What fluxgrid does beyond the separatrix. S.E. Kruger J.R. King

Transcription

1 What fluxgrid does beyond the separatrix S.E. Kruger J.R. King

2 Outline Brief review of fluxgrid Beyond the separatrix (fg)nimeq with SOL currents

3 Two main methods for solving Grad- Shafranov equations exist Two methods for solving equilibrium: Direct, inverse Direct: Y(R,Z) Obvious approach: Callen and Dory in early 70s, EFIT: Lao 1985 Inverse: R(Y,Q), Z(Y,Q) Idea: Used mapped variables (DeLucia and Jardin, 1985) Mapping only works within separatrix (mapping becomes non-invertable for fixed points) Conventional wisdom for linear MHD codes: Way more accurate Main workhorse for tokamak computational studies throughout the 80s and 90s Linear MHD codes written in flux coordinates Prominent examples: TOQ (Miller, GA), JSolver (Jardin, PPPL) Simulations that have: Wall on separatrix: fixed-boundary Wall past separatrix: free-boundary Fixed-boundary simulations were common in 80s/90s. Useless generally. Fixed-boundary stabilizing. De-stabilize by raising pressure. Make high-n modes unstable. Made equilibrium useless. Inverse codes have limited usage.

4 Fluxgrid is a Grad-Shafranov mapping code Mapping code: Direct representation Y(R,Z)[,Q(R,Z)] to Inverse: R(Y,Q), Z(Y,Q) Basic method follows Grimm, Green, Johnson 1976 Overly complicated (3 coordinate systems) because of lack of good numerical interpolators/integrators Fluxgrid: Primary developers: Glasser, Gianakon, Kruger, King Only mapper to go beyond separatrix Only GATO mapper is as accurate based on comparisons Now irrelevant for NIMROD users because you should use nimeq anyway

5 Key to accuracy: Create grid, then interpolate From a linear MHD point of view: Derive everything from R(Y,Q), Z(Y,Q) Fields, metric elements, weird linear MHD quantities, etc. Tangles up mesh and quantities More direct view R(Y,Q), Z(Y,Q) is convenient for creating grid, analyzing equilibrium After grid created, put needed quantities associated with grid using interpolation based on original equilibrium For direct equilibria for NIMROD: read_direct.f, direct.f, grid.f Goal: Fill dir derived type (see global.f for definitions)

6 Key algorithm ideas for grid generation Create radial grid inside psihigh(subroutine psigrid) Radial coordinate to use: R outboard Worry about: Grid packing, high-order elements Create 2D grid inside separatrix For each psi surface: Integrate B pol to get surface Hard part is to get theta points while using lsode (interior nodes: PITA) Intermediate coordinate system also makes it hard to follow Create grid in vacuum region For each radial ray, create radial vacuum grid that uses geometric progression Geometric progression: dr(i+1)/dr(i)=c dr(psihigh)=dr(first vac point) [Continuity of grid for robustness]

7 Once grid is known, interpolate fields Inside psihigh: Near end of subroutine process_structured DO ipsi=0,mpsi DO itheta=0,mtheta r=twod%fs(1,itheta,ipsi) z=twod%fs(2,itheta,ipsi) CALL get_bfield(r,z,bf,1_i4)... ENDDO ENDDO Outside psihigh: subroutine calc_vacuum needs to know if in separatrix or not DO ivac=vacst,mpsi+mvac rnew= dir%fs(1,0,ivac) radc2=(rnew-ro)/(twod%fs(1,0,mpsi)-ro)*sqrt(sq%xs(mpsi)) DO itheta=0,mtheta rnew= dir%fs(1,itheta,ivac);znew= dir%fs(2,itheta,ivac) CALL get_bfield(rnew,znew,bf,1_i4) CALL sep_distance(rnew,znew,itheta,radc,dsep) IF (radc < 0..AND..NOT. sepmpsi) THEN! Inside separatrix btor=bf%f/rnew jr = -bf%f1*bf%br/mu0/psio jz = -bf%f1*bf%bz/mu0/psio jt = -(bf%f*bf%f1+bf%p1*rnew**2)/psio/(mu0*rnew**2) ELSE! Outside separatrix bf%p=psep; bf%pe=pesep; bf%nd=nsep IF (vacflows.and.psin<psi_flow0) THEN <Jake s new stuff>... ENDIF ENDIF ENDDO ENDDO

8 Key algorithm is sep_distance Distance of the R,Z point to the separatrix along the theta array Acts a pseudo-radial coordinate (useful for other stuff) Negative inside separatrix To calculate in-or-out, use cross-product of vector of point to separatrix, and the clockwise tangential vector of the separatrix. Does not work for arbitrary point (see gen_sep_distance) but obviously fast Algorithm Find intersection of R outboard and separatrix (subroutine position) Find contour of y sep -e, save points, order CW

9 Jake s latest development adds new region: Scrape-Off-Layer currents Detailed write up in King, et.al. submitted to Physics of Plasmas Recommended for ALL NIMROD simulations of H-mode plasmas With nimeq re-solve, of course More physical, no discontinuities, more accurate If you do not have data, use parameterized functions

10 Comparing methods Mapping: Convergence issues for edge modes SOL currents: Do not affect linear stability Do make nonlinear runs better (continuity is always good)

11 Quantitative accuracy: Dominant source of inaccuracy is hold y boundary fixed from EFIT Density, temperature improve (by definition) Edge coil measurements get worse Movement of separatrix: Up to 3 cm for high current discharges (like QH mode) EFIT: y=y coils +y plasma Should hold y coils fixed and solve as free-boundary