Transcript No Slide Title

The Magnetopause
• Back in 1930 Chapman and Ferraro foresaw that a
planetary magnetic field could provide an effective
obstacle to the solar-wind plasma.
• The solar-wind dynamic pressure presses on the outer
reaches of the magnetic field confining it to a
magnetospheric cavity that has a long tail consisting of two
antiparallel bundles of magnetic flux that stretch in the
antisolar direction.
• The pressure of the magnetic field and plasma it contains
establishes an equilibrium with the solar wind.
• The solar wind is usually highly supersonic before it
reaches the planets. The wind velocity exceeds the velocity
of any pressure wave that could act to divert the flow
around the obstacle and a shock forms.
Lecture 8
Magnetopause
Magnetosheath
Bow shock
Fore Shock
Homework:
6.5, 6.10, 6.11*, 8.1, 8.3, 8.7, 8.2*
A Digression on the Dipole Magnetic Field
• To a first approximation the magnetic field of the Earth can
be expressed a that of the dipole. The dipole moment of the
Earth is tilted ~110 to the rotation axis with a present day
value of 8X1015Tm3 or 30.4x10-6TRE3where RE=6371 km
(one Earth radius).
• In a coordinate system fixed to this dipole moment
Br  2Mr 3 cos
B  Mr 3 sin 
B  Mr (1  3 cos  )
3
2
1
2
where  is the magnetic colatitude, and M is the dipole
magnetic moment.
The Dipole Magnetic Field
• Alternately in cartesian coordinates
Bx  3xzM z r 5
By  3 yzM z r 5
Bz  (3z 2  r 2 ) M z r 5
• The magnetic field line for a dipole. Magnetic field lines
are everywhere tangent to the magnetic field vector.
dr
d
d  0
r
Br
B
• Integrating r  r0 sin 2  where r0 is the distance to equatorial
crossing of the field line. It is most common to use the
magnetic latitude  instead of the colatitude
r  L cos2 
where L is measured in RE.
Properties of the Earth’s Magnetic Field
• The dipole moment of the Earth presently is ~8X1015T m3
(3 X10-5TRE3).
• The dipole moment is tilted ~110 with respect to the rotation axis.
• The dipole moment is decreasing.
– It was 9.5X1015T m3 in 1550 and had decreased to 7.84X1015T m3 in
1990.
– The tilt also is changing. It was 30 in 1550, rose to 11.50 in 1850 and has
subsequently decreased to 10.80 in 1990.
• In addition to the tilt angle the rotation axis of the Earth is inclined by
23.50 with respect to the ecliptic pole.
– Thus the Earth’s dipole axis can be inclined by ~350 to the ecliptic pole.
– The angle between the direction of the dipole and the solar wind varies
between 560 and 900.
The Magnetosphere
The Magnetopause
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In the simples t approximation the
magnetopause can be considered to be the
boundary between a vacuum magnetic field
and a plasma.
Charged particles in the solar wind approach
the Earth’s magnetic field B which is pointed
upward in the equatorial plane
The Lorentz force q(V x B) on the particles
deflects protons to the right (left hand
gyration), and electrons to the left (right hand)
The opposite motion of the charges produces a
sheet current from left to right (dawn to dusk)
Solar
Magnetic perturbations from this current
Wind
reduce the Earth’s field Sunward of the current
and increase the field Earthward
Above the pole the field points in the opposite
direction so the current does as well. This is the
return current
Return
Current
North
Magnetopause
B
Dusk
V
F=q(VxB)
Chapman-Ferraro
Current
The Magnetosphere
A Particle View of the Magnetopause

 penetrates
When an electron or ion
the
uB
boundary they sense a
force. After half
an orbit they exit the boundary.
•
The electrons and ions move in opposite
directions and create a current. The ions
move farther and carry most of the current.
The number of protons per unit length in the
z-direction that enter the boundary and cross
y=y0 per unit of time is 2rLpnu . (Protons in a
band 2rLp in y cross the surface at y=y0.)
Since each proton carries a charge e the
current per unit length in the z-direction
crossing y=y0 is
2nm p 2
I  2rLp nue 
u
Bz
where rLp  (um p ) (eBz )
I   jdx
•
Applying Ampere’s law and noting
•
Bz   0 I
B z2
 nm p u 2  
2 0
2
sw sw
u
A Fluid Picture of the Magnetopause
• The location of the boundary can be calculated by
requiring the pressure on the two sides of the boundary to
be equal. The pressure in the magnetosphere which is
mostly magnetic must match the pressure of the
magnetosheath which is both magnetic and thermal.
• The magnetosheath pressure is determined by the solar
wind momentum flux or dynamic pressure.
2
 sw usw
 
• The current on the boundary must provide a j  B force
sufficient to change the solar wind momentum (divert the
flow).
• The change in momentum flux into the boundary is 2 swusw2
(we are assuming perfect reflection at the boundary)
 
2
2 u  I  B  B
2
sw sw
0
Current Continuity
From
  E= e /  0
  B   0 J   0 0
E
t
We have
 e
J  0
t
In long time scales (for MHD)
e
is very small.
t
J  0
Current has no source nor sink. Current lines are continuous.
The Magnetosphere
Currents on the Magnetopause
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Near the pole there is a singular point in
the field where |B| = 0. This is called
the neutral point.
The Chapman-Ferraro (C-F) current
circulates in a sheet around the neutral
point
This current is symmetric about the
equator with a corresponding
circulation around the southern neutral
point
The C-F current completely shields the
Earth’s field from the solar wind
confining it to a cavity called the
magnetosphere
North
Magnetopause
Field Line
Neutral
Point
Solar
Wind
Dusk
Chapman-Ferraro
Current
The Magnetosphere
The Location of the Magnetopause
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The standoff distance to the subsolar
magnetopause is determined by a balance
between the solar wind dynamic pressure and
the magnetic field inside the boundary
The collisions of particles with the boundary
may not be completely elastic hence a factor k
is introduced
The magnetic field inside the boundary is the
total field from dipole and boundary current.
For an infinite planar sheet current the field
would be exactly doubled. Inside a spherical
boundary the multiplication factor is 3. The
factor f must lie in this range .
Equate and substitute for the dipole strength
variation with distance
Solve for the dimensionless standoff distance
Ls.
pdyn  pB
pdyn  kmnu2
pB
f 2 BD2
kmnv 
2 0
2
2
(
fBD )

BD  B0 æç RE ö÷
è Rs ø
3
2 0
Where k is the elasticity of particle collisions
and f is the factor by which the magnetospheric
magnetic field is enhanced by the boundary
current. Rs is the subsolar standoff distance.
æR
Ls  çç s
è RE
öù
ö éæ f 2 öæ
B02
÷
÷÷  êçç
÷÷çç
2 ÷ú
ø ëè k øè 2  0 mnu øû
1
6
The Magnetosphere
The Shape of the Magnetopause
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Half of the noon-midnight meridian
plane is shown above the axis and half
of the equatorial plane is shown below
Dashed lines show different solutions
while the solid line shows the final
shape obtained by iteration
The equatorial section is quite simple
with no indentations. The subsolar
point is at ~10 R e for the most probable
solar wind conditions
The equatorial boundary crosses the
terminators at 15 R e
The meridian boundary is indented at
the neutral points where the Earth’s
magnetic field is too weak to stand off
the solar wind
The Magnetosphere
The Effect of the Magnetopause Currents
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Close to the Earth the dipole field
dom inates and there is little distortion
Further away there is a significant
change in the s hape of the field lines
with all field lines passing through the
equator closer to the Earth than dipole
field lines from the same latitude.
All dipole field lines that originally
passed through the equator more than
10 Re sunward of the Earth are bent
back and close on the night side
The neutral point separates the two
types of field lines
The Magnetosphere
The Shape of the Nightside Magnetosphere
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At every point along the magnetopause the
component of dynamic pressure normal to the
boundary must be balanced by the pressure of
the tangential magnetic field interior to the
boundary
Far downstream the solar wind velocity
becomes parallel to the magnetopause and the
normal component of dynamic pressure
becomes zero
This would lead to a cylindrical tail
But both the thermal and magnetic pressure of
the solar wind exert a transverse pressure that
eventually becomes important
At the distance where the dipole field pressure
equals the sum of the solar wind thermal and
magnetic pressure the magnetosphere should
close giving it a tear-drop shape
•The solution in the previous view
graph treated the normal stresses
correctly but did not include
tangential stresses.
Tangential Stresses on the Boundary
• Tangential stresses (drag) transfers momentum to the
magnetospheric plasma and causes it to flow tailward. The
stress can be transferred by diffusion of particles from the
magnetosheath, by wave process on the boundary, by the
finite gyroradius of the magnetosheath particles and by
reconnection. Reconnection is thought to have the greatest
effect.
• Assume that one tail lobe is a 2semicircle, then the magnetic
flux in that tail lobe is T   RT BT where RT is the lobe
2
radius, and BT is the magnetic field strength.
• The asymptotic radius of the tail is given by RT  2T2 ( 2  0 psw )
where psw included both the thermal and magnetic pressure
of the solar wind.
1
4
The Tail (Magnetopause) Current
• The stretched field
configuration of the magnetotail
is naturally generated by a
current system.
– The relationship between the
current and the magnetic field
is given by Ampere’s law



B

d
s


0  j  dA

c
where C bounds surface with
area A
2BT  0 I
where I is the total sheet
current density (current per
unit length in the tail)
– For a 20nT field I=30 mA/m or
2X105A/RE
The Magnetosphere
Observing the Magnetopause
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Boundary normal coordinates are
frequently used to study the magnetopause
The boundary normal coordinates have one
component normal to the boundary ( nˆ )
and two tangential ( Lˆ nothward and Mˆ
azimuthal).
The dayside m agnetopause can be
approximated as a tangential discontinuity
when IMF Bz >0. In this case there will be
no field normal to the boundary on either
side and the normalized cross product of
the two fields defines the normal.
When IMF Bz < 0 the boundary is a
rotational discontinuity with a small normal
component.
In this case minimum variance analysis
defines the directions of maximum,
intermediate, and m inimum variance with
the m inimum variance determining the
normal.
The Magnetosphere
Observing the Magnetopause
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Data from two spacecraft show two crossings of the
boundary.
Initially both spacecraft are inside the magnetosphere
(strong field).
The boundary moves inward and crosses first the
ISEE-1 spacecraft (thick line) and later the ISEE-1
spacecraft (thin line).
Some time later the boundary reverses and moves
outward appearing first at ISEE-2 and later at ISEE1.
Assume a planar boundary moving with constant
velocity along the average normal during each
crossing.
The spacecraft separation along the average normal
divided by the time delay gives the boundary
velocity.
The time profile scaled by the velocity gives the
spatial profile of the boundary
The thickness of the magnetopause varies from 200
to 18000 km with a most probably thickness of 700
km.
Structure of Magnetopause
(theory)
Structure of the Magnetopause
Northward IMF
Southward IMF
Magnetopause Crossings
Magnetopause Shape
Model
Bow shock and magnetosheath divert the solar wind flow around the
magnetosphere: computer simulation
Formation of Sonic Shock
• A shock is a discontinuity separating two different regimes in a
continuous media.
– Shocks form when velocities exceed the signal speed in the medium.
– A shock front separates the Mach cone of a supersonic jet from the
undisturbed air.
• Characteristics of a shock :
– The disturbance propagates faster than the signal speed. In gas the signal
speed is the speed of sound, in space plasmas the signal speeds are the
MHD wave speeds.
– At the shock front the properties of the medium change abruptly. In a
hydrodynamic shock, the pressure and density increase while in a MHD
shock the plasma density and magnetic field strength increase.
– Behind a shock front a transition back to the undisturbed medium must
occur. Behind a gas-dynamic shock, density and pressure decrease, behind
a MHD shock the plasma density and magnetic field strength decrease. If
the decrease is fast a reverse shock occurs.
• A shock can be thought of as a non-linear wave propagating faster than
the signal speed.
– Information can be transferred by a propagating disturbance.
– Shocks can be from a blast wave - waves generated in the corona.
– Shocks can be driven by an object moving faster than the speed of sound.
• The Shock’s Rest Frame
– In a frame moving with the
shock the gas with the larger
speed is on the left and gas
with a smaller speed is on the
right.
– At the shock front irreversible
processes lead the the
compression of the gas and a
change in speed.
– The low-entropy upstream side
has high velocity.
– The high-entropy downstream
side has smaller velocity.
• Collisionless Shock Waves
– In a gas-dynamic shock
collisions provide the required
dissipation.
– In space plasmas the shocks
are collision free.
• Microscopic Kinetic effects
provide the dissipation.
• The magnetic field acts as a
coupling device.
• MHD can be used to show
how the bulk parameters
change across the shock.
Shock Front
Upstream
(low entropy)
vu
Downstream
(high entropy)
vd
• Shock Conservation Laws
– In both fluid dynamics and MHD conservation equations for mass, energy


and momentum have the form: Q    F  0 where Q and F are the
t
density and flux of the conserved quantity.
– If the shock is steady ( t  0 ) and one-dimensional Fn  1 or that
n
 
( Fu  Fd )  nˆ  0 where u and d refer to upstream and downstream and nˆ is
the unit normal to the shock surface. We normally write this as a jump
condition[ Fn ]  0.
– Conservation of Mass  (  vn )  0 or [  vn ]  0. If the shock slows the
n
plasma then the plasma density increases.
2
æ
ö

v

p

B
n
– Conservation of Momentum  vn
÷÷  0 where the first term
  çç
n n n è 2 0 ø
is the rate of change of momentum and the second and third terms are the
gradients of the gas and magnetic pressure in the normal direction.
é 2
B2 ù
ê  vn  p 
ú0
2

0û
ë
é
 Bn  ù
v v 
B 0
– Conservation of momentum ê n t  t ú
. The subscript t refers
0
ë
û
to components that are transverse to the shock (i.e. parallel to the shock
surface).
é
æ1 2
 pö
B 2   Bn ù
÷÷  vn
– Conservation of energy ê  vn çç 2 v 
vB ú  0


1


 0 úû
êë
è
ø
0
The first two terms are the flux of kinetic energy (flow energy and internal
energy) while the last two terms come form the electromagnetic energy
 
flux E  B 0

– Gauss Law   B  0 gives Bn   0


– Faraday’s Law   E   B t gives




vn Bt  Bnvt  0
• The jump conditions are a set of 6 equations. If we want to find the
downstream quantities given the upstream quantities then there are 6
unknowns (  ,vn,,vt,,p,Bn,Bt).
• The solutions to these equations are not necessarily shocks. These
conservations laws and a multitude of other discontinuities can also be
described by these equations.
Types of Discontinuities in Ideal MHD
Contact Discontinuity
vn  0 ,Bn  0
Density jumps arbitrary,
all others continuous. No
plasma flow. Both sides
flow together at vt.
Tangential Discontinuity
vn  0 , Bn  0
Complete separation.
Plasma pressure and field
change arbitrarily, but
pressure balance
Rotational Discontinuity
vn  0 , Bn  0
Large amplitude
intermediate wave, field
and flow change direction
but not magnitude.
vn  Bn ( 0  )
1
2
Types of Shocks in Ideal MHD
Shock Waves
vn  0
Flow crosses surface
of discontinuity
accompanied by
compression.
Parallel Shock
Bt  0
B unchanged by
shock.
Perpendicular
Shock
Bn  0
P and B increase at
shock
Oblique Shocks
Bt  0, Bn  0
Fast Shock
P, and B increase, B
bends away from
normal
Slow Shock
P increases, B
decreases, B bends
toward normal.
Intermediate
Shock
B rotates 1800 in
shock plane, density
jump in anisotropic
case
•Configuration of magnetic field lines for fast and slow shocks. The lines
are closer together for a fast shock, indicating that the field strength
increases. [From Burgess, 1995].
Bow Shock and Magnetopause Crossings
Bow Shock Crossings with
Location Front Orientation
Functions of Magnetosheath
Diverts the solar wind
flow and bends the
IMF around the
magnetopause
Observations of Density Enhancements in the Sheath
Internal Structure of the Magnetosheath
Bow
Shock
Magneto
pause
Postbow
shock
density
Slow Shock in the Magnetosheath
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Particles can be accelerated in the
shock (ions to 100’s of keV and
electrons to 10’s of keV).
Some can leak out and if they have
sufficiently high energies they can
out run the shock. (This is a unique
property of collisionless shocks.)
At Earth the interplanetary
magnetic field has an angle to the
Sun-Earth line of about 450. The
first field line to touch the shock is
the tangent field line.
– At the tangent line  Bn the angle
between the shock normal and the
IMF is 900.
– Lines further downstream have Bn 900
Particles have parallel
 motion
along the field line ( v )and
cross


field drift motion ( v  (E  B) / B ).
2
d

– All particles have the same vd
– The most energetic particles will
move farther from the shock
before they drift the same distance
as less energetic particles
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The first particles observed behind
the tangent line are electrons with
the highest energy electrons closest
to the tangent line – electron
foreshock.
A similar region for ions is found
farther downstream – ion
foreshock.
Ion Foreshock
Upstream Waves
Summary of Foreshock:
shock-field angle determines the features in
the sheath and upstream