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Structure Formation and Particle Mixing in a Shear Flow Boundary Layer

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. Fabian Heitsch University of Michigan. Ellen Zweibel University of Wisconsin Center for Magnetic Self Organization.

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Structure Formation and Particle Mixing in a Shear Flow Boundary Layer

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  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 Fabian Heitsch University of Michigan Ellen Zweibel University of Wisconsin Center for Magnetic Self Organization SINS meeting

  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 ts, the MHD instability decays into a new ‘equilibrium flow’. • 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. II. Mixing SINS meeting

  3. Kelvin-Helmholtz Instability Kelvin-Helmholtz vortexes are seen in clouds… …and Vincent Van Gogh’s “Starry Night” SINS meeting

  4. Yj+1/2 F qi,j=q(xi,yi) F F Right Cell Left Cell F yj-1/2 Xi-1/2 Xi+1/2 The Numerical MethodMHD-Xu (1999) Gas-Kinetic Method: All of the MHD equations can be written in the form: 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. BGK Approximation: g The collision term can be approximated as: (relaxation approximation) f0,right f0,left Therefore, the distribution function is: SINS meeting

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

  6. 1 ts = 1/cs ≈ 0.77 0 -1 0 1 Initial Conditions • 2-dimensional Grid Resolution - 512x1024 X : [0,1], Y : [-1,1] BC - X : Periodic, Y : Outflow • 0=1.0, P0=1.0, =5/3 Cs = √(5/3) ≈ 1.29 SINS meeting

  7. 1 v0/2 ts = 1/cs ≈ 0.77 0 vx -vs- y v0/2 v0/2 -1 -v0/2 0 1 Initial Conditions • 2-dimensional Grid Resolution - 512x1024 X : [0,1], Y : [-1,1] BC - X : Periodic, Y : Outflow • 0=1.0, P0=1.0, =5/3 Cs = √(5/3) ≈ 1.29 • Ms = V0/cs = 1.0 Layer width ~ 0.1 SINS meeting

  8. 1 v0/2 ts = 1/cs ≈ 0.77 0 v0/2 -1 0 1 Initial Conditions • 2-dimensional Grid Resolution - 512x1024 X : [0,1], Y : [-1,1] BC - X : Periodic, Y : Outflow • 0=1.0, P0=1.0, =5/3 Cs = √(5/3) ≈ 1.29 • Ms = V0/cs = 1.0 B0 0.0 HD •  = ca/cs = Ms/Ma = 0.1 MHD Defines B0 SINS meeting

  9. 1 v0/2 ts = 1/cs ≈ 0.77 0 v0/2 -1 0 1 Initial Conditions • 2-dimensional Grid Resolution - 512x1024 X : [0,1], Y : [-1,1] BC - X : Periodic, Y : Outflow • 0=1.0, P0=1.0, =5/3 Cs = √(5/3) ≈ 1.29 • Ms = V0/cs = 1.0 B0 0.0 HD •  = ca/cs = Ms/Ma = 0.1 MHD Defines B0 • Add perturbation to y-velocity Amplitude damped with y-position SINS meeting

  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 ~ 0.00002 (Rm~5x103) - In the ISM, the Rm ~ 1015 • Need to run convergence study. - Magnetic Energy, Kinetic Energy, Velocity Profile Decay SINS meeting

  11. Saturation Non-linear Phase Linear Phase Total Y-Kinetic Energy SINS meeting

  12. Energy Partition in MHD 1) Instability Saturates. Saturation mechanism is dynamic instead of magnetic. 2 2) Magnetic field winds up until its length scale reaches the dissipation length scale. Reconnection occurs. Here kinetic energy is at a minimum. 3 4 3) After reconnection, smaller vortexs start to form and wind up, leading to a build-up of kinetic energy. 1 5 4) The smaller vortexs wind up the field until it then reconnects. 5) 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. SINS meeting

  13. 1.10 0.71 HD - DensityAfter Saturation 0.5 In the HD model, no small scale structure develops, only a single vortex in the middle of the grid. -0.5 0 1 SINS meeting

  14. Energy Partition in MHD 2 1 3 4 SINS meeting

  15. 1.10 0.78 MHD - Density1) Saturation As the MHD model reaches saturation, the magnetic field evacuates tiny tubes in the vortex. SINS meeting

  16. Energy Partition in MHD 2 1 3 4 SINS meeting

  17. 1.05 0.64 MHD - Density2) ME Peak, KE Trough When the magnetic field is at the point of reconnectin, the stucture is more elongated. SINS meeting

  18. Energy Partition in MHD 2 1 3 4 SINS meeting

  19. 1.04 0.71 MHD - Density3) KE Peak, ME Trough As the smaller vortexs are formed, the structure begins to get more complex. SINS meeting

  20. Energy Partition in MHD 2 1 3 4 SINS meeting

  21. 1.01 0.84 MHD - Density4) End of the Run At the end of the run, most of the structure is gone, leaving a new equilibrium flow with a wider layer. SINS meeting

  22. Velocity Profile • Layer width defined as position where the x-velocity is v0/2√2 Is the MHD model mixed significantly more than the HD? SINS meeting

  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: • Calculate the RMS y-position of the particles - level of convective momentum transport - Is the mixing layer the same as the velocity layer? • Calculate the average y-separation between particle pairs that are initially close together - Measure of chaos in the flow SINS meeting

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

  25. Mixing II:Pair Separation Ave. Standard Deviation: HD: 0.0045 MHD: 0.0062 • The MHD model is more mixed than the HD model SINS meeting

  26. Velocity Profile-vs-Particle Mixing Layer • The stress force is: ∂y(< vxvy>x-<BxBy>x) • The total stress force is much greater in the MHD model than in the HD model • The velocity profile is determined by the total stress force • Which component of the stress force contributes the most to the profile decay? SINS meeting

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

  28. What’s Next? • Resistivity Study - Can we extrapolate to ISM Rm? • 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. SINS meeting

  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 ts, the MHD instability decays into a new ‘equilibrium flow’. • 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. II. Mixing SINS meeting

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