The Physics of Field-Aligned Currents
Andrew N. Wright
U NIVERSITY OF S T A NDREWS
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Magnetospheric Current Circuit
There is a rich structure of currents flowing parallel (jk ) and perpendicular (j⊥ ) to the
magnetic field (B).
●
●
●
Magnetopause current
Ring current
Ionospheric currents
Region 1 and 2
currents
● ULF Alfvén waves
●
Magnetopause current
Region 1
current
Region 2
current
Ionospheric
Pedersen current
Partial ring current
[Figure from NASA GSFC.]
View from tail
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Quasi-Neutrality and Current Continuity
●
The Ampère-Maxwell equation states
∇ × B/µ0 = j + ε0
●
Taking the divergence yields
0 = ∇ · j + ε0
●
∂E
.
∂t
∂∇ · E
.
∂t
Using Gauss’s Law, ∇ · E = ρ∗ /ε0 (where ρ∗ = net charge density) we find
∂ρ∗
∇·j+
= 0.
∂t
●
Quasi-neutrality (ρ∗ ≈ 0) ⇒
❍
❍
❍
❍
∇ · j ≈ 0: Currents are self-closing.
The displacement current (ε0 ∂ E/∂t) is neglected.
ρ∗ ≈ 0 means (np
e − ni )/ne 1.
Debye length is ε0 kTe /e2 ne ∼ 100 m (magnetosphere), 0.01 m (ionosphere).
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Some Magnetohydrodynamic Driving Mechanisms
Fundamental standing (ULF)
Alfvén wave
● Axisymmetric oscillation of
magnetic shells
● bφ and uφ components only
● jk and j⊥ currents
●
●
Imposed velocity shear [Rönnmark, GRL,
1998]
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Generation of jk (MHD description)
●
Single fluid MHD momentum equation
ρ
●
●
Solve for j⊥
dv
= j × B − ∇p + F
dt
dv
B
j⊥ = − ρ + ∇p − F × 2
dt
B
Since ∇ · j = 0 we find jk
Z
∂jk
+ ∇⊥ · j⊥ = 0,
⇒
jk = −
∇⊥ · j⊥ds
∂s
along B
●
The field-aligned current (jk ) flows to maintain quasi-neutrality (ρ∗ ≈ 0)
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What is jk microscopically?
●
Net drift of charged particles parallel to B
●
Consider the parallel component of the equation of motion for a charged particle (uniform B),
dvk qEk
=
.
dt
m
Ek can be used to establish a field-aligned current.
●
Since me /mi P
1 the electrons are more mobile, and carry most of the parallel
current: jk ≈
−evek
●
In MHD me /mi → 0: electrons are represented by a massless charge-neutralizing
fluid (Ek → 0)
●
To generate jk requires Ek
❍
❍
How does Ek arise?
Causal interpretation in ideal MHD difficult since ∂ E/∂t has been neglected
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Guiding centre description of particle motion
●
Particles execute a circular trajectory about B whose centre drifts. Valid when
❍
❍
●
Gyroradius background scale length
Gyroperiod background timescale
Guiding centre drifts parallel to B arise from
❍
Ek and magnetic mirror force
m
❍
●
dvk
= qEk + µ∇kB
dt
The magnetic moment µ = mv⊥2 /2B is a constant of motion.
Guiding centre drifts perpendicular to B (can depend upon q , m and energy) arise from
❍
❍
❍
❍
Grad B drift
B curvature drift
Polarization drift (E(t))
E × B drift
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Generation of jk (particle description)
●
In general, given E(r, t) and B(r, t), electrons and ions drift relative to one
another ⇒
❍
❍
●
current can flow
net charge density is likely to develop ρ∗ 6= 0
As ρ∗ becomes non-zero, Ek and E⊥ change to satisfy
∇ · E = ρ∗/ε0
and influence the particle motion
●
Effect of even a small Ek :
❍ electrons are accelerated parallel to B
❍ jk is established
❍ parallel electron motion acts to reduce ρ∗
❍ the plasma remains quasi-neutral (ρ∗ ≈ 0)
●
Interestingly, E still satisfies
∇ × B/µ0 = j + ε0
∂E
,
∂t
even though the last term is still negligible.
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Ring Current and Standing Alfvén Wave jk
ve||
ve||
E
||
E
||
E
j
E
−ve
+ve
vi
j
j
j||
electrons
electrons
j||
ions
j
E
||
E
||
ve||
B0
●
●
●
j||
electrons
ve||
electrons
B0
Looking earthward from the tail
Warm ion cloud drifts westward
Slight charge imbalance generates
E, electron motion and jk
j||
Snapshot of Alfvén wave current circuit
E(t) in equatorial region produces polarization drift
● Ions drift across L-shells leading to charge
imbalance and subsequent jk
●
●
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Magnetosphere-Ionosphere Equilibrium
Near the Earth
●
A simple equilibrium has a total ion number density comprising:
❍
❍
●
constant magnetospheric contribution (nM ∼ 1 cm−3 )
exponentially decreasing ionospheric contribution (nI ∼ 104 − 106 cm−3 ,
scale height ∼ 100 − 200 km)
If n = nI + nM and B is dipolar, B/n has a peak at a few thousand km altitude
❍
❍
below B/n peak: B/n ∼ exp(+r/h)
above B/n peak: B/n ∼ 1/r3 •First •Prev
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Upward and Downward Currents and the Auroral
Acceleration Region
●
Upward Current:
❍
●
Downward Current:
❍
●
magnetospheric
electrons
precepitated
⇒ visible
aurora
ionospheric electrons evacuated to magnetosphere
Current-Voltage (energy) relations for upward and downward
currents?
[Marklund et al., Nature, 2001.]
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Gyrotropic Electron Vlasov Equation: f (s, vk, v⊥, t)
●
Retains information of ionospheric and magnetospheric electron populations
∂f
∂f
+ vk
−
∂t
∂s
eEk v⊥2 ∂B
·
+
me
2B ∂s
vkv⊥ ∂B ∂f
∂f
·
·
+
=0
∂vk
2B ∂s ∂v⊥
s = field-aligned coordinate.
●
For a steady current E = −∇φ ⇒ Ek = −∂φ/∂s. Constants of motion are
total energy: W = 21 me v 2 − eφ
2
❍ magnetic moment: µ = me v⊥
/2B
❍
●
Solve for f (s, vk , v⊥ ) with vk (W, µ), v⊥ (W, µ)
❍
❍
Liouville’s Theorem (f is constant along an electron trajectory)
ne = ni ⇒ φ(s) and
Z
jk = −e
●
vkf d3v
Solution relates current flow to potential drop along the field line
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Upward Current (downgoing electrons)
●
Assuming a potential drop φm : map magnetospheric electrons down to the
ionosphere using W and µ to identify the loss cone.
●
Calculate jk at the ionosphere in terms of φm :
r
jk ≈ n 0 e
kT
2πme
eφm
1−
kT
The “Knight” relation [Planet. Space Sci., 1973].
❍
❍
❍
❍
❍
Useful for interpreting data (electron energies)
Good for incorporating in global modelels
Knight’s solution says little about quasi-neutrality or φ(s) variation
Ek needed to overcome mirror force
ve||
E||
Where does acceleration occur?
ve||
acceleration
region
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Upward Current Quasi-Neutral Solution
●
Quasi-neutral solution for previous B/n variation gives [Cran-McGreehin, 2006]
●
●
●
Plots of normalized potential and Ek along B. (Ionosphere at s = 0)
Below B/n peak, ambipolar Ek traps ionospheric electrons
Above B/n peak, Ek
❍
❍
helps precitating electrons overcome the mirror force
adjust mirroring magnetospheric electrons to maintain quasi-neutrality
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Downward Current (upgoing electrons)
●
Some ionospheric electrons are accelerated upward into the magnetosphere
Velocity parallel to B
Ionosphere
Magnetosphere
Beam
lm
lc le lp
l0
[Cran-McGreehin and Wright, JGR, 2005a,b]
●
Field-aligned coordinate is `. Important locations are
❍ `p :
❍ `c :
location of the B/n peak
ionospheric electron trapping point/beam emergence height
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Downward Current Quasi-Neutral Solution
●
Quasi-neutral solution for previous B/n variation gives [Cran-McGreehin
and Wright, JGR, 2005a,b]
●
●
●
Plots of normalized potential and Ek along B. (Ionosphere at s = 0)
Electron acceleration is centred around the B/n peak
Ionospheric jk of ∼ µAm−2 correspond to potential drops along B of ∼keV
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Analytical Current-Voltage Relation
(Downward Current)
●
The potential drop (φm ) depends upon jk and n at the B/n peak, as well as the magnetospheric electron temperature (T ) [Cran-McGreehin and Wright, JGR, 2005b]
❍
2
me/2kT n2pe2 < 1.7 then
If jkp
3
−φm ≈
2
❍
2
jkp
m1/2
e kT
npe5/2
4
jkp
m2e kT
n4pe7
1
+
2
! 31
+
2
jkp
me
6n2pe3
2
If jkp
me/2kT n2pe2 > 1.7 then
kT
ln
−φm ≈
e
●
●
! 23
2
jkp
me
n2pe2kT
!
+
2
jkp
me
2n2pe3
+
kT
e
Approximations accurate to at least 5%
May only need one or two terms
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Conclusion and Summary
●
Field aligned currents
❍
❍
❍
are an integral part of the magnetosphere
arise from both large scale driving and internal particle drifts
carried mainly by electrons (accelerated by Ek )
●
Downgoing electrons excite the aurora
●
Details of φ(s) and Ek (s) are different for upward and downward currents
(but B/n peak location is important)
●
Analytical Current-Voltage relations available
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Future Work
●
Allow ions to move and modify background density
●
Address time-dependence
●
Combine large and small scale physics of acceleration region (FAST)
●
Other acceleration mechanisms (wave-particle interactions?)
●
Interpret with governing equations
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