Structure Formation and Particle Mixing in a Shear Flow Boundary Layer

May 23, 2006 SINS meeting

Structure Formation and Particle Mixing in a Shear

Flow Boundary Layer

Matthew [email protected] of Wisconsin

Center for Magnetic Self Organization

Ellen ZweibelUniversity of Wisconsin

Center for Magnetic Self Organization

Fabian HeitschUniversity of Michigan


Main Points

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.

I. Structure Formation

II. Mixing


Kelvin-Helmholtz Instability

Kelvin-Helmholtz vortexes are seen in clouds…

…and Vincent Van Gogh’s “Starry Night”


The Numerical MethodMHD-Xu (1999)

Xi-1/2 Xi+1/2

yj-1/2

Yj+1/2

qi,j=q(xi,yi)

F

F

FF

Right Cell

Left Cell

€

∂tq+∇ ⋅(r u q) = 0

€

f = f0e− ttc + 1− e

− ttc

( )g

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:The collision term can be approximated

as:

Therefore, the distribution function is:

f0,right

g

f0,left

€

∂t f =g− f

tc

(relaxation approximation)


Initial Conditions

2-dimensional Grid Resolution - 512x1024X : [0,1], Y : [-1,1]BC - X : Periodic, Y : Outflow

0 1-1

1

0


Initial Conditions


0 1-1

1

0

0=1.0, P0=1.0, =5/3 Cs = √(5/3) ≈ 1.29

ts = 1/cs ≈ 0.77

May 23, 2006 SINS meeting 0 1-1

1

0

Initial Conditions


0=1.0, P0=1.0, =5/3 Cs = √(5/3) ≈ 1.29

ts = 1/cs ≈ 0.77

Ms = V0/cs = 1.0

v0/2

v0/2

vx -vs- y v0/2

-v0/2

Layer width ~ 0.1


1

0

Initial Conditions


0=1.0, P0=1.0, =5/3 Cs = √(5/3) ≈ 1.29

ts = 1/cs ≈ 0.77

Ms = V0/cs = 1.0 = ca/cs = Ms/Ma = 0.0 HD

0.1 MHDDefines B0

v0/2

v0/2

BB00


1

0

Initial Conditions


0=1.0, P0=1.0, =5/3 Cs = √(5/3) ≈ 1.29

ts = 1/cs ≈ 0.77

Ms = V0/cs = 1.0 = ca/cs = Ms/Ma = 0.0 HD

0.1 MHDDefines B0

v0/2

v0/2

BB00

Add perturbation to y-velocityAmplitude damped with y-position


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


Total Y-Kinetic Energy

Linear Phase

Non-linear Phase

Saturation


Energy Partition in MHD

1

2

3

4

5

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.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.


HD - DensityAfter Saturation

1.10

0.710 1

-0.5

0.5

In the HD model, no small scale structure develops, only a single vortex in the middle of the grid.


1 3 4

2



MHD - Density1) Saturation

1.10

0.78

As the MHD model reaches saturation, the magnetic field evacuates tiny tubes in the vortex.


1 3 4

2



MHD - Density2) ME Peak, KE Trough

1.05

0.64

When the magnetic field is at the point of reconnectin, the stucture is more elongated.


1 3 4

2



MHD - Density3) KE Peak, ME Trough

1.04

0.71

As the smaller vortexs are formed, the structure begins to get more complex.


1 3 4

2



MHD - Density4) End of the Run

1.01

0.84

At the end of the run, most of the structure is gone, leaving a new equilibrium flow with a wider layer.


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?


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


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


Mixing II:Pair Separation

Ave. Standard Deviation:

HD: 0.0045

MHD: 0.0062

o The MHD model is more mixed than the HD model


Velocity Profile-vs-

Particle Mixing Layero The stress force is: ∂y(< vxvy>x-<BxBy>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?


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


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.


Main Points

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.

I. Structure Formation

II. Mixing

Structure Formation and Particle Mixing in a Shear Flow Boundary Layer

Documents

magnetic energy

bc x

weak magnetic field

magnetic stresses

grid cell

magnetic field winds

buildup of kinetic energy

gaskinetic method