A Simulation Model for Drift Resistive Ballooning Turbulence Examining the Influence of Selfconsistent Zonal Flows *


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1 A Simulation Model for Drift Resistive Ballooning Turbulence Examining the Influence of Selfconsistent Zonal Flows * Bruce I. Cohen, Maxim V. Umansky, Ilon Joseph Lawrence Livermore National Laboratory Livermore, CA Ben Dudson University of York Heslington, York YO10 5DD United Kingdom Transport Task Force Meeting Denver, Colorado March 29April 1, 2016 LLNLPOST This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract DEAC5207NA Lawrence Livermore National Security, LLC
2 A Simulation Model for Drift Resistive Ballooning Turbulence Examining the Influence of Selfconsistent Zonal Flows 1. Introduction Overview and motivation 2. Previous edge turbulence simulations with selfconsistent zonal physics Toroidal simulations by the Maryland group (closed field lines) Cylindrical simulations of LAPD device (UCLALLNL) Toroidal simulations of ISTTOK (Dudson) 3. Previous work on BOUT simulations of DIIID Lmode shots # & 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 selfconsistently 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
3 1. Motivation for Including Selfconsistent Zonal Flow Physics in Simulation Model for Tokamak Edge Turbulence There are many examples of the stabilizing influence of selfconsistent zonal flow physics on drifttype turbulence in tokamak core and edge plasmas. Turbulent transport in tokamak edge plasmas is not fully understood. Ron Waltz [ APS DPP 2012 DI : Search for the Missing L mode Edge Transport and Possible Breakdown of Gyrokinetics ]: While GYRO simulations of typical core (0<r/a<0.7) DIIID Lmodes seem to be in good agreement with experiment, simulated lowk (k θ ρ s <1) transport and turbulence intensity is more than 5fold lower than experimentally inferred levels in the near edge Lmode (r/a= ) DIIID shot [1]. Global edge slice GYRO simulations of this and the wellstudied discharge [2] are presented here to document the shortfall. TGLF transport code simulations over a large Lmode database indicate this short fall is not atypical so that Lmode edges transit to Hlike pedestal profiles contrary to experiment. B. Cohen, et al., APS DPP
4 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 selfconsistent 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 Slablike 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 driftresistive ballooning turbulence with selfconsistent zonal modes and shear flows. The SOL and divertor region were not included. B. Cohen, et al., APS DPP
5 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, (2010); P. Popovich, M. V. Umansky, T. A. Carter, and B. Friedman,Physics of Plasmas 17, (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 selfconsistently 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, (2012); B. Friedman, T. A. Carter, M. V. Umansky, D. Schaffner, and I. Joseph Physics of Plasmas 20, (2013). B. Cohen, et al., APS DPP
6 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 3field 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
7 in LAPD simulation B. Cohen, et al., APS DPP
8 BOUT++ simulation of ISTTOK with an alternative toroidal coordinate system succeeds in including zonal physics 2 has both / x and / z Driftreduced Braginskii equations B. Cohen, et al., APS DPP
9 3. BOUT Simulations of Resistive Drift Ballooning Turbulence in Edge Region for DIIID LMode Shots #119919/..21/..30/..34 B.I. Cohen, M.V. Umansky, et al., Phys. Plasmas 20, (2013) Simulations of electromagnetic resistive drift ballooning in DIIID Lmode shots #119919, , , and , 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 shearflow effects BOUT has obtained steadystate turbulence with fluctuation amplitudes and transport that compare reasonably to DIIID probe and BES data. Sheared rotation due to imposed E radial (r) is stabilizing, at least linearly. Selfconsistent zonal flows were not included. B. Cohen, et al., APS DPP
10 d N i dt BOUT06 Equations for Resistive Drift Ballooning with Magnetic Flutter (Cohen et al. PoP 20, (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 ) ν 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 DIIID 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# DIIID  like fixed background profiles for shots # and # 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
11 There was reasonable agreement between BOUT simulation and Langmuir probe data for DIIID # 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
12 Reasonable Agreement between BOUT Simulation and Beam Emission Spectroscopy Data for DIIID # with respect to Peak Fluctuation Amplitude, Localization, Spatial Correlation Width, and Spectral Width No zonal physics ~ n/n Amplitude Profile Spatial Correlation Density Fluctuation 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
13 Imposed E 0 xb Shearing Reduces Both Linear Growth Rates and Saturated Turbulent Amplitudes in Simulations of # Probe data BOUT No zonal physics B. Cohen, et al., APS DPP
14 Inclusion of Imposed Radial Electric Field Reduces Growth Rates and Saturated Fluctuation Levels in Simulation of Shot # DIIID expt  BOUT no Er  BOUT w t=0.6ms Case 5 & 6a 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
15 4. Including selfconsistent evolution of n=0 modes in conventional BOUT++ tokamak geometry so far leads to various numerical problems Example: Including the selfconsistent 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 shearalfvén 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
16 5. Fieldline 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 For n=0, is the lack of y derivatives in this simplified perpendicular Laplacian a source of trouble? B. Cohen, et al., APS DPP
17 Laplacian in BOUT Coordinates I. Joseph B. Cohen, et al., APS DPP
18 Laplacian in BOUT Coordinates (cont d)!eq.(39) on p. 16 B. Cohen, et al., APS DPP
19 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
20 Laplacian in BOUT Coordinates (cont d) B. Cohen, et al., APS DPP
21 6. Inversion of vorticity for n=0 modes in BOUT++: LaplaceXY  B. Dudson z = 0 see p. 19 B. Cohen, et al., APS DPP
22 Inversion of vorticity for n=0 modes: LaplaceXY (cont d) B. Cohen, et al., APS DPP
23 Inversion of vorticity for n=0 modes: LaplaceXY (cont d) B. Cohen, et al., APS DPP
24 Inversion of vorticity for n=0 modes: LaplaceXY (cont d) B. Cohen, et al., APS DPP
25 Inversion of vorticity for n=0 modes: LaplaceXY (cont d) B. Cohen, et al., APS DPP
26 Summary A Simulation Model for Drift Resistive Ballooning Turbulence Examining the Influence of Selfconsistent Zonal Flows Including zonal flows selfconsistently in fluid turbulence simulations using conventional BOUT++ fieldlinefollowing 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++ fieldlinefollowing 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
27 Sign Up for Copies of This Poster Name and address B. Cohen, et al., APS DPP
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