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
Top Related