Next Generation of MagnetosphereIonosphere-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
J   E b   (E  u  B )   b  (E  u  B )
coupling
||
||
P

n
H
– 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
n
Field-aligned Current Coupling Models
Full
dynamics
Electrostatic
Steady state
(density and
neutrals time
varying)
• 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
M-I Coupling (Conventional)
• Ohm’s law in the neutral frame: the key to coupling
J   || E ||b   P ( E   u n  B )   H b  ( E  u n  B )
• Magnetospheric Approach
–
–
–
–
–
A
 J ||    ds   J  
A
 ds

  p (E  u n  B )
A'
A'
Height-integrated ionosphere
     p (E  u n  B )
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
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
J   || E ||b   P ( E   u n  B )   H b  ( E  u n  B )
A
• Magnetospheric Approach
 J ||    ds   J  
A'
A
 ds

  p (E  u n  B )
A'
– Height-integrated ionosphere
     p (E  u n  B )
– Neutral wind velocity is not a function of height and time
• Ionospheric Approach
 in  i ( E   u n   B )   i b  ( E   u n   B )
2
v i 
2
2
– Structured ionosphere
B ( in   i )
– 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
 u n
Theoretical Basis for Conventional
Coupling Models
• B0 >>δB and B0 is treated as time independent in the approach, and δB is
produced to compare with observations
• J  B  J  B0
not a bad approximation
•   E    B     B  0 questionable for short time scales: dynamics
t
t
• =>   =  E
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!
Ionospheric Dynamic Processes
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.
Huang et al, 2009
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
sun
• 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, iniV2, is highest at compression region of
dayside and nightside cusps and strong along the noon-midnight meridian
Consequence of Heating
• Energy equation
Q  R     T 
3
2
N kT
d 
p 
log

5/3 
dt 
 
• Neglecting radiative loss, R, and heat conduction
d 
p 
2
N kT
log

Q



S



V
in i i

5/3 
2
dt 
 
3
• Enhanced temperature and upward motion are
expected
Basic Equations
• Continuity equations
ns
t
s = e, i or n, and es = -e, e or 0
P
   (ns v s ) 
ss '
 ns L s
s'
• Momentum equations
 (ns v s )
t
   (ns v s v s 
n s k BTs
I) 
ms
ns es
(E s  v s  B )
ms
 n s (  G r  2 Ω r  v s  Ω r  ( Ω r  r )) 
Field-aligned flow
allowed
 n
s
st
(vt  v s ) 
t
v
s
Pss '  n s v s L s
s'
• Temperature equations
 Ts
t
 v s   Ts 
2
3
Ts (  v s ) 
2
1
3 ns k B
  q 
t
m s st
m s  mt
[2(Tt  T s ) 
 Q s  CL s
• Faraday’s Law and Ampere's Law
B
t
   E
1
0
  B  0
E
t
J 
 n ev
i
i
i
 ne ev e
2 mt
3 kB
(vt  vs ) ]
2
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
 (ns v s )
t

ns es
ms
   (ns v s v s 

ts
ms
I) 
n s k B Te  n e  n e  ( k B Te )
ne
ms
( v e  B  v s  B )  n s (  G r  2 Ω r  v s  Ω r  ( Ω r  r ))
m ,n ,e

n s k B Ts
m ,n
n s st ( v t  v s ) 

t
ns
me
ms
 et ( v t  v e ) 
v
s'
s'
Pss '  n s v s L s
Numeric Consideration
Large collision frequencies make equations strongly stiff
 ( ni v i )
t
 in
   ( ni v i v i 
n i k B Ti
mi
I) 
ni e
mi
( v e  B  v i  B )  n i in ( v i  v n )
is very large at low altitude, e.g., at 80 km  in
~ 10
6
s-1
Extremely small time step (< 10-6 s) is required for explicit algorithms
to be numerically stable. Implicit algorithms are necessary
19
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
2000 km
field-aligned flow
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 Flayer can be large
• During the transition, Elayer 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 reversal
Song 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
ztop
defined as
tA 

zbottom
dz / V A
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.
Tu et al., 2011
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.
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.
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:
• Implemented in simulations:
– Dynamic Coupling:
not specified,
~0
1~2 min (Alfvén transient)
t A

 t  10 ~ 20 t A 30 min (M-I equilibrium)
1 / 
1~3 hours (neutral acceleration)
ni

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
i ( t )

V  V inc  V ref ~ V inc 1   ( t ) e

i ( t )

 B   B inc   B ref ~  B inc   1  ( t ) e

i ( t )

E  E inc  E ref ~  V inc  B 0 1   ( t ) e

• 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
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:
V  E 0 D 0 / BD   B 0 D 0 / BD  V 0
~  B0 / B 
1/ 2
V0
– Dynamic Coupling: For single A-wave, parallel
propagation, weakly damped (there are reflected
waves)
S0
1
1
2
2
A0  S0 

B0
  0  i0 
V 



i 

2 B0
 i 0V 0 V A 0 
2B
 iV V A
1/ 4
V 0 ,  is neutral inertia loading factor
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)
 B  VB /VA   i0 V
Local along B, from
1/ 4
1/ 4
2

B




V


/

 B0


 i i0 0  0
i
i0
B0, V0, δB0, ρi0,
• Electric field perturbation E
 B 
E  E0 D0 / D ~ 

B
 0 
1/ 2
E
0
– Steady-state Coupling:
– Dynamic Coupling: For single A-wave parallel propagation weakly damped
1/ 4
1/ 2
(there are reflected waves)
   B 
E   i0  
 E0
 i   B0 
1 / 4
1/ 2




– Dynamic with reflection:  i 0
B
i ( t )

E ~
 
 E inc 0 1   ( t ) e

  i   B0 
Comparison of Steady-state Coupling
and Dynamic Coupling, cont.
• Current J
J     (E   u n  B )
– Steady-state Coupling:
J  0
– Dynamic Coupling: (derived from δB, current continuity satisfied)
J    B / 0
• 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;
S
– Dynamic with reflection;
1B
2 0
2
VA 
1
2
 iV V A
i ( t )

S  S inc  S ref ~ S inc 1   ( t ) e

2
Comparison of Steady-state Coupling
with Dynamic Coupling, cont.
• Heating Rate q
– Steady-state Coupling:
q   p (E  u n  B )
2
1
2



 
   1  in2   i e
i 
 e in
 
  in  ( V  u n )
1
2

  en

e
2


   in  ( E / B  u n  b )
  e  en e  
2
– Dynamic Coupling: For single A-wave parallel propagation weakly damped;
The perturbations include incident and reflected waves
q  J   E  V  B    in  ( u n  V )
2

 e in 
1
2
 1 
  in  ( V  u n )     V
 e i 
2


~  in  ( V  u n )
2
2
  

2
 / t
u
n n 
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