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Magnetosphere-Ionosphere Coupling:
Alfven Wave Reflection, Transmission and
Mode Conversion
P. Song and V. M. Vasyliūnas
Center for Atmospheric Research
University of Massachusetts Lowell
• Conventional Models: Steady-state coupling between magnetosphere and
ionosphere
• Dynamic Coupling:
• Inductive: B changes with time
• Dynamic: in particular ionospheric horizontal acceleration
• Multi fluid: allowing upflows and outflows of different species
• Wave propagation/reflection: overshoots
• What and How Much is Coupled at the interface between magnetosphere
and ionosphere
• Wave reflection: law of reflection generalized
• Transmission: Snell’s law specified
• Mode conversion: Fresnel conditions modified
•Summary
M-I Coupling
• Explain the observed ionospheric responses to solar wind
condition/changes, substorms and auroras etc. and feedback to
the magnetosphere (not simply to couple codes)
• Conventional: Ohm’s law in the neutral frame=> the key to
J   || E||b   P (E  u n  B)   H b  (E  u n  B)
coupling
– Derived from steady state equations (no ionospheric acceleration)
– Conductivities are time constant (not including )
– 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)
– Maxwell’s equations not solved
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
A
J||    ds   J    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)
– Maxwell’s equations not solved
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)
• Magnetospheric Approach
– Height-integrated ionosphere
– Neutral wind velocity is not a function of height and time
• Ionospheric Approach
–
–
–
–
–
–
–
–
 in i (E  u n  B)  i2b  (E  u n  B)
vi  
 un
Structured ionosphere
B( in2  i2 )
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)
Maxwell’s equations not solved (J and B are not self consistent)
No wave reflection
No propagation in ionosphere (force imbalance cannot propagate
horizontally)
No ionospheric acceleration
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 Parameters at Winter North Pole
Time scale
of interest
•
•
•
•
Weakly ionized ( =Ne/Nn: ionization fraction)
Collisions are dominant below 120 km
MHD regime
A large density increase at the top of the ionosphere
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
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
Ionosphere Reaction to Magnetospheric Motion
• Slow down wave propagation (neutral inertia loading)
• Partial reflection
• Drive ionosphere convection and feedback to magnetosphere
– 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)
Left-hand mode
Collisional
MHD
Dispersion
Relation
=Ne/Nn
Right-hand mode
Song et al., 2005
1-D, B  Ionosphere
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
Heating rate per particle is
peaked in the F layer of the
ionosphere, around about 300
km in this case.
Time variation of height integrated
heating rate. Overshoot in dynamic
stage
Tu et al., 2011
M-I Coupling via Waves
• The interface between magnetosphere and ionosphere is idealized as a
discontinuity with possible small deformation with wave oscillations
• Alfven waves from the magnetosphere incident onto the ionospheric interface
• Reflected waves feedback to the magnetosphere
• Fast mode waves penetrate into the ionosphere and drive ionospheric convection
• Ionospheric motion
feedback to the
magnetosphere
• B  k
• Polarizations:
• Alfven mode
B,u  k-B0 plane
• Fast/slow modes
B,u in k-B0 plane
• Antisunward ionospheric motion
=>fast/slow modes
Determination of Reflection and
Refraction Alfven Wave Vectors
•
•
•
•
Alfven and slow modes are highly anisotropic: reflection law=? (45 incidence=?)
The wave vector is normal to the wave front
A wave front is formed (in 2-D) by the line connecting equal-phase surfaces
For Alfven mode,
equal-phase surface is
a circle with radius of CA/2
and B0 along a diameter
• Ratio of radii of circles is
proportional to ratio of
Alfven speeds
PS 
Vi
V 'a
V
 r 
sin  i
sin  r sin  'a
Determination of Fast and Slow
Refraction Wave Vectors
• For slow mode (<<1), equal-phase surface is a circle with radius of Cs/2 and B0
along a diameter
• For <<1 fast mode, equal-phase surface is a circle with radius of CA and
centered at the point of incidence
V 'f
Vi
V 's
PS 


sin  i sin  ' f sin  's
Snell’s Law and Generalized Law of Reflection
for an Alfven Wave Incident onto Ionosphere
For parallel Alfven
incidence
tan  r  
sin  ' f 
tan  's 
tan  'a 
sin 2 i
2(1  sin  i )
V 'f
VA
sin  i 
C 'A
sin  i
CA
sin 2 i
C

2  A  sin  i 
 C 's

sin 2 i
 C

2  A  sin  i 
 C 'A

Reflection angle is not equal to
incident angle!!!
Fresnel Conditions: Amplitudes of Reflection
and Refraction for an Incident Alfven Wave
• Alfven mode
(perturbation normal to incident plane)
• Tangential E-field continuous
• Total Poynting flux conserved
 u iy   u ry   B 0    u 'ay  B 0 

T
T
 u iy  B 0    Biy   kˆ i   u ry  B 0    B ry   kˆ r   u 'ay  B 0    B 'ay   kˆ 'a






• Fast/slow modes (perturbation in incident plane)
• Normal velocity continuous
• Total Poynting flux conserved
 uip  nˆ   u ' f   u 's   nˆ
 uip  B 0    Bip   kˆ i   u ' f  B 0    B ' f   kˆ ' f   u 's  B 0    B 's   kˆ 's




Magnetosphere-Ionosphere Coupling
• B, u  k-B0 plane
• Reflection: Alfven mode
• Transmission: Alfven mode

C'
Ra  1  2 A
CA


 cos( i   r )

• Reflection dominates
• B, u in k-B0 plane
• Reflection: not necessary
• Transmission: fast and slow
1/ 2
 C '2A

2
2
sin


cos

 2
i
i 
2
C
'

C
'
s
 A

C 's sin 2  i
ts   C A  C ' A t f 
C '2A cos  i
 C 
tf   A 
 C 'A 
Tf  1
• Fast mode dominates
1/ 2
Summary
• When including the inductive and dynamic effects
– The magnetosphere-ionosphere is coupled via waves (not necessarily sinesoidal)
– Dispersion relation and attenuation rate are derived for the collisional Alfven mode
– 1-D self-consistent simulations with continuity, momentum, energy conservation, and
Maxwell’s equations and photochemistry have been performed with vertical magnetic
field
• Transient time for M-I equilibrium: not Alfvén travel time, but 10-20  tA ~ 20-30 min.
• Reflection effect: enhanced (overshooting) 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
– For inclined magnetic field, in the noon-midnight meridian, for an incident Alfven
wave,
• Velocity and magnetic perturbations in the meridian plane penetrate into the ionosphere as fast modes
• Velocity and magnetic perturbations in dawn-dusk direction partially reflect and partially transmit
– The coupling is not accomplished by an imposed E-field nor imposed field-aligned
current
• The 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.