Presentation on theme: "P.W. Terry K.W. Smith University of Wisconsin-Madison Outline"— Presentation transcript:
1 A Theory for the dynamical origin of intermittency in kinetic Alfvén wave turbulence P.W. TerryK.W. SmithUniversity of Wisconsin-MadisonOutlineMotivationAlfvénic turbulence and kinetic Alfvén wave turbulenceIntermittency in decaying KAW turbulenceTheoretical description/dynamical originsCMSO Workshop on Intermittency in Magnetic Turbulence, June 20-21, 2005
2 MotivationPulsar signal broadening is consistent with intermittent turbulence in ISM (Boldyrev et al.)Pulsar signal dispersed by electron density fluctuations in ISMBroadened signal width scales as R (R: distance from source)If pdf of electron density fluctuations is Gaussian, signal width ~ R2If pdf is Levy distributed (long tail), can recover R4 scalingKey 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?
3 Study intermittency in kinetic Alfvén wave turbulence model MotivationStudy intermittency in kinetic Alfvén wave turbulence modelReduced MHD + ñeLarge scale (k<<ri-1): system evolves as MHD + passive ñeSmall scale (k>>ri-1): flow decoupleskinetic Alfvén wavesCan be modeled by coupled ñe and B2-field intermittency study (Craddock, et al., 1991)Decaying turbulenceResistivity << diffusivityObserved: coherent current filaments, high kurtosisQuestionsCan density structures form? Effect of high diffusivity?Steady state?
4 Pursue a physical approach to intermittency MotivationPursue a physical approach to intermittencyCommon 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 stretchingMHD Self pinch by Lorentz force (dv/dt~JB)2D NS No inward flow on stable structures, but shear ofazimuthal flow suppresses mixingKAW No flow, no Lorentz force, no vortex tube stretchWhat is mechanism?
5 How does electron density evolve in magnetic turbulence? Kinetic Alfvén wave turbulenceHow does electron density evolve in magnetic turbulence?Basic model: RMHD + compressible electron continuitywhereFieldTerm (large scale)Turbulent Alfvén waveTerm (small scale)Kinetic Alfvén waveElectron densityElec. advection ( flux): vn|| compr. (|| flux):B JFlowLorentz force: B JIon advection: vvMagnetic fieldParallel electric field: B fElec. pressure: B n
6 Kinetic Alfvén wave turbulence Fluctuations change character across the gyroradius scale (Fernandez et al.)Large scales (kri << 1): Alfvén wavesCoupling: B and vDensity: advected passively (no reaction back on B or v)Intermittency: Primary structure is current filamentAncillary structure in vorticityDensity tracks flow; flow is integral of vorticity density not strongly intermittentSmall scales (kri > 0.1 in ISM): Kinetic Alfvén wavesCoupling: B and nFlow: Dominated by self advection, decouples from B and nIntermittency: Under study
7 Kinetic Alfvén wave turbulence Electron density fluctuations increase in kinetic Alfvén regime as n equipartitions with BSpectral energy transfer:kri << 1: transfer dominated by v Bkri 1: transfer dominated by n BLow k:v and B equipartitionedDensity at level dictated byHigh k:n and B equipartitioned, even if nolinear or external drive of densityv and B decoupleLow k - high k crossover at kri < 1
8 Kinetic Alfvén wave (KAW) modeled by two-field system for B and n Kinetic Alfvén wave turbulenceKinetic Alfvén wave (KAW) modeled by two-field system for B and nwhereNo ion flow; ions are fixed, neutralizing backgroundDamping is ad hoc in n, allows regimes analogous to low, high magnetic Prandtl number of MDHPrevious study (Craddock et al.)2DDecaying turbulenceu>> Coherent structures observed in current
9 Intermittent current structures emerged as turbulence decayed IntermittencyIntermittent current structures emerged as turbulence decayedKurtosis of currentCurrent contoursCuts acrossstructureCurrent structures are localized, small scale, circularNo intermittency in flux (not localized, kurtosis of 3)Density intermittency not described, but damping was large
10 IntermittencyIntermittency in KAW turbulence has similarities with decaying 2D Navier Stokes turbulenceDecaying 2D NS turbulence (McWilliams):Structures emerged from Gaussian start upKurtosis in vorticity increased to 30Stream function stayed GaussianCascade connection:enstrophy (large kurtosis)small scaleenergy (small kurtosis)large scaleStructures:Gaussian curvature profile CT(r)(mean sq. shear stress - vorticity sq.)Core: CT <<1Edge: CT >>1
11 Consider coherent vortex: DynamicsTheory: 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 azimuthalLocalized vorticity (zero at some radius) shear is large at edgeVortex long lived it is equilibrium for turbulent eddiesClosure + WKB: In strong vortex shear, turbulence is localized to periphery of vortex mixing only in edge of vortex vortex decays very slowly relative to turbulenceLocalization of turbulence at edge of vortex vortex + turbulence: CT << coreCT >> edge
12 Similar process operates in KAW turbulence (with certain differences) DynamicsSimilar 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 structureKAW2D Navier StokesLocalized coherent structure (origin at center of structure)Inhomogeneous azimu-thal “flow” of structureTurbulenceKinetic Alfven wavesTurbulent eddiesTurbulence sourceAgent that localizes turbulenceInhomogeneity of field Bqin which KAW propagateShear flow Vq of vortex
13 Describe with two time scale analysis DynamicsDescribe with two time scale analysisSlow time scale:Evolution of coherent current filament under mixing produced by inhomogeneous KAW turbulenceRapid time scale:Radial structure of KAW turbulence in quasi stationary magnetic field of coherent current filamentAssume coherent structure is azimuthally symmetric, turbulence is not azimuthally symmetric (origin at center of structure)Look at large scales where NL times exceed dissipation timesShow:KAW turbulence is localized away from coherent structureEvolution of coherent structure under mixing by KAW turbulence is slowShow if there is coherent structure in n, in addition to J
14 Use Fourier-Laplace transform to distinguish two time scales DynamicsUse Fourier-Laplace transform to distinguish two time scales Turbulence: rapidly evolvingAzimuthal variationWhereDescribe rapid evolution with Fourier-Laplace transform:Recover slow evolution from average over t: Structure: Slowly evolvingAzimuthally symmetric
15 DynamicsSlow time evolution is governed by turbulent mixing of rapidly evolving KAW fluctuationsSolve for turbulent quantities (in vicinity of structure) as nonlinearresponse to structure gradientswhere P-1 is a turbulent response
16 DynamicsFast time equations: nonlinear Alfvén waves propagating in magnetic field of structure, driven by gradients of structureAlfvén waves propagate in inhomogeneous medium of structureWhen structure has strong variation, Alfvénic turbulence acquires radial envelope with narrow widthSeen from balance
17 DynamicsAs variation in structure field becomes large, turbulence localizes to narrow layerIn circular structure, waves propagate without distortion of phase fronts if B varies linearly with rDeviation from B~ r describes shear fieldthat distorts propagationExpand B:If shear becomes very large, turbulence must become localized in narrow layer r = (r-r0) :B
18 DynamicsLocalization strongly diminishes turbulent mixing relative to mixing rate outside structure Long-lived structures are those with strong shear in BTurbulent 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 narrowerDiminishing of mixing governed by balances , coherent structure form in both j and n
19 ConclusionsSmall scale kinetic Alfvén turbulence can be modeled with two-field equationApplicable for ki > 0.1Simulations 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 mixingSuppression of mixing applies both to current and densityKAW turbulence dynamics non Gaussian density fluctuations