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. 595Keating & 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 Rem-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 Rem.

In a nutshell…

Statement of the problem

Magnetic potential: B = r A £ ey

Induction equation: t A + u¢r A = r2 A

where A(x, z, t) = hAi + a(x, z, t)

Introduce turbulent flux: hu ai = - T r hAi

then t hAi2 = 2 ( +

T) hBi2

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 energydensity

(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

Rem = U L / c

X2 = B02 / h u2 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 n

orm

aliz

ed c

ross

-pha

se

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 A2 to small scales

inverse cascade of A2 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 eddies dispersive waves:

“Wavy MHD” = MHD + dispersive waves

Coriolis forcebuoyancy• MHD + additional body forces:

Rossby wavesinternal 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 waveAlfven wave

dominant for large scales (small k)

dominant for small scales (large k)

dispersivenon-dispersive

small scales large scalesl*

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 waveAlfven wave

dominant for large scales (small k)

dominant for small scales (large k)

dispersivenon-dispersive

small scales large scales

Brunt-Vaisala frequency

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 (I3 > I2 > I1)

• for an ensemble of triads, origin of irreversibility is chaos induced by multiple overlapping wave resonances

Landau &

Lifschitz, Mechanics, 1

960

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

Hor

izon

tal p

hase

vel

ocity

(v p

h,x)

Vrms

B0

Strong turbulence Weak turbulence

Non-dispersive Dispersive

Scale (k-1)l*

I. Alvenic Range

II. IntermediateRange

III. WavyRange

L*

Spectral ranges

I. Alvenic Range

II. IntermediateRange

III. WavyRange

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 towave 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 orderin 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 |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’, + ’):

A useful analogy I

Fourth-order flux

~

Second-order flux

~

A useful analogy II

NL wave interactions: Basic paradigms I

• Free asymmetric top (I3 > I2 > I1)

Landau & Lifschitz, Mechanics, 1960perturbations about stableaxes 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

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:

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