ESS 200C, Lecture 8The 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.

vu vd

Shock Front

Upstream(low entropy)

Downstream(high entropy)

• Shock Conservation Laws– In both fluid dynamics and MHD conservation equations for mass,

energy and momentum have the form: where Q and

are the density and flux of the conserved quantity.

– If the shock is steady ( ) and one-dimensional or

where u and d refer to upstream and downstream and

is the unit normal to the shock surface. We normally write this as a

jump condition .

– Conservation of Mass or . If the shock slows

the plasma then the plasma density increases.

– Conservation of Momentum where the first

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:

0 F

tQ

F

0 t 0

nFn

0ˆ)( nFF du

n̂

0][ nF

0)(

nvn 0][ nv

02 0

2

tn

nB

nnp

nvv

02 0

22

Bpvn

– In the transverse direction: . The subscript t

refers to components that are transverse to the shock (i.e. parallel

to the shock surface).

– Conservation of energy

There we have used

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

– Gauss Law gives

– Faraday’s Law gives

00

t

ntn BBvv

01 00

22

21

nnn

BBvBvpvv

0BE

0 B

tBE 0nB

0 tntn vBBv

.constp

• 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 non-linear, dissipative versions of those waves, so they have similar types.

Types of Discontinuities in Ideal MHDContact Discontinuity , Density jumps arbitrary,

all others continuous. No plasma flow. Both sides flow together at vt.

Tangential Discontinuity , Complete separation. Plasma pressure and field change arbitrarily, but pressure balance

Rotational Discontinuity , Large amplitude intermediate wave, field and flow change direction but not magnitude.

0nB

0nv

0nv

0nB

21

0nn Bv

0nv 0nB

Types of Shocks in Ideal MHD

Shock Waves Flow crosses surface of discontinuity accompanied by compression.

Parallel Shock ( along shock normal)

B unchanged by shock.(Transverse momentum and electric field continuity result in [Bt]=0 ). Hydrodynamic-like.

Perpendicular Shock

P and B increase at shock.There is no slow perp. shock.

Oblique Shocks

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.

0nv

0tB

0nB

0,0 nt BB

B

•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 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.

B

• Examples of the change in plasma parameters across the bow shock

– The solar wind is super-magnetosonic 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 the angle between the shock normal and the IMF is 900.

– Lines further downstream have• Particles have parallel motion along

the field line ( ) and cross field drift motion ( ).

– All particles have the same – 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.

Bn

090Bn

v

2/)( BBEvd

dv

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)

• The transverse component of the momentum equation can be written as

and Faraday’s Law gives

• Therefore both and are parallel to and thus are parallel to each other.

• Thus . Expanding

• If and must be parallel.

• The plane containing one of these vectors and the normal contains both the upstream and

downstream fields.

• Since this means both and are perpendicular to the normal and

0)(ˆ ud BBn

00

t

ntn BBvv

0 tntn vBBv

tB tnBv

tv

0 tnt BvB

0))(( ,,,, dtutdnun BBvv

dnun vv ,, utB ,

dtB ,

)()(/)()(ˆ dudududu BBBBBBBBn

0ˆ)( nBB du

0 utdtundtutdndtdtdnututun BBvBBvBBvBBv

ud BB

du BB

• 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:

• Using either of the above and the divergenceless constraintwill also result in a high fidelity shock normal.

• Finally, the shock may be traveling at speeds of 10-100km/s andwith a single spacecraft it is possible to determine its speed. Using thecontinuity equation in shock frame we get: [(vn-vsh)]=0 which gives:

0ˆ)(

0ˆ)(

nvB

nvB

d

u

0ˆ)( nBB du

nvvvud

uuddsh ˆ

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 thosewhose thermalization occurs due to ionbeams reflected from the shock. Theyhave a foot large enough to reflect moreions which result in higher energytransfer to backstreaming particles.

The backstreaming and re-appearanceof particles in the shock is very fast, andresults in heating because the ions havespread 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

222 sincos PuK

Empirical magnetosphere of Tsyganenko (1989) with realistic boundary shape and implicit plasma content

• 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:

Where Ψ is the angle of themagnetopause normal to the solar windflow, P∞ is the thermal pressure at ∞ andK accounts for stream tubeexpansion.

)1(1

21 2

Mpp

0][ 2 pun

121

11

,

sMr

puu

tu

sPuK 2

• 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.

• We can determine the parameter K subject to the conservation laws at the shock and to the fluid equations through the magnetosheath.

– At shock and MA i.e., compression ratio, r, is

and we get: and

– At sheath: and considering and

we get Bernoulli’s equation (along the streamline): from which we get

ratio between pressure at stagnation point and downstream of shock:

– From the above we get:

i.e., K~0.89, and a weak function of M

Pressure by Solar wind on Magnetopause

p uuuuu 2

21)(

.12

2

constpu

1/2

211

Mpps

12

122

22

MMM

)1/(12

1/)1(

2/11

21

MK

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

npnuu ˆ)ˆ(

• The momentum flux and thermal pressure in the solar wind confine the size of the magnetosphere.

• 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:

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

where nsw is the solar wind proton number density in cm-3 and the usw is the solar wind bulk speed in kms-1.

160

20 )2()(2

mpLaBuK

6/12 )(4.107 swswmp unL

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.

• 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.

)()(

)()(/

)()(/

)(//)()(/

)()(/

2

AmpereBBJ

FaradayBButB

pressureupputp

momentumBJpuutu

continuityut

d

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 quasi-perpendicular 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.