Dissipation Scales & Small Scale Structure

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1 Dissipation Scales & Small Scale Structure Ellen Zweibel Departments of Astronomy & Physics University of Wisconsin, Madison and Center for Magnetic Self-Organization in Laboratory and Astrophysical Plasmas Dissipation Scales &Small Scale Structure p.1/26

2 Basic Problem What are the smallest possible scales for fluctuations of v, B, n α, & T? The Plan of this Talk Passive scalars & dynamical quantities Diffusion, especially in the ISM Sources & sinks Nonlinear diffusion Missing pieces Dissipation Scales &Small Scale Structure p.2/26

3 Dual Nature of Turbulent Diffusion Smooths large scale gradients with effective diffusivity D turb LV which depends only weakly on molecular diffusivity Produces intermittent small scale structure Dissipation Scales &Small Scale Structure p.3/26

4 An Example Diffusion of a centered blob of tracer by a single scale, 2D, time dependent chaotic flow. From Heitsch et al Dissipation Scales &Small Scale Structure p.4/26

5 Transport of a Passive Scalar Q is a passive scalar if v is independent of Q. where Q t + v Q = Q v + D Q Q + S Q, D Q is the diffusion tensor S Q represents sources & sinks. Dissipation Scales &Small Scale Structure p.5/26

6 Solution of Transport Equation When D Q & S Q 0 this has an exact solution. If x(x 0,t) is position of a fluid element at t that was at x 0 at t = 0, and D ij x i x 0j then Q(x,t) = D 1 Q(x 0, 0); Q = D 1 0 Q The gradients become large as fluid parcels wander away from each other. What are statistics of Q distribution? Dissipation Scales &Small Scale Structure p.6/26

7 Statistics of Q Assume viscously damped Kolmogorov turbulence which cuts off at l K. Energy input rate Ė ρv 3 Constant energy throughput v l V ( l L ) 1/3; E v (k) k 5/3. Turnover rate γ l v l l V ( ll ) 2/3 L Inertial range ends at l K where Γ = γ l L l K L ( Dv LV ) 3/4. Dissipation Scales &Small Scale Structure p.7/26

8 Three Regimes of Q Spectra Define Prandtl number Pr Q D v /D Q & cutoff length l Q l K Pr 3/4 Q. min (l Q,l K ) < k 1 < L: E Q (k) k 5/3 (Oboukhov - Corrsin). Pr Q 1; l K < k 1 < l Q : E Q (k) k 4 E v (k) k 17/3 (Batchelor - Howell - Townsend). Pr Q 1; l Q < k 1 < l K : E Q (k) k 1 (Batchelor). Dissipation Scales &Small Scale Structure p.8/26

9 Spatial Structure of Q Mixing of a scalar by Kraichnan model: sharp fronts between excess & deficit, lots of structure. From Shraiman & Siggia Dissipation Scales &Small Scale Structure p.9/26

10 Ramp & Cliff Structure 1D slice through a turbulent mixing model, described as ramps" & cliffs". Time trace at a point is similar. From Holzer & Siggia Dissipation Scales &Small Scale Structure p.10/26

11 Statistics of Q PDF of fluctuation gradient along concentration gradient. Diffusivity D Q decreases upward. From Holzer & Siggia Dissipation Scales &Small Scale Structure p.11/26

12 Heuristics of Structure Formation Fast flows between heteroclinic stagnation points give strong coherent transport, act like jets. In the model shown the stagnation points sweep through at a fxed pattern speed; in a real, multiscale flow they form at random. Dissipation Scales &Small Scale Structure p.12/26

13 Collisional Dissipation: I Ohmic diffusion Viscosity In CNM In WIM D m δ2 e τ e T 3/2 cm 2 s 1 D v v H λ H n 1 cm 2 s 1 D v v i λ i 10 8 T 5/2 n 1 cm 2 s 1. D v v i min(r i,λ i ) v i r i B 1 µ cm 2 s 1. P r D v /D m 1. Dissipation Scales &Small Scale Structure p.13/26

14 Collisional Dissipation: II Ambipolar diffusivity D AD v 2 A τ ni (n n /n i ) ( Bµ n n ) 2 cm 2 s 1. In CNM, D v /D AD 1.5n i /B 2 µ < 1. Diffusion of density In CNM, D ρ D v In WIM, D ρ D v ; D ρ βd m 0. Dissipation Scales &Small Scale Structure p.14/26

15 Collisionless Dissipation Landau damping of fluctuations with δe B 0. Cyclotron damping of fluctuations with kr i 1. Maximum cross field D ρ v i r i D v Unlikely that cascade survives below kinetic damping scales. Smallest possible scale is electron MHD, on scales << ion skin depth, 200 n 1/2 i km. Dissipation Scales &Small Scale Structure p.15/26

16 Sources & Sinks Can minority ions in CNM be advected away from equilibrium? τ rec (α R n e ) 1. Transport rate > recombination rate for l < ( V L 1/3 α R n e ) 3/2.001( V 6 n e L 1/3 100 ) 3/2 pc Yes, on small scales. Dissipation Scales &Small Scale Structure p.16/26

17 Radiation & Thermal Conduction Radiative cooling damps compressive modes when turnover time cooling time & takes a bite out of the density cascade. Thermal diffusivity in WIM is D T 2 3 κ nk B 10 9T5/2 n cm2 s 1. At T 10 4, Pr T 0.1, so T spectrum steepens at lower k than v spectrum. Conduction suppressed to B. Pr T Pr T ω ci τ i (m i /m e ) 10 7 (B µ /n)pr T 1. Could have a k 1 T spectrum below viscous cutoff. Dissipation Scales &Small Scale Structure p.17/26

18 Dissipation of Waves Ambipolar drift: Nonpropagation (strong friction) for k l < k < k u ; k l 2ν ni /v A ; k u ν in /2ν ni. Γ in k2 v 2 A 2ν ni ; k < k l Γ in ν in /2; k > k u. Viscosity: Γ v = k D k Ohmic diffusion: Γ m = k 2 D m Γ v & Γ m also saturate on short scales. Dissipation Scales &Small Scale Structure p.18/26

19 Magnetic Fluctuations for l < l K Magnetic fields are sheared by the smallest eddies into folded structures, down to the resistive scale. From Schekochihin et al Dissipation Scales &Small Scale Structure p.19/26

20 Spectrum from Numerical Simulation Kinetic energy spectrum is steeper than magnetic spectrum beyond the viscous cutoff (arrow). From Cho et al The spectrum is reminiscent of the Batchelor process. Dissipation Scales &Small Scale Structure p.20/26

21 Ambipolar Diffusion D AD > D v replace l K with larger ambipolar cutoff? Is the cascade terminated by AD? Increase of γ l & saturation of damping with l could permit re-emergence of cascade, involving only magnetic fields & plasma, at small l. Turbulence transports B, & plasma with it, with respect to matter: turbulent ambipolar diffusion". Dissipation Scales &Small Scale Structure p.21/26

22 Turbulent Ambipolar Drift In a turbulent flow, small scale ion - neutral drifts separate the field from the gas on an eddy turnover time (EGZ 2002, Heitsch et al 2004). Dissipation Scales &Small Scale Structure p.22/26

23 Nonlinear Diffusion When the diffusion coefficient depends on the quantity being diffused, as D AD B 2 does, diffusion can create sharp fronts. At a magnetic null, plasma slides down the magnetic pressure gradient, carrying the field along and sharpening the profile. If P = 0, B x 1/3 in a steady state. From Heitsch & Zweibel Dissipation Scales &Small Scale Structure p.23/26

24 Density Profile Profiles of n i & v i in a magnetic neutral sheet. From Heitsch & Zweibel Dissipation Scales &Small Scale Structure p.24/26

25 Summary - CNM CNM l K cm for n=10, L = 100 pc, V = 10 km/s. Magnetic structure survives below viscous/ambipolar scale w k 1 Batchelor spectrum & folded structure Plasma is tied to fieldlines Best prospect for k 5/3 spectrum is re-emergence of plasma cascade well below ion-neutral decoupling scale. WNM Suppression of viscosity allows joint cascade of v & passive tracers to small scales. Dissipation Scales &Small Scale Structure p.25/26

26 Missing Pieces Theory/simulation of turbulent cascade in weakly ionized gas: does ion-neutral friction set the cutoff? Theory/simulation of magnetized turbulence with anisotropic viscosity: what is the spectrum? Theory/simulation of passive scalar advection with anisotropic diffusivity: what are the statistics, where is the cutoff? Dissipation Scales &Small Scale Structure p.26/26