Download 5.14 The magnetospheric ring current. The largest component of the

5.14 The magnetospheric ring current. The largest component of the external variations in Earth’s magnetic field
comes from a ring of current circulating westward at a distance of 2-9 Earth radii. To understand why this might be
we need to consider the nature of charged particles moving in a plasma. A charged particle moving with velocity v in
a magnetic field B will be acted on by the Lorentz force
f = ev × B
where here we assume an electron with a charge of −e. If v is perpendicular to B then the particle moves in a circular
path whose radius can be obtained by balancing the centripetal force with the Lorentz force:
mv 2
= evB
r
mv
r=
.
eB
Consider a particle thus spiraling around a magnetic field line near the equator. The main magnetic field, originating
inside the core, has a radial gradient and is larger closer to Earth. The radius of curvature of a particle will thus be
smaller closer to Earth. Looking down from the North pole, a positive particle will circle clockwise and drift westward;
an electron will do the opposite, and together they make a westward directed current that circles Earth between 2 and
9 Earth radii. The strength of the ring current is characterized by the ‘Dst’ (disturbance storm time) index, an index of
magnetic activity derived from a network of low- to mid-latitude geomagnetic observatories that measures the intensity
of the globally symmetrical part of the equatorial ring current.
Magnetic storm of June 1982
sudden commencement
0
-50
Dst, nT
-100
-150
recovery
-200
-250
-300
1982.53
1982.535
1982.54
Year
1982.545
1982.55
Figure 5.28: A typical magnetic storm as recorded by the Dst index.
The solar wind injects charged particles into the ring current, mainly positively charged oxygen ions. Sudden increases
in the solar wind associated with coronal mass ejections cause magnetic storms (Figure 5.28), characterized by a sudden
commencement, a small but sharp increase in the magnetic field associated with the sudden increased pressure of the
solar wind on Earth’s magnetosphere. Following the commencement the main phase of the storm is associated with a
large decrease in the magnitude of the fields as the ring current is energized – the effect of the ring current is to cancel
Earth’s main dipole field slightly. Finally, there is a recovery phase in which the field returns quasi-exponentially back
to normal. All this can happen in a couple of hours, or may last days for a large storm.
On the 13th March 1989 a 600 nT magnetic storm caused the entire power grid in Quebec, Canada, to collapse.
5.15 Electromagnetic induction. One of the key concepts in geomagnetic induction is that of the skin depth, the
characteristic length over which electromagnetic fields attenuate. We can derive the skin depth starting with Faraday’s
Law:
∂B
∇×E=−
∂t
154
and Ampere’s Law:
∇×H=J
where J is current density (A/m2 ), E is electric field (V/m), B is magnetic flux density or induction (T), H is magnetic
field intensity (A/m). We can use the identity ∇ • ∇ × A = 0 to show that
∇•B=0
∇•J=0
and
in regions free of sources of magnetic fields and currents. B and H are related by magnetic permeability µ and J and
E by conductivity σ:
B = µH
J = σE
(the units of σ are S/m, where S = 1/Ω; the latter equation is Ohm’s law), and so
∇ × E = −µ
∂H
∂t
∇ × H = σE
.
If we take the curl of these equations and use ∇ × (∇ × A) = ∇(∇ • A) − ∇2 A
∇2 E = µ
∂E
∂
(∇ × H) = µσ
∂t
∂t
∇2 H = σ(∇ × E) = µσ
∂H
∂t
.
(5.27)
(5.28)
You will recognize these as diffusion equations. Now if we consider sinusoidally varying fields of angular frequency
ω and a uniform conductivity σ
∂E
= iωE
E(t) = Eo eiωt
∂t
H(t) = Ho eiωt
∂H
= iωH
∂t
and so
∇2 E = iωµσE
∇2 H = iωµσH
(5.29)
.
(5.30)
These are the equations describing propagation of electric and magnetic fields in a conductive medium. In air and very
poor conductors where σ ≈ 0, or if ω = 0, the equations reduce to Laplace’s equation.
If we now consider fields that are horizontally polarized in the xy directions and are propagating vertically into a
half-space, these equations decouple to
∂2E
+ k2 E = 0
∂z 2
∂2H
+ k2 H = 0
∂z 2
with solutions
E = Eo e−ikz = Eo e−iαz e−βz
(5.31)
H = Ho e−ikz = Ho e−iαz e−βz
(5.32)
where we have defined a complex wavenumber
k=
p
iωµσ = α − iβ
155
(5.33)
and an attenuation factor, which is called a skin depth
s
zs = 1/α = 1/β =
2
σµo ω
.
(5.34)
The skin depth is the distance that field amplitudes are reduced by 1/e, or about 37%, and the phase progresses one
radian, or about 57◦ . In practical units
p
zs ≈ 500m 1/(σf )
where circular frequency f is defined by ω = 2πf .
The skin depth concept underlies all of inductive electromagnetism in geophysics. Substituting a few numbers into
the equation shows that skin depths cover all geophysically useful length scales from less than a meter for conductive
rocks and kilohertz frequencies to thousands of kilometers in mantle rocks and periods of days (see table). Skin depth
is a reliable indicator of maximum depth of penetration.
material
σ, S/m
1 year
1 month
1 day
1 hour
1 sec
1 ms
core
lower mantle
seawater
sediments
upper mantle
igneous rock
3×105
10
3
0.1
0.01
1×10−5
4 km
900 km
1600 km
9000 km
3×104 km
106 km
770 m
170 km
470 km
1700 km
104 km
2×105 km
200 m
46 km
85 km
460 km
1500 km
4×104 km
40 m
9 km
17 km
95 km
300 km
9500 km
71 cm
160 m
280 m
1.6 km
5 km
160 km
23 mm
5m
9m
50 m
158 m
5 km
Skin depths for a variety of typical Earth environments and a large range of frequencies. One can see that the core is
effectively a perfect conductor into which even the longest period external magnetic field cannot penetrate. However,
1-year variations can penetrate into the lower mantle. The very large range of conductivities found in the crust indicates
the need for a corresponding large range of frequencies in electromagnetic sounding of that region.
5.16 Geomagnetic depth sounding. Geomagnetic depth sounding, or GDS, is used to derive the electrical conductivity
inside Earth using measurements of the magnetic field. The geomagnetic ring current generates a field that is largely
dipolar in morphology (it is like being inside a big solenoid), and if one chooses a coordinate system defined by the
dipole field (a geomagnetic coordinate system), then the field can be described as P10 in morphology. Equation 5.6
reduces to
2 0
0 r
0 a
V (r, θ) = aP1 (cos θ) (ge )1
+ (gi )1
(5.35)
a
r
(notice that we have kept the internal AND external Gauss coefficients, since we will be interested in the internally
induced fields).
One can define a geomagnetic response at a given frequency ω as simply the ratio of induced (internal) to external
fields:
gi (ω)
.
(5.36)
Q(ω) =
ge (ω)
In certain circumstances, such as satellite observations, one has enough data to fit the ge and gi directly. However, most
of the time one just has horizontal (H) and vertical (Z) components of B as recorded by a single observatory. We can
obtain H and Z from the appropriate partial derivatives of V to obtain:
1 ∂V
−H = µo
r ∂θ r=a
156
= µ(ge + gi )
∂ 0
∂
P1 (cos θ) = µ(ge + gi ) (cos θ)
∂θ
∂θ
= µAH sin θ
and
(5.37)
∂V
−Z = µ
∂r r=ao
= µ(ge − 2gi )P10 (cos θ)
= µAZ cos θ
(5.38)
where we have defined new expansion coefficients
AH = ge + gi
AZ = ge − 2gi
.
Thus we can define a new electromagnetic response
W (ω) =
AZ
Z(ω) sin θ
=
H(ω) cos θ AH
(5.39)
where θ is the co-latitude of the observatory and we have included the ω dependence to remind us that this is all for a
particular frequency. W is related to Q by
Q(ω) =
gi (ω) 1 − AZ /AH
=
ge (ω) 2 + AZ /AH
.
The admittance or inductive scale length c provides a practical unit for interpretation, and is given by
c(ω) =
a
W (ω).
2
(5.40)
Note that for causal systems, the real part of c is positive, and the imaginary part is always negative. For a uniform
earth of conductivity σa , the resistivity is given by
ρa = 1/σa = ωµo |ic|2
.
(5.41)
3000
2500
2000
C, km
1500
Real (C)
1000
500
0
−500
−1000
Imag (C)
1 hour
3
1 day
4
5
1 month
6
Log (Period, s)
157
1 year
7
11 years
8
9
Figure 5.29: The admittance, c, based on an analysis of geomagnetic observatory data. The red line is a fit to the data
for the model shown in Figure 5.30.
Conductivity, S/m
Core
LOWER MANTLE
10 1
Perovskite at 2000°C
UPPER MANTLE
10 2
TRANSITION ZONE
10 3
Bounds for monotonic models
10 0
Ringwoodite/
Wadsleyite at 1500°C
10
Bounds on all models
−1
Olivine at 1400°C
10 −2
10 −3
0
500
1000
1500
Depth, km
2000
2500
3000
Figure 5.30: Electrical conductivity in Earth’s mantle derived from the GDS data shown in Figure 5.29. The blue line
shows one model that fits the data (the fit is shown in Figure 5.29), but the yellow regions give upper and lower bounds
on average conductivity for the three mineralogical mantle regimes. Black dots are conductivities of representative
mantle minerals at appropriate pressures and temperatures.
158