Zoe Cournia 25 October 2004 Cholesterol vs. Ergosterol / Lanosterol in Membrane S(t)imulations.
Island s uppression s imulations for SWIM Physics results and coupling issues
description
Transcript of Island s uppression s imulations for SWIM Physics results and coupling issues
![Page 1: Island s uppression s imulations for SWIM Physics results and coupling issues](https://reader034.fdocuments.net/reader034/viewer/2022051116/56814f9d550346895dbd5abe/html5/thumbnails/1.jpg)
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
![Page 2: Island s uppression s imulations for SWIM Physics results and coupling issues](https://reader034.fdocuments.net/reader034/viewer/2022051116/56814f9d550346895dbd5abe/html5/thumbnails/2.jpg)
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
![Page 3: Island s uppression s imulations for SWIM Physics results and coupling issues](https://reader034.fdocuments.net/reader034/viewer/2022051116/56814f9d550346895dbd5abe/html5/thumbnails/3.jpg)
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.
![Page 4: Island s uppression s imulations for SWIM Physics results and coupling issues](https://reader034.fdocuments.net/reader034/viewer/2022051116/56814f9d550346895dbd5abe/html5/thumbnails/4.jpg)
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
![Page 5: Island s uppression s imulations for SWIM Physics results and coupling issues](https://reader034.fdocuments.net/reader034/viewer/2022051116/56814f9d550346895dbd5abe/html5/thumbnails/5.jpg)
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
![Page 6: Island s uppression s imulations for SWIM Physics results and coupling issues](https://reader034.fdocuments.net/reader034/viewer/2022051116/56814f9d550346895dbd5abe/html5/thumbnails/6.jpg)
Island width data obtained from field line traces – expensive.
Marked reduction in the size of saturated islands can be achieved – proofof principle.
![Page 7: Island s uppression s imulations for SWIM Physics results and coupling issues](https://reader034.fdocuments.net/reader034/viewer/2022051116/56814f9d550346895dbd5abe/html5/thumbnails/7.jpg)
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.)
![Page 8: Island s uppression s imulations for SWIM Physics results and coupling issues](https://reader034.fdocuments.net/reader034/viewer/2022051116/56814f9d550346895dbd5abe/html5/thumbnails/8.jpg)
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
![Page 9: Island s uppression s imulations for SWIM Physics results and coupling issues](https://reader034.fdocuments.net/reader034/viewer/2022051116/56814f9d550346895dbd5abe/html5/thumbnails/9.jpg)
Pack grid around (2,1) and (3,1) rational surfaces
Begin with equilibrium where the dominant instability is the (2,1) resistive tearing mode
![Page 10: Island s uppression s imulations for SWIM Physics results and coupling issues](https://reader034.fdocuments.net/reader034/viewer/2022051116/56814f9d550346895dbd5abe/html5/thumbnails/10.jpg)
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)].
![Page 11: Island s uppression s imulations for SWIM Physics results and coupling issues](https://reader034.fdocuments.net/reader034/viewer/2022051116/56814f9d550346895dbd5abe/html5/thumbnails/11.jpg)
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.
![Page 12: Island s uppression s imulations for SWIM Physics results and coupling issues](https://reader034.fdocuments.net/reader034/viewer/2022051116/56814f9d550346895dbd5abe/html5/thumbnails/12.jpg)
Full μ profile and closeup.Relaxes to a quasilinearlyflattened equilibrium atlong times.
![Page 13: Island s uppression s imulations for SWIM Physics results and coupling issues](https://reader034.fdocuments.net/reader034/viewer/2022051116/56814f9d550346895dbd5abe/html5/thumbnails/13.jpg)
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).
![Page 14: Island s uppression s imulations for SWIM Physics results and coupling issues](https://reader034.fdocuments.net/reader034/viewer/2022051116/56814f9d550346895dbd5abe/html5/thumbnails/14.jpg)
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.
![Page 15: Island s uppression s imulations for SWIM Physics results and coupling issues](https://reader034.fdocuments.net/reader034/viewer/2022051116/56814f9d550346895dbd5abe/html5/thumbnails/15.jpg)
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?
![Page 16: Island s uppression s imulations for SWIM Physics results and coupling issues](https://reader034.fdocuments.net/reader034/viewer/2022051116/56814f9d550346895dbd5abe/html5/thumbnails/16.jpg)
Images from C. Sovinec, SWIM Slow MHD campaign meetup, December 2007.
![Page 17: Island s uppression s imulations for SWIM Physics results and coupling issues](https://reader034.fdocuments.net/reader034/viewer/2022051116/56814f9d550346895dbd5abe/html5/thumbnails/17.jpg)
![Page 18: Island s uppression s imulations for SWIM Physics results and coupling issues](https://reader034.fdocuments.net/reader034/viewer/2022051116/56814f9d550346895dbd5abe/html5/thumbnails/18.jpg)
Most terms die off on resistivetimescale, with shorter-wavelengthspatial scalesdamping morequickly
![Page 19: Island s uppression s imulations for SWIM Physics results and coupling issues](https://reader034.fdocuments.net/reader034/viewer/2022051116/56814f9d550346895dbd5abe/html5/thumbnails/19.jpg)
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
![Page 20: Island s uppression s imulations for SWIM Physics results and coupling issues](https://reader034.fdocuments.net/reader034/viewer/2022051116/56814f9d550346895dbd5abe/html5/thumbnails/20.jpg)
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
![Page 21: Island s uppression s imulations for SWIM Physics results and coupling issues](https://reader034.fdocuments.net/reader034/viewer/2022051116/56814f9d550346895dbd5abe/html5/thumbnails/21.jpg)
In particular,what happensnear the (2,1)rational surface(where we willmost likely wantthe RF to be)for a tightlytoroidallylocalized RF perturbation?
![Page 22: Island s uppression s imulations for SWIM Physics results and coupling issues](https://reader034.fdocuments.net/reader034/viewer/2022051116/56814f9d550346895dbd5abe/html5/thumbnails/22.jpg)
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