Next Generation of Magnetosphere-Ionosphere-Thermosphere Coupling Models

P. Song

University of Massachusetts Lowell

Acknowledgments: V. M. Vasyliūnas, and J. Tu

• Conventional Models: Steady-state coupling between magnetosphere and ionosphere

• (Steady state) Ohm’s law with constant conductivities• Electrostatic potential• Constant magnetic field: self-consistency breaks when there are currents and spatially varying electric field• Dynamics in the magnetosphere does not couple dynamically to the ionosphere• Ionospheric horizontal motion is not derived with dynamic effects • Observationally, difficult to explain the overshoot of an onset (< 30 min)

• New generation models:• Inductive: B changes with time • Dynamic: in particular ionospheric motion perpendicular to B • Multi fluid: allowing upflows and outflows of different species• Wave propagation/reflection: overshoots

• Summary

M-I Coupling• Explain the observed ionospheric responses to solar wind

condition/changes, substorms and auroras etc. and feedback to the magnetosphere (not to simply couple codes)

• Conventional: Ohm’s law in the neutral frame=> the key to coupling

– Derived from steady state equations (no ionospheric acceleration)– Conductivities are time constant– J and E are one-to-one related: no dynamics

• Magnetospheric Approach– Height-integrated ionosphere– Neutral wind velocity is not a function of height and time

• Ionospheric Approach– Structured ionosphere– Magnetosphere is a prescribed boundary– Not self-consistent: steady state equations to describe time dependent

processes (In steady state, imposed E-field penetrates into all heights)– Do not solve Maxwell equations

|| || ( ) ( )P n H nE J b E u B b E u B

Field-aligned Current 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

Full dynamics

Electrostatic

Steady state(density and neutrals time varying)

M-I Coupling (Conventional)• Ohm’s law in the neutral frame: the key to coupling

• Magnetospheric Approach– Height-integrated ionosphere– Current conservation– Neutral wind velocity is not a function of height and time– No self-consistent field-aligned flow– No ionospheric acceleration

• Ionospheric Approach– Structured ionosphere– Magnetosphere is a prescribed boundary– Not self-consistent: steady state equations to describe time dependent

processes (In steady state, imposed E-field penetrates into all heights)– Do not solve Maxwell equations

||

' '

( )

( )

A A

p n

A A

p n

J ds ds

J E u B

E u B

|| || ( ) ( )P n H nE J b E u B b E u B

M-I coupling model: Driven by imposed E-field in the polar cap

Conventional Model Results: Penetration E-field

M-I Coupling (Conventional)• Ohm’s law in the neutral frame: the key to coupling

• Magnetospheric Approach– Height-integrated ionosphere– Neutral wind velocity is not a function of height and time

• Ionospheric Approach– Structured ionosphere– Magnetosphere is a prescribed boundary– When upper boundary varies with time, the ionosphere varies with time:

(misinterpreted as dynamic coupling)– Not self-consistent: steady state equations to describe time dependent processes

(In steady state, imposed E-field penetrates into all heights)– Do not solve Maxwell’s equations– No wave reflection– No fast and slow modes in ionosphere (force imbalance cannot propagate

horizontally)– No ionospheric acceleration

||

' '

( )

( )

A A

p n

A A

p n

J ds ds

J E u B

E u B

|| || ( ) ( )P n H nE J b E u B b E u B

2

2 2

( ) ( )

( )in i n i n

i nin iB

E u B b E u B

v u

• B0 >>δB and B0 is treated as time independent in the approach, and δB is produced to compare with observations

• not a bad approximation

• questionable for short time scales: dynamics

• questionable for short time scales

• Time scale to reach quasi-steady state δt~δLδB/δE

• given δL, from the magnetopause to ionosphere, 20 Re

• δB, in the ionosphere, 1000 nT

• δE, in the ionosphere, for V~1 km/s, 6x10-2 V/m

• δt ~ 2000 sec, 30 min, substorm time scale!

• Conventional theory is not applicable to substorms, auroral brightening!

Theoretical Basis for Conventional Coupling Models

0

=> = t t

B BE

E

0 J B J B

Ionospheric Dynamic Processes

Huang et al, 2009

An overshoot lasting 40 min was seen on ground but not in geosynchronous orbits;indicating the overshoot is related to the ionospheric processes

Epoch analysis showing on average an overshoot in ionospheric velocity for 30 min.

North Pole, Winter Solstice

Ion-neutral Interaction

• Magnetic field is frozen-in with electrons• Plasma (red dots) is driven with the magnetic field (solid line) perturbation from above• Neutrals do not directly feel the perturbation while plasma moves• Ion-neutral collisions accelerate neutrals (open circles), strong friction/heating • Longer than the neutral-ion collision time, the plasma and neutrals move nearly together with a small

slippage. Weak friction/heating• On very long time scales, the plasma and neutrals move together: no collision/no heating

Ionosphere Reaction to Magnetospheric Motion

• Slow down wave propagation (neutral inertia loading)

• Partial reflection

• Drive ionosphere convection – Large distance at the magnetopause corresponds to small distance in

the ionosphere

– In the ionosphere, horizontal perturbations propagate in fast mode speed

– Ionospheric convection

modifies magnetospheric

convection

(true 2-way coupling)

Global Consequence of A Poleward Motion

• Antisunward motion of open field line in the open-closed boundary creates– a high pressure region in the open field region (compressional wave), and – a low pressure region in the closed field region (rarefaction wave)

• 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.

• Ionospheric convection will drive/modify magnetospheric convection

Expected Heating Distribution

• For uniform conductivity, velocity pattern coincides with the magnetic perturbation.

• FAC forms at the center of the convection cells• Poynting flux is proportional to V2, weakest at the center of convection cells• Neglecting the heating from precipitation particles,

– Conventional model (EJ paradigm) predicts heating, J2/p, is highest at the FAC – New model (BV paradigm) predicts heating, iniV2, is highest at compression region of

dayside and nightside cusps and strong along the noon-midnight meridian

sun

Consequence of Heating• Energy equation

• Neglecting radiative loss, R, and heat conduction

• Enhanced temperature and upward motion are expected

5/3

3log

2

d pQ R T NkT

dt

25/3

3log

2 in i i

d pNkT Q S V

dt

Basic Equations• Continuity equations

• Momentum equations

• Temperature equations

• Faraday’s Law and Ampere's Law

22 2 1 2( ) [2( ) ( ) ]

3 3 3

Q CL

s s st ts s s s t s t s

ts B s t B

s s

T m mT T T T

t n k m m k

v v q v v

t

B

E0

0

1i i e e

i

n e n et

EB J v v

s = e, i or n, and es = -e, e or 0

s ''

( )( ) ( )

( 2 ( )) ( ) P L

s s s B s s ss s s s s

s s

s r r s r r s st t s ss s s st s

n n k T n en

t m m

n n n v

vv v I E v B

G Ω v Ω Ω r v v v

''

( ) P Lss s ss s s

s

nn n

t

v

Field-aligned flow allowed

Simplifying Assumptions (dt > 1sec)

• Charge quasi-neutrality– Replace electron continuity with

• Neglecting the electron inertial term in the electron momentum equation– Electric field, E, can be eliminated in other equations;

– electron velocity will be calculated from current definitions.

Momentum equations without electric field E

, , ,

' ''

( ) ( )( )

( ) ( 2 ( ))

( ) ( ) P L

s s s B s s B e e e B es s s

s e s

s se s s r r s r r

s

m n e m ne

s st t s s et t e s ss s s st s t ss

n n k T n k T n n k Tn

t m n m

n en

m

mn n n

m

vv v I

v B v B G Ω v Ω Ω r

v v v v v v

19

is very large at low altitude, e.g., at 80 km s-1

( )( ) ( ) ( )i i i B i i

i i i e i i in i ni i

n n k T n en n

t m m

vv v I v B v B v v

Large collision frequencies make equations strongly stiff

Extremely small time step (< 10-6 s) is required for explicit algorithms to be numerically stable. Implicit algorithms are necessary

6~ 10inin

Numeric Consideration

1-D Stratified Ionosphere/thermosphere

• Equation set is solved in 1-D (vertical), assume B<<B0.

• Neutral wind velocity is a function of height and time

• The system is driven by a change in the motion at the top boundary

• No local field-aligned current; horizontal currents are derived

• No imposed E-field; E-field is derived.

• test 1: solve momentum equations and Maxwell’s equations using explicit method

• test 2: use implicit method (increasing time step by 105 times)

• test 3: include continuity and energy equations with

field-aligned flow

2000 km

500 km

Dynamics in 2-Alfvén Travel Time

x: antisunward; y: dawnward, z: upward, B0: downward

On-set time: 1 sec

Several runs were made: the processes are characterized in Alfvén time

Building up of the Pedersen current Song et al., 2009

30 Alfvén Travel Time

• The quasi-steady state is reached in ~ 20 Alfvén time.

• During the transition, antisunward flow in the F-layer can be large

• During the transition, E-layer and F-layer have opposite dawn-dusk flows

• There is a current enhancement for ~10 A-time, more in “Pedersen” current

Song et al., 2009

Neutral wind velocity

•The neutral wind driven by M-I coupling is strongest in F-layer

•Antisunward wind continues to increase Song et al., 2009

After 1 hour, a flow reversal at top boundary

•Antisunward flow reverses and enhances before settled

•Dawn-dusk velocity enhances before reversing (flow rotates)

•The reversal transition takes slightly longer than initial transition

•Larger field fluctuations

Song et al., 2009

After 1 hour, a flow reversal at top boundary

“Pedersen” current more than doubled just after the reversalSong et al., 2009

Electric field variations

Not Constant!

Electric field in the neutral

wind frame E’ = E + unxB

Not Constant!

Song et al., 2009

Heating rate q as function of Alfvén travel time and height. The heating rate at each height becomes a constant after about 30 Alfvén travel times. The Alfvén time is the time normalized by tA, which is defined as

If the driver is at the magnetopause, the Alfvén time is about 1 min.

Height variations of frictional heating rate and true Joule heating rate at a selected time. The Joule heating rate is negligibly small. The heating is essentially frictional in nature.

/ztop

A Azbottomt dz V

Tu et al., 2011

Time variation of height integrated heating rate. After about 30 Alfvén travel times, the heating rate reaches a constant. This steady-state heating rate is equivalent to the steady-state heating rate calculated using conventional Joule heating rate J∙(E+unxB) defined in the frame moving with the neutral wind. In the transition period, the heating rate can be two times larger than the steady-state heating rate.

Heating rate divided by total mass density (neutral mass density plus plasma mass density) as function of Alfvén travel time and height. The heating rate per unit mass is peaked in the F layer of the ionosphere, around about 300 km in this case.

Tu et al., 2011

Summary• A new scheme of solar wind-magnetosphere-ionosphere-thermosphere coupling is

proposed– Including continuity, momentum equation, and energy equation for each species of multi fluids– Including Maxwell’s equations– Including photochemistry– No imposed E-field is necessary, and no imposed field-aligned current is necessary– 1-D studies: steady state, wave dispersion relation and attenuation, time dependence,

ionospheric heating, coronal heating

• An implicit numerical scheme has been developed to make the time step large (5 orders) enough for global simulations

• In 1-D simulations, there are 4 major differences between the dynamic (and inductive) coupling and the steady-state coupling

– Transient time for M-I equilibrium: not Alfvén travel time, but 10-20 tA ~ 20-30 min.– Reflection effect: enhanced Poynting flux and heating rate during the dynamic transient period can be a

factor of 1.5 greater than that given in of steady-state coupling – Plasma inertia effect: velocity, magnetic field, and electric field perturbations depend on density profile

during the transition period– Field-aligned upflow allowed

• In 2-D and 3-D: ionosphere can be an active player in determining magnetospheric convection. It can be the driver in some regions.

• Using Ohm’s law in the neutral wind frame in conventional M-I coupling will miss– the dynamics during the transition < 30 min– neutral wind acceleration > 1 hr.

Comparison of Steady-state Coupling with Dynamic Coupling

• Coupling speed Vphase

– Steady-state Coupling• Original model (Vasyliunas, 1970, Wolf, 1970): not specific,

presumed to be VA

• Implemented in simulations: (instantaneously)

– Dynamic Coupling: Vphase ~ α1/2 VA ( is neutral

inertia loading factor)

• Coupling time δt– Steady-state Coupling

• Original model: not specified,• Implemented in simulations: ~0

– Dynamic Coupling:

10 ~ 20

1/

A

A

ni

t

t t

1~2 min (Alfvén transient) 30 min (M-I equilibrium)

1~3 hours (neutral acceleration)

Comparison of Steady-state Coupling with Dynamic Coupling, cont.

• Reflection– Steady-state Coupling

• Original model: Multiple reflections assumed, V,B = final result, (depends on ionospheric conductivity)

• Implemented in simulations: No reflection,

E=Einc, V and δB are derived

– Dynamic Coupling: Total=I+R for both δB and V

• Reflection coefficient γ: depends on gradient (height) and frequency (time lapse);

• Reflection may be produced continuously over height• Incident perturbation may consist of a spectrum: dispersion effect• A phase delay φ due to propagation to and from the reflection point

( )

( )

( )0

~ 1 ( )

~ 1 ( )

~ 1 ( )

i tinc ref inc

i tinc ref inc

i tinc ref inc

t e

t e

t e

V V V V

B B B B

E E E V B

Comparison of Steady-state Coupling with Dynamic Coupling, cont.

• Velocity perturbation V– Steady-state Coupling

• Original model: Include final result of multiple reflections

• Implemented in simulations:

– Dynamic Coupling: For single A-wave, parallel propagation, weakly damped (there are reflected waves)

0 0 0 0 01/ 2

0 0

/ /

~ /

V E D BD B D BD V

B B V

2 200 0 0 0 0

0 01/ 4

0 00

1 1

2 2

, is neutral inertia loading factor

i A i A

i

i

SV V

B B B

V V

A S V V

Comparison of Steady-state Coupling with Dynamic Coupling, cont.

• Magnetic perturbation δB– Steady-state Coupling: not included as part of model evolution,

calculated from J

– Dynamic Coupling: For single A-wave parallel propagation weakly damped (there are reflected waves)

Local along B, from

B0, V0, δB0, ρi0,

• Electric field perturbation E– Steady-state Coupling:

– Dynamic Coupling: For single A-wave parallel propagation weakly damped (there are reflected waves)

– Dynamic with reflection:

0

1/ 4 1/ 420 0 0 0 0

/

/A i

i i i i

B V

B V B

B V V

1/ 2

0 0 00

/ ~B

E E D D EB

1/ 4 1/ 2

00

0

i

i

BE E

B

1/ 4 1/ 2

( )00

0

~ 1 ( ) i tiinc

i

BE E t e

B

Comparison of Steady-state Coupling and Dynamic Coupling, cont.

• Current J– Steady-state Coupling:

– Dynamic Coupling: (derived from δB, current continuity satisfied)

• Poynting vector S– Steady-state Coupling: Not considered explicitly,

DC part included implicitly in dissipation

– Dynamic Coupling: For single A-wave parallel propagation weakly damped;

– Dynamic with reflection;

( )

0n

J E u B

J

�

0/ J B

22

0

1 1

2 2A i A

BV

S V V

( )~ 1 ( ) i tinc ref inc t e S S S S

Comparison of Steady-state Coupling with Dynamic Coupling, cont.

• Heating Rate q– Steady-state Coupling:

– Dynamic Coupling: For single A-wave parallel propagation weakly damped;

The perturbations include incident and reflected waves

 

2

1 12 22

2

2

( )

1 ( / )

( )

p n

in i e en ein n

i e in e en e

in n

q

B

E u B

E u b

V u

2

2 2 2

2

( )

11 ( ) /

2

~ ( )

in n

e inin n n n

e i

in n

q

V u t

J E V B u V

V u

V u

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