Inductive-Dynamic
Magnetosphere-Ionosphere
Coupling via MHD Waves
Jiannan Tu
Center for Atmospheric Research
University of Massachusetts
Collaborators: Paul Song, Vytenis Vasyliunas
Boston College Seminar 11/5/2013
Outline
• Background
• Governing Equations
• Numerical Method
• Simulation Results
• Conclusion
2
Background
• Coupling must be through waves in addition to flow
• Transient responses of ionosphere/thermosphere to
changes in solar wind/IMF variations
• Conventional approach: electrostatic, quasi-stead state
B
B B 0
=    E  0 Because  B  B 0 , B  B 0 , 

0
t
t
t
 B  (B 0   B)  B
(In fact
=

 0    E  0)
t
t
t

=0 in plasma momentum equations and J  σ  (E  u n  B)
t
Implication: time averaging or quasi-steady state. Missing
wave or oscillation information
(Detailed discussion in Vasyliunas, Angeo., 30, 357, 2012)
3
Governing Equations
Plasma and neutral continuity, momentum equations

    u   S   L
t
(1)
 n
    nu n    L  S
t
(2)
m
 u
   (  uu)  p  J  B   in (u  u n )  e ( en  in )J
t
e
  g  Su n   Lu
 nun
m
   (  nu nu n )  pn   in (u  u n )  e ( en  in )J
t
e
  n g   Lu  Su n
(3)
(4)
4
Plasma and neutral dissipation equations [Vasyliunas and
Song, JGR, 110, A02301, 2005]

3  p
1
2
2
2





(
p
u
)

p


u



q

J

E


(
u

u
)


(
w

w
in 
n
n )


2  t
2

Q
(5)

3  pn
1
2
2
2





(
p
u
)

p


u



q


(
u

u
)


(
w

w
n n 
n
n
n
in 
n
n )

2  t
2

 Qn  Cn
(6)
Generalized Ohm’s law
E=  (u  B) 
m (  ei )
1
J  B+ e en2
J
ene
e ne
(7)
Faraday’s law and Ampere’s law
B
=    E,
t
  B=0 J
(8)
5
Eliminate J with Ampere’s law and E with generalized
Ohm’s law, and write eq. (1)-(6) and Faraday’s law in a
compact form
W F
+
R
t z
(9)
where
W  (  ,  n ,  u x ,  u y ,  u z ,  nunx ,  nuny ,  nunz , Bx , By , p, pn )T
u





u
n n


2


B  BB 
  uu   p 

I 
2 0 
0 


F
,

 nu nu n  pn I




uB  Bu


p
u




pnu n


S  L




L  S




me
 
( en  in )  B   g  Su n  L  u


i in (u  u n ) 
e 0




me

( en  in )  B   n g  Su  L  nu n


i in (u  u n ) 
e 0


R 

ne
1
1 
    (  B)   2 B  


B


B

(


B
)
B

(


B
)

B



 en  
ne
 0 
e 0 




pn p 

1
2


2
 p  V  2   B   in (u n  u)   (  )   Q    q


0
2
n  







p
1
p
 pn  u n   in (u n  u) 2   ( n  )   Qn  Cn    q n


2


n




6
Numerical Method
• One-dimensional geometry for the central polar cap
(northern hemisphere) with a uniform background
magnetic field B   B0 zˆ
z = 1000 km
z - vertical
A scaled M-IT
system
y
z = 80 km
x – antisunward
• Eq. (9) is a system of stiff partial differential equations
because of very large (up to 106 s-1) ion-neutral collision
frequency vin at low altitudes
7
Altitude distribution of ion-neutral, electron-neutral, and electron-ion
collision frequencies
8
• Fully implicit difference scheme
W jn1  W jn
t

Fjn1/1 2  Fjn1/1 2
z
 R nj1
(10)
• Implicit difference overcomes the stiffness so that 5-6
order of magnitude larger time step (up to tens of sec),
compared to explicit difference scheme, can be used while
still obtaining stable solutions
• Eq. (10) is a set of nonlinear algebraic equations, which is
solved by an iterative Newton-Krylov subspace method
• An implicit framework has been developed based on a
portable, extensible toolkit for scientific computation
(PETSc) package (http://www.mcs.anl.gov/petsc), which
includes various kinds of Newton-Krylov solvers
9
Main Routine
Initialization:
background &
runtime param
Time Stepping
Photoionization
Chemistry
PETSc Nonlinear Solver (SNES)
Collisionfrequencies
Optional
PETSc Krylov
Linear Solver
Functions
Evaluation
Large time step (up to tens of
seconds) can be used with
implicit solver – quite efficient
Jacobian
Matrix
Flexible: 1-D, 2-D or 3-D,
easy to add variables
Postprocessing
Components of implicit framework for the inductive-dynamic
ionosphere/thermosphere model
10
• At the top boundary (1000 km), the density, velocity,
pressure are linearly extrapolated, the free boundary
condition (zero first order derivative) is applied to the
perturbation magnetic field.
• At the bottom boundary (80 km) the plasma and neutral
velocities, and perturbation magnetic field is set to zero.
The plasma and neutral mass densities are determined by
photo-chemical equilibrium. The plasma and neutral
pressures are assumed fixed in time.
• The initial ionosphere/thermosphere is specified by IRI
2011
ionospheric
model
and
NRLMSISE00
thermospheric model for the solar minimum night time
polar cap.
11
• Spatial cell size Δz = 5 km and time step Δt = 0.01s.
Choose Δt = 0.01 s so that Δt is less than the transition
time (0.1 s) of the convection velocity imposed at the top
boundary, and also less than the time for the Alfven wave
to propagate from the top to bottom boundary. If the
simulation domain is extended to the magnetopause the
Alfven travel time will be about 1 min and we may use a
larger time step, say 10 sec.
• The system is driven by an antisunward convection
velocity (Vx) at the top boundary, changing from 0 to 600
m/s in 0.1 s, keeping at 600 m/s for 2 min, and then to
2000 m/s in 0.1 s, keeping at 2000 m/s for 1 min.
Simulation results for the last 1 min are shown.
12
Transient
time 0.1 s
Time variation of an antisunward convection (Vx) imposed at the
top boundary
13
Simulation Results
Oscillations & overshoots. Also Vy & By 14
1000 km
80 km
The transient to quais-steady state
takes ~18 sec (~40 Alfven travel
times) for scaled M-IT system but
will be proportionally longer with
higher altitude of the top
boundary so that the Alfven travel
time tA is used to scale the time
Bristow et al., 2003
15
Superposition of
incident and
reflected waves
causes oscillations
and overshoots
Alfven travel time tA ~ 0.46 s. t = 0 equilibrium state. Topside Vx
increases from 600 m/s to 2000 m/s in 0.1 s, velocity perturbation
propagates downward along the field line. Localized enhancement
and all-altitude overshoot.
16
Dynamic Hall effect
not present in
electrostatic models
B
=  zˆ  E
t
0 J  zˆ  B
E=  (u  B)  J  B / ene + J
High Alfven speed (~10,000 km/s) at high altitude, decreases to
~3000 km/s in the F region. Topside Vx increases from 600 m/s to
2000 m/s in 0.1 s. Prompt perturbation in Vx at 600 km, about one tA
to reach 120 km. Perturbations in Vy begins at 120 km when Vx
perturbation arrives and propagates upward.
17
Ionosphere behaves as a
damped resonator in response
to magnetospheric
perturbations
The oscillation period depends
on the ionospheric mass or
inertia.
 u 

 u   u    J  B
 t


  in (u   u n ,  ) 
me
( en  in ) J 
e
18
Compressional waves
similar to acoustic-gravity
waves but with much faster
propagation speed because
of the different parameter
range.
Pressure oscillations with
progressive phase delay with
altitudes. The phase velocity is
~ 200 km/s at lower altitudes
and ~400 km/s at higher
altitudes, and correlate with
the plasma temperature
variations
19

3  p
1



(
p
u
)

p


u



q

J

E

 in (u  u n ) 2   ( w2  wn2 ) 


2  t
2


3  pn
1



(
p
u
)

p


u



q

 in (u  u n ) 2   ( w2  wn2 ) 
n n 
n
n
n

2  t
2

q  J  E   in (u  u n ) 2
Variation of total heating rate q with time and height. Strong heating
below about 250 km. Heating rate included both frictional and true
Joule heating rates, with dominant contribution from the frictional
heating rate.
20
Conclusions
• The M-I coupling is through MHD waves and produces
strong variation of the ionospheric state variables during the
transient stage, which persists for 20-40 Alfven travel times
(20-30 minutes).
• Inductive-dynamic (non-electrostatic and retaining time
derivative terms in the momentum equations) approach is
necessary to correctly to describe transient variations
• The overshoot and oscillation is caused by the superposition
of the incident and reflected Alfven waves with the period
dependent on IT inertial.
• The dynamical Hall effect is an inherent aspect of the M-I
coupling.
• The ionosphere-thermosphere responds to magnetospheric
driving forces as a damped oscillator.
21