Download The Inner Magnetosphere

The Inner Magnetosphere
Nathaniel Stickley
George Mason University
1
Overview
• Particle populations
– Radiation belts, plasmasphere, ring current
• Particle injection and energization
– Diffusion, wave-particle interaction
• Electric fields and drift paths
– Shielding, co-rotational electric field, Alfvén layer
• DPS relation
– Derivation, discussion
• Modeling
– Rice Convection Model (RCM)
2
Particle populations
Radiation belts (Van Allen, 1958)
Inner belt
Located at L ≈ 1.1-3.3
Primarily cosmic ray albedo protons of high energy (>10MeV)
Very stable
Outer belt
Located at L≈3-9
Primarily high energy electrons with energy up to 10MeV
Population is unstable (particles are not trapped as efficiently)
3
Particle populations
Radiation belts
Electron “slot” region
Located at L ≈ 2.2
Apparently due to increased
wave-particle interactions
There is no corresponding slot
for ions
4
Particle populations
Plasmasphere
Cool particles (~1eV-1keV)
High particle density (~103 cm-3)
Extends to L=3-6
Distinct from radiation belts but shares same region of space
Primarily ExB drifting particles (because of low temperature)
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Particle populations
Ring current
Mostly indistinguishable from trapped radiation belts
Gradient and curvature drift
Composed of mostly 20-300keV ions
Typically in the range L=3-6
O+ is dominant Ion in terms of abundance
H+ begins to dominate > few keV
Total energy density dominated by O+ and H+
Energy midpoint: 85keV
6
Particle injection and energization
How are particles injected into the inner
magnetosphere?
Cosmic rays
Ionosphere injection
Substorm and storm particle injections
Diffusion (adiabatic invariants do not strictly hold).
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Brief theoretical aside (1)
Sample derivation of the 1st adiabatic invariant: 
Definition of adiabatic invariant:
S
 p  dq
•p and q are canonical momentum and coordinates
respectively.
•Integration is performed over one cycle
•If system changes slowly during each cycle, the action
S is a constant.
We could use this definition to show that μ is an adiabatic
invariant, but we will use a less direct approach in order
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to illustrate a point.
Brief theoretical aside (2)
Starting with Faraday’s law:
B
   E
t
and the equation of motion:
dv 
m
 q  E  v   B
dt
where v  = dl /dt
Taking the scalar product of this with v 
 dv  
m
  v   qE  v 
 dt 
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Brief theoretical aside (3)
 dv  
m
  v   qE  v 
 dt 
Left hand side is rate of change of KE:
d 1 2
 dl 
 mv    qE   
dt  2

 dt 
Variation in KE is:
2 / 
1 2
 dl
  mv     qE  
2

 dt
0

d t

Key point: if the field changes slowly, this is the
same as
1 2
  mv     qE  dl
2

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Brief theoretical aside (4)
1 2
  mv   
2

 qE  dl
Re-write using Stokes’ theorem:
1 2 
  mv     q(  E)  dS
2
 S
Now we use Faraday’s law:
B
1 2 
  mv    q 
 dS
t
2

S
B  dS  0 for ions B  dS  0 for electrons
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Brief theoretical aside (5)
2
 mv   B
B
1 2 
2 B
  mv    q 
 dS  q rL
 q 

t
t
2

S
 q B  t
mv 2
Factoring out  
2B
2
1 B
 1 2   mv    2  B
  mv    


  B
 
T t
2
  2 B     t
Thus:    B    B   = const
1 2
Since  B  mv  and    B   B   B
2
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Particle injection and energization
How particles in the inner magnetosphere become
energized:
Electric fields
Wave-particle interaction
• Whistlers (review)
Play whistler audio in Adobe Audition:
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Orientation Reminder
Dawn
f
Convection Electric Field
Sun
Dusk
14
Polarization field
In the absence of the convection
electric field, the plasma sheet
appears as illustrated:
What happens when the
convection electric field is
included?
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Polarization field
Positive charge collects on duskward edge of
plasma sheet while negative charge collects
on the dawnward edge.
--Resulting electric field is called
the “polarization electric field”.
The inner magnetosphere is
thus shielded from convection
electric field.
Other features:
+++
Over-shielding
Partial ring current
Region 2 Birkeland current
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Overview of current systems
17
Co-rotation Electric field (review)
Ecorotate  (ω  r )  B  
Dominates for highenergy particles
E B0 RE3
r
2
rˆ
Dominates for low-energy
particles
Small due to
shielding
Etotal
3 B0 RE3
E B0 RE3
 Ey yˆ 
rˆ 
rˆ
4
2
qr
r
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The Alfvén Layer
For cool particles, E  B drift dominates
E  B drift
19
The Alfvén Layer
Recall the shape of the plasmasphere
Compare with the shape of low-temperature Alfvén layer.
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The Alfvén Layer
Hot ions vs. Hot electrons
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The Alfvén Layer
Variability in position:
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Dessler-Parker-Schopke (DPS) Relation
• Derived originally in 1959 by Dessler and Parker
• Relates the total energy in the ring current to the
magnetic field perturbation at the center of the Earth
B(0)
2E
  RC
B0
3U E
 B(0)[nT]  3.98 1030  ERC [keV]
B(0) : Magnetic field perturbation at center of Earth
B0 : Equatorial surface magnetic field strength
ERC : Total energy of ring current
U E : Magnetic energy of dipole field beyond Earth's
surface
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Derivation of DPS Relation (1)
mv 2 W
Bd  Bd
Starting with  

and gradient drift: v g  
,
2
2 Bd Bd
qBd
0
where Bd 
3(μ  rˆ )rˆ  μ 
3
4 r
3
B0
 RE 
 Bd  B0 
  3
L
 r 
at equatorial plane
r : distance from center of Earth, (r  LRE )
Bd : dipole field
μ : dipole moment of Earth's field
W : kinetic energy of charged particle
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Derivation of DPS Relation (2)
we want to find the magnetic field perturbation of a single charge,
so we first calculate the current due to one charge:
I drift
q Bd  Bd
 Bd  Bd



2
2 r 2 r qBd
2 r
Bd2
qvg
3
3B0 RE3
 RE 
Now, since Bd  B0 
 then Bd   4
r
 r 
3Bd B0 RE3
and Bd  Bd  Bd Bd  
r4
since Bd  Bd
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Derivation of DPS Relation (3)
I drift
 Bd  Bd 3 Bd B0 RE3


2
2 r
Bd
2 r 5 Bd2
W
B0
Using substitutions:  
, Bd  3 , r  LRE
Bd
L
I drift
3WL

2 B0 RE2
The field at the center of the Earth due to this current is:
B
drift
r 0

0 I drift
2r
zˆ  
30W
zˆ
3
4 B0 RE
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Derivation of DPS Relation (4)
There is also a contribution due to gyration of the charge:
Brgy0
0  gy
0W
ˆ

z
zˆ
3
3
4 r
4 B0 RE
The total field perturbation for the charge is:
B(0)  B
gy
r 0
 B
drift
r 0
 0W

zˆ
3
2 B0 RE
The perturbation for all ring current particles combined is:
 0 ERC
B(0) 
zˆ
3
2 B0 RE
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Derivation of DPS Relation (5)
 0 ERC
From previous slide: B(0) 
zˆ
3
2 B0 RE
The total dipole field energy outside of the Earth's surface is:
UE 
1
1
B

H
dV

2 r RE
2 0

B 2 dV
r  RE
4 2 3
Skipping integration details... U E 
B0 RE
30
B(0)
2 ERC
Writing B(0) in terms of U E :

B0
3U E
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Problems with the DPS relation
False assumptions:
•
•
•
•
Ring current is circular and concentric with Earth
Ring current is azimuthally symmetric (no partial ring)
Magnetic field is purely dipolar
Field perturbation due to ring current is not important
– Nonlinear “feedback” is not accounted for.
• Earth is non-conducting
• Assumes ring current is confined to the equatorial plane.
– However Schopke proved that the relation holds for arbitrary
pitch angle
29
Testing the DPS relation
Experimental:
•
DPS typically estimates Dst index to within 20% (the relation does
not pretend to include affects from other current systems, so this is rather
impressive)
Computational:
•
Liemohn (2003) used computational model of ring current to test
DPS relation
1.
2.
3.
4.
•
Calculate realistic particle distributions
Calculate pressures from particle distributions (include non-zero
pressure outside of volume of integration)
Calculate currents using pressure information
Calculate ΔB(0) from currents using Biot-Savart
DPS systematically over-estimates ΔB(0) for isotropic pressure
distribution.
BBSI P

BDPS P||
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Modeling the Inner Magnetosphere
• Goals
–
–
–
–
Calculate-particle distributions / drifts
Self-consistently calculate electric fields
Self-consistently calculate magnetic fields
Couple inputs and outputs to global MHD and
ionosphere models
• What physics should be included in such a
model?
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Rice Convection Model
Multi-fluid model (typically 100 fluids)
Self-consistently computes electric fields
Isotropic particle distribution
Calculates adiabatic drifts
Requires specified magnetic field
Input:
Magnetic field model
Polar cap potential distribution
Initial plasma density
Plasma boundary conditions
Some possible outputs:
velocity distribution
particle fluxes
potential distribution (and therefore electric fields)
Ionospheric precipitation
32
Rice Convection Model
Fundamentals
Flux tube volume: V 

field line
ds
B
ds
Energy Invariant: k  WkV 2/ 3
1/B
E  1q Wk   B
Drift: v k 
(1)
2
B
Particles per unit magnetic flux: k


Conservation law:   v k  k  S (k )  L(k ) (2)
 t

33
Rice Convection Model
Fundamentals
Electric field: E  -  v  B
Potential:    convection   corotation   field aligned
2
Pressure: P  V
k k

3 k
j|| N  j||S
B
Field-Aligned Current:
 2 V  P (3)
Bionisphere B
5/ 3
Algorithm:
Iterate through eq(2) and eq(3), updating velocities with
eq(1).
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Coupling the models
RCM requires as boundary conditions:
• Ionosphere conditions (conductivities, etc)
• Magnetic field
• Outer plasma conditions
Solution:
• Couple RCM with MHD model for outer magnetosphere
• Couple RCM with ionosphere model.
• Self-consistently compute magnetic field with MHD
model.
• There are difficulties in actually implementing this.
36