A Theory for the dynamical origin of intermittency in kinetic Alfvén wave turbulence P.W. Terry K.W. Smith University of Wisconsin-Madison Outline Motivation Alfvénic turbulence and kinetic Alfvén wave turbulence Intermittency in decaying KAW turbulence Theoretical description/dynamical origins CMSO Workshop on Intermittency in Magnetic Turbulence, June 20-21, 2005

Motivation Pulsar signal broadening is consistent with intermittent turbulence in ISM (Boldyrev et al.) Pulsar signal dispersed by electron density fluctuations in ISM Broadened signal width scales as R4 (R: distance from source) If pdf of electron density fluctuations is Gaussian, signal width ~ R2 If pdf is Levy distributed (long tail), can recover R4 scaling Key question: is there dynamical path to intermittent density fluctuations in magnetic turbulence? How do density fluctuations evolve in magnetic turbulence? Can density fluctuations become large? At what scale? Can they become intermittent? What is the physics? Recover scaling of broadened pulse from dynamical intermittency?

Study intermittency in kinetic Alfvén wave turbulence model Motivation Study intermittency in kinetic Alfvén wave turbulence model Reduced MHD + ñe Large scale (k<<ri-1): system evolves as MHD + passive ñe Small scale (k>>ri-1): flow decoupleskinetic Alfvén waves Can be modeled by coupled ñe and B 2-field intermittency study (Craddock, et al., 1991) Decaying turbulence Resistivity << diffusivity Observed: coherent current filaments, high kurtosis Questions Can density structures form? Effect of high diffusivity? Steady state?

Pursue a physical approach to intermittency Motivation Pursue a physical approach to intermittency Common approaches: Characterize observations (structure functions, etc.) Statistical (statistical ansatz or theory for pdf) Physical questions: How do coherent structures avoid turbulent mixing? What differentiates them from surrounding turbulence? 3D NS Vortex tube stretching MHD Self pinch by Lorentz force (dv/dt~JB) 2D NS No inward flow on stable structures, but shear of azimuthal flow suppresses mixing KAW No flow, no Lorentz force, no vortex tube stretch What is mechanism?

How does electron density evolve in magnetic turbulence? Kinetic Alfvén wave turbulence How does electron density evolve in magnetic turbulence? Basic model: RMHD + compressible electron continuity where Field Term (large scale) Turbulent Alfvén wave Term (small scale) Kinetic Alfvén wave Electron density Elec. advection ( flux): vn || compr. (|| flux):B J Flow Lorentz force: B J Ion advection: vv Magnetic field Parallel electric field: B f Elec. pressure: B  n

Kinetic Alfvén wave turbulence Fluctuations change character across the gyroradius scale (Fernandez et al.) Large scales (kri << 1): Alfvén waves Coupling: B and v Density: advected passively (no reaction back on B or v) Intermittency: Primary structure is current filament Ancillary structure in vorticity Density tracks flow; flow is integral of vorticity  density not strongly intermittent Small scales (kri > 0.1 in ISM): Kinetic Alfvén waves Coupling: B and n Flow: Dominated by self advection, decouples from B and n Intermittency: Under study

Kinetic Alfvén wave turbulence Electron density fluctuations increase in kinetic Alfvén regime as n equipartitions with B Spectral energy transfer: kri << 1: transfer dominated by v  B kri  1: transfer dominated by n  B Low k: v and B equipartitioned Density at level dictated by High k: n and B equipartitioned, even if no linear or external drive of density v and B decouple Low k - high k crossover at kri < 1

Kinetic Alfvén wave (KAW) modeled by two-field system for B and n Kinetic Alfvén wave turbulence Kinetic Alfvén wave (KAW) modeled by two-field system for B and n where No ion flow; ions are fixed, neutralizing background Damping is ad hoc in n, allows regimes analogous to low, high magnetic Prandtl number of MDH Previous study (Craddock et al.) 2D Decaying turbulence u>>  Coherent structures observed in current

Intermittent current structures emerged as turbulence decayed Intermittency Intermittent current structures emerged as turbulence decayed Kurtosis of current Current contours Cuts across structure Current structures are localized, small scale, circular No intermittency in flux (not localized, kurtosis of 3) Density intermittency not described, but damping was large

Intermittency Intermittency in KAW turbulence has similarities with decaying 2D Navier Stokes turbulence Decaying 2D NS turbulence (McWilliams): Structures emerged from Gaussian start up Kurtosis in vorticity increased to 30 Stream function stayed Gaussian Cascade connection: enstrophy (large kurtosis)small scale energy (small kurtosis)large scale Structures: Gaussian curvature profile CT(r) (mean sq. shear stress - vorticity sq.) Core: CT <<1 Edge: CT >>1

Consider coherent vortex: Dynamics Theory: Strong shear in edge of intense localized vortex disrupts mixing by ambient turbulence (Terry et al.) Consider coherent vortex: Stability  vortex is circular  flow is azimuthal Localized vorticity (zero at some radius)  shear is large at edge Vortex long lived  it is equilibrium for turbulent eddies Closure + WKB: In strong vortex shear, turbulence is localized to periphery of vortex  mixing only in edge of vortex  vortex decays very slowly relative to turbulence Localization of turbulence at edge of vortex  vortex + turbulence: CT <<1 - core CT >>1 - edge

Similar process operates in KAW turbulence (with certain differences) Dynamics Similar process operates in KAW turbulence (with certain differences) No flow, but localized J of coherent structure creates inhomogen-eous B (VA) that localizes turbulence away from structure KAW 2D Navier Stokes Localized coherent structure (origin at center of structure) Inhomogeneous azimu-thal “flow” of structure Turbulence Kinetic Alfven waves Turbulent eddies Turbulence source Agent that localizes turbulence Inhomogeneity of field Bq in which KAW propagate Shear flow Vq of vortex

Describe with two time scale analysis Dynamics Describe with two time scale analysis Slow time scale: Evolution of coherent current filament under mixing produced by inhomogeneous KAW turbulence Rapid time scale: Radial structure of KAW turbulence in quasi stationary magnetic field of coherent current filament Assume coherent structure is azimuthally symmetric, turbulence is not azimuthally symmetric (origin at center of structure) Look at large scales where NL times exceed dissipation times Show: KAW turbulence is localized away from coherent structure Evolution of coherent structure under mixing by KAW turbulence is slow Show if there is coherent structure in n, in addition to J

Use Fourier-Laplace transform to distinguish two time scales Dynamics Use Fourier-Laplace transform to distinguish two time scales  Turbulence: rapidly evolving Azimuthal variation Where Describe rapid evolution with Fourier-Laplace transform: Recover slow evolution from average over t:  Structure: Slowly evolving Azimuthally symmetric

Dynamics Slow time evolution is governed by turbulent mixing of rapidly evolving KAW fluctuations Solve for turbulent quantities (in vicinity of structure) as nonlinear response to structure gradients where P-1 is a turbulent response

Dynamics Fast time equations: nonlinear Alfvén waves propagating in magnetic field of structure, driven by gradients of structure Alfvén waves propagate in inhomogeneous medium of structure When structure has strong variation, Alfvénic turbulence acquires radial envelope with narrow width Seen from balance

Dynamics As variation in structure field becomes large, turbulence localizes to narrow layer In circular structure, waves propagate without distortion of phase fronts if B varies linearly with r Deviation from B~ r describes shear field that distorts propagation Expand B: If shear becomes very large, turbulence must become localized in narrow layer r = (r-r0) :  B

Dynamics Localization strongly diminishes turbulent mixing relative to mixing rate outside structure  Long-lived structures are those with strong shear in B Turbulent mixing is weaker because: 1) Turbulence localized in narrow layer to periphery of structure where structure has strongest variation (strongest turbulent source n0/r) For a given source strength n0/r, turbulence becomes weaker as r becomes narrower Diminishing of mixing governed by balances ,  coherent structure form in both j and n

Conclusions Small scale kinetic Alfvén turbulence can be modeled with two-field equation Applicable for ki > 0.1 Simulations show formation of coherent current filaments in decaying turbulence (large electron diffusivity) Theory shows: •Coherent structures form because mixing is suppressed if structure has large magnetic field shear •Turbulence in structure is kinetic Alfvén wave propagating in inhomogeneous medium •Turbulence is weak and localized to structure periphery when shear in structure field is large  structures avoid mixing Suppression of mixing applies both to current and density KAW turbulence dynamics  non Gaussian density fluctuations