Geophysical wave interactions in turbulent magnetofluids Shane R. Keating University of California San Diego Collaborators: Patrick H. Diamond (UCSD) L. J. Silvers (Cambridge) Keating & Diamond (Jan 2008) J. Fluid Mech. 595 Keating & Diamond (Nov 2007) Phys. Rev. Lett. 99 Keating, Silvers & Diamond (Submitted) Astrophys. J. Lett. Courant Institute for Mathematical Sciences March 12, 2008

Turbulent resistivity is “quenched” below its kinematic value by an Re m -dependent factor in 2D MHD turbulence. Mean-field electrodynamics offers little insight into the physical origin of small-scale irreversibility, relying upon unconstrained assumptions and a free parameter (  ). … Borrowing some ideas from geophysical fluid dynamics… 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 wave interactions. This flux is manifestly independent of Re m. In a nutshell…

Statement of the problem Magnetic potential: B = r A £ e y Induction equation:  t A + u ¢r A =  r 2 A where A(x, z, t) = h A i + a(x, z, t) Introduce turbulent flux: h u a i = -  T r h A i then  t h A i 2 = 2 (  +  T ) h B i 2 Statement of the problem: How fast are magnetic fields dissipated in 2D turbulence? Nomenclature:

In the presence of even a weak mean magnetic field, the dissipation of magnetic fields is strongly suppressed below its kinematic value in 2D MHD: Resistivity quenching in 2D MHD I Time Magnetic energy density (normalized) Cattaneo & Vainshtein (1991) Increasing initial magnetic field strength

kinematic resistivity Initial magnetic energy in units of the kinetic energy Magnetic Reynolds number Resistivity quenching in 2D MHD II  kin ¼ U L Re m = U L /  c X 2 = B 0 2 / h u 2 i Quasi-linear theory gives (Gruzinov & Diamond 1994): Implicit assumption: the spatial transport of magnetic potential is suppressed, not the turbulence intensity…

Is the quench due to a suppression of the spatial transport of magnetic potential or a suppression of the turbulence itself? Useful diagnostic: the normalized cross-phase Quench then becomes Ln normalized cross-phase Ln square of initial field Keating, Silvers & Diamond (2008) Resistivity quenching in 2D MHD III

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

EDQNM calculation of the vertical flux of magnetic potential  (Gruzinov & Diamond 1994, 1996) Turbulent resistivity: Quasi-linear response: Quasi-linear closure: here be dragons Quenching: Closure calculation

What sets the timescale τ? 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. Motivated to explore extensions of the theory for which the correlation time is unambiguous. The key question

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

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

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

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 buoyancy term density gradient Illustration 2: stratified MHD I mean magnetic field

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

describes the slow transfer of energy among a triad of waves satisfying the resonance conditions: analogous to the free asymmetric top (I 3 > I 2 > I 1 ) for an ensemble of triads, origin of irreversibility is chaos induced by multiple overlapping wave resonances Landau & Lifschitz, Mechanics, 1960 stable unstable Wave turbulence theory

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

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

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

Horizontal phase velocity (v ph,x ) V rms B0B0 Strong turbulence Weak turbulence Non-dispersive Dispersive Scale (k -1 ) l*l* I. Alvenic Range II. Intermediate Range III. Wavy Range L*L* Spectral ranges

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

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

The wave-interaction-driven flux I Calculate the flux of A due to wave-interactions: Linear response Response to wave interactions Flux due to molecular collisions Flux due to wave-wave interactions

The wave-interaction-driven flux II Expand  A in powers of the wave-slope  = k  < 1 First contribution to wave-interaction-driven flux is O(  4 ) Formal solution:

geometrical factor coupling coefficient fourth order in wave-slope 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 The wave-interaction-driven flux III Response time:  § = Re i (  ’ +  ’’ §  k’ + k’’ + i 0 + ) -1 =  (  ’ +  ’’ §  k’ + k’’ ) Resonance condition

Analysis of the result:  -plane MHD

Analysis of the result: stratified MHD

Within the wavy range, origin of irreversibility is chaos induced by overlapping wave resonances For this triad class there exists a rigorous and transparent route to irreversibility based upon ray chaos 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 Irreversibility

Within the wavy range, the origin of irreversibility is unambiguous: ray chaos induced by overlapping resonances 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. Conclusions

References

Nonlinear wave-particle interactions in a Vlasov plasma: Diffusive flux of particles in velocity space can be expanded in powers of |E k,  | 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’,  +  ’): A useful analogy I

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

NL wave interactions: Basic paradigms I Free asymmetric top (I 3 > I 2 > I 1 ) Landau & Lifschitz, Mechanics, 1960 perturbations about stable axes are localized perturbations about unstable axis wander around entire ellipsoid Energy is slowly transferred from x 2 to x 1 or x 3 However, motion is ultimately reversible

NL wave interactions: Basic paradigms II Three nonlinearly coupled oscillators Formally equivalent to the asymmetric top Assuming weak coupling: A i (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 A i (t) are: As in the asymmetric top, one of the modes can pump energy into the other two at a rate: d ~ Basic timescale for nonlinear transfer of energy However, also like the asymmetric top, such transfer is reversible!

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