A Simulation Model for Drift Resistive Ballooning Turbulence Examining the Influence of Self-consistent Zonal Flows *

A Simulation Model for Drift Resistive Ballooning Turbulence Examining the Influence of Self-consistent Zonal Flows * Bruce I. Cohen, Maxim V. Umansky, Ilon Joseph Lawrence Livermore National Laboratory Livermore, CA 94551 Ben Dudson University of York Heslington, York YO10 5DD United Kingdom Transport Task Force Meeting Denver, Colorado March 29-April 1, 2016 LLNL-POST-686483 This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344. Lawrence Livermore National Security, LLC

A Simulation Model for Drift Resistive Ballooning Turbulence Examining the Influence of Self-consistent Zonal Flows 1. Introduction Overview and motivation 2. Previous edge turbulence simulations with self-consistent zonal physics Toroidal simulations by the Maryland group (closed field lines) Cylindrical simulations of LAPD device (UCLA-LLNL) Toroidal simulations of ISTTOK (Dudson) 3. Previous work on BOUT simulations of DIII-D L-mode shots #119919 & 119934 and comparisons to plunging probe and BES with model E radial (r) fitted to probe and CER data, but with n=0 modes suppressed 4. Including n=0 modes self-consistently in BOUT++ with conventional fieldline coordinates has numerical problems 5. There are subtleties in the perpendicular Laplacian operator in conventional BOUT++ coordinates and in alternative coordinates 6. Progress has been made in solving 2 φ =U including n=0 modes B. Cohen, et al., APS DPP 2015 2

1. Motivation for Including Self-consistent Zonal Flow Physics in Simulation Model for Tokamak Edge Turbulence There are many examples of the stabilizing influence of self-consistent zonal flow physics on drift-type turbulence in tokamak core and edge plasmas. Turbulent transport in tokamak edge plasmas is not fully understood. Ron Waltz [ APS DPP 2012 DI3.00002 : Search for the Missing L- mode Edge Transport and Possible Breakdown of Gyrokinetics ]: While GYRO simulations of typical core (0<r/a<0.7) DIII-D L-modes seem to be in good agreement with experiment, simulated low-k (k θ ρ s <1) transport and turbulence intensity is more than 5-fold lower than experimentally inferred levels in the near edge L-mode (r/a=0.7 0.95) DIII-D shot 128913 [1]. Global edge slice GYRO simulations of this and the well-studied discharge 101391 [2] are presented here to document the shortfall. TGLF transport code simulations over a large L-mode database indicate this short fall is not atypical so that L-mode edges transit to H-like pedestal profiles contrary to experiment. B. Cohen, et al., APS DPP 2015 3

2. Previous Edge Turbulence Models With Zonal Mode Physics: Toroidal Geometry With Closed Field Lines The Maryland group 20 years ago performed 3D fluid simulations of tokamak edge turbulence including self-consistent zonal modes and shear flows. Refs. Guzdar, Drake, McCarthy, Hassam, and Liu, Phys. Fluids B 5, 3712 (1993). Zeiler, Biskamp, Drake, and Guzdar, Phys. Plasmas 3, 2951 (1996). Reduced Braginskii equations for three fields: density, electric potential, and parallel current, with v = c φ b / B Slab-like representation of toroidal coordinates for closed field lines: r,θ,ζ x = r a, y = aθ, z = Rζ x' = x y' = y εz / q a ( ) z' = z + ( ε y / q a ) periodic in y' and z' where q a is the safety factor at a reference rational surface. z' +ε(1 q 1 q a ) y' 2 d 2 / dx' 2 2 k y' Results were obtained for saturated drift-resistive ballooning turbulence with selfconsistent zonal modes and shear flows. The SOL and divertor region were not included. B. Cohen, et al., APS DPP 2015 4

Previous Edge Turbulence Models With Zonal Mode Physics: Simulation of LAPD in Cylindrical Geometry BOUT++ has been used to simulate turbulence in the LAPD device UCLA in cylindrical geometry with a straight uniform magnetic field. ref. P. Popovich, M. V. Umansky, T. A. Carter, and B. Friedman, Phys. Plasmas 17, 102107 (2010); P. Popovich, M. V. Umansky, T. A. Carter, and B. Friedman,Physics of Plasmas 17, 122312 (2010) Coordinates: (r,θ, z), B = B 0 ẑ, 2 = 1 r r r r + 1 r 2 2 θ 2 In nonlinear simulations of LAPD, zonal flows have been included self-consistently with density and temperature sources to keep azimuthally averaged profiles from relaxing. The azimuthal averaged electric potential is allowed to evolve, which allows zonal flows to arise. ref. B. Friedman, T. A. Carter, M.V. Umansky, D. Schaffner, and B. Dudson, Phys. Plasmas 19, 102307 (2012); B. Friedman, T. A. Carter, M. V. Umansky, D. Schaffner, and I. Joseph Physics of Plasmas 20, 055704 (2013). B. Cohen, et al., APS DPP 2015 5

Evidence of Stabilizing Effects of Sheared Radial Electric Field with Reynolds Stress Zonal Flow Effects in Simulation of LAPD Edge Turbulence We simulate edge turbulence in LAPD cylindrical geometry with a 3-field electrostatic model supporting drift resistive instability (density, vorticity, electron temperature) including E r and with/without zonal flow effects on the electric potential only, with density & temperature axisymmetric modes held constant to maintain equilibrium profiles fitting the experiment. Including all zonal flows in electric potential reduces the saturated turbulence in the edge by a factor of 2 in the rms amplitudes over the time sampled. B. Cohen, et al., APS DPP 2015 6

in LAPD simulation B. Cohen, et al., APS DPP 2015 7

BOUT++ simulation of ISTTOK with an alternative toroidal coordinate system succeeds in including zonal physics 2 has both / x and / z Drift-reduced Braginskii equations B. Cohen, et al., APS DPP 2015 8

3. BOUT Simulations of Resistive Drift Ballooning Turbulence in Edge Region for DIII-D L-Mode Shots #119919/..21/..30/..34 B.I. Cohen, M.V. Umansky, et al., Phys. Plasmas 20, 055906 (2013) Simulations of electromagnetic resistive drift ballooning in DIII-D L-mode shots #119919, 119921, 119930, and 119934, with full geometry and magnetic shear, crossing the separatrix BOUT Nonlinear BOUT equations for ion density, vorticity, electron and ion velocities, electron and ion temperatures, Ohm s law, and Maxwell s equations. Simulation results for various physics models and validation against probe and BES data with imposed E r shear-flow effects BOUT has obtained steady-state turbulence with fluctuation amplitudes and transport that compare reasonably to DIII-D probe and BES data. Sheared rotation due to imposed E radial (r) is stabilizing, at least linearly. Self-consistent zonal flows were not included. B. Cohen, et al., APS DPP 2015 9

d N i dt BOUT06 Equations for Resistive Drift Ballooning with Magnetic Flutter (Cohen et al. PoP 20, 055906 (2013))# Consider the following simplified Braginskii + reduced Maxwell eqns with drift ordering in the BOUT06 framework: + N i V = # 2c & % ( b 0 κ ( P e N i e ϕ ) + ( j /e) $ eb ' dϖ = 2ω ci b 0 κ P + N i0 Z i e 4πV A 2 dt c 2 d V e t dv i dt dt e,i dt E = 1 c = e m e E = 1 N i0 M i P, j 1 N i0 m e (T e0 N i ) + 0.51ν ei j = 2 κ e,i T e,i 3N i0 ( ), κ e = 3.2 N i0t e0 τ e0, κ i =... m e t A ϕ, 2 A = 4π c j, B = A + B 0 [ ] + 2 P i ez i N i0 2 ϕ = b 0 + Electromagnetic with # # = b# 0 + b ## in φ and Actual DIII-D geometry# Radial bdry conditions: Von Neumann on fluid fluctuations, Dirichlet on A & φ #Fluctuations decay to 0 at outer bdry & not necessarily at inner bdry# DIII-D - like fixed background profiles for shots #119919 and 119934# ϖ = ez i N i ϕ b Z i = 1 Simulations with a fixed impose Er d dt = t + (V E 0 + V E ) N i = N i0 + N i, T s = T s0 + T s,... were conducted# P = N i0 ( T e + T i ) + N i (T e0 +T i0 ), T i0 = T e0, V s0 = 0 n=0 modes are suppressed" j B. Cohen, et al., APS DPP 2015 10

There was reasonable agreement between BOUT simulation and Langmuir probe data for DIII-D #119919 with respect to peak fluctuation amplitudes, particle and thermal flux, and localization with no zonal physics BOUT with T e & T i fluct ns, electron parallel thermal conduction, convective nlrity, = b 0 + b Radius at midplane Probe signals decrease below noise levels for R > 231 cm, and stop for R < 225 cm Typical experimental rms δn e and δt e fluctuations at the separatrix exceed ~20% & ~50% δn e, δt e and the probe fluxes in the midplane usually peak near the separatrix BOUT simulations and Langmuir probe data agree within factors of 2 in peak amplitudes and localization for 2.25m R 2.31m B. Cohen, et al., APS DPP 2015 11

Reasonable Agreement between BOUT Simulation and Beam Emission Spectroscopy Data for DIII-D #119921 with respect to Peak Fluctuation Amplitude, Localization, Spatial Correlation Width, and Spectral Width No zonal physics ~ n/n Amplitude Profile Spatial Correlation Density Fluctuation Spectrum @outer midplane Spatial filtering (1D or 2D) is required in simulation diagnostics to model the 1 cm limit on spatial resolution in the BES grid in R and Z. Spatial filtering of the BOUT diagnostics reduces and spatially spreads peaks There is agreement between BOUT and BES to within factors of two or three, or better B. Cohen, et al., APS DPP 2015 12

Imposed E 0 xb Shearing Reduces Both Linear Growth Rates and Saturated Turbulent Amplitudes in Simulations of #119919 Probe data BOUT simulation @midplane No zonal physics B. Cohen, et al., APS DPP 2015 13

Inclusion of Imposed Radial Electric Field Reduces Growth Rates and Saturated Fluctuation Levels in Simulation of Shot #119934 -- DIII-D expt -- BOUT no Er -- BOUT w Er @ t=0.6ms Case 5 & 6a #119934 @separatrix No zonal physics Inclusion of imposed E r reduces linear growth rates and saturated fluctuation amplitudes less so; the finite E r saturated amplitudes tend to recover to the levels of the E r =0 case when run t > 1 ms Simulation agreement with probe for relevant radii, 2.25m R 2.31m, remains fair B. Cohen, et al., APS DPP 2015 14

4. Including self-consistent evolution of n=0 modes in conventional BOUT++ tokamak geometry so far leads to various numerical problems Example: Including the self-consistent n=0 toroidal modes in the resistive ballooning model in full toroidal geometry with single X point leads to a fastgrowing unphysical instability localized near the SOL target plates in some cases Electric potential fluctuations vs. (x,y) at successive time steps The resistive ballooning and simple shear-alfvén models in a toroidal slab model with no X point and with conducting boundary conditions in y (parallel to the magnetic field) exhibit unphysical behavior when n=0 toroidal modes are included. B. Cohen, et al., APS DPP 2015 15

5. Field-line Following Coordinates and Representation of Perpendicular Laplacian in BOUT/BOUT++ For n = 0, z = 0 and 2 (RB θ ) 2 2 x 2 with no y terms Conventional BOUT++ perpendicular Laplacian used in pgs. 10-15 For n=0, is the lack of y derivatives in this simplified perpendicular Laplacian a source of trouble? B. Cohen, et al., APS DPP 2015 16

Laplacian in BOUT Coordinates I. Joseph B. Cohen, et al., APS DPP 2015 17

Laplacian in BOUT Coordinates (cont d)!eq.(39) on p. 16 B. Cohen, et al., APS DPP 2015 18

Laplacian in BOUT Coordinates (cont d) (as in p. 21) Connects to Maryland, LAPD & ISTTOK examples on pgs. 4, 5, & 8 B. Cohen, et al., APS DPP 2015 19

Laplacian in BOUT Coordinates (cont d) B. Cohen, et al., APS DPP 2015 20

6. Inversion of vorticity for n=0 modes in BOUT++: LaplaceXY -- B. Dudson z = 0 see p. 19 B. Cohen, et al., APS DPP 2015 21

Inversion of vorticity for n=0 modes: LaplaceXY (cont d) B. Cohen, et al., APS DPP 2015 22

Inversion of vorticity for n=0 modes: LaplaceXY (cont d) B. Cohen, et al., APS DPP 2015 23

Inversion of vorticity for n=0 modes: LaplaceXY (cont d) B. Cohen, et al., APS DPP 2015 24

Inversion of vorticity for n=0 modes: LaplaceXY (cont d) B. Cohen, et al., APS DPP 2015 25

Summary A Simulation Model for Drift Resistive Ballooning Turbulence Examining the Influence of Self-consistent Zonal Flows Including zonal flows self-consistently in fluid turbulence simulations using conventional BOUT++ field-line-following coordinates for toroidal geometries requires solving for n=0 modes which has been problematic BOUT++ toroidal simulations have no trouble with n=0 modes suppressed, and there has been significant successes in validation against experimental data BOUT++ simulations of cylindrical plasmas (e.g., LAPD) including zonal physics and n=0 modes are successful Toroidal simulations with closed field lines by the Maryland group including zonal physics using a different coordinate scheme were successful 20 years ago Analysis of the perpendicular Laplacian in conventional BOUT++ field-linefollowing coordinates and in alternative coordinates illustrates some of the issues Recent work by Dudson examining the inversion of the perpendicular Poisson equation in BOUT++ including the n=0 modes is showing progress in resolving numerical problems B. Cohen, et al., APS DPP 2015 26