF/I0 < 0.04%, so current profile modification is negligible.

In agreement withgeneral conclusions ofHegna/Callen [PoP 4,2940 (1997)].

Pletzer/Perkins find that for IRF/I0 < 4%, offset from rationalsurface is destabilizing –simulations in progressto verify this effect.

Full μ profile and closeup.Relaxes to a quasilinearlyflattened equilibrium atlong times.

To study current profilemodification, turn on RF after resistive tearing modehas saturated; examine theisland width reduction and μprofile (IRF/I0 ~ 1%).

Without the RF, quasilinearflattening leads to increasedcurrent on outward side of rational surface; decreasedcurrent on inward side.

RF adds current across therational surface – destabilizinginside, but stabilizing outside(yielding net stabilization).

The island cannot shrink appreciably below the width of the poloidal cross-section of the RF drive. In agreement with Hegna/Callen

When RF is offset from rational surface, investigating Δ’ and current profile modifications relative to non-offset case; compare with Pletzer/Perkins conclusions (in progress). Size of RF cross-section relative to saturated islandsize also important.

RF effects entering NIMROD are not assumed to be flux-surface averaged,we rely on the dynamics of the plasma to distribute them over the flux surface.

Previous simulations have suggestedthat equilibration occurs over the fluxsurface after a few Alfvén times. Here,we have a cylindrical plasma with az-independent, poloidally localizedRF drive.

Can we analytically capture anyof the physics of equilibration?

Images from C. Sovinec, SWIM Slow MHD campaign meetup, December 2007.

Most terms die off on resistivetimescale, with shorter-wavelengthspatial scalesdamping morequickly

In this geometry the unperturbed field lines are straight, so that single fieldlines in z are the equivalent of a `flux surface’

The z-independent (n=0) magnetic field perturbations grow up from zeroon a resistive timescale;

γl,m,n = √(n2k2vA2 – Ωl,m,n

2) so that

In the previous model, boundary conditions in z are periodic – waveslaunched from the middle of the cylinder propagate in z toward bothendcaps, cross the boundary, and begin to interfere with one another

Similar effects occur in the periodic θ-coordinate of the previous movie;the RF is uniform in z, so the waves begin nonlinearly interfering after onlypropagating halfway around poloidally.

Bz Bθ

In particular,what happensnear the (2,1)rational surface(where we willmost likely wantthe RF to be)for a tightlytoroidallylocalized RF perturbation?

Full coupling of GENRAY and NIMROD is well under way and bugs arerapidly being worked out (beginning phase 3 of our 5-phase plan, as outlined in the proposal) – Scott Kruger’s talk (next).

First simulation results with ad hoc JRF term being written up for publication, demonstrating proof-of-principle concepts and agreeing well with existing literature