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Boundary Conditions In the NIMROD Code

S.E. Kruger

Tech-X Corporation

D.D. Schnack

U.W. - Madison

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Experimental Evidence Demonstrates Important Effects of Boundary Conditions on Long-Wavelength Instabilities

• Resonant Magnetic Perturbation (RMP) Experiments– Apply perturbation to plasma using coils. Has demonstrated

suppression of ELMs

• Locked modes– Rotating islands slow down due to interaction with field errors,

grow, and cause problems

• Resistive Wall Modes– Ideal modes that are stable to a perfectly conducting wall, are

allowed to grow due to the finite resistivity of the walls

– Near ideal boundary, intrinsic fields become larger (Field Error Amplfiication) and interact with tearing modes

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Physics of Stability Depend Upon Time Scale of Perturbation

• Let:RMP= time scale of external perturbation

rec= reconection time scale (depends on diffusivities in resistive MHD model)

• Different orderings:ARMPrectransport Forced ReconectionArecRMPtransport Field-Error Penetration ArectransportRMP Static Field Error

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Different Experimental Issues in Different Regimes

• Resistive wall modes– Experimentally static

error fields known to be important

• RMP Experiments– Field-error penetration

problem • NTM Seeding

– Forced reconnection problems

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Taylor Problem Canonical Problem For Studying Effects of Boundary Perturbations

• Start with stable slab plasma

• Perturb boundary by adding sinusoidal displacement

• Island forms

• Well studied in literature. A sampling:

Hahm and Kulsrud, Phys. Fluids 28, 2412 1985.• Analytic studies of problem

Wang and Bhattacharjee, Phys. Fluids B 4, 1795 1992.• Analytic studies of problem with flow

Ma, Wang, and Bhattacharjee, Phys. Plasmas 3, 2427 1996. • Numerical studies of problem with flow

Fitzpatrick, Phys. Plasmas 11, 937 2004.• Analytic and numerical studies with Hall MHD

Fitzpatrick, Phys. Plasmas 10, 1782 2003.• Extended problem to study field-error penetration problem

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Need to specify boundary conditions consistent with primitive variable, finite-element formulation with NIMROD

• Faraday’s law:

• Finite-element formulation:

• Need to specify electric field on boundary.What is electric field consistent with displacement?

• At boundary and using Fitzpatrick’s notation for displacement:

€

∂B

∂t= −∇× E ,

€

dBq

dtαp(x)αq(x)d3x

V∫q

∑ = − αp(x)∇× Ed3xV

∫ ,

= ∇αp(x)× Ed3xV

∫ − αp(x) ˆ n × EdSS

∫ ,

€

E = −V × B +ηJ

€

E z = −VxBy = −˙ Ξ By

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Tangential electric field is insufficient numerically for boundary condition

• Recalling finite-element formulation:

• Normal component has no boundary condition; For Taylor problem, Bx does not have boundary condition directly

• Numerical problem that NIMROD really solves is:

• Div-B correction term changes order of equation.

• Need to specify Dirichlet boundary condition for magnetic field

€

dBq

dtαp(x)αq(x)d3x

V∫q

∑ = − αp(x)∇× Ed3xV

∫ ,

= ∇αp(x)× Ed3xV

∫ − αp(x) ˆ n × EdSS

∫ ,

€

∂B

∂t= −∇× E +κ∇∇⋅B .

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Summary of Boundary Conditions Needed for Consistency Within NIMROD

• Velocity Field Eq:

• Magnetic Field Eq.:

• Density Equation:

• Temperature Equation:

€

Vx = ˙ Ξ

€

E z = −VxBy = −˙ Ξ By

€

Bx =∂Ξ

∂yBy

€

n = −Ξ∂n

∂x

€

T = −Ξ∂T

∂x

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Code Implemented Using New Data Structures

vert: DO iv=0,rb(ibe)%mxe

tang=rb(ibe)%tang%fs(:,iv)

btangeq=SUM(rb(ibe)%beq_edge%fs(1:2,iv)*tang)

btang=SUM(rb(ibe)%b_edge%fs(1:2,iv,imode)*tang)

vnorm=dbydt*rb(ibe)%displacement%fs(1,iv,imode)

! E_tor=-V_n X B_t

rb(ibe)%e_surface%fs(3,iv,imode)=-vnorm*(btangeq)

!rb(ibe)%e_surface%fs(3,iv,imode)=-vnorm*(btangeq+btang)

! Interior nodes

IF(seam(ibe)%vtype(iv)=='v') THEN

ENDIF

ENDDO vert

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… but doesn’t seem to work

• Div(B) errors persist• Bug? Formulation problem?

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Conclusions

• Since APS various problems with new edge finite element reformulation have been found

• Loop voltage benchmark with nimuw worked well– No obvious bugs

• Resistive wall still needs the pre-processor and matrix-vector multiply work, but waiting on this benchmark to pass

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Development

• Regression tests have been completely redone– Nate’s stuff is out. David Wade-Stein’s stuff is in

• Much more convenient for developer’s and responsive

• Want to have in place and working for upcoming merges

• Merging is worrying me. Changes from:– Carl

– Eric

– Scott

– Val

• Unlikely to be as bad as last year, but …

• Targeting July visit so that I can do it in person