A hybrid statistics/amplitude approach to the theory of turbulent states of drift waves and zonal flows

Transcription

1 A hybrid statistics/amplitude approach to the theory of turbulent states of drift waves and zonal flows Jeff Parker John Krommes (PPPL) Gyrokinetic Theory Working Group Meeting June 25, 2012 Madrid, Spain

2 Overview New perspective for detailed analytic study of drift wave zonal flow interactions Involves keeping only the DW statistics but working with the full ZF amplitude Potentially useful for clarifying how ZFs regulate DW turbulence Utility: suggest some numerical diagnostics for simulations provide insights useful for modeling in simulations

3 Outline Background, Motivation Statistical Descriptions of Turbulence and Closures Methodology of Stochastic Structural Stability Theory Prospects for analytic understanding

4 Outline Background, Motivation Statistical Descriptions of Turbulence and Closures Methodology of Stochastic Structural Stability Theory Prospects for analytic understanding

5 Zonal Flow (ZF) Diamond (2011) ExB flow due to poloidally and toroidally symmetric radial electric fields Length scale: usually comparable to that of turbulence (seen in both simulations and measurements) Time scale: typically f < 10 khz, small compared to f DW ~ 50 khz ZFs believed to play a role in the L-H transition

6 Turbulence Saturation Mechanism by Zonal Flow? eddy shearing vs. damped eigenmodes (not well understood) Hatch (2011)

7 Hasegawa-Wakatani Model 2D (slab) Hasegawa-Wakatani model for vorticity & density Resistive term Background density gradient Has an unstable branch and stable branch of eigenmodes (contains coupling to damped modes) One-field model (Generalized Hasegawa-Mima) is also useful for seeing the basic structure of the equations

8 Zonal Flow Hasegawa-Wakatani

9 Driving Questions 1. How do zonal flows regulate drift wave turbulence? 2. What sets the zonal flow length scale and amplitude? Examples of previous work Physical picture of DW packets moving in an inhomogeneous medium of random long-wavelength ZFs, causing DWs to diffuse to high wavenumber Coupling to damped eigenmodes ZF generation by secondary/modulational instability (not self-consistent)

10 Zonal Flows in Other Contexts Earth s ocean and atmosphere Jupiter s atmosphere Plasma physics can borrow insight from these other fields

11 Outline Background, Motivation Statistical Descriptions of Turbulence and Closures Methodology of Stochastic Structural Stability Theory Prospects for analytic understanding

12 Statistical Closures Why statistical closures? Want to understand the nature of turbulence without the details Look at smoothly varying, averaged quantities But equations of motion are nonlinear, leading to an infinite hierarchy of moments and a closure problem.

13 Statistical Closures Example of a famous closure: the Direct Interaction Approximation (DIA). Complicated, but it motivates simpler models Conserves nonlinear quadratic invariants No adjustable parameters Statistically realizable Coherent nonlinear damping Internal forcing

14 Statistical Realizability Statistical quantities have various constraints, such as positivity for certain quantities Naive closures violate these, and, e.g, may predict that energy spectra become negative Realizability: condition that all constraints on statistics are satisfied eliminates many unsatisfactory closures Can show realizability of a statistical equation if a specified random amplitude equation exhibits the same statistics

15 Outline Background, Motivation Statistical Descriptions of Turbulence and Closures Methodology of Stochastic Structural Stability Theory Prospects for analytic understanding

16 Stochastic Structural Stability Theory (SSST) a realizable closure Developed by Farrell & Ioannou in the atmospheric sciences community since 90 s Takes a given turbulent model, produces a simplified set of equations for zonal flow amplitude and 2 nd order drift wave statistics Has been used to study emergence of zonal jets in baroclinic & barotropic turbulence, channel flow, and Earth s midlatitude jets F&I explored SSST for HW system (2009) My interest: understand their method and evaluate its usefulness; possibly extend its use in plasmas

17 Schematic of SSST Zonal component DW component Look for single-time, 2 nd -order statistics and apply a (time) average: Assume timescale separation, and statistical homogeneity and ergodicity in the direction

18 Schematic of SSST Closure term: treat DW self-interactions as white noise Properties of SSST equations Linear dynamics inherited from original equation DW-ZF interactions treated exactly (with appropriate nonlinear conservation properties) White noise forcing does not obey conservation properties of DW-DW interactions ZF amplitude + DW covariance F&I have argued that this simple closure captures the essential details of the equilibration between turbulence and zonal flows (in atmospheric settings)

19 SSST for Hasegawa-Mima Equation Background density gradient = DW component SSST Equations: Linear + zonal terms DW selfinteractions Generalizes appropriately for the HW system

20 Linear Stability vs. Stochastic Structural Stability Linear stability of the statistically averaged system, distinct from linear stability of the original system Later example: loss of stability of equilibrium (using L operator) as a parameter is changed

21 SSST in Relation to Inhomogeneous Turbulence Description Inhomogeneous turbulence: statistically-averaged quantities can depend on position HW simulations: stable, steady ZF implies non-ergodicity in the radial direction homogeneous description may not be appropriate. Inhomogeneous description is more suitable: mean field (ZF) and fluctuation statistics (DW) are positiondependent SSST has the same structure (with a particular closure of the 3 rd moment)

22 Zonal Flows found in SSST (Barotropic Turbulence) F&I(2007) x

23 Simulation Work in Progress Reproduced a number of F&I s SSST simulation results of the HW system HW direct numerical simulation code available for comparing with SSST Understanding is not yet at the stage where such comparisons are meaningful

24 Measured Forcing vs F&I s forcing Can measure an appropriate forcing (amplitude and wavenumber dependence) from the DNS Both forcings maintain similar zonal flow in nearly-steady state Saturation in SSST simulation is problematic Still a work in progress

25 Outline Background, Motivation Statistical Descriptions of Turbulence and Closures Methodology of Stochastic Structural Stability Theory Prospects for analytic understanding

26 Analytic, Steady State Solutions Possible? Simple models: find analytically a steady-state, nonlinear equilibrium between the DW and ZF as a foundation to build upon With an analytic steady state, one could begin to provide deep answers to questions about saturation mechanisms Simplified cases: Disparate Scale Limit Bifurcation to a zonal flow state

27 No Solution in the Disparate Scale Limit Simplest model: (Generalized) Hasegawa-Mima SSST in disparate-scale (weak inhomogeneity) limit: Wave-kinetic form Group velocity Modulation by ZF Forcing; internal DW interactions This simple model does not seem to have a self-consistent steady-state solution

28 Bifurcation Analysis Looking close to a bifurcation: small parameter for simplification. Limited validity, but allows glimpse of analytic nonlinear equilibrium involving zonal flow Simple case: without explicit linear instability DW+ZF DW, no ZF Q (forcing) HW system with linear drive set to 0 numerical results (F&I 2009) ZF ZF Q Q

29 Bifurcation with Linear Drive More realistic: system with linear drive Analyze bifurcation from a turbulent equilibrium without zonal flow to a state where zonal flows are present In progress for HW system; distinction between the homogeneous and inhomogeneous part of the DW spectrum seems to be important DW, no ZF DW+ZF (More complete closure than SSST required here)

30 Summary Simple statistical descriptions (like the SSST) may help in understanding nonlinear balance between ZFs and DWs in a steady state as simple as possible, but no simpler Comparison between arbitrary and measured (from DNS) forcing show similar behavior for ZF generation, but understanding is incomplete and some issues need to be resolved In simple models, no steady state between disparate-scale ZFs and DWs Work progressing on distinguishing turbulence saturation mechanism by ZF (eddy shearing vs. coupling to damped eigenmodes)