Transcript Shock_and_Magnetosheath_v1

ESS 200C, Lecture 8
The Bow Shock and Magnetosheath
Reading: Ch 5, 6 Kivelson & Russell
• 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.
• Shocks can form when an obstacle moves
with respect to the unshocked gas.
• Shocks can form when a gas encounters an
obstacle.
• 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 to 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,



Q
energy and momentum have the form:
   F  0 where Q and F
t
are the density and flux of the conserved quantity.
– If the shock is steady (  t  0 ) and one-dimensional Fn  0 or
n
 
( Fu  Fd )  nˆ  0 where u and d refer to upstream and downstream and
n̂
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
n
the plasma then the plasma density increases.
2
vn p   Bt 
  0 where the first
– Conservation of Momentum  vn

 
n n n  2  0 
term is the rate of change of momentum and the second and third
terms are the gradients of the gas and (transverse) magnetic pressures
in the normal direction. Remembering [ vn]=0 and using [Bn]=0 from
Gauss’s law (below), we get:
 2
B2 
  vn  p 
0
2 0 

– In the transverse direction:

 Bn  
Bt   0 . The subscript t
  vn vt 
0 

refers to components that are transverse to the shock (i.e. parallel
to the shock surface).
– Conservation of energy

1 2
 p
B 2   Bn 
  vn
v B   0
  vn  2 v 


1


 0 



0
There we have used p   const .
The first two terms are the flux of kinetic energy (flow energy and
internal energy) while the last two terms come from 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 conservation laws are 6 equations. If we want to find the downstream
quantities from upstream ones, we have 6 unknowns: (,vn,,vt,p,Bn,Bt).
• The solutions to these equations are not necessarily shocks. A multitude
discontinuities can also be described by these equations. Low-amplitude
step waves (F,S,I MHD waves) also satisfy them. Shocks are the nonlinear, dissipative versions of those waves, so they have similar types.
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  2
1
Types of Shocks in Ideal MHD
Shock Waves

Parallel Shock ( B
vn  0
Flow crosses surface of
discontinuity accompanied by
compression.
along shock
normal)
Bt  0
B unchanged by shock.
(Transverse momentum and
electric field continuity result in
[Bt]=0 ). Hydrodynamic-like.
Perpendicular
Shock
Bn  0
P and B increase at shock.
There is no slow perp. 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.
[p]=0 non-compressive,
propagates at uA, 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.
• Quasi-perpendicular and quasi-parallel shocks.

– Call the angle between B and the
–
–
–
–
normal θBn .
Quasi-perpendicular shocks have
θBn> 450 and quasi-parallel have
θBn< 450.
.Perpendicular shocks are sharper
and more laminar.
Parallel shocks are highly turbulent.
The reason for this is that
perpendicular shocks constrain the
waves to the shock plane while
parallel shocks allow waves to leak
out along the magnetic field
– In these examples of the Earth’s
bow shock – N is in the normal
direction, L is northward and M is
azimuthal.
• Examples of the change in
plasma parameters across
the bow shock
– The solar wind is supermagnetosonic so the purpose
of the shock is to slow the
solar wind down so the flow
can go around the obstacle.
– The density and
temperature increase.
– The magnetic field (not
shown) also increases.
–The maximum compression
at a strong shock is 4 but 2 is
more typical.
THEMIS-C crossing(s)
of bow shock: overview
THEMIS-C crossing(s)
of bow shock: detail
•
•
•
•
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 90 0
Particles have parallel motion along
the field line (v ) and cross field drift
 
motion ( vd  ( E  B) / B 2).

– 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
•
•
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.
THEMIS-C crossing(s)
of ion/electron foreshock
• For compressive fast-mode and slow-mode oblique shocks the
upstream and downstream magnetic field directions and the
shock normal all lie in the same plane. (Coplanarity Theorem)
 
nˆ  ( Bd  Bu )  0
• The transverse component of the momentum equation can be

 Bn  

written as   vn vt 
Bt   0 and Faraday’s Law gives vn Bt  Bnvt  0
0 




• Therefore both vn Bt and Bt are parallel to vt  and thus are


 
parallel to each other.
•
•










Thus Bt  vn Bt  0 . Expanding
 vunBut  But  vdn Bdt  Bdt  vdn But  Bdt  vun Bdt  But  0
(vn,u  vn,d )( Bt ,u  Bt ,d )  0
If vn ,u  vn ,d Bt ,u and Bt ,d must be parallel.
  
• The plane containing one of these vectors and the normal
contains both the upstream and downstream fields.
 
 
 
ˆ
(
B

B
)

n

0
• Since u d
this means both Bd  Bu and Bu  Bd are
perpendicular to the normal and
 
 
 
 
nˆ  ( Bu  Bd )  ( Bu  Bd ) / ( Bu  Bd )  ( Bu  Bd )
•
•
•
•
Another way of determining the shock normal direction is by using
velocity or electric field data.
We note that velocity change is coplanar with Bu, Bd
Then either one of them crossed into the velocity change will lie on the
shock plane:


( Bu  v )nˆ  0


( Bd  v )nˆ  0
 
ˆ 0
Using either of the above and the divergenceless constraint ( Bu  Bd )  n
will also result in a high fidelity shock normal.
•
Finally, the shock may be traveling at speeds of 10-100km/s and
with a single spacecraft it is possible to determine its speed. Using the
continuity equation in shock frame we get: [(vn-vsh)]=0 which gives:


 v  u vu
vsh  d d
nˆ
 d  u
NIF: vt,u=0
HTF: Et=0
• Structure of the bow shock.
– Since both the density and B increase this is a fast mode shock.
– The field has a sharp jump called the ramp preceded by a gradual
rise called the foot.
– The field right behind the shock is higher than its eventual
downstream value. This is called the overshoot.
Super-critical shocks (MA>2.7) are those
whose thermalization occurs due to ion
beams reflected from the shock. They
have a foot large enough to reflect more
ions which result in higher energy
transfer to backstreaming particles.
The backstreaming and re-appearance
of particles in the shock is very fast, and
results in heating because the ions have
spread in phase space.
As MA  Bd  J  J2 but shock
jump is limited to 4, so current increase
and resistive heating is limited. Then ion
reflection becomes important mechanism.
Subsonic Versus Supersonic
Interaction
•
•
•
If a flowing magnetized plasma
encounters an obstacle to that flow
such as a magnetosphere or
ionosphere, it is deflected around the
obstacle by a standing wave.
In a gas-dynamic flow, a pressure
gradient forms that slows and deflects
the flow. This is possible because the
thermal speed (temperature) of the
particles is large enough that the
sound speed is greater than the flow
speed. The Mach number is less than
1.
When the particles are cold, the flow
speed exceeds the sound speed, a
shock forms heating and slowing the
flow so that a pressure gradient can
develop sufficient to deflect the flow.
The Shape of the Magnetic Cavity
•
The pressure of the solar wind is applied to
the magnetosphere along the normal to its
surface, the magnetopause.
•
The direction of the magnetopause normal
varies with position and the pressure applied
drops as one moves away from the subsolar
point.
•
There is always some pressure applied even
when the boundary is aligned with the
asymptotic flow. A good approximation to the
pressure is:
Ku2 cos 2   P sin 2 
Where Ψ is the angle of the
magnetopause normal to the solar wind
flow, P∞ is the thermal pressure at ∞ and
K accounts for stream tube
expansion.
Empirical magnetosphere of Tsyganenko (1989) with
realistic boundary shape and implicit plasma
content
Pressure by Solar wind on Magnetopause
•
Consider the stagnation streamline. Assume Bsw=0. The flow is supersonic upstream
and the momentum flux is conserved through the shock, so the dynamic pressure is
converted to thermal pressure and some (slower) flow. That pressure then increases (as
flow decreases further) through the magnetosheath towards the boundary at the
stagnation point.
Ku2  Ps
•
We can determine the parameter K subject to the conservation laws at the shock and to
the fluid equations through the magnetosheath.
–
At shock
[ un2  p ]  0 and MA i.e., compression ratio, r, is
and we get:
–
p
2
 1
( M 2  1)
p
 1
2    1M 2
and M 
2M 2    1

1
1  2

   1

r   1  M s ,

2

u  
p
  1



2
 u u  
At sheath:
and considering p   and (u )u  u  u    u 
t

u2
 p 2
we get Bernoulli’s equation (along the streamline):

 const. from which we get
2
 1 
p   1 2 
ratio between pressure at stagnation point and downstream of shock: s  1 
M 
p 
2

  1
– From the above we get: K  

 2 
(  1) /  1
i.e., K~0.89, and a weak function of M
1
     1 / 2M 2 
1 /( 1)
 /  1
Mirror Dipole Magnetosphere
Image Dipole Magnetosphere
Dipole Field in a Vacuum with Superconducting Shell
• If you bring a flat superconducting sheet close to a magnetic
dynamo, it mirrors the magnetic field to produce a very simple
magnetosphere with two neutral points and a doubled magnetic field
strength at the nose.
• If you wrap a superconducting sheet around the magnetosphere like
the solar wind envelops the Earth, you change the subsolar field to
2.4 times the dipole strength and preserve the topology.
Pressure Exerted by the Solar
Wind on the Magnetosphere
•
The momentum flux and thermal pressure in the solar wind confine the size of the
magnetosphere.
u (u  nˆ )  pnˆ
•
The magnetosheath causes streamlines to diverge so there is a pressure drop across the
magnetosheath. This depends on Mach number and the polytropic index.
The magnetic field pushes back with the magnetic pressure. The field falls off in strength as
r-3 and the magnetic pressure as r-6
The field is enhanced by a factor, a, at the nose where a depends on the geometry or the
curvature of the boundary.
The pressure balance is:
•
•
•
K u  (aB 0 ) 2 (2 0Lmp ) 1
2
6
where K is determined from Bernoulli’s law, Bo is the field at the equator on the
surface of the planet, μo is the magnetic permeability of free space and Lmp is the
distance to the magnetopause in planetary radii.
•
For the Earth, ~2.44 (based on typical values of Lmp and K) and we get
2 1/ 6
Lmp  107.4(nswusw
)
where nsw is the solar wind proton number density in cm-3 and the usw is the
solar wind bulk speed in kms-1.
Gasdynamic Simulations of the
Solar Wind Interaction
• Gasdynamics has only one
wave mode, the compressional
wave to slow and deflect the
incoming flow.
• Since there is no magnetic
force, the obstacle cannot be a
magnetosphere.
• This diagram shows
streamlines around a
cylindrically symmetric
magnetospherically shaped
obstacle for a Mach number of
8, and a polytropic index of
5/3.
Gasdynamic Interaction Continued
• Density contours show that density jumps close to a
factor of 4 across the shock. It increases behind the
shock in the subsolar region and decreases elsewhere.
• In gasdynamics, the velocity and temperature ratios are
proportional. The velocity increases along the flank and
the temperature drops.
Gasdynamics Concluded
•
•
The mass flux is equal to the product of the density ratio and the velocity
ratios and parallels the streamlines. The bow shock positions itself so that
all the shocked mass can flow past the obstacle. The mass flux is highest
near the shock than near the obstacle.
The magnetic field cannot exert a force in gasdynamics, but it can be
approximated as threads being carried by the flow. Here we show the
magnetic field lines behind the shock for two upstream orientations.
Magnetohydrodynamics
• When there is a magnetic force as there is in a magnetized
plasma, the interaction becomes much more complicated but we
can solve for more quantities such as the size and shape of the
magnetosphere.
 / t    (u )
(continuity )
u / t  (u  )u  (p) /   J  B / 
p / t  (u  ) p  p  u
( pressure )
B / t    (u  B)   2 B
( Faraday )
J    ( B  Bd )
(momentum)
( Ampere)
• These equations can provide a solution that allows the flow to go
by the obstacle, the magnetic field to be convected around and
over the obstacle, and the magnetic field to stretch. This is done
with three standing waves: fast compressional, Alfven or shear,
and slow compressional modes.
MHD Forces Affecting the Flow
•
•
•
•
•
Here we show the thermal
pressure gradient force and the
magnetic force separately on a
background contour plot of
density.
At the shock, the pressure
gradient and magnetic forces
(arrows) both act to slow the flow.
Well away from the shock, the
pressure gradient forces
turnaround as the plasma density
and pressure is lower, close to the
boundary.
The magnetic forces still push
away from the magnetosphere.
The net result is a total force that
slows and turns the flow parallel to
the magnetopause.
Hybrid Simulations: Importance of
Scale Size
•
•
•
•
•
MHD simulations ignore the underlying
motions of the charged particles. The
plasma is treated as a fluid. Hybrid
simulations treat the ion motion but not
the electrons.
If the solar wind (top) encounters a
magnetic obstacle much smaller than
the ion inertial length, only a whistler
mode wake is formed.
If the obstacle is larger, it causes a
pile-up in density ahead of the
obstacle and decreases the density in
the wake.
For obstacles larger than the ion
inertial length, a compressional wave
forms upstream that heats,
compresses, and diverts the flow. A
plasma sheet forms downstream
For obstacles the size of Mercury, you
obtain a shock and a magnetosphere
and tail similar to those of Earth.
Hybrid Simulations: Quasi-Parallel
Shock
• When the interplanetary
field is near the Parker
spiral direction, it creates
a quasi-parallel shock on
one side and a quasiperpendicular shock on
the other.
• The quasi-parallel shock
is highly fluctuating.
• The parallel shock is
even more fluctuating.
Hybrid Simulations: Upstream
Waves
• Ions can move back upstream
from the bow shock in the
quasi-parallel region of the
bow shock.
• The counter-streaming of the
solar wind and the back
streaming ions provides free
energy that generates waves.
• The waves scatter the particles
and thermalize them.
• Thus the foreshock
preprocesses the plasma,
altering it upstream of the main
shock.
• Observations of the magnetic
field near the magnetopause
from the ISEE satellites.
• The magnetosphere is on
either end of the figure. The
region in between is the
magnetosheath.
• The magnetic field of the
magnetosheath is
characterized by oscillations in
the magnetic field.