Introduction
Transport equations
Examples
Inner Magnetosphere Models
John Haiducek
June, 2014
Summary
Introduction
Transport equations
S INGLE PARTICLE MOTION
Examples
Summary
Gyro motion:
Governed by Lorentz force law:
qv×B
F = q(E + v × B)
B
Mirroring:
v
qv×B
Drift:
B
qv×B
E+g+∇B
B
v
v
Introduction
Transport equations
Examples
T HE B OLTZMANN EQUATION
∂F
δF
+ (v · ∇)F + (a · ∇v )F =
∂t
δt
I
F = F(r, v, t) is a 7-dimensional probability density
function
I
For most space physics problems this cannot be solved
directly
Summary
Introduction
Transport equations
Examples
MHD E QUATIONS
Applying a number of simplifications to the Boltzmann
equation leads to the following fluid equations1 :
∂ρ
+ ∇ · (ρu) = 0
∂t
∂u
+ ρ(u · ∇)u + ∇p − ρg − j × B = 0
∂t
5
3 ∂p 3
+ (u · ∇)p + p(∇ · u) = j · (E + uB)
2 ∂t
2
2
ρ
The most important simplification here is
ZZZ
1
vF(r, v, t)d3 v
v = u + c, where u(r, t) =
n(r, t)
∞
1
Gombosi, Physics of the Space Environment.
Summary
Introduction
Transport equations
Examples
Summary
A SIMPLIFIED KINETIC TREATMENT
A model that incorporates the cyclotron physics of the inner
magnetosphere can be constructed with the following
assumptions2 :
I
Guiding center approximation (cyclotron motion averaged
out)
I
Motion of the guiding centers is bounce averaged
I
Fields and particle distributions change much more slowly
than the bounce period
2
Jordanova et al., “A bounce-averaged kinetic model of the ring current
ion population”.
Introduction
Transport equations
Examples
Summary
T RANSPORT EQUATIONS FOR THE INNER
MAGNETOSPHERE
Coordinate system corresponds to a dipole field:
R0 : Radial distance in equatorial plane (L-shell)
φ: Geomagnetic longitude
s: Distance along field line (eliminated by bounce averaging)
E: Kinetic energy of guiding center
µ0 : Cosine of equatorial pitch angle
Introduction
Transport equations
Examples
This results in a phase-space distribution function
Q = Q(R0 , φ, E, µ0 , t), with the transport equation3 :
1 ∂
∂Q
2 dR0
+ 2
Q
R0
∂t
dt
R0 ∂R0
∂
dφ
1 ∂ √
dE
√
+
Q +
Q
E
∂φ
dt
∂E
dt
E
1
∂
dµ0
δQ
+
f (µ0 )µ0
Q =
f (µ0 )µ0 ∂µ0
dt
δt
3
Angle brackets denote bounce averaging
Summary
Introduction
Transport equations
Examples
Summary
RAM/HEIDI
I
RAM (Ring current
Atmosphere interaction
Model), developed in the
early-mid 1990’s at
University of Michigan4
I
HEIDI (Hot Ion Drift
Integrator), developed
from RAM in 2000’s5
Liemohn et al. 2004
4
Fok et al., “Decay of equatorial ring current ions and associated
aeronomical consequences”.
5
Liemohn et al., “Dependence of plasmaspheric morphology on the
electric field description during the recovery phase of the 17 April 2002
magnetic storm”.
Introduction
Transport equations
Examples
Summary
R ICE C ONVECTION M ODEL (RCM)
I
Assumes pitch angle
distribution is isotropic
I
Treats drift as an advection
process6
δQ
∂
+ vD · ∇ Q(r, t) =
,
∂t
δt
E × B B × ∇W
vD =
+
,
B2
qB2
1
W = mv2
2
6
Toffoletto, 2007
Toffoletto, “Inner Magnetospheric Modeling with the Rice Convection
Model”.
Introduction
Transport equations
Examples
D IFFERENCES BETWEEN I NNER M AGNETOSPHERE
AND MHD MODELS
MHD models
Spatial grid
Solved for
directly
Solved for
directly
3-D
Velocities/
energies
Guiding
center drifts
Bulk velocity
per cell
Not well
captured
Magnetic
fields
Electric field
Inner
Magnetosphere
models
Assumed dipolar, or imposed externally
Empirically modeled, or
imposed externally
2-D (in equatorial plane),
usually polar
Multiple energy bins
Explicitly included
Summary
Introduction
Transport equations
Examples
Summary
The distribution function Q has volume elements of the form
√
dV = R20 Eµ0 f (µ0 )dR0 dφdEdµ0
Z s0m
1
ds
p
where f (µ0 ) =
.
2R0 sm
1 − B(s)/Bm
In order to satisfy conservation it must be that dV = const. This
means that (the Liouville equation)7 :
∂dV ∂R0 ∂dV ∂φ ∂dV ∂E ∂dV ∂µ0
d(dV)
=
+
+
+
=0
dt
∂R0 ∂t
∂φ ∂t
∂E ∂t
∂µ0 ∂t
The derivatives of dV become (using the R0 derivative as an
example):
∂dV
∂ 2√
=
R0 Ef (µ0 )dR0 dφdEdµ0
∂R0
∂R0
√
∂
= Ef (µ0 )dR0 dφdEdµ0
R20
∂R0
7
Jordanova et al., “A bounce-averaged kinetic model of the ring current
Introduction
Transport equations
Examples
Summary
Doing the same for each term in the Liouville equation, we can
divide out most of the coefficients outside the derivatives and
get the following8 :
1 ∂
∂
dφ
1 ∂ √
dE
2 dR0
R0
+
+√
E
2
dt
∂φ
dt
dt
R0 ∂R0
E ∂E
1
∂
dµ0
+
f (µ0 )µ0
=0
f (µ0 )µ0 ∂µ0
dt
8
Jordanova et al., “A bounce-averaged kinetic model of the ring current
ion population”.