1. Modeling of plasma stability in advanced divertor configurations with ArbiTER D. A. Baver, J. R. Myra Lodestar Research Corporation M. V. Umansky Lawrence Livermore National Laboratory Lodestar

2. Modeling of plasma stability in advanced divertor configurations with ArbiTER D. A. Baver, a) J. R. Myra, a) M. V. Umansky b) a) Lodestar Research Corporation b) Lawrence Livermore National Laboratory ArbiTER is a flexible code for linear fluid or kinetic plasma models that can handle problems of various dimensionality and topology. It permits run-time specification of a particular linearized physics model, geometry, and grid connectivity using suitable equation and topology parsers to determine how a particular equation set will be discretized. The resulting matrix form of the problem is then solved using the SLEPc [1] eigensolver package. Although originally built for solving eigenvalue problems such as finding fluid and kinetic instabilities in edge plasma, with a recent upgrade ArbiTER can now solve problems of the linear response kind, i.e., funding the response of plasma parameters driven by a given source function. This expands the code capabilities to include important edge-relevant problems such as the resonant magnetic perturbations (RMP’s). One particular are of application of the code is calculations pertaining to ballooning stability of plasma in advanced divertor configurations. The flexibility of the code with respect to physics models allows its use for calculation of both ideal and resistive MHD stability. On the other hand, its topological flexibility allows its application to configurations with higher-order nulls, such as the snowflake [2] and cloverleaf [3] divertor configurations. In the present study, analyzing the ballooning stability thresholds and mode structures in snowflake-like configurations, we develop insights into the effects of these topologies on important edge phenomena such as ELMs and edge turbulence. [1] http://www.grycap.upv.es/slepc/ [2] D.D. Ryutov, Phys. Plasmas 14, 064502 (2007). [3] D.D. Ryutov and M.V. Umansky Phys. Plasmas 20, 092509 (2013). Work supported by the U.S. DOE. Lodestar

3. Outline Introduction Motivation Equation parser Topology parser Procedure Results Conclusions Lodestar

4. Introduction The Arbitrary Topology Equation Reader (ArbiTER) is an eigenvalue code for solving linear PDE’s in diverse geometries. PDE’s are discretized using finite difference methods. – Recent upgrade also permits finite element methods. Model equations are defined using a specialized equation parser. – Inherited from the edge fluid eigenvalue code 2DX*. ArbiTER expands these capabilities by adding a topology parser. – Permits arbitrary connectivity. – Permits variable number of dimensions. Resulting discretized equations are then solved using the SLEPc eigensolver package. *D. A. Baver, J. R. Myra and M.V. Umansky, Comp. Phys. Comm. 182, 1610, (2011) Lodestar

5. Motivation Interest has been increasing in alternative divertor configurations. – Snowflake – Cloverleaf This creates a need for codes capable of handling these advanced magnetic topologies. ArbiTER has capabilities that permit rapid reconfiguration for new topologies. – Topology parser allows radical topological changes by editing an input file. – Modularity between topology and equation sets allows models to be benchmarked on more familiar topologies. Test case needed to demonstrate these capabilities. Lodestar

6. Equation parser ArbiTER uses an input file to define what equations are to be solved. – Inherited (with minor modifications) from 2DX. – Python scripts allow conversion between edit (human-readable) and machine (program-readable) forms. Equation input file has four main parts: – Input language: defines format of data input file (gridfile). Gridfile defines all constants, profile functions, dimension sizes, and geometry functions for a particular instance of a model equation set. – Element language: derives functions, constants, and operators from input data and topology parser instructions. Profile functions from gridfile can be processed to create more complicated functions. Differential operators from topology parser can be combined with functions to form more complicated differential operators. – Formula language: assembles elements into equation sets. – Hard constant list: numerical constants (i.e. i, pi, 2, etc.) that should not be modified by gridfile. Lodestar

7. Example: resistive ballooning model, density equation – Equation defined in formula language gg*(1+0j)*N=(-1+0j)*kbrbpx*n0p*PHI – Derived function defined by element language xx=bigrbpxx=-1j*xx xx=kb*xxkbrbpx=xx xx=xx/bbmag – Profile functions defined by input language kb kb ind0bigrbp bigrbp ind0 bbmag bbmag ind0n0p n0p ind0 – Coefficients defined by hard constant list (1+0j) (1.000000,0.000000)-1j (0.000000,-1.000000) (-1+0j) (-1.000000,0.000000) where Lodestar

8. Topology parser ArbiTER uses an input file to define the topology on which equations are to be solved. – Python scripts allow conversion between edit (human-readable) and machine (program-readable) forms. Topology input file has two main parts: – Integer language: processes integer inputs in order to calculate sizes of subregions. – Topology language: generates domains and operators for use in equation parser. Topology language creates four basic types of objects: – Bricks (blocks): simple meshes that can be arranged into more complicated domains. – Domains: spaces on which a function or variable can be defined, created by assembling bricks and linkages. – Linkages: sparse arrays relating points between two domains or bricks. Can be used to define how bricks are assembled into domains, or can be used to construct operators. – Operators: sparse arrays used to calculate derivatives, interpolation, etc. by the equation parser. Lodestar

9. Example: 2DX emulation topology Seven total bricks – Edge, SOL, right private, left private – SOL and right private come in indented, non-indented varieties – 1 x nx brick for periodic boundary condition Three domains – Indented (for staggered grids), non-indented, and pbc 24 linkages – 9 for building domains – 15 for building operators PBC domain used to organize q function – q used to generate phase-shift periodic linkage for the edge – every function must have a domain Remaining domains used to organize fields and functions Lodestar

10. Procedure File containing magnetic geometry information in UEDGE format imported to Mathematica. Mathematica interacts with special version of ArbiTER to import topology data. – Ensures that topology information is encapsulated in topology parser language. – Allows Mathematica script to be used for multiple topologies. Mathematica script exports grid file to Arbiter. Mathematica script reads Arbiter output for processing and display. Lodestar

11. Procedure (cont.) Model equations: – Model equations for resistive ballooning mode. – Identical to equations used in 2DX benchmark case. – Provides convenient model to test capabilities on new geometries. Lodestar

12. Results Test case: – 30x80 resolution. – Temperature set to constant value (61.57 eV). – Density set to tanh function plus floor function. Scale length 16% of total flux width. Peak density is 3x floor density. Floor density is 1.4x10 13 cm -3. – Varying flux surface with peak density gradient allows control of dominant eigenmode position. – Demonstrates interaction between eigenmode and features of given magnetic topology. Lodestar

13. Results (cont.) |Phi| component of eigenfunctions as a function of density profile for n=20 (linear palette): Lodestar

14. Results (cont.) Close-up of case 4, log plot, 2x resolution Lodestar

15. Results (cont.) Growth rate as a function of maximum gradient position, n=20 Lodestar Growth rate as a function of toroidal mode number, density profile from case 3

16. Conclusions ArbiTER is a linear solver capable of rapid reconfiguration for new models and topologies. The capabilities of this code have been demonstrated on snowflake-minus topology with two x-points nearby in divertor region. Generality of code and modularity of physics models suggests other snowflake-related topologies can be modeled with little additional work. Lodestar