Ionospheric electrodynamics and its coupling to the magnetosphere

Akimasa Yoshikawa

International Center for Space Weather Science and Education(ICSWSE)

Department of Earth and Planetary SciencesKyushu University, Japan

Outline (1) Fluid description (B,V) for partially ionized system

(2) M-I coupling via shear Alfven wave

・wave generation and propagation and their relation to velocity shear

・Plasma Equation of motions

・Generalized Ohm’s law

・atmospheric dynamo

・magnetospheric dynamo

・3D current closure inside ionosphere

Fundamental evolution equations of physical

quantities Qk

Qk = B,E,J,V,Vn

evolution of B-field

evolution of E-field∂∂tE =

∇×B( )− µ0 jε0µ0

∂∂tV = ?evolution of plasma velocity

evolution of current density∂∂tj = ?

evolution of neutral velocity∂∂tVn = ?

What’s drive What in the weakly ionized system?

momentum equations in the collisional system

mini∂∂tv i = eni E+ v i ×B( )−∇⋅ pi −miν inni v i − vn( )−meνeini v i − ve( )

for ion fluid

for electron fluid

for neutral fluid

(1)

(2)

(3)

Lorentz force kinetic tensormomentum exchangebetween plasma and

neutral

momentum exchange between electron and ion

ρn∂vn∂t

+ (vn .∇)vn⎡⎣⎢

⎤⎦⎥= −∇pn − ρnνni (vn −Vi )− ρnνne(vn −Ve )+ Fn

mene∂∂tve = −ene E+ ve ×B( )−∇⋅ pe −meνenne ve − vn( ) +meνeini v i − ve( )

momentum exchangeBetween ion and neutral

momentum exchangeBetween electron and

neutral

One fluid−description for global phenomena (Hall MHD approximation)

j = en v i − ve( ) V = miv i +mevemi +me

ρ = mi +me( )n

v i = V + me

ρe⎛⎝⎜

⎞⎠⎟j ve = V − mi

ρe⎛⎝⎜

⎞⎠⎟j

current density barycentric velocity mass density

ion velocity electron velocity

n = ni = neplasma density

P = Pi + Pepressure gradient

＋ - << ＋ - >> ＋ - >> mi >> me

One-fluid description of ion and electron�(motion and current)

for ion minddtV + me

ρe⎛⎝⎜

⎞⎠⎟j

⎡

⎣⎢

⎤

⎦⎥ = en E+V ×Β( ) + me

mi +me

⎛⎝⎜

⎞⎠⎟j×B−minν in V − vn −

me

ρe⎛⎝⎜

⎞⎠⎟j

⎡

⎣⎢

⎤

⎦⎥ −

miν ie

e⎛⎝⎜

⎞⎠⎟ j−∇pi

for electron menddtV − mi

ρe⎛⎝⎜

⎞⎠⎟j

⎡

⎣⎢

⎤

⎦⎥ = −en E+V ×Β( ) + mi

mi +me

⎛⎝⎜

⎞⎠⎟j×B−menνen V − vn +

mi

ρe⎛⎝⎜

⎞⎠⎟j

⎡

⎣⎢

⎤

⎦⎥ −

meνei

e⎛⎝⎜

⎞⎠⎟ j−∇pe

current density (charge weighted form of velocity evolution)

equation of motion (mass weighted form of velocity evolution)

en ddtV + me

ρe⎛⎝⎜

⎞⎠⎟j

⎡

⎣⎢

⎤

⎦⎥ =

e2nmi

⎛⎝⎜

⎞⎠⎟E+V ×Β( ) + e me

mi

⎛⎝⎜

⎞⎠⎟j×Bmi +me

− enν in V − vn −me

ρe⎛⎝⎜

⎞⎠⎟j

⎡

⎣⎢

⎤

⎦⎥ −ν ie j−

e∇pimi

−en ddtV − mi

ρe⎛⎝⎜

⎞⎠⎟j

⎡

⎣⎢

⎤

⎦⎥ =

e2nme

⎛⎝⎜

⎞⎠⎟E+V ×Β( )− e mi

me

⎛⎝⎜

⎞⎠⎟j×Bmi +me

+ enνen V − vn +mi

ρe⎛⎝⎜

⎞⎠⎟j

⎡

⎣⎢

⎤

⎦⎥ +νei j+

e∇peme

ionic current

electron current

dominant partcancellation part

inertial part Lorentz force Ampere force momentum exchange to neutrals

momentum exchangebetween ion and electrons

ion : main portion of plasma dynamics electron : main portion of current carier

Generalized Equation of motion for one-fluid plasma

ρ ∂∂tV = j×B−∇P − n miν in +meνen( ) V −Vn( )−me ν in −νen( ) j

e

inertial

Ampere force

pressure gradient

momentum exchange between neutral andcharged particles

friction force along j

electromotive e-field

Hall e-field

am-bipolar field

emf

= V ×B+B ν in −νen( )

Ωe

V − vn( )− ∇pene

resistive e-field

convection electric field

electric field induced by momentum exchange effect

Generalized Ohm’s law

!R ⋅ j = B

ne⎛⎝⎜

⎞⎠⎟

νen +νei

Ωe

⎛⎝⎜

⎞⎠⎟j+ j× b( )⎡

⎣⎢

⎤

⎦⎥

∂∂tj = ε0ω pe

2 E+ emf! "!!

−#R ⋅ j( )

reactive-field

Distribution of collision frequency

controls the momentum exchange between neutral and charged particles

miν in > meνen

νen >>ν in

All altitudes

180km

νen >νei

80km

νen

Ωe

>1

ν in

Ωi

>1

140km

ν in >νei

110kmmax(νei )

νen

Ωe

<1ν in <νei

ν in

Ωi

<1νen <νei

From Song, et al., [2001]

Equation for motion of plasma (dominant)

inertial Ampere force pressure gradient forcemomentum exchange

between ion and neutralsfriction force along j

miν in >> meνen ν in <<νen

everywhere importantdominant

in the ionosphere important below 80kmimportant ω>~νin

Important near plasma sheet

∂∂tV = −ν in V −Vn( ) + j×B

ρ+ Bρ

νen

Ωe

⎛⎝⎜

⎞⎠⎟j− ∇P

ρ

Force balance Equation below �E-layer

inertial Ampere forcemomentum exchange between ion and neutrals

friction force along j

miν in >> meνen ν in <<νen

everywhere importantdominant

in the ionosphere important below 80km

0 ! −ν in V −Vn( ) + j×B

ρ+ Bρ

νen

Ωe

⎛⎝⎜

⎞⎠⎟j

ω <<ν inWith cold plasma plasma approximation

Ionospheric current is� a force balance current

j×B = en ken2

1+ ken2

⎛⎝⎜

⎞⎠⎟kin−1 V −Vn( ) + ken

1+ ken2

⎛⎝⎜

⎞⎠⎟kin−1 V −Vn( )×B⎧

⎨⎩⎪

⎫⎬⎭⎪

j×B ! kin−1en V −Vn( )

For 90km~140km Important only for 80km Atmospheric dynamo through collision via electron

(dynamo region-> 70~90km)

Atmospheric dynamo through collision via ion(dynamo region-> 90~140km)

kin ≡

ν inΩi

⎛

⎝⎜⎞

⎠⎟ ken ≡

νenΩe

⎛

⎝⎜⎞

⎠⎟

plasma – neutral interaction j×B ≅ ρν in V − un( )

j×B( ) ⋅V" ≅ ρν in V" 2 V ' ≅ V − un

In the neutral frame

j×B( ) ⋅v ≅ ρν in V − un( ) ⋅v = j ⋅E

j ⋅E > 0 for v > un j ⋅E < 0 for v < un

Work done by Ampere force = work done by momentum exchange by collision

Atmospheric load Atmospheric dynamo

Electromagnetic energy is converted into mechanical energy

Electromagnetic energy is generated from mechanical energy

Equivalence of electromagnetic load and mechanical load

Work done by Ampere force = heating by plasma

In the rest (plasma) frame

Atmospheric dynamo

neutral wind

Plasma flow (within the dynamo region)

Magnetic perturbation, current

Electric field (within the dynamo region)

Momentum exchange by Ion-neutral collision

Generalized Ohm”s law

Non-zero curl E (different E within and outside dynamo region along B0)

Plasma flow out side dynamo region Ampre froce (magnetic tension acting plasma )

To the magnetosphere Alfven wave generation

Atomospheric dynamo j×B( ) ⋅V ! kin−1en V −Vn( ) ⋅V

Work done by Ampere force acting on ion Work done by momentum exchange between ion and neutral on ion

j×B( ) ⋅V ! kin−1en V −Vn( ) ⋅V > 0Case (A) if

Ampere force can accelerate neutrals through collision(atmospheric load)

j×B( ) ⋅V ! kin−1en V −Vn( ) ⋅V < 0Neutral particle can produces Ampere force through

Collision （electromagnetic energy is produced by kinetic energy of Neutral particles) (atmospheric dynamo)

Case (B) if

Generalized Ohm’s law (dominant)

Electromotive e.m.f. e-field

∂∂tj = ε0ω pe

2 E+ emf! "!!

−#R ⋅ j( )

Hall e-fieldresistive e-field

convection e- field e-field induced by

momentum exchange between electron and neutral

!R ⋅ j = B

ne⎛⎝⎜

⎞⎠⎟

νen

Ωe

⎛⎝⎜

⎞⎠⎟j⊥ + j× b( )⎡

⎣⎢

⎤

⎦⎥

Current evolution equation

e-field

reactive-field

emf! "!!

= V ×B− B νen

Ωe

⎛⎝⎜

⎞⎠⎟V − vn( )

Electromagnetic wave Radiationevolution of B-field evolution of E-field

∂∂tE =

∇×B( )− µ0 jε0µ0

1c2

∂2

∂t 2E+∇× ∇×E( ) = −µ0

∂∂tj evolution eq of J?

no!! electromagnetic field radiations eq by changing of current(like a dipole antenna)

we need to know how J is produced by electro-dynamics (not -magnetics)

∂∂tj = ε0ω pe

2 E− emf

−Rj( )

Rj = B

ne⎡⎣⎢

⎤⎦⎥

νen +νei

Ωe

⎛⎝⎜

⎞⎠⎟j+ j× b

⎡

⎣⎢

⎤

⎦⎥

Ohm’s law in ion-collisional system

j×B ! kin−1en V −Vn( )

emf! "!!# −V ×B

E = −V ×B+ kin−1B V − vn( )

!R ⋅ j " B

ne⎛⎝⎜

⎞⎠⎟ j× b( )

= kin−1B v i − vn( )

Eq of motion (force balance) Generalized Ohm’s law

Convection e-field

E+ vn ×B = − V − vn( )×B+ kin−1B V − vn( )

E* = −V* ×B+ kin−1BV*

Hall e-field

Ion-collisional resistive e-field

Ohm’s law in the neutral frame

E = emf! "!!

+#R ⋅ j

Ohm’s law in the rest frame

In the ionosphere electric field is induced not only in–VxB (convection electric field) direction but also

in the direction of plasma flow !!(very different from magnetosphere)

E* = E+ vn ×BV* = V − vn

E = emf! "!!

+#R ⋅ j

Magnetospheric Generator

“Generator” region can be defined as a region enable to production of velocity shear (vorticity) both parallel and perpendicular direction to B0 by the mechanical balance.

∇× v( )// ∝ ∇⋅E⊥( ) eB

v⊥ = E× eBB0

Parallel shear induces (div E⊥)

perpendicular shear produces (rot E⊥) .∂∂sv⊥ ∝ ∇× v( )⊥ ∝ ∇×E⊥( )⊥

∇× v( )// ∇× v( )⊥

Note: ∇⋅ ∇ × v( ) = 0 ∇ // ⋅ ∇ × v( )// = −∇⊥ ⋅ ∇ × v( )⊥Vorticity conservation as like a current

B0

∇⋅ j = 0 ∇ // ⋅ j// = −∇⊥ ⋅ j⊥

j⊥ = B0ΣA ∇× v( )⊥

j// = B0ΣA ∇× v( )//

∇⋅ j⊥ = 0 ∇⋅ ∇ × v( ) = 0

Shear Alfven wave

Always inductive!!

Induction process: generation and propagation shear Alfven wave

(1) Generator ：drives enhanced convection velocity by balance of acceleration by pressure gradient force and deceleration by the magnetic tension force

(2) velocity shear along corresponds to rot(E) → development of magnetic field

　　 → development of tension force→　acceleration of

plasma flow at wave front

・ role of shear Alfven wave　→　disappearance of velocity shear along B0

･　Generator acts until difference of convection velocity along B0 is disappeared

∂∂sV⊥ ∝ ∇×E( )⊥ ∝ ∂

∂tb⊥ → j×B0 → ρ ∂

∂tV⊥

V0

V

Generator

E0

B0

j×B0( ) ⋅v0 > 0 −∇p ⋅v0 < 0

∇ // ⋅S// > 0

VAAlfven Induction loop responsible for A.Y

Magnetospheric dynamo

Magnetic perturbation, current

Electric field (within the dynamo region)

Generalized Ohm”s law

Non-zero curl E (different E within and outside dynamo region along B0)

Plasma flow out side dynamo region Ampre froce (magnetic tension force acting plasma )

Enhanced convection velocity (plasma flow within the dynamo region)

Alfven induction loop

velocity shear along corresponds to rot(E) → development of magnetic field

　　 → development of tension force→　acceleration of

plasma flow at wave front

To the ionosphere

Propagation of electromagnetic field inside the ionosphere�and production of reflection field

Ampere force

Ion collision Alfven-induction loop

velocity shear along B0 corresponds to rot(E)⊥ 　→ development of magnetic field　　 → development of tension force →　acceleration of ion flow →　braking of ion flow by collision

attenuation of vi No attenuation of ve

→　no braking of electron flow

Motion of ion induces two types electric field(1) Convection electric field

(2) Hall electric field Attenuation of vi through collisionCorresponds to attenuation of type (1) e-field

Equivalent to generation of oppositelydirected e-field

Magnetic perturbation, current

Shear (rot) of e-field along B-field

Generalized Ohm’s law

Plasma flow out side dynamo region

To the magnetosphere

(1) reflection Alfven-induction loop

From the magnetosphere

Process (1)

Propagation of electromagnetic field inside the ionosphere�and production of reflection field

Ampere force

Ion collision Alfven-induction loop

velocity shear along B0 corresponds to rot(E)⊥ 　→ development of magnetic field　　 → development of tension force →　acceleration of ion flow →　braking of ion flow by collision

attenuation of vi No attenuation of ve

→　no braking of electron flow

Generation of e-field in the direction vi

Shear of e-field along B0

Magnetic perturbation, current

This e-filed perpendicular to convection type e-field

Generalized Ohm’s law

Plasma flow out side dynamo region

To the magnetosphere

Hall dynamo Alfven-induction loop

From the magnetosphere

Motion of ion induces two types electric field(1) Convection electric field

(2) Hall electric field Process (2)