Structure Formation and Particle Mixing in a Shear Flow Boundary Layer

Transcription

1 Structure Formation and Particle Mixing in a Shear Flow Boundary Layer Matthew Palotti palotti@astro.wisc.edu University of Wisconsin Center for Magnetic Self Organization Ellen Zweibel University of Wisconsin Center for Magnetic Self Organization Fabian Heitsch University of Michigan

2 Main Points I. Structure Formation In the purely hydrodynamic regime, the Kelvin-Helmholtz Instability develops only a large(er) scale eddy. A weak magnetic field breaks the eddy and leads to structure on smaller scales. After ~20 t s, the MHD instability decays into a new equilibrium flow. II. Mixing The average convective transport layer is about the same for both MHD and HD. Momentum transport by magnetic stresses substantially broadens the flow profile. Efficient mixing accompanies fine structure in MHD model.

3 Kelvin-Helmholtz Instability Kelvin-Helmholtz vortexes are seen in clouds and Vincent Van Gogh s Starry Night

4 The Numerical Method MHD-Xu (1999) Gas-Kinetic Method: All of the MHD equations can be written in the form: " t q + # $ ( r u q) = 0 F q i,j =q(x i,y i ) Y j+1/2 To solve numerically for a grid cell, the flux at each boundary needs to be calculated. To get the flux, first one needs the distribution function. F F F y j-1/2 Left Cell g Right Cell " f t f = 0,right f 0,left X i-1/2 f = f 0 e " t t c + ( 1" e " t t c ) g X i+1/2 BGK Approximation: The collision term can be approximated as: g # f t c (relaxation approximation) Therefore, the distribution function is:

5 Initial Conditions 2-dimensional Grid Resolution - 512x1024 X : [0,1], Y : [-1,1] BC - X : Periodic, Y : Outflow

6 Initial Conditions 2-dimensional Grid Resolution - 512x1024 X : [0,1], Y : [-1,1] BC - X : Periodic, Y : Outflow ρ 0 =1.0, P 0 =1.0, γ=5/3 C s = 5/3) 1.29 t s = 1/c s

7 Initial Conditions 2-dimensional Grid Resolution - 512x1024 X : [0,1], Y : [-1,1] BC - X : Periodic, Y : Outflow ρ 0 =1.0, P 0 =1.0, γ=5/3 C s = 5/3) v 0 /2 t s = 1/c s 0.77 M s = V 0 /c s = 1.0 v x -vs- y v 0 /2 Layer width ~ v 0 /2 v 0 /

8 Initial Conditions 2-dimensional Grid Resolution - 512x1024 X : [0,1], Y : [-1,1] BC - X : Periodic, Y : Outflow ρ 0 =1.0, P 0 =1.0, γ=5/3 C s = 5/3) v 0 /2 t s = 1/c s 0.77 M s = V 0 /c s = 1.0 α = c a /c s = M s /M a = 0.0 HD 0.1 MHD 0 B 0 Defines B 0 v 0 /

9 Initial Conditions 2-dimensional Grid Resolution - 512x1024 X : [0,1], Y : [-1,1] BC - X : Periodic, Y : Outflow ρ 0 =1.0, P 0 =1.0, γ=5/3 C s = 5/3) v 0 /2 t s = 1/c s 0.77 M s = V 0 /c s = 1.0 α = c a /c s = M s /M a = Defines B HD 0.1 MHD Add perturbation to y-velocity Amplitude damped with y-position B 0 v 0 /2

10 Resistivity The weak-field MHD KHI is characterized by the winding up of the magnetic field to the point of reconnection (Malagoli et al. 1996, Frank et al. 1996) - Resistivity is an important parameter!!!!! The numerical resistivity in the gas-kinetic scheme varies from cell to cell and from timestep to timestep and does not scale with resolution. We have added a physical resistivity to our models so that we know that all reconnection events are physical in nature. - Use a resistive length scale ~ (R m ~5x10 3 ) - In the ISM, the R m ~ Need to run convergence study. - Magnetic Energy, Kinetic Energy, Velocity Profile Decay

11 Total Y-Kinetic Energy Saturation Linear Phase Non-linear Phase

12 Energy Partition in MHD 1) Instability Saturates. Saturation mechanism is dynamic instead of magnetic. 2) Magnetic field winds up until its length scale reaches the dissipation length scale. Reconnection occurs. Here kinetic energy is at a minimum. 3) After reconnection, smaller vortexs start to form and wind up, leading to a build-up of kinetic energy. 4) The smaller vortexs wind up the field until it then reconnects ) At the end of the run, both the kinetic and magnetic energy have decayed down to ~0, leaving an equilibrium flow in the x- direction.

13 HD - Density After Saturation In the HD model, no small scale structure develops, only a single vortex in the middle of the grid

14 Energy Partition in MHD

15 MHD - Density 1) Saturation 1.10 As the MHD model reaches saturation, the magnetic field evacuates tiny tubes in the vortex. 0.78

16 Energy Partition in MHD

17 MHD - Density 2) ME Peak, KE Trough 1.05 When the magnetic field is at the point of reconnectin, the stucture is more elongated. 0.64

18 Energy Partition in MHD

19 MHD - Density 3) KE Peak, ME Trough 1.04 As the smaller vortexs are formed, the structure begins to get more complex. 0.71

20 Energy Partition in MHD

21 MHD - Density 4) End of the Run 1.01 At the end of the run, most of the structure is gone, leaving a new equilibrium flow with a wider layer. 0.84

22 Velocity Profile Layer width defined as position where the x- velocity is v 0 /2 Is the MHD model mixed significantly more than the HD?

23 Tracer Particles: Mixing At t=0, tracer particles are placed in the layer ( y < 0.05) Particles are advected through the velocity field Characteristics of mixing: 1) Calculate the RMS y-position of the particles - level of convective momentum transport - Is the mixing layer the same as the velocity layer? 2) Calculate the average y-separation between particle pairs that are initially close together - Measure of chaos in the flow

24 Mixing I: RMS Y-Position Ave. Standard Deviation: HD: 0.10 MHD: 0.12 o Convective momentum transport in y-direction is about the same for HD and MHD models o Mixing layer does not follow the velocity profile

25 Mixing II: Pair Separation Ave. Standard Deviation: HD: MHD: o The MHD model is more mixed than the HD model

26 Velocity Profile -vs- Particle Mixing Layer o The stress force is: y (< ρv x v y > x -<B x B y > x ) othe total stress force is much greater in the MHD model than in the HD model o The velocity profile is determined by the total stress force o Which component of the stress force contributes the most to the profile decay?

27 Velocity Profile Decay: Components of Stress o Magnetic stress force dominates o Magnetic stress tends to spread out the velocity layer o Particle tracers are not affected by magnetic stress o shear stress for MHD ~ shear stress for HD. o Shear stress tends to determine particle mixing layer

28 What s Next? Resistivity Study - Can we extrapolate to ISM R m? Mach Number Study - change the compressibility in the flow. Maybe more structure formation? Magnetic Field Study - Magnetic tension will resist the instability. How does this affect the structure formation and mixing? Move to 3D - Is there anything new to learn in 3D? Two Fluids - How does ion-neutral friction affect structure formation and mixing.

29 Main Points I. Structure Formation In the purely hydrodynamic regime, the Kelvin-Helmholtz Instability develops only a large(er) scale eddy. A weak magnetic field breaks the eddy and leads to structure on smaller scales. After ~20 t s, the MHD instability decays into a new equilibrium flow. II. Mixing The average convective transport layer is about the same for both MHD and HD. Momentum transport by magnetic stresses substantially broadens the flow profile. Efficient mixing accompanies fine structure in MHD model.