Space Plasma Physics ? Problems ? 4 Frozen-in magnetic flux

Jacobs University Bremen
Joachim Vogt
Spring 2010
4 Frozen-in magnetic flux
Space Plasma Physics
— Problems —
4.1 Perpendicular magnetic field in two-dimensional flows (Q)
Recall the hydromagnetic theorem:
∂B
∂t
= ∇ × (V × B) .
Show that in planar flows where the magnetic field is straight and vertical, it satisfies a twodimensional
continuity equation. More specifically, assume Vz ≡ 0, Bx ≡ By ≡ 0, ∂/∂z = 0, and
then demonstrate that
∂Bz
∂t + ∇ · (BzV) = 0 .
4.2 Geoeffective electric field and transpolar potential (Q)
The so-called geoeffective electric field is the motional electric field associated with the southward
component (negative Bz) of the interplanetary magnetic field in the solar wind flow.
(a) How large is the geoeffective electric field for Bz = −5 nT and regular solar wind flow
conditions (V = 400 km/s) ?
(b) What is the associated potential drop across the diameter (∼ 40 RE) of the magnetosphere ?
Which value do you get if due to reconnection at the dayside magnetopause about 10% of
this potential couples into the magnetosphere ?
4.3 Walén equation (E)
In this problem you are supposed to combine the hydromagnetic theorem and the mass continuity
equation to find an expression for the variation of B/ϱ along streamlines.
(a) Rearrange the hydromagnetic theorem to obtain an equation for the variation dB/dt of
the magnetic field B along the flow.
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(b) Rearrange the mass continuity equation to find the rate of change of the inverse mass
density along streamlines, i.e., write down an equation for dϱ −1 /dt.
(c) Combine your results from parts (a) and (b) to derive the Walén equation:
d
dt
B
ϱ
=
B
· ∇V
ϱ
that gives the rate of change of B/ϱ in ideal plasmas.
Hints and explanations
In ideal plasmas, the evolution of the magnetic field B is governed by the hydromagnetic theorem
∂B/∂t = ∇×(V × B) where V denotes the plasma bulk velocity. The substantial (or convective
or total) time derivative is given by
d
dt
∂
= + V · ∇ .
∂t
The mass continuity equation is usually written in the form
where ϱ denotes mass density.
∂ϱ
∂t
+ ∇ · (ϱV) = 0
4.4 Vector fields, streamlines, and flow (E)
Consider the following vector fields in two dimensions.
V(r) =
V(r) =
V(r) =
1
, (1)
y
1
2xy/(1 + x2
, (2)
)
U(y)
, (3)
0
V(r) = W (r) ˆϕ . (4)
Here (x, y) denote cartesian coordinates, and (r, ϕ) are polar coordinates. U(y) and W (r) are
arbitrary functions of y and r, respectively.
(a) Construct the flows of the vectors fields.
(b) Sketch a representative set of streamlines. Here you may assume U(y) = y and W (r) = r.
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Hints and explanations
Recall that the streamlines (= field lines = integral curves) r(t) of a vector field V(r) are
solutions of ˙r(t) = V(r(t)). This system of ODEs is usually supplemented by an initial condition
of the form r(t = 0) = r0. The initial value problem corresponds to the task of finding the
streamline that passes through the point r0.
The collection of streamlines constitutes the so-called flow F = F(r, t) of the vector field V
defined through
∂F
= V(r(t)) and F(r, 0) = r .
∂t
For brevity, we write ˙ F = ∂F/∂t.
If the flow F(r, t) is known, the solution of the initial value problem
can be conveniently written in the form:
˙r(t) = V(r(t)) and r(t = 0) = r0
r(t) = F(r0, t) .
4.5 Interplanetary magnetic field lines (H-12p)
The field lines of the average interplanetary magnetic field (IMF) are usually referred to as
Parker spirals. In the equatorial plane, the IMF is conveniently written as
B = B(r) = Br(r)ˆr + Bλ(r) ˆ λ
where λ denotes the azimuth and r is radial distance. We are interested in the scaling behavior
of Br and Bλ with r.
In the following, we construct Parker spirals through an essentially stationary solution of the
hydromagnetic theorem. The radial solar wind outflow combined with the effect of solar rotation
yields an effective plasma velocity in the equatorial plane with components Vr and Vλ in radial
and in azimuthal direction, respectively. The radial component is assumed to be constant and
equal to the solar wind speed Vsw.
(a) What does ∇ · B = 0 imply for the radial component of B = B(r) ? Write down Br in
terms of r and the field value Br0 at a reference distance R0.
(b) Assuming that the Sun rotates at an angular rotation rate Ω, how does Vλ scale with radial
distance ?
(c) With the help of the frozen-in flux concept, express Bλ in terms of the supposedly constant
radial component Vr = Vsw and the azimuthal component Vλ(r) from part (b).
(d) Construct and then solve a differential equation for the interplanetary field lines. Sketch
their geometrical shape.
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4.6 Open and closed magnetosphere (H-20p)
The scalar potential of a magnetic dipole in spherical coordinates (r, ϑ, λ) is given by
3 cos ϑ
Φd(r, ϑ) = Bd∗R
r2 where Bd∗ is a reference field value. The parameter R denotes a reference radius, e.g., the radius
of a magnetized planet. The dipole moment is assumed to be aligned with the polar (z) axis.
In this problem we consider the superposition of such a dipole field and a uniform field
Bh = Bh∗ˆz ,
so |Bh∗| is the field strength. The sign of Bh∗ controls the orientation of Bh.
(a) Compute the dipole field Bd = −∇Φd. If Bd is supposed to correspond to the geomagnetic
field, and R = RE is the radius of the Earth, what is the meaning of the parameter Bd∗ ?
Should it be positive or negative ?
(b) Write the uniform field in spherical coordinates and find a scalar potential Φh = Φh(r, ϑ)
so that Bh = −∇Φh. If Bh is supposed to correspond to the North-South component
of the interplanetary magnetic field, should the parameter Bh∗ be positive or negative
to yield reconnection at the dayside magnetopause ? In the following the reconnection
case is referred to as an open magnetosphere, and the opposite orientation gives a closed
magnetosphere.
(c) Construct the superposed potential and field in spherical coordinates:
Φ = Φd + Φh , B = Bd + Bh .
(d) In the closed magnetosphere case, there exist a sphere where the radial field component
vanishes. What are the implications for the field line topology ? Express the radius of
the sphere in terms of the parameters Bd∗, Bh∗ and R. If Bd∗ and R are chosen to be
consistent with the geomagnetic field, what is the numerical value of Bh∗ that yields the
actual dayside magnetopause stand-off distance for the radius of that sphere ? Discuss.
(e) Plot the field lines of the combined field B for the closed magnetosphere case.
Hint: A convenient approach is through the construction of an appropriate stream function
Ψ = Ψ(r, ϑ) for the axisymmetric field B. Try the ansatz B = ∇Ψ × ∇λ. Field lines are
contours of Ψ.
(f) Plot the field lines of the combined field B for the open magnetosphere case.
(g) In the open magnetosphere case, the area where surface magnetic flux connects to interplanetary
space defines the so-called polar cap. Determine the opening angle of the polar
cap in terms of the model parameters Bd∗, Bh∗ and R.
Submission due date for homework problems: 8 March 2010
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