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The Inner Magnetosphere

Nathaniel Stickley George Mason University

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Overview

• Particle populations– Radiation belts, plasmasphere, ring current

• Particle injection and energization– Diffusion, wave-particle interaction

• Electric fields and drift paths– Shielding, co-rotational electric field, Alfvén layer

• DPS relation– Derivation, discussion

• Modeling– Rice Convection Model (RCM)

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Particle populations

Radiation belts (Van Allen, 1958)

Inner beltLocated at L ≈ 1.1-3.3

Primarily cosmic ray albedo protons of high energy (>10MeV)Very stable

Outer beltLocated at L≈3-9

Primarily high energy electrons with energy up to 10MeVPopulation is unstable (particles are not trapped as efficiently)

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Particle populations

Radiation beltsElectron “slot” region

Located at L ≈ 2.2Apparently due to increased wave-particle interactions There is no corresponding slot for ions

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PlasmasphereCool particles (~1eV-1keV)

High particle density (~103 cm-3)

Extends to L=3-6

Distinct from radiation belts but shares same region of space

Primarily ExB drifting particles (because of low temperature)

Particle populations

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Particle populations

Ring currentMostly indistinguishable from trapped radiation belts

Gradient and curvature drift

Composed of mostly 20-300keV ions

Typically in the range L=3-6

O+ is dominant Ion in terms of abundance

H+ begins to dominate > few keV

Total energy density dominated by O+ and H+

Energy midpoint: 85keV

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Particle injection and energization

How are particles injected into the inner magnetosphere?

Cosmic rays

Ionosphere injection

Substorm and storm particle injections

Diffusion (adiabatic invariants do not strictly hold).

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Brief theoretical aside (1)

Sample derivation of the 1st adiabatic invariant:

Definition of adiabatic invariant:

S p dq •p and q are canonical momentum and coordinates respectively. •Integration is performed over one cycle•If system changes slowly during each cycle, the action S is a constant.

We could use this definition to show that μ is an adiabatic invariant, but we will use a less direct approach in order to illustrate a point.

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Brief theoretical aside (2)

Starting with Faraday’s law:

B

Et

and the equation of motion:

E B

td

m qd

vv

where

Taking the scalar product of this with t= d /dv l

v

Et

dm q

d

vv v

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Brief theoretical aside (3)

Et

dm q

d

vv v

Left hand side is rate of change of KE:

21

2

Evt

d dm q

dt d

l

Variation in KE is: 2 /

0

21

2

Ev t

td

m q dd

l

Key point: if the field changes slowly, this is the same as

21

2

Ev m q dl

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Brief theoretical aside (4) 21

2

Ev m q dl

Re-write using Stokes’ theorem:

21( )

2

E Svm q d

S

Now we use Faraday’s law:

21

2

B

SvtS

m q d

0 B Sd 0 B Sdfor ions for electrons

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Brief theoretical aside (5)2

2 21

2

B B BSv

t t tL

S

mm q d q r q

q B

v

Factoring out 2

2

m

B

v

221 2 1

2 2

m B Bm B

B T

vv

t t

Thus: = constB B

Since 21

2 and B m B B B v

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Particle injection and energization

How particles in the inner magnetosphere become energized:

Electric fields

Wave-particle interaction• Whistlers (review)

Play whistler audio in Adobe Audition:

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Sun

Dawn

Dusk

f

Orientation ReminderC

onvection Electric F

ield

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In the absence of the convection electric field, the plasma sheet appears as illustrated:

What happens when the convection electric field is included?

Polarization field

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Polarization field

Positive charge collects on duskward edge of plasma sheet while negative charge collects on the dawnward edge.

Resulting electric field is called

the “polarization electric field”.

The inner magnetosphere isthus shielded from convection

electric field.

Other features:Over-shielding

Partial ring current

Region 2 Birkeland current

+ + +

- - -

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Overview of current systems

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3 30 0

24

3ˆ ˆ ˆ

t

Ey

E Eotal

B R

q rE

R

r

BE y r r

Co-rotation Electric field (review)3

02

ˆ( ) E Ecorotate

B R

r

E ω r B r

Small due to shielding

Dominates for high-energy particles Dominates for low-energy

particles

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The Alfvén Layer

driftE B

For cool particles, drift dominatesE B

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Recall the shape of the plasmasphere

Compare with the shape of low-temperature Alfvén layer.

The Alfvén Layer

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Hot ions vs. Hot electrons

The Alfvén Layer

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Variability in position:

The Alfvén Layer

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Dessler-Parker-Schopke (DPS) Relation

• Derived originally in 1959 by Dessler and Parker• Relates the total energy in the ring current to the

magnetic field perturbation at the center of the Earth

0

(0) 2

3RC

E

B E

B U

(0) : Magnetic field perturbation at center of EarthB

0: Equatorial surface magnetic field strengthB

: Total energy of ring currentRCE

: Magnetic energy of dipole field beyond Earth's

surfaceEU

30(0)[nT] 3.98 10 [keV]RCB E

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Derivation of DPS Relation (1)2

2 ,2

Starting with and gradient drift: d dg

d d d

m W B

B B qB

B

vv

3

0 003 3

ˆ ˆ3( )4

where Ed d

R BB B

r r L

B μ r r μ

: )

:

distance from center of Earth, (

dipole field

: dipole moment of Earth's field

: kinetic energy of charged particle

E

d

r r LR

B

W

μ

at equatorial plane

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Derivation of DPS Relation (2)we want to find the magnetic field perturbation of a single charge,

so we first calculate the current due to one charge:

drift 2 22 2 2d d d d

d d

q q B BI

r r qB r B

B Bgv

3 30

0 4

3Now, since then E E

d d

R B RB B B

r r

30

4

3and d E

d d d d

B B RB B B

r B

since d dB B

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Derivation of DPS Relation (3)3

0drift 2 5 2

3

2 2d d d E

d d

B B B RI

r B r B

B

03Using substitutions: , , d E

d

W BB r LR

B L

drift 20

3

2 E

WLI

B R

The field at the center of the Earth due to this current is:

drift 0 drift 00 3

0

3ˆ ˆ

2 4rE

I W

r B R

B z z

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Derivation of DPS Relation (4)

There is also a contribution due to gyration of the charge:

0 00 3 3

0

ˆ ˆ4 4

gygyr

E

W

r B R

B z z

The total field perturbation for the charge is:

drift 00 0 3

0

ˆ(0)2

gyr r

E

W

B R

B B B z

The perturbation for all ring current particles combined is:

03

0

ˆ(0)2

RC

E

E

B R

B z

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Derivation of DPS Relation (5)

03

0

ˆ(0)2

From previous slide: RC

E

E

B R

B z

The total dipole field energy outside of the Earth's surface is:

2

0

1 1

2 2E E

E

r R r R

U dV B dV

B H

2 30

0

4

3Skipping integration details... E EU B R

(0)Writing in terms of :EB U0

(0) 2

3RC

E

B E

B U

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Problems with the DPS relation

False assumptions:• Ring current is circular and concentric with Earth• Ring current is azimuthally symmetric (no partial ring)• Magnetic field is purely dipolar• Field perturbation due to ring current is not important

– Nonlinear “feedback” is not accounted for.

• Earth is non-conducting• Assumes ring current is confined to the equatorial plane.

– However Schopke proved that the relation holds for arbitrary pitch angle

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Experimental:• DPS typically estimates Dst index to within 20% (the relation does

not pretend to include affects from other current systems, so this is rather impressive)

Computational:• Liemohn (2003) used computational model of ring current to test

DPS relation1. Calculate realistic particle distributions2. Calculate pressures from particle distributions (include non-zero

pressure outside of volume of integration)3. Calculate currents using pressure information4. Calculate ΔB(0) from currents using Biot-Savart

• DPS systematically over-estimates ΔB(0) for isotropic pressure distribution.

||

BSI

DPS

B P

B P

Testing the DPS relation

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Modeling the Inner Magnetosphere

• Goals– Calculate-particle distributions / drifts– Self-consistently calculate electric fields– Self-consistently calculate magnetic fields– Couple inputs and outputs to global MHD and

ionosphere models

• What physics should be included in such a model?

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Rice Convection ModelMulti-fluid model (typically 100 fluids)Self-consistently computes electric fieldsIsotropic particle distributionCalculates adiabatic driftsRequires specified magnetic fieldInput:

Magnetic field modelPolar cap potential distributionInitial plasma densityPlasma boundary conditions

Some possible outputs:velocity distributionparticle fluxespotential distribution (and therefore electric fields)Ionospheric precipitation

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Fundamentals

Rice Convection Model

ds

VB

field line

Flux tube volume:

ds

1/B

2/3Energy Invariant: k kW V

Particles per unit magnetic flux: k

( ) ( ) (2)Conservation law: k k k kS Lt

v

1

2(1Drif )t:

kq

k

W

B

E B

v

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Fundamentals

Algorithm:Iterate through eq(2) and eq(3), updating velocities with eq(1).

Rice Convection Model

-Electric field: E Bv

Potential: convection corotation field aligned

5/3 2

3Pressure: k k

k

P V || ||

2Field-Aligned Current: ( ) 3N S

ionisphere

j jV P

B B

B

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Coupling the modelsRCM requires as boundary conditions:• Ionosphere conditions (conductivities, etc)• Magnetic field• Outer plasma conditions

Solution:• Couple RCM with MHD model for outer magnetosphere• Couple RCM with ionosphere model.• Self-consistently compute magnetic field with MHD

model.• There are difficulties in actually implementing this.