Waves, irreversibility, and turbulent diffusion in MHD turbulence

1. Waves, irreversibility, and turbulent diffusion in MHD turbulence Shane R. Keating Patrick H. Diamond Center for Astrophysics and Space Sciences University of California San Diego High Altitude Observatory, Jan 23 2008 Keating & Diamond (Jan 2008) J. Fluid Mech. 595 Keating & Diamond (Nov 2007) Phys. Rev. Lett. 99

2. In a nutshell… • Turbulent resistivity is “quenched” below its kinematic value by an Rm-dependent factor in 2D MHD turbulence. • Theoretical models offer little insight into the physical origin of small-scale irreversibility, relying upon unconstrained assumptions and a free parameter (). • Introduce a simple extension of the theory which does possess an unambiguous source of irreversibility: three-wave resonances. • Rigorously calculate the spatial transport of magnetic potential induced by nonlinear wave interactions. This flux is manifestly independent of Rm.

3. Resistivity quenching in 2D MHD I • In the presence of even a weak mean magnetic field, turbulent resistivity is strongly suppressed below its kinematic value in 2D MHD (Cattaneo & Vainshtein 1991): kin¼ U L • kinematic resistivity • Alfven velocity (squared) • Magnetic Reynolds number Rm = U L / c

4. Resistivity quenching in 2D MHD II • Examine three increasingly detailed descriptions of turbulent resistivity: • Global: magnetic potential balance; Zel’dovich relation • Local: competing couplings / cascades • Microscopic: Closure calculation of flux

5. Global magnetic potential balance • Up to dissipation and boundary fluxes,2 is conserved in 2D MHD (Zel’dovich relation): • Suggestive of quenching; however • requires to be independent of • spectral exponent of not known in general • valid only for stationary turbulence

6. Local: Competing couplings/cascades • Physically, expect: turbulence strains and chops up a scalar field, generating small-scale structure magnetic potential tends to coalesce on large scales: is not passive forward cascade of 2 to small scales inverse cascade of 2 to large scales “positive viscosity” effect “negative viscosity” effect

7. here be dragons Turbulent resistivity: Microscopic: Closure calculation • EDQNM calculation of the vertical flux of magnetic potential  (Gruzinov & Diamond 1994, 1996) • Quasi-linear closure: • Quasi-linear response:

8. The key question • These closure calculations offer no insight into the detailed microphysics of resistivity quenching because the microphysics has been parametrized by , a free parameter in EDQNM / quasi-linear closures. What sets the timescale τ? • Motivated to explore extensions of the theory for which the correlation time is unambiguous.

9. “Wavy MHD” in 2D Coriolis force buoyancy • MHD + additional body forces: Rossby waves internal waves • Large-scale eddies dispersive waves: “Wavy MHD” = MHD + dispersive waves • (c.f. Moffatt (1970, 1972) and others) • When the wave-slope < 1, wave turbulence theory is applicable. • Origin of irreversibility is in three-wave resonances, which are present even in the absence of c

10. latitudinal gradient in locally vertical component of planetary vorticity mean magnetic field Illustration 1: plane MHD I • 2D MHD turbulence on a rotating spherical shell ( plane) • “Minimal model” of solar tachocline turbulence (Diamond et al. 2007) • symmetry between left and right is broken by  N

11. l* Illustration 1: plane MHD II •  modifies the linear modes: Alfven wave Rossby wave dominant for small scales (large k) dominant for large scales (small k) non-dispersive dispersive small scales large scales

12. buoyancy term mean magnetic field density gradient Illustration 2: stratified MHD I • 2D MHD in the presence of stable stratification • additional dynamical field: density • Boussinesq approximation: appears only in buoyancy term • Minimal model of stellar interior just below solar tachocline N

13. Brunt-Vaisala frequency l* Illustration 2: stratified MHD II • stratification modifies the linear modes: Alfven wave Internal gravity wave dominant for small scales (large k) dominant for large scales (small k) non-dispersive dispersive small scales large scales

14. stable unstable Wave turbulence theory • describes the slow transfer of energy among a triad of waves satisfying the resonance conditions: • analogous to the free asymmetric top (I3 > I2 > I1) Landau & Lifschitz, Mechanics, 1960 • for an ensemble of triads, origin of irreversibility is chaos induced by multiple overlapping wave resonances

15. Wave turbulence theory: validity • Wave turbulence theory requires a broad spectrum of dispersive, weakly interacting waves

16. Wave turbulence theory: validity • Wave turbulence theory requires a broad spectrumof dispersive, weakly interacting waves • broad enough for triad to remain coherent during interaction

17. Wave turbulence theory: validity • Wave turbulence theory requires a broad spectrum of dispersive, weakly interacting waves • broad enough for triad to remain coherent during interaction • resonance manifold is empty for non-dispersive waves

18. Wave turbulence theory: validity • Wave turbulence theory requires a broad spectrum of dispersive, weakly interactingwaves • turbulent decorrelation doesn’t wash out wave interactions: • broad enough for triad to remain coherent during interaction • resonance manifold is empty for non-dispersive waves < 1 • for dispersive waves, this is unity for a cross-over scale:

19. Strong turbulence Weak turbulence Dispersive Non-dispersive I. Alvenic Regime II. Intermediate Regime III. Wavy Regime Spectral regimes Vrms Horizontal phase velocity (vph,x) B0 L* l* Scale (k-1)

20. I. Alvenic Regime II. Intermediate Regime III. Wavy Regime dispersive waves are washed out by turbulence molecular diffusion strongly interacting eddys and Alfven waves molecular diffusion waves are dispersive and weakly interacting molecular diffusion; nonlinear wave interactions character of turbulence sources of irreversibility Spectral regimes

21. The wavy regime • Within the wavy regime the turbulent resistivity can be expanded in powers of the small parameter: Lowest order contribution is tied to c and so will depend upon Rm, Re, Pm… At higher order, nonlinear wave interactions make Rm-independent contribution • Overall turbulent resistivity has twoasymptotic parameters: Rm and wave-slope

22. A useful analogy I • Nonlinear wave-particle interactions in a Vlasov plasma: • Diffusive flux of particles in velocity space can be expanded in powers of |Ek,|2 • Second-order flux driven by interaction between electron (v) and plasma wave (k,): • Fourth-order flux driven by interaction between electron (v) interacting with beating of two plasma waves (k+k’, + ’):

23. Fourth-order flux ~ Second-order flux ~ A useful analogy II

24. linear wave displacement: The wave-interaction-driven flux I • The vertical flux of the jth scalar field is given by • In the wavy range the fluid/field response can be expanded in powers of the wave-slope: • The linear response is simply due to wave oscillations: • The wave-interaction-driven flux is then

25. The wave-interaction-driven flux II • third-order nonlinearities will depend upon first and second-order fields • formal solution of the third-order equations of motion • solve for higher-order responses iteratively • everything is ultimately expressed in terms of the linear fields

26. The wave-interaction-driven flux III • The flux driven by wave interactions is calculated to fourth order in the wave-slope • This is the same order as in wave-kinetic theory; however, we are interested in spatial transport rather than spectral transfer gradient of the mean field sum over left- and right-moving waves resonance condition coupling coefficient beat condition fourth order in wave-slope

27. Analysis of the result: -plane MHD

28. Analysis of the result: stratified MHD

29. Irreversibility • Within the wavy range, origin of irreversibility is chaos induced by overlapping wave resonances • triads with one short, almost vertical leg dominate the coupling coefficient • “induced diffusion” triad class (McComas & Bretherton 1977) • short leg acts as a large-scale adiabatic straining field on other two modes • directly analogous to wave-particle resonances in a Vlasov plasma • For this triad class there exists a rigorous and transparent route to irreversibility based upon ray chaos

30. Conclusions • Within the wavy range, the origin of irreversibility is unambiguous and  is set by the spectral auto-correlation time • In the presence of dispersive waves, T does not decay asymptotically as Rm-1 for large magnetic Reynolds number • The presence of an additional restoring force can actually increase the transport of magnetic potential in 2D MHD • If so, concerns about resistivity quenching in real magnetofluids may be moot.

31. References

32. NL wave interactions: Basic paradigms I • Free asymmetric top (I3 > I2 > I1) Landau & Lifschitz, Mechanics, 1960 perturbations about stable axes are localized perturbations about unstable axis wander around entire ellipsoid • Energy is slowly transferred from x2 to x1 or x3 • However, motion is ultimately reversible

33. NL wave interactions: Basic paradigms II • Three nonlinearly coupled oscillators • Formally equivalent to the asymmetric top • Assuming weak coupling: Ai(t) are slow functions of t • Equations of motion will be dominated by slow, secular drive of one oscillator on the other two if oscillator frequencies satisfy “three-wave resonance” condition: • In this case, equations governing amplitudes Ai(t) are: • As in the asymmetric top, one of the modes can pump energy into the other two at a rate: Basic timescale for nonlinear transfer of energy d ~ • However, also like the asymmetric top, such transfer is reversible!

34. NL wave interactions: Basic paradigms III • Wave turbulence theory: • ensemble of many wave triads (statistical theory) • infinite moment hierarchy (closure problem) • key timescales: • three-wave mismatch • triad lifetime : tied to dispersion • nonlinear transfer rate • expand in small parameter (turbulentintensity) Wave turbulence requires a broad spectrum of dispersive waves • Origin of irreversibility: • Mechanically: Random phase approximation • moment hierarchy is truncated • Physically: ray chaos induced by resonance overlap