1. Tech-X Corporation1 Boundary Conditions In the NIMROD Code S.E. Kruger Tech-X Corporation D.D. Schnack U.W. - Madison

2. Tech-X Corporation2 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

3. Tech-X Corporation3 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:  A  RMP  rec  transport Forced Reconection  A  rec  RMP  transport Field-Error Penetration  A  rec  transport  RMP Static Field Error

4. Tech-X Corporation4 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

5. Tech-X Corporation5 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

6. Tech-X Corporation6 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:

7. Tech-X Corporation7 Tangential electric field is insufficient numerically for boundary condition Recalling finite-element formulation: Normal component has no boundary condition; For Taylor problem, B x 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

8. Tech-X Corporation8 Summary of Boundary Conditions Needed for Consistency Within NIMROD Velocity Field Eq: Magnetic Field Eq.: Density Equation: Temperature Equation:

9. Tech-X Corporation9 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

10. Tech-X Corporation10 … but doesn’t seem to work Div(B) errors persist Bug? Formulation problem?

11. Tech-X Corporation11 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

12. Tech-X Corporation12 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