Relation Between Electric Fields and Ionospheric/magnetospheric Plasma

Flows at Very Low Latitudes

Paul SongCenter for Atmospheric Research

University of Massachusetts Lowell

Vytenis M. VasyliūnasMax-Planck-Institut für Sonnensystemforschung,

Katlenburg-Lindau, Germany

2006 AGU Fall Meeting

San Francisco, 11-15 December

Paper SA41A-1395

Conventional Model

Can Electric Field Drive Magnetosphere/Ionosphere?

• Imposing an E-field (without flow): charge separation at boundaries in plasma oscillation period, nearly no E-field inside. Most E-field is concentrated in the sheath near the boundary

• Imposing a flow at the top boundary: perturbation propagates along the field (Alfven wave), E-field is created accordingly.

• Finite collisions result in leakage current and small E-field inside • Flow is driven by forces and not by E-field!

Equations for SW-M-I-T Coupling(neglecting photo-ionization, horizontally

uniform)Faraday’s law

Ampere’s law

Generalized Ohm’s law

Plasma momentum equation

Neutral momentum equation

Energy equations

t

B

E

0 2

1

c t

E

B j

( )( ) ( )n n en e i in e en n en in n

N mm N m m

t e

uU u j F

0 ( )( ) ( )e ei e i in e en n en in

N mm N m m

t e

U

j B U u j F

0 0( ) ( )( ) ( )e e e ee e e en in n in en ei e i

i i

m m m mN e N m

e t e m m

jj B E U B U u j F F

2 2 25/3

2 2 25/3

2 2 21 2

3 1[log ] ' [ ] [( ) ( )]

2 2

3 1[log ] [( ) ( )]

2 2

( ) /[ ( ) ] ~ 1

, : thermal speeds; ' = .

in n n

nn in n n

n

n n n

n q

d PP w wdt c

PdP w wdt

c w w c w w

w w

Uj E B u U

u U

u U

j j U

Time Evolution of a QuantityBasic Equations: gyro-averaged, valid on most time and spatial scales

20

0 0

0

( )

[ ( ) ( )( ) ...]

1( ) [ ( )( ) ...]

1[ ( )( ) ...]

e e e e en in ne

e i in e en n qi e

ne i in e en n n

n n

t

ct

eN e N m

t m

p N m mt m N

t

N m mt m N

BE

EB j

jj B E U B j U u

UU U j B U u E

U

uU u F

•For given values on the right at one time, the system evolves continuously. (No time derivatives on the right.)

•Right-hand-side terms are the drivers of left-hand-side variable

Plasma Flow and Electric Field:Primary vs. Derived

• In MHD (Alfven, dynamic) time scales

• B and U are determined (primary), E and j then can be derived (secondary).

• Time variations of E and j cannot cause changes in B and U because they are results of B and U changes.

• In quasi-equilibrium, E and U appear to be mutually determined.

0

0 0

0

0

0 ( ) ( )( )

1( ) [ ( )( )]

e e e en in n

e i in e en ni e

t

N e N m

p N m mt m N

t

BE

B j

j B E U B U u

UU U j B U u

U

Solar Wind-Magnetosphere Coupling: Conventional Steady State Convection

• magnetosphere is coupled with interplanetary electric field via reconnection

• magnetospheric convection: electric drift

= = sw msph

M-I Coupling Models

• coupled via field-aligned current, closed with Pedersen current

• Ohm’s law gives the electric field and Hall current

• electric drift gives the ion motion

Steady State Height-integratedM-I Coupling

0 ||

0 0 0

0

0 => = = = = const

0 => = 0 => j = ( )

0 ( ) ( )( ) ...

10 ( ) [ (

sw msph isph

e e e en in n

e i in ei e

N e N m

p N m mm N

E

B j j Σ E

j B E U B U u E U B

U U j B )( ) ...] = ( )

0 ( )( ) ...

en n n

e i in e en n nN m m

t

U u j Σ E u B

U u F

U

•Time variations are introduced as boundary conditions in the solar wind. All quantities respond instantaneously, except density.

•E and U cannot be distinguished as to which is the cause.

•Neutral wind velocity is independent of height and time

•Some models introduce time dependence by (t) through all heights: not self-consistent

Sunward Convection on Closed Field Lines (after an IMF southward turning)

• Convection of a flux tube can be cause by a force imbalance either in equator or ionosphere

• Simplified momentum equation is, x-component, equatorial plane

• Dayside force balance before the turning

• Southward turning: reconnection creates outflow UMP at the magnetopause, which goes to the 3rd dimension. • The outflow lowers the pressure at the magnetopause • Magnetospheric plasma is accelerated in the sunward direction

• Nightside: jxB force

2 20( / 2 ) /x

x

UU p B x

t

200 ( / 2 ) /p B x

2 2x MP x

MP

U U U

t x x

• Static mapping:

• Dynamic mapping: Poynting flux conservation

• Consider both incident and reflected perturbations

• If the phase difference between the two is not important (120 km ~ 3 Re)

• Perturbation velocity is related to local density

• Potential change is a function of height

Magnetosphere-Ionosphere Mapping:Collisionless

||

0

0 , E =0 => = = = => =0sw msph isph

MP n MP PC isphL v B L v B

E

2 2

1/ 20, av B B

W constB

E A S

1 2

1 2

B B B

U U U

1 1

2 2

A

A

V

BV

B

U B

U B

1/ 4

1/ 4

1/ 4

isph isph

isph isph

isph isph

B

U

Ionospheric Parameters at Winter North Pole

Proposed Model

• Distortion of the field lines result in current• Continuity requirement produces convection cells through fast mode waves in

the ionosphere and motion in closed field regions.• Poleward motion of the feet of the flux tube propagates to equator and produces

upward motion in the equator.

Dynamic M-I Coupling: Collisional

•Neutral wind velocity is a function of height and time

•Neutral wind responds over a long time period => plasma and B

0

0 0

0

0

0 ( ) ( )( ) ...

1( ) [ ( )( ) ...]

1[ ( )( ) ...]

e e e e en in n

e i in e en ni e

ne i in e en n n

n n

t

N e N m

p N m mt m N

t

N m mt m N

BE

B j

j B E U B j U u

UU U j B U u

U

uU u F

•Magnetospheric energy input: j • E

•Joule heating: j • E* frame dependent

•Conventional interpretation:

Comments:

• Ohm’s law is derived assuming cold gases, no energy equation is used.• Ohm’s law is defined in a given frame• In multi-fluid, there are multiple frames: plasma and neutral wind.• The behavior at lowest frequencies indicates a drag process, not Joule heating

Energy equations show:• Joule heating (electromagnetic dissipation) is near zero.• Heating is through ion-neutral collisions: frictional• Thermal energy is nearly equally distributed between ions and neutrals

Joule Heating

Heating Mechanical work

( ) '

'

' ( )

n

n

n

j σ E u B σE

E E u B

j E j E u j B

��

2 2 25/3

2 2 25/3

2 2 21 2

3 1[log ] ' [ ] [( ) ( )]

2 2

3 1[log ] [( ) ( )]

2 2

( ) /[ ( ) ] ~ 1

, : thermal speeds; ' = .

pn n n

nn pn n n

n

n n n

n q

d PP w wdt c

PdP w wdt

c w w c w w

w w

Uj E B u U

u U

u U

j j U

Evolutionary Equations (time derivative determined by present values):

Divergence equations:

4

( , , )

( )

a a a a

coll

a a a

ct

ctf t f f f

t t

m q mc

BE

Ej B

r vv v

r v

vv E B g

•

•

0 4 4 c G B E g

Definition of current density:

Generalized Ohm’s Law:

Plasma momentum equation:

Collision terms (ionosphere):

3

2

( )

( ) ( )

1

( ) ( )( )

( ) ( )

a aa

a a a aa a a

a colla a

coll

eei en in e en in n

coll i

i in e en e ncoll

q f d v

q n qq n

t m c m t

t c t

men

t m

m m nt

J v v

VJ JE B κ g

V Vκ J B

JJ V V

VV V ( )e in enm

e

J

Simplified overview of key equations

2

4

difference between & ( /4 ) produces change of .

( / 4 )[ / /( )] ....

deviation of from value given by generalized Ohm's law

produces change of , on time scale

p e

ct

c

c n ect

Ej B

J B E

JE V B J B

E

J -1p~ .

change of produced only if there is spatial variation.

... / ( )

change of bulk flow produced only by stress imbalance.

in n

ct

ct

BE

B

VJ B V V

Implications• J is determined by the motion of all the charged

particles, and there is no a priori reason why it should equal (c/4)B.

• The equality of the two is established as consequence of the E/t (“displacement current”) term.

• In a large-scale plasma (p >>1, Lp/c >> 1), this occurs primarily by changing J to match the existing (c/4)B, while E takes the value implied by the generalized Ohm’s law (LH side = 0), both on time scale of order ~ p

-1.

• V is changed by stress imbalance, while B changes as consequence of changing B to achieve stress balance, both on time scale typically of order ~ L/VA .

Summary• When dynamic processes are considered, B and U are

primary/causes and E and j are derived/results.

• Sunward magnetospheric convection is driven by pressure forces and not by E-field. It produces an E-field.

• Dynamic mapping indicates that the amplitude of the ionospheric velocity/E-potential) varies with height/density.

• Neutral wind velocity should be treated as a function of height and time in M-I coupling.

• Energy equations are derived for the thermal energy. The term “Joule heating” has been misused in M-I coupling.

Conclusions• Throughout the magnetosphere and the ionosphere, large-

scale plasma flows and magnetic field deformations are determined by stress considerations. Tangential stress from the solar wind is transmitted predominantly by Alfven (shear) waves along open magnetic field lines and by fast-mode (compressional/rarefactional) waves across closed magnetic field lines. Large-scale electric fields and currents are determined as consequences of the above.

• Within the poorly conducting atmosphere below the ionosphere, electromagnetic propagation at nearly the speed of light can occur, but the resulting fields have only a minor effect on the ionosphere.

• Magnetospheric convection propagates from the polar cap to low latitudes on a time scale set by the fast-mode speed ( Alfven speed) just above the ionosphere.

References• Vasyliūnas, V. M.: Electric field and plasma flow: What

drives what?, Geophys. Res. Lett., 28, 2177–2180, 2001.• Vasyliūnas, V. M.: Time evolution of electric fields and

currents and the generalized Ohm’s law, Ann. Geophys.,23, 1347–1354, 2005.

• Vasyliūnas, V. M.: Relation between magnetic fields andelectric currents in plasmas, Ann. Geophys., 23, 2589–2597, 2005.

• Song, P., Gombosi, T. I., and Ridley, A. J.: Three-fluidOhm’s law, J. Geophys. Res., 106, 8149–8156, 2001.

• Vasyliūnas, V. M., and Song, P.: Meaning of ionosphericJoule heating, J. Geophys. Res., 110, A02301,doi:10.1029/2004JA010615, 2005.