A Theory for the dynamical origin of intermittency in kinetic Alfvén wave turbulence

P.W. TerryK.W. SmithUniversity 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

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?

Motivation

Study intermittency in kinetic Alfvén wave turbulence model

Reduced MHD + ñe

Large scale (k<<i-1): system evolves as MHD + passive ñe

Small scale (k>>i-1): flow decoupleskinetic Alfvén waves

Can be modeled by coupled ñe and B

2-field intermittency study (Craddock, et al., 1991)

Decaying turbulenceResistivity << diffusivityObserved: coherent current filaments, high kurtosis

QuestionsCan density structures form? Effect of high diffusivity?

Steady state?

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~JB) 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?

Motivation

How does electron density evolve in magnetic turbulence?

where

Basic model: RMHD + compressible electron continuity

Field Term (large scale)Turbulent Alfvén wave

Term (small scale)Kinetic Alfvén wave

Electron density Elec. advection ( flux): vn

|| compr. (|| flux):B J

Flow Lorentz force: B J Ion advection: vv

Magnetic field Parallel electric field: B Elec. pressure: B n

Kinetic Alfvén wave turbulence

Fluctuations change character across the gyroradius scale (Fernandez et al.)

Large scales (ki << 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 (ki > 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:

ki << 1: transfer dominated by v B

ki 1: transfer dominated by n B

€

v ⋅∇n0

Low k: v and B equipartitioned Density at level dictated by

High k: n and B equipartitioned, even if nolinear or external drive of density

v and B decouple

Low k - high k crossover at ki < 1 1

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 backgroundDamping is ad hoc in n, allows regimes analogous to low,

high magnetic Prandtl number of MDHPrevious study (Craddock et al.)

2DDecaying turbulence

>> Coherent structures observed in current

€

∂ψ∂t

− B ⋅∇n = η∇2ψ

∂n∂t

+ B ⋅∇∇2ψ = υ∇2n

€

B = B0 +∇ψ × ˆ z

€

J = ∇2ψ ˆ z €

∂ /∂z → 0( )

Kinetic Alfvén wave turbulence

Intermittent current structures emerged as turbulence decayed

Kurtosis of current Current

contours

Cuts acrossstructure

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 upKurtosis in vorticity increased to 30 Stream function stayed GaussianCascade connection: enstrophy (large kurtosis)small scale energy (small kurtosis)large scaleStructures:

Gaussian curvature profile CT(r)

(mean sq. shear stress - vorticity sq.)

Core: CT <<1

Edge: CT >>1

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

Intermittency

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 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 <<1 - core

CT >>1 - edge

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

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 B

in which KAW propagate

Shear flow V of vortex

KAW 2D Navier Stokes

€

Jz(r) = μ 0(∇× B)z

(current)

€

ωz (r) = (∇× V)z

(vorticity)

€

Bϑ (r) → VA(r)

€

Vϑ (r)

€

˜ b r , ˜ b ϑ , ˜ n

€

˜ v r , ˜ v ϑ

€

n0(r), Jz (r) (structure)

€

ωz (r) (structure)

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 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

Dynamics

Use Fourier-Laplace transform to distinguish two time scales

Where

Describe rapid evolution with Fourier-Laplace transform:

Recover slow evolution from average over t:

€

ψ =ψ0(r,τ ) + ˜ ψ (r,ϑ , t)

n = n0(r,τ ) + ˜ n (r,ϑ , t)

Structure: Slowly evolvingAzimuthally symmetric

Turbulence: rapidly evolvingAzimuthal variation

€

˜ j z (r,ϑ , t) =∇ 2 ˜ ψ =1

r

∂

∂rr

∂ ˜ ψ

∂r

⎛

⎝ ⎜

⎞

⎠ ⎟+

1

r2

∂ 2 ˜ ψ

∂ 2ϑ

€

˜ j z (r,ϑ , t) =1

2πidγ

−i∞+γ 0

i∞+γ 0∫m=1

∑ ˜ j γ ,m (r)e imϑ eγt

€

ψ0(r,τ ) = ψt= ψ dt∫ =

1

2πiψ γ (τ ) dγ

−i∞+γ 0

i∞+γ 0

∫

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

€

∂n0

∂τ+ ˜ b ⋅∇˜ j z t

= υ∇ 2n0

€

∂ψ0

∂τ+ ˜ b ⋅∇ ˜ n

t= η∇ 2ψ 0

€

∂ψγ',m '

∂r˜ j γ −γ ',m−m' →

∂ψ γ ',m'

∂rPγ −γ ',m−m'

−1 ∂ψ γ −γ ',m−m'

∂r ∂n0

∂r

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

€

γ ˜ ψ γ ,m + B(r) ⋅∇ ˜ n γ ,m + ˜ b ⋅∇ ˜ n + ˜ b ⋅∇n0(r) = 0

€

γ˜ n γ ,m + B(r) ⋅∇ ˜ j γ ,m + ˜ b ⋅∇˜ j + ˜ b ⋅∇j0(r) = 0

€

Bϑ (r)1

r

∂

∂ϑ˜ n + ˜ b r

∂

∂r˜ n

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) :

€

Bϑ (r) = rBϑ (r0)

r0

+ (r − r0)∂

∂r

Bϑ

r

⎛

⎝ ⎜

⎞

⎠ ⎟+ ...

⎡

⎣ ⎢

⎤

⎦ ⎥

€

mΔr∂

∂r

Bϑ

r

⎛

⎝ ⎜

⎞

⎠ ⎟r0

˜ n ≈ ˜ b r1

Δr˜ n

€

r2 =˜ b r

m∂

∂r

Bϑ

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€

˜ n ≈ Δr∂n0

∂r

€

Bϑ

1

r

∂

∂ϑ˜ n = br

∂

∂r˜ n

€

Bϑ

1

r

∂

∂ϑ˜ j z = br

∂

∂r˜ j z

Dynamics

Conclusions

Small scale kinetic Alfvén turbulence can be modeled with two-field equation

Applicable for ki > 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 mixingSuppression of mixing applies both to current and density

KAW turbulence dynamics non Gaussian density fluctuations