The SWIM “Slow” MHD Campaign: Overview, MHD Plans and Accomplishments

D. D. Schnack, T. Jenkins, J. Callen. C. Hegna. C. Sovinec, F. Ebrahimi University of Wisconsin,

Madison, WI, E. Held, J.-Y. Ji, Utah State University, Logan, UT, S. Kruger, J. Carlson, Tech-X Corp.,

Boulder, CO

What to Get Done this Week

• Begin dialog between MHD, RF, and CS researchers on “Slow MHD” problem

• Provide update on MHD plans and progress• Get feedback from RF/ CS folks• Agree on technical plan and timeline• Produce report for PM addressing the

following issues:

0. What Problem are We Trying to Solve? O(1/))

• Demonstrate proof of principle of stabilization of resistive islands with some RF model?

• Demonstrate coupling between RF and MHD physics codes?

• Demonstrate proof of principle of stabilization of resistive islands with coupled RF/MHD models?

• Demonstrate stabilization of pre-existing (finite amplitude) NTM island with RF?

• Demonstrate seeding, growth, and eventual RF stabilization of NTM island?

• Understand physics of island stabilization (inner or outer regions)?

• Evaluate efficacy of various experimental RF deposition scenarios?

• Determine how much RF power is required for ITER?

• All of the above? (Do we have the resources?)

• Some of the above? (Must match to resources)

• None of the above? (Then, what?)

What are MINIMUM Requirements to Satisfy

SOW? O(1) • What is the technical plan for meeting

this goal? Is it reasonable?• What resources are required, and are

they available?• What is the timeline for meeting this

goal? Is it reasonable?

What is Required to do the Problem “Better” O())

• Reasonable technical plan?• Resources?• Reasonable timeline?

What is Required to do the Problem “Right” O(22))

• Reasonable technical plan?• Resources?• Reasonable timeline?

RF/MHD Dialog Issues - 1

• Is the MHD plan reasonable?– What information does MHD want from RF?– What information can MHD give to RF?

• What is the RF physics plan?– What information does RF want from MHD?– What information can RF give to MHD?

• What RF codes are involved?• Are substantial modifications to RF codes

required?– Island geometry?

RF/MHD Dialog Issues - 2• Are there algorithmic issues that might affect

numerical stability?– Can they be mitigated?

• How do we envision the RF/MHD code coupling?– Strong coupling? (RF called from MHD)– Weak coupling? (Separate code execution)

• What is the most efficient means of passing data?

• Are there any CS obstacles? – Can they be mitigated?

Technical Approach

• Recognize that problem is additive– Modified Rutherford equation:

• Can approach problems in parallel• Don’t need to get NTM “right” before attacking RF

– Can start with RF/tearing coupling

dwdt

=ηnc

μo

k1[Δ'+ Δnc(w)+ ...+ ΔRF(w)Pηysics is additiv

1 24 4 4 4 34 4 4 4 ]

Implementation Plan (MHD)• I: Add RF source module and interface to NIMROD• II: Define “simple” RF closure (force on electrons)• III: ad hoc analytic source term (force on electrons) in

Ohm’s law – Tearing mode stabilization => proof of principle

• IV:Evolutionary equation for RF source• V:Revisit NTM calculations

– Kruger/Hegna heuristic closure?– Collisional Braginskii nonlinear electron viscous stress tensor?

• V: Couple to RF codes• VI: Throughout, refine theoretical closure models and

implementations

Physics Issues (Kinetic Equation)

• How to incorporate RF effects into fluid models?

• Start with kinetic equation

dfdt

=C(f)Collisiona{ + Q(f)

RF quasi-linar oprator{    ,     

ddt

=∂∂t

+rv⋅∇+

qs

μ s

(rE +

rv×rB)

∂∂rv

Q( f ) =

∂∂rv

⋅tΔ ⋅

∂ f∂rv

,

Quasi-linear diffusion tensorOutput of RF codes

Physics Issues (Closures)• Take moments• RF sources modify fluid equations

∂ns

∂t+∇ ⋅(ns

rvs ) = 0   ,

msns (∂ rvs

∂t+ rvs ⋅∇

rvs ) = nsqs (rE + rv ×

rB) −∇ps −∇ ⋅ t

π s +rRs +

rFs0

rf    ,

32

ns (∂Ts

∂t+ rvs ⋅∇Ts ) + nsTs∇ ⋅

rvs = −∇ ⋅ rqs −tπ s :∇rvs + Qs + Ss0

rf

Force on electrons

Heating

rFs0

rf = d 3rvμ srvQ(fs)∫ ,

Ss0rf = d 3rv

12μ sv'

2Q(fs)∫

RF closures

€

Q( f ) = ∂∂

r v ⋅

r J v / rf ⇒ d3r

v Q( f )∫ = 0.

RF produces no particles

Existing closures are also modified!

Information (RF <==> MHD)• Chapman-Enskog-Like theory can

produce closures, i.e.,

rFs0

rf = d 3rvμ srv∫ Q(fs)≅ d 3rvμ s

rv∫ Q(fMs),

Ss0rf = d 3rv

12μ sv'

2∫ Q(fs)≅ d 3rv12μ sv'

2∫ Q(fMs)

Maxwellian

• Moments can be computed by numerical integration in velocity space once DQL is specified

• DQL(v) produced at each mesh point by RF codes

RF/MHD Coupling

1. NIMROD produces n(r,t), T(r,t), B(r,t) => RF codes

2. RF codes produce DQL(r,v) on rays => interpolated to NIMROD grid

3. NIMROD integrates in v => moments of Q => force on electrons and modified moments

4. NIMROD produces n(r,t+Δt), T(r,t+Δt), B(r,t+Δt)

5. t +Δt => t6. Go to 1 RF module. Presently has model Jrf(r,t).

Issues• What are the most appropriate fluid moments?• What is the most efficient way to compute them?• Are there any new numerical stability issues?• Can NTM growth and saturation be demonstrated?• How often does RF need to be updated?• How should data be passed between MHD and RF

codes?• Do RF codes need to be modified (islands, etc)?• Do we need to go beyond ECCD?• ???

MHD Results• Electron force localized in poloidal

plane, toroidally symmetric• Current spreads over flux surfaces• Demonstrated in cylinder and DIII-D

geometry

QuickTime™ and a decompressor

are needed to see this picture.

Tearing Mode Stabilization by RF in DIII-D geometry

(Tom Jenkins)

Linear

Rutherford

MHD Status• Closure procedure defined• Spitzer-like problem formulated• Data required from RF modules defined• General source modules and interfaces

added to NIMROD• Simple axi-symmetric source implemented• Stabilization of 2/1 resistive tearing mode

demonstrated in DIII-D geometry• Conversations with RF physicists beginning

More About the Workshop

• Informal!• Purpose is to have a dialog, not to impress

with latest results!– 2-way communication

• Allow plenty of time for discussion• Discussion leaders: Take notes!• Thursday morning: Produce coordinated

technical plan and timeline• Will all be input for report to PM