Solar Wind-Magnetosphere-Ionosphere Coupling: Dynamics in

Report
Relation Between Electric Fields and
Ionospheric/magnetospheric Plasma Flows
at Very Low Latitudes
Paul Song
Center for Atmospheric Research
University of Massachusetts Lowell
Vytenis M. Vasyliūnas
Max-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
B
t
1 E
  B  0 j  2
c t
E  
Generalized Ohm’s law

me j
m m
m
 j  B0  Ne e(E  U  B0 )  Ne me ( en  in )(U  un )  e ( e  in  en  ei ) j  Fe  e Fi
e t
e mi
mi
Plasma momentum equation
Neutral momentum equation
Energy equations
mi
Ne U
m
 j  B0  Ne (mi in  me en )(U  un )  e ( en  in ) j  F
t
e
mn
N nu n
m
 N e (mi in  me en )(U  u n )  e ( en  in ) j  Fn
t
e
3 d
P
U
1
P [log 5/ 3 ]  j ' [E   B]   in [(u n  U) 2   ( wn2  w2 )]
2 dt

c
2
P
3 d
1
Pn [log 5/n 3 ]   in  [(u n  U) 2   ( wn2  w2 )]
2 dt
n
2
  (u n  U) 2 /[c1 ( wn2  w2 )  c2 wn w] ~ 1
wn , w : thermal speeds; j' = j   q U.
Time Evolution of a Quantity
Basic Equations: gyro-averaged, valid on most time and spatial scales
B
   E
t
E
 c 2 (  B  0 j)
t
j
e

[ j  B 0  N e e(E  U  B 0 )  e j  N e me ( en  in )(U  u n )  ...]
t
me
U
1
 ( U   ) U 
[p  j  B 0  N e ( mi in  me en )( U  u n )   q E  ...]
t
mi N e

    U
t
u n
1

[ N e (mi in  me en )( U  u n )  Fn  ...]
t
mn N n
•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
   E
t
0    B  0 j
0  j  B 0  N e e(E  U  B 0 )  N e me ( en  in )(U  u n )
U
1
 ( U   ) U 
[ p  j  B 0  N e (mi in  me en )( U  u n )]
t
mi N e

    U
t
• 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.
Solar Wind-Magnetosphere Coupling:
Conventional Steady State Convection
• magnetosphere is coupled with interplanetary electric field via
reconnection
 = sw = msph
• magnetospheric convection: electric drift
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-integrated
M-I Coupling
0  E
=>  =  sw =  msph =  isph = const
0    B  0 j
=>   j = 0
=>
j|| =    (Σ  E) 
0  j  B 0  N e e(E  U  B 0 )  N e me ( en  in )(U  u n )  ...  E  U  B 0
0  ( U   ) U 
1
[p  j  B 0  N e ( mi in  me en )(U  u n )  ...]  j = Σ  (E  u n  B)
mi N e
0  N e (mi in  me en )(U  u n )  Fn  ...

    U
t
•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

U x
  ( U x2  p  B 2 / 2 0 ) / x
t
• Dayside force balance before the turning
0  ( p  B2 / 20 ) / x
• 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
2
in the sunward direction U x U MP
 U x2
t
• Nightside: jxB force

xMP  x
Magnetosphere-Ionosphere Mapping:
Collisionless
• Static mapping:
0    E, E|| =0 =>  =  sw =  msph = isph => =0
LMP vn BMP  LPC visph B0
• Dynamic mapping: Poynting flux conservation
va B 2  B2
 E  0,
W  AS 
 1/ 2  const
B

• Consider both incident and reflected perturbations
V
 U1  A  B1
 B   B1   B 2
B
V
 U   U1   U 2
U   A B
2
B
2
• If the phase difference between the two is not important (120 km ~ 3 Re)
1/ 4
 Bisph  isph
• Perturbation velocity is related to local density
1/ 4
 U isph  isph
• Potential change is a function of height
1/ 4
isph  isph
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
B
   E
t
0    B  0 j
0  j  B 0  N e e(E  U  B 0 )  e j  N e me ( en  in )(U  u n )  ...
U
1
 ( U   ) U 
[p  j  B 0  N e ( mi in  me en )(U  u n )  ...]
t
mi N e

    U
t
u n
1

[ N e (mi in  me en )( U  u n )  Fn  ...]
t
mn N n
•Neutral wind velocity is a function of height and time
•Neutral wind responds over a long time period => plasma and B
Joule Heating
•Magnetospheric energy input: j • E
j  σ(E  u n  B)  σE '
•Joule heating: j • E* frame dependent
E '  E  un  B
•Conventional interpretation:
j  E  j  E ' u n  ( j  B)
Heating Mechanical work
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
3 d
P
U
1
P [log 5/ 3 ]  j ' [E   B]   pn [(u n  U) 2   ( wn2  w2 )]
2 dt

c
2
P
3 d
1
Pn [log 5/n 3 ]   pn [(u n  U) 2   ( wn2  w2 )]
2 dt
n
2
  (u n  U) 2 /[c1 ( wn2  w2 )  c2 wn w] ~ 1
wn , w : thermal speeds; j' = j   q U.
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
Evolutionary Equations (time derivative determined by present
values):
B
 c  E
t
E
 4 j  c  B
t
f a (r , v, t )
f
f
  fa 
 v  a  v•  a  

t
r
v   t coll
v
ma v•  qa (E   B )  ma g
c
Divergence equations:
B  0
  E  4c
  g  4 G 
Definition of current density:
J   qa  vf a ( v )d 3v
a
Generalized Ohm’s Law:
 qa2 na
 J 
Va
qa
J
 
(E 
 B) 
(  κ a )  qa na g   

t
c
ma

t

coll
a  ma

Plasma momentum equation:
 V
1
  V 
   κ  J  B  

t
c
  t coll
Collision terms (ionosphere):
me
J 
 in )J  ene ( en  in )(V  Vn )

  ( ei   en 

t
m

coll
i
J
  V 


(
m


m

)
n
(
V

V
)

m
(



)
i in
e en
e
n
e
in
en


e
  t coll
Simplified overview of key equations
E
 4 j  c  B
t
difference between J & (c/4 )   B produces change of E.
J
 ( p2 / 4 )[E  V  B / c  J  B /(ne ec)]  ....
t
deviation of E from value given by generalized Ohm's law
produces change of J , on time scale ~ p-1.
B
 c  E
t
change of B produced only if there is spatial variation.
 V
 ...  J  B / c  in  ( V  Vn )
t
change of bulk flow produced only by stress imbalance.
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, Lp/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
and
electric currents in plasmas, Ann. Geophys., 23,
2589–
2597, 2005.
• Song, P., Gombosi, T. I., and Ridley, A. J.: Three-fluid
Ohm’s law, J. Geophys. Res., 106, 8149–8156,
2001.

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