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The Theory of Gyrokinetic Turbulence: a Multiple-Scales Approach

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The Theory of Gyrokinetic Turbulence: a Multiple-Scales Approach UNIVERSITY OF CALIFORNIA Los Angeles The theory of gyrokinetic turbulence: A multiple-scales approach A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Physics by Gabriel Galad Plunk arXiv:0903.1091v2 [physics.plasm-ph] 24 Apr 2009 2009 © Copyright by Gabriel Galad Plunk 2009 The dissertation of Gabriel Galad Plunk is approved. James McWilliams George Morales Troy Carter Steven Cowley, Committee Chair University of California, Los Angeles 2009 ii For Nicole, with love iii Table of Contents 1 Introduction :::::::::::::::::::::::::::::::: 1 1.1 Gyrokinetics . 1 1.2 The Tokamak Transport Problem . 2 1.2.1 Confinement and Fusion . 3 1.2.2 Classical transport . 5 1.2.3 Anomalous transport . 7 1.2.4 Weak turbulence . 10 1.2.5 Strong turbulence . 12 1.3 Results . 15 1.3.1 Chapter 2: Gyrokinetics as a transport theory . 16 1.3.2 Chapter 3: Primary and secondary mode theory . 19 1.3.3 Chapter 4: The phase space cascade in two dimensional gyrokinetics . 22 2 Gyrokinetics as a transport theory ::::::::::::::::: 27 2.1 Introduction . 27 2.2 Scale separation . 28 2.2.1 Characteristic scales . 29 2.2.2 Method of multiple scales . 30 2.3 Gyrokinetic ordering . 32 2.3.1 Electron and ion orderings . 33 iv 2.4 Averaging and scale separation . 33 2.4.1 Gyro-average and single particle motion . 34 2.4.2 Spatial averages . 35 2.4.3 Time average . 37 2.5 Maxwell's equations and potentials . 37 2.6 Ordered Fokker-Planck equation for ion species . 39 2.6.1 O(−1) equation: . 39 2.6.2 O(1) equation: . 40 2.6.3 O(): Gyrokinetic equation . 43 2.6.4 Kinetic and classical equations for electron species . 45 2.7 Maxwell's equations for gyrokinetics . 46 2.7.1 Quasi-neutrality . 46 2.7.2 Parallel Ampere's law . 46 2.7.3 Perpendicular Ampere's law . 47 2.7.4 Pressure balance . 47 2.8 O(2): Transport equations in the slab limit . 48 2.8.1 Particle transport ..................... 49 2.8.2 Heat transport equation . 53 2.8.3 Entropy balance . 59 2.9 Poynting's theorem . 60 3 Primary and Secondary Mode Theory ::::::::::::::: 64 3.1 Introduction . 64 v 3.2 Normalized equations . 65 3.2.1 The primary mode . 68 3.3 Introduction to secondary instability theory . 71 3.3.1 Motivation . 72 3.3.2 Content . 75 3.3.3 Secondary instability . 77 3.4 Secondary instability for toroidal branch primary mode . 80 3.4.1 ETG . 81 3.4.2 ITG . 88 3.4.3 Parametric dependence and contour plots . 88 3.5 Transport and mixing length theory . 92 3.6 Secondary instability for the slab branch primary mode . 94 3.7 ETG secondary with kinetic ion response . 95 3.8 Conclusion . 100 4 The phase space cascade in two dimensional gyrokinetics ::: 103 4.1 Introduction . 103 4.2 Equations . 106 4.3 Collisionless invariants . 112 4.4 Charney{Hasegawa{Mima/Euler turbulence . 114 4.4.1 Collisional limit, Navier{Stokes and viscous Charney{Hasegawa- Mima equations . 118 4.5 Phenomenology . 120 vi 4.5.1 The nonlinear phase-mixing range, k 1 . 121 4.5.2 The cascade through multiple scales . 125 4.6 Statistical theory . 127 4.6.1 Symmetries of the gyrokinetic system . 128 4.6.2 The ensemble average . 130 4.6.3 third-order Kolmogorov relations . 130 4.6.4 The E third-order Kolmogorov relation . 133 4.6.5 The CHM/Euler limit . 135 4.7 The gyrokinetic turbulence cascade . 136 4.7.1 Self-similarity hypothesis . 136 4.7.2 The kinematics of gyrokinetic turbulence . 137 4.7.3 The forward cascade of free energy Wg . 139 4.7.4 Spectral scaling laws . 140 4.7.5 Scales smaller than the Larmor radius: k 1 . 140 4.7.6 The two-dimensional spectrum Wg(k; p) . 144 4.7.7 The one-dimensional spectra Wg(k) and Wg(p) . 145 4.7.8 The inverse cascade of E . 146 4.7.9 Some comments on the two dimensional spectrum Wg(k; p) 149 4.8 Conclusion . 149 4.8.1 Summary . 149 4.8.2 A note on forcing and universality . 150 4.8.3 Future work . 151 vii 5 Conclusion ::::::::::::::::::::::::::::::::: 154 5.1 Highlights . 154 5.2 Final note . 155 A Appendix: Gyrokinetics transport in toroidal geometry :::: 157 A.1 Introduction . 157 A.2 Equilibrium field geometry . 158 A.2.1 Axi-symmetry . 158 A.2.2 Magnetic coordinates . 158 A.2.3 Annulus volume average . 159 A.3 Proving F0 is a flux surface quantity . 161 A.4 O(2): Transport equations . 163 A.4.1 Particle transport . 163 A.4.2 Electron particle transport . 165 A.4.3 Heat transport . 166 A.4.4 Electron temperature transport . 167 A.5 Closure of the moment equations . 167 A.5.1 Flux surface motion . 168 A.5.2 Evolution of the toroidal field function I( ) . 171 A.5.3 The neoclassical equation and obtaining hB0 · E0ii . 172 A.5.4 Closure . 174 A.6 Entropy balance . 175 A.7 Particle transport calculation details . 177 viii A.8 Heat transport calculation details . 180 A.8.1 Heat transport manipulation . 185 References . 188 ix List of Figures 1.1 Transport by turbulent mixing: The radial mixing length is set by the strength of the flow (the saturation amplitude) but limited by the dominant wavenumber in the transport spectrum −1 ξx . πkx ................................ 13 1.2 Flux tube simulation domain: A thin flux tube volume surrounding an equilibrium field line is rendered in green. Also, a small segment of the full annulus volume is included on the left side of the torus. A larger segment of an annulus volume is pictured in figure A.1. (courtesy of G. Stantchev) . 18 2.1 Patch Volume and Scale Separation: The patch volume Vp (colored brown) is chosen to be small enough such that the equilibrium geometry is flat but large enough to contain many turbulent scale (correlation) lengths. 36 3.1 Primary growth rate comparison: Taking η = 3:14, T = :14 and τ = 1, the local dispersion relation is solved and compared with results from linear run(Wan06) from GYRO.(CW03) (color figure) 70 3.2 Rogers-Dorland secondary instability: Growth rates for the toroidal (R-D) case are plotted as a function of primary wavenum- ber kp. The secondary growth rates correspond to the fastest growing kx mode for each value of kp and are given relative to the amplitude of the primary mode. 82 x 3.3 Secondary growth rate comparison: For the R-D secondary, the growth rate from gyorkinetic secondary theory, Eqn (3.12), is compared with values computed from the gyro-fluid model of Ref (DJK00). For the gyrokinetic case parameters are: τ = 1, η = 3:14 and T = 0:14. 84 3.4 ETG kx spectrum: Growth rates for the R-D secondary are plotted as a function of radial wavenumber kx for three values of kp. Parameters are τ = 1, η = 3:14 and T = 0:14 . 85 3.5 Resolution in ky needed for convergence of !s to within 50%, 10% and 1% of it's fully converged value for R-D ETG secondary in- stability. (color figure) . 86 3.6 Rogers-Dorland secondary eigenfunctions: Re['(~y)] (black- solid) and Im['(~y)] (grey-solid) are plotted for the fastest growing modes. The primary mode 'p = 'p0 cos(kpy) appears red-dashed. Parameters are τ = 1, η = 3:14 and T = 0:14. (color figure) . 87 3.7 Rogers-Dorland secondary parameter scan: The growth rate γs='p0 is plotted for τ = 1 and kp = 0:21. The red line gives the approximate marginal stability contour and the black region outside this contour corresponds to γp ≤ 0. Also at the top of the plot the primary growth rate goes to zero at the critical marginal stability parameter which is roughly T;crit ∼ 0:3 (as calculated from the local kinetic dispersion relation). Secondary growth rates are calculated for the region to the right of the red contour where γp > 0. The color scales linearly for ITG from black < 0:035 to white > 0:05. ETG has a more sensitive parametric dependence with black < 0:0003 and white > 0:003. 89 xi 3.8 Rogers-Dorland kx;max parameter scan: Contour plot para- metric dependence of kx;max, the value of kx corresponding to the fastest growing secondary. Here we have kp = 0:21. The red line gives the approximate marginal stability contour and the black re- gion outside this contour corresponds to γp ≤ 0. Also at the top of the plot the primary growth rate goes to zero at the critical marginal stability parameter which is T;crit ∼ 0:3 (as calculated from the local kinetic dispersion relation). 90 3.9 Effective aspect ratio parameter scan: Constant contours of the effective aspect ratio at saturation ξsat/λp are plotted. The primary wavenumber and temperature ratio are held constant: kp ∼ 0:21 and τ = 1. As expected, ETG exhibits the signa- ture large aspect ratio \streamers," while the ITG aspect ratio is kept below unity aspect ratio. Both cases indicate a strong sup- pression of turbulent transport near the linear marginal stability cutoff (LT =R)crit (top of plots) giving a theoretical understanding of the Dimits shift. 91 3.10 Cowley (slab branch) secondary: η = 3:14. The primary growth rate is maximized over kk. The secondary growth rate is maximized over both kks and kx.................... 94 3.11 Rogers-Dorland secondary instability with kinetic ions: Parameters are ηi = ηe = 3:14, T i = T e = :14 and τ.
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