“Harris” Equilibrium:
Initial State for a Broad Class of
Symmetric Reconnection Simulations
Bx=B0
Bx(y)/B0
nH(y)/nH0
Bx=-B0
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
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
Diamagnetic Drift
Electrons contributing to f(v) at origin for uniform out-of-plane B in pressure gradient
(e.g., density gradient with uniform temperature)
Velocity-space view
Guiding-center 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)
Explicit Form of Harris Equilibrium
d is current-sheet thickness
(=1/2 in simulation example)
From (static) Ampere’s Law
Bx(y)/B0
nH(y)/nH0
Pressure Balance
(magnetic pressure)
Plasma Pressure [Ptot – PB]
Pressure Force
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.
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]
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
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
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 (zcomponent 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
Harris Equilibrium Redux:
Kinetic Theory
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???
Harris as “Seen” by Electrons and Ions
Electron rest frame
Ey(y) [diverging]
r=dEy/dy > 0
Harris frame
z
Ey(y) [converging]
Ion rest frame
r=dEy/dy < 0
Lorentz contraction!
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
“Generalized” Harris-Like Equilibria
Numerical Construction via ODEs
[Quasineutral]
[Electric field In rest frame of species “s”]
;
;
[Boltzmann]
Examples
Drifting and Background Electrons and Ions
dvz=-0.5viz(Harris)
No drifting ions
B,f
Density
Standard Harris
(Ti=5Te)
(all examples have the same maximum current)
Initial Harris B-Field Profile Without Harris Current
Results in Alternative Equilibrium in PIC Simulation
Further Generalize to Non-Parallel Drifts in vz-vx Plane
Note: Boltzmann distributions support local – but not global -- asymmetry
Non-Uniform Potential Provides Route to
Asymmetry
f(y)
Low-energy electrons reflected by potential barrier
Example
f(|v|)
3D-isotropic
Upstream
At minimum f
(Boltzmann)
Downstream
(low-energy orbits vacant)
Different upstream distributions on opposite sides of current sheet can break
the symmetry
f(y)
Asymmetric Six-Population Model Used to Model
Magnetopause Crossing