plasma interactions with electromagnetic...

Plasma Interactions with Electromagnetic Fields

Roger H. Varney

SRI International

June 21, 2015

R. H. Varney (SRI) Plasmas and EM Fields June 21, 2015 1 / 23

1 Introduction

2 Particle Motion in Fields

3 Generation of Electric Fields in PlasmasAmbipolar Electric FieldsDynamo TheoryElectrodynamical Magnetosphere-Ionosphere Coupling


Introduction

The Ionosphere and Thermosphere


Introduction

Magnetic Structure of the Ionosphere and Magnetosphere


Particle Motion in Fields

Particle Motion in a Uniform B field

mdv

dt= qv × B

Separate by components

mdvx

dt= qvyBz

mdvy

dt= −qvxBz

Solution to coupled system with v0 = v0x

vx = v0 cos (Ωt)

vy = −sgn (q)v0 sin (Ωt)

Gyrofrequency: Ω = |qB|m

x

y

z

B = Bz z

Electrons

Ions



The E× B Drift

VD

B

E

vD =E× B

B2



Electric Fields in Different Frames of Reference

Lorentz Force: F = q [E+ v × B]

In a different frame of reference moving with velocity u

F′ = q[E′ + (v − u)× B

]

The force must be the same in all reference frames: F = F′

E′ = E+ u× B

The frame moving at the E× B drift velocity is special:

E′ = E+E× B

B2× B

= E−E⊥B

2

B2

= 0 Assuming: E‖ = 0


Generation of Electric Fields in Plasmas Ambipolar Electric Fields

Ambipolar Electric Fields and Ambipolar Diffusion

Steady state parallel electron momentum equation:

me

[∂

∂t(neue) +∇‖ ·

(neu

2e

)]

= −∇‖pe − neeE‖ −→ E‖ = −1

ene∇‖pe

Substitute into parallel ion momentum equation:

mi

[∂

∂t(niui) +∇‖ ·

(niu

2i

)]

= −∇‖pi −ni

ne∇‖pe −minig‖

−mini∑

j

νij (ui − uj)+

+

+

+

+

-

-

-

-

-

ER. H. Varney (SRI) Plasmas and EM Fields June 21, 2015 8 / 23

Generation of Electric Fields in Plasmas Dynamo Theory

Fundamentals of Ionospheric Electrodynamics

Electrostatic Limit of Maxwell’s Equations:

∇× B = µ0J+

01

c2∂E

∂t−→ ∇ · J = 0

∇× E = −0

∂B

∂t−→ E = −∇Φ

Ohm’s Law for the ionosphere:

J = σ · E+ J0

Putting everything together yields a boundary value problem:

∇ · σ · ∇Φ = ∇ · J0



Ohm’s Law for the Ionosphere

Steady-state momentum equation for each species (zero neutral windcase):

0 = nαqα (E+ uα × B)− ναnmαnαuα

Resulting Ohm’s Law:

J =∑

α

nαqαuα −→ J =

σP −σH 0σH σP 00 0 σ0

· E



Conductivity Profiles



Other Kinds of Current

Substitute F for qαE in steady state momentum equation.

Wind drag: F = ναnmαun −→ J = σ · (un × B)

Gravity: F = mαg −→ J = Γ · g

Pressure Gradients (Diamagnetic Currents):F = − 1

nα∇pα −→ J = D · ∇

∑

αpα

Complete Dynamo Equation:

∇ · σ · ∇Φ = ∇ ·

(

σ · (un × B) + Γ · g+D · ∇∑

α

pα

)



Slab Model of the F-region Dynamo

J = σP (E+ un × B)

Two ways to achieve ∇ · J = 01 Parallel currents which close elsewhere2 J = 0

J = 0 −→ E = −un × B

VD =E× B

B2=

−un ×B× B

B2= un



Slab Model of the E-region Dynamo

Suppose Ex is the eastward component of un × B in the E-region.

A vertical electric field forms to oppose the vertical Hall current.

σHEx = σPEz =⇒ Ez =σHσP

Ex

The Hall current from this new Ez adds to the existing Pedersen currentfrom Ex

Jx = σHEz + σPEx =[(σH/σP)

2 + 1]σPEx ≡ σCEx



Equatorial Fountain Effect

−20 −10 0 10 20 30 400

500

1000

Latitude

Alti

tudeNe (cm−3)

3

4

5

6

7

00 06 12 18 00 06 12 18 00−20

0

20

Ver

tical

Drif

t (m

/s)

Local Time



Influences of Atmospheric Tides (Immel et al. 2006)


Generation of Electric Fields in Plasmas Electrodynamical Magnetosphere-Ionosphere Coupling

Current Systems in the Ionosphere and Magnetosphere



Closure of Field Aligned Currents in a Slab Ionosphere

3D potential equation with magnetospheric currents:

∇ · σ · ∇Φ = ∇ · Jmag

Integrate over altitude, assume equipotential field lines:

∇⊥ · Σ · ∇⊥Φ =

∫

∇ · Jmag dz

Expand the divergence:

∇ · Jmag = ∇⊥ · J⊥ +∂J‖

∂z

J⊥ goes to 0 above ionosphere, thus:∫

∇ · Jmag dz = J‖

2D slab ionosphere potential equation:

∇⊥ · Σ · ∇⊥Φ = J‖



High Latitude Convection Patterns



Joule Heating

Weimer, 2005.



Conductivity Effects on Magnetosphere (Lotko et al., 2014)



Summary of Ionospheric Electrodynamics

∇ · σ · ∇Φ = ∇ ·

(

σ · (un × B) + Γ · g+D · ∇∑

α

pα + Jmag

)

The ionospheric potential, and thus the E× B drifts, depend on:

Neutral winds (driving from below)

Magnetospheric currents (driving from above)

Ionospheric conductivities (chemistry)

Ionospheric pressure gradients (energetics)


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