1D VlasovMaxwell Equilibria of Sheared Current Sheets

1
1D Vlasov-Maxwell Equilibria of Sheared Current
Sheets

• Mike Harrison
• Thomas Neukirch
• May 9, 2007

School of Mathematics and Statistics The
University of St. Andrews
2
Motivation
Collisionless Simulation of Reconnection
Solar Corona
Harris Sheet, Harris Sheet Constant Guide
Field used as Initial Configuration
Force-Free
Sheared Magnetic Fields
Is this the correct initial configuration for
studying collisionless reconnection in the
corona?
Aim to carry out collisonless reconnection
simulations using Vlasov-Maxwell Equilibria with
sheared magnetic fields to investigate
differences in Reconnection process
3
(No Transcript)
4
MHD
5
Vlasov Theory
- The single particle distribution
functions for each species
-Probability of finding particles
within the 6 dimensional volume element
centred at the point
Distribution functions depend only on constants
of motion
6
Constants of Motion
Canonical momentum in direction
Particle energy for each species
Canonical momentum in direction
Distribution function may depend on all three
constants of motion
7
Maxwells Equations
We must solve this coupled set of equations
8
Important Restrictions
Force Balance
We have the following relationships

must always satisfy
Existence of a potential P
9
Quasineutrality
Assume length scales much larger than Debye
length
Amperes law reduces to
Set charge density to zero
Hamiltonian equations for a pseudo particle with
coordinates moving in a 2D potential
10
Harris Sheet
One to one Correspondence to MHD
Constant drift velocity in direction
Constant guide field can be added
Constant guide field-adds no free energy
Spatially varying no. density maintains force
balance
Current density perp to magnetic field
11
Force-Free?
Spatially varying drift velocity
Free energy increases if you increase guide
field
Current related to the shear of magnetic field
Is there a distribution function which
gives force-free Harris sheet?
Field aligned currents
12
(No Transcript)
13
(No Transcript)
14
(No Transcript)
15
(No Transcript)
16
Harris sheet one to one correspondence in MHD
and Vlasov-Maxwell theory.
Simple linear force-free solution possible but
highly special case (Bobrova et. al. 2001).
Conclusions
Distribution function for non-linear force-free
equilibria still not known in Vlasov-Maxwell
theory.
Possible to find Vlasov equilibria that have
properties closer to non-linear force-free
equilibria
17
Write 1D PIC code to investigate example
Vlasov-Maxwell Equilibria.
Perform MHD simulations with Lare2D using the
Harris sheet and force-free Harris sheet as
initial conditions.
Perform 2D PIC simulations using example Vlasov
equilibria.
Investigate the differences in reconnection
process compared to current studies using Harris
sheet/Harris sheet constant guide field
WHAT NEXT?
Compare MHD simulations to PIC simulations to
find differences in reconnection processes.
Find additional Vlasov-Maxwell equilibria with
properties closer to that of the force-free
Harris sheet.
Investigate the possibilities of additional
constants of motion.