Simulating Interchange Turbulence in a Dipole Confined Plasma

CTX Simulating Interchange Turbulence in a Dipole Confined Plasma B.A. Grierson, M.W. Worstell, M.E. Mauel Columbia University CTX American Physical Society DPP 2007 Orlando, FL 1

Abstract The dipole magnetic field is a simple, shear-free configuration. Strong, low-freqency interchange mixing, with k = 0, allows plasma cross-field dynamics to be!bounce-averaged". When dipole confined plasma is produced with ECRH, fast Hot Electron Interchange (HEI) instabilities appear at low densities, and lower-frequency turbulent fluctuations occur at higher densities. The global mode structure of the HEI and centrifugal interchange are understood, with good agreement between laboratory measurements and nonlinear simulations. However, the turbulent fluctuations are much less understood. They exhibit a power-law like spectrum, and require a spatially refined computational grid. To study the interchange turbulence, a fully parallel simulation has been developed to examine these fluctuations. The simulation includes a distributed Fast- Poisson solver for the ion polarization drift, a particle source and sink. 2

Interchange Modes Exist in a Non-uniform Plasma Classic fluid instability, similar to the gravitational Rayleigh-Taylor instability. Fluting mode, with kll! 0. ŷ, g 4 2 ẑ, B ˆϕ n n(y) B z y 1 y 8 6 n y φ 4 CTX 0.3 0.4 0.5 0.6 DPP 2007 φ 0.7 0.8 0.9 1 φ 3

Interchange Instabilities Occur in Fluids and Plasmas Interchange instabilities are ubiquitous in fluids and plasmas. Rayleigh-Taylor instability in fluids. Edge SOL dynamics in tokamak are interchange-like. Equatorial spread-f layer and Jupiter s magnetosphere. 4

Questions This Research Addresses What are the characteristics of turbulent interchange modes in a dipole-confined plasma? What are the average plasma parameters (n, T, ") and their profiles in this turbulent state? What is the interplay between the plasma profiles and the interchange turbulence in a dipole plasma? 5

Experimental Motivation Probe #1 for Simulations Mach Probe #1 Probe #2 Probe #4 1.0kW @2.45GHz ECRH Hydrogen Plasma. Bo=875G Biasing Array Vacuum Range! 1-2*10-7 Torr!Wave Power C L #c>>#b>>#d Probe #5 Cyclotron Resonance 67 cm Multiple movable probes for Isat, "float, Particle Flux, Mach Number. Probe #3 1 m Mach Probe #2 Probe location measured by equatorial L. 6

Gas Fueling Causes a Transition To Turbulence! f (V) +10 0 Floating Potential Low Density HEI 10-2 10-3 10-4 Power Spectrum -10-20 0.198 0.200 0.202 0.204 Time(s) +10 0 Floating Potential Gas Puff 10-5 10-6 0.1 1.0 10.0 100.0 1000.0 Freq (khz) High Density Turbulence! f (V) -10-20 0.20120 0.20173 0.20227 0.20280 Time(s) 7

Dipole Interchange Turbulence Characteristics Intensity and frequency of fluctuations depend on density profile. Freq (khz) 40 4.4e-02 20 2.9e-02 10 2.5 1016 Time Signal (S-<S>)/Var 15 1.5 1016 1.0 10 16 5.0 10 15 0.0e+00 0.45 0.50 0.55 Time(s) 0.60 0.65 0.70 Edge density PDFs have long tails (blobs). 2.0 1016 1.5e-02 0 40 45 50 L (cm) 55 60 10 5 0-5 0.2 0.3 0.4 Time (s) Intermittency and coherent structures, modes. 0.6 PDF 0.010 0.001 Bin Size=0.01000000 s -15 CTX 0.5 0.100 PDF Edge Isat Normalized <Density> 5.8e-02 30 Amplitude (AU) C1,2 (t) n (m-3) -10-5 0 5 Signal Amplitude 10 15 DPP 2007 8

Adaptation of Initial Value HEI Simulation For Steady State Turbulence Time-explicit Leapfrog method in 2D. Includes ion polarization drift. Fully parallelized using the PETSc package for distributed computation. Requires parallel solver developed in collaboration with ANL (H. Zhang). 9 9

The Computational Domain 1.4 1.2 The Ion Profile Global Domain Local Domain Proc2 1 6 Proc2 y 0.8 y 7 Proc1 My=20 Proc1 0.6 0.4 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Ni φ 7 Proc0 Mx=20 Square 2D domain in azimuthal angle $ and radial coordinate normalized flux y=%/%0. Proc0 Periodicity in $ motivates spectral methods, and distributed array geometry into horizontal bands. 10

Calculating The Ion Polarization Drift in a Loop 2 y 2 t ɛ ˆΦ = ˆρ t = Ĵ ϕ ɛ ϕ(y) ˆΦ ϕ + 4 y ɛ y(y) ˆΦ y = 0.91 L2 0 λ 2 ρ D0 Poisson s Equation In dipole magnetic flux coordinates Resulting non-symmetric tri-diagonal system. b(l)ˆφ (m, l 1) + a(l)ˆφ (m, l) + c(l)ˆφ (m, l + 1) = R(l) 11

Ax = b A x Parallel Partition Method 1 Solver: Ax=b = à + A = à + VET = A 1 b = (à + VET ) 1 b = à 1 b à 1 V(I + E T à 1 V) 1 E T à 1 b A A V E = The matrix A is decomposed based on the array distribution. à x = b ÃY = V Zy = h h = E T x Z = I + E T Y x = Yh x = x x 1 Xian-He Sun, Hong Zhang, and Lionel M. Ni, Efficient Tridiagonal Solvers on Multicomputers, IEEE Transactions on Computers Vol. 41, No. 3, March 1992. T Solution Concurrent Solves Dense Solve 12

Particle Source and Sink Are Required For Steady State Turbulence A diffusive particle flux, as well as a particle source is added to the continuity equation. ˆN i t + ( ˆN i ˆV i ) + Γ D = S The diffusion is modeled as crossfield only, with a radial functional form. The source is also axisymmetric. The parameter ˆD is the strength of the particle recycling. The continuity equation in normalized magnetic flux coordinates. ˆN ˆt Γ D = D ˆN i D = D(y)(I ˆbˆb) + ϕ ( ˆN i V i,ϕ ) + y ( ˆN i V i,y ) D(y) = ˆDh D (y) S = D S ˆDhS (y) 1.8 ˆDh D y 2 2 ˆNi ϕ 2 3.2 y [ ] ˆDh D y (y4 ˆNi ) The integrated diffusive particle loss provides the particle conserving source coefficient. = ˆDD S h S D S = 1.8h D y 2 2 ˆNi ϕ 2 + 3.2 hs d 2 x y (h D y (y4 ˆNi ))d 2 x 13 13

Diffusion and Source Profiles Strong edge diffusion is used. Ions and electrons are added at the experimental ECRH resonance. 14

Example Ni and diffusive &Ni. Given Ni and profiles hd(y), hs(y), compute $ and y losses. Add axi-symmetric source hs(y) and scale by diffusion coefficient. + = Loss dominated by the divergence of the y diffusive flux. Steep hd(y) gradient + y 4 term. CTX Steep hd(y) gradient. DPP 2007 15

Steady State Turbulence is Achieved for the First Time in a Dipole Simulation D=0.001 D=0.0001 D=0.00001 Diffusion too strong. Steep edge gradients and charge densities. Diffusion just right. Steady-state turbulence achieved. Diffusion too weak. Not enough edge loss to facilitate recycling. 1 Density Profile 0.8 CTX 0.6 0.4 0.2 DPP 2007 Note: When ˆN(ψ)=cnst, n(l) 1/L 4 0.2 0.3 0.4 0.5 0.6 n L n0 L m 16

Power-law Spectrum Reproduced As recycling is increased, the fluctuation amplitude increases. The density spectrum takes on a power law slope. Power AU 0.1 0.001 0.00001 Ni Fluctuation 0.06 0.05 0.04 0.03 0.02 N i N i at y 0.9517 0.00005 0.0001 0.00015 0.0002 0.00025 0.0003 D coefficient Frequency Spectra at y 0.9517 0.005 0.01 0.05 0.1 f t CTX DPP 2007 17

Non-Symmetric PDF is Reproduced The simulation also reproduces the PDF with a strong positive tail. Simulation 20 15 10 7 5 3 2 1.5 1 Density PDF -2-1 0 1 2 3 4 This is indicative of intermittent `blob transport events. Experiment Amplitude (AU) 15 10 5 0-5 0.100 Signal (S-<S>)/Var 0.2 0.3 0.4 0.5 0.6 Time (s) PDF PDF 0.010 0.001 Bin Size=0.01000000 s -15-10 -5 0 5 10 15 Signal Amplitude 18

Coherent Structure is Observed and Simulated, L=0.40m Observed and simulated azimuthal mode structure is dominated by low m-number. 19

Summary The basic characteristics of dipole interchange turbulence have been reproduced by simulation. The particle source/sink drive determines the intensity and spectra of the turbulent fluctuations. A power-law spectra is seen in the simulation results. 20

Future Work Simulate dipole interchange turbulence on a more refined computational grid, and on a massively parallel architecture. Vary the source profiles, diffusion profiles, and heating electron energy. Further comparison between experiments, observations, and simulation. 21