1. “Harris” Equilibrium: Initial State for a Broad Class of
Symmetric Reconnection Simulations
Bx=B0
Bx=-B0
Bx(y)/B0
nH(y)/nH0
Small perturbation at center (x=100di) triggers reconnection
Equilibrium is unaffected by addition of uniform background plasma and out-of-plane “guide” magnetic field Bz0

2. Outline
First treat Harris equilibrium in context of fluid and MHD descriptions
Balance of fluid and magnetic pressure
Current responsible for reversal of magnetic field is localized to a narrow “sheet” and results from counterstreaming electrons and ions undergoing “diamagnetic” drift
Later show that Harris current sheet is also an exact “kinetic” equilibrium (i.e., a solution of the time-independent Vlasov equation)
Solution is simplified after transforming into frame where the diamagnetic drift vanishes (different for each species)
Interpretation requires special relativity
Many detours along the way

3. Diamagnetic Drift General Expression for Uniform B
Electrons contributing to f(v) at origin for uniform out-of-plane B in pressure gradient
(e.g., density gradient with uniform temperature)
Guiding-center view
Velocity-space view
More electrons move in +vx than –vx direction resulting in net rightward drift (opposite for ions)
General Expression for Uniform B
(analogous to ExB drift)
(force per particle)
(isotropic pressure in CM frame)

4. Explicit Form of Harris Equilibrium
nH(y)/nH0
Bx(y)/B0
d is current-sheet thickness
(=1/2 in simulation example)
From (static) Ampere’s Law
Pressure Balance
(magnetic pressure)
Plasma Pressure [Ptot – PB]
Pressure Force

5. Is the Current Diamagnetic?
Diamagnetic Drift of Species s
[note extra minus sign from cross product]
Diamagnetic Current of Species s
ne≈ni≈nH (quasineutrality)
Total Current Consistent with Value from Ampere’s Law
The diamagnetic drift is independent of y so the current is modulated by the density only. Hotter species (typically ions in magnetosphere) contributes most to the current.
[consistent despite approximations (e.g., B gradient). Need kinetic analysis for confirmation]
A complementary model is the “force-free” equilibrium in which B rotates through the sheet with constant magnitude in such a way that J is everywhere parallel to B. The density is also uniform across the current sheet.

6. Transport of Electromagnetic Energy
Electromagnetic Transport from Maxwell’s Equations
Evaluate following expression
to get transport equation
relating the rate of change of energy density to the divergence of a flux
and a source/sink term coupling fields and particles
[Useful vector identity]
[vector form]
[component form]

7. Relation Between EM and Kinetic Energy Transport
Note that the “Joule” term (J·E) is a source for kinetic energy but a sink for electromagnetic energy. This term is sometimes referred to as “Joule dissipation,” but dissipation implies irreversibility (i.e., entropy increase), which does not always apply.
The kinetic energy flux Q includes coherent flux, enthalpy flux, and heat flux. The kinetic energy density UKE contains contributions from both thermal and ram pressure. However, in general the pressure is a rank-two tensor, with additional transport equations describing the evolution of the individual components.
Global energy conservation follows from assumption that there is no flux through boundaries enclosing the entire system

8. Frame Transformation of Electromagnetic Fields
Lorentz Transformation
The following term (from MHD) can be interpreted as the electric field in the co-moving frame
The correction to B in the co-moving frame is usually ignored in the magnetosphere because |E|<<|B| for typical field strengths when converted into commensurate (e.g., Gaussian) units.
Neglect of electric pressure relative to magnetic pressure is similarly justified

9. How Do You Follow Magnetic Field Lines As System Evolves?
Difficult in 3D, but analysis simplifies for 2D geometry
Identify in-plane B-field lines with contours of constant Az (z-component of vector potential)
In-plane B normal to gradient of Az
Density of contours proportional to field strength
Need location where field line is assumed stationary (e.g., corner of simulation) to set constant of integration and anchor the plotted set of in-plane field lines
Even in a 2D simulation, the out-of-plane Bz (initially uniform) develops inhomogeneity (e.g., Hall-B), which influence the orientation of the field lines in 3D

10. Harris Equilibrium Redux: Kinetic Theory

11. Transform to Frame Co-Moving with Species ‘s’
Reminders from Previous Slides
Fluid Harris
Relativistic Frame Transformation
Transform Electric Field into Diamagnetically Drifting Frame
Charge Neutrality Is Violated in Co-Moving Frame
How can charge density be frame dependent???

12. Harris as “Seen” by Electrons and Ions
Electron rest frame
Ey(y) [diverging]
r=dEy/dy > 0
Lorentz contraction! z
Harris frame
Ion rest frame
Ey(y) [converging]
r=dEy/dy < 0

13. Does Distribution in Co-Moving Frame Satisfy Vlasov Equation?
Electrostatic:
Assume f is isotropic in CM frame:
Boltzmann Distribution Is Valid Solution of Vlasov for These Conditions
Kinetic Solution Is Identical to Fluid Solution

14. “Generalized” Harris-Like Equilibria
Numerical Construction via ODEs
[Quasineutral]
[Electric field In rest frame of species “s”] ;
[Boltzmann]

15. Drifting and Background Electrons and Ions
Examples
Drifting and Background Electrons and Ions
Standard Harris
(Ti=5Te)
dvz=-0.5viz(Harris)
No drifting ions
Density
B,f
(all examples have the same maximum current)

16. Initial Harris B-Field Profile Without Harris Current Results in Alternative Equilibrium in PIC Simulation

17. Further Generalize to Non-Parallel Drifts in vz-vx Plane
Note: Boltzmann distributions support local – but not global -- asymmetry

18. Non-Uniform Potential Provides Route to Asymmetry
Low-energy electrons reflected by potential barrier
f(y)
Upstream
At minimum f
(Boltzmann)
Downstream
(low-energy orbits vacant)
Example
f(|v|)
3D-isotropic
Different upstream distributions on opposite sides of current sheet can break the symmetry
f(y)

19. Asymmetric Six-Population Model Used to Model Magnetopause Crossing