1. ESS 154/200C Lecture 11 Collisionless Shocks

2. Date Day Topic Instructor Due
ESS 200C Space Plasma Physics ESS 154 Solar Terrestrial Physics M/W/F 10:00 – 11:15 AM Geology Instructors: C.T. Russell (Tel. x-53188; Office: Slichter 6869) R.J. Strangeway (Tel. x-66247; Office: Slichter 6869)
Date Day Topic Instructor Due
1/4 M A Brief History of Solar Terrestrial Physics CTR
1/6 W Upper Atmosphere / Ionosphere CTR
1/8 F The Sun: Core to Chromosphere CTR
1/11 M The Corona, Solar Cycle, Solar Activity
Coronal Mass Ejections, and Flares CTR PS1
1/13 W The Solar Wind and Heliosphere, Part 1 CTR
1/15 F The Solar Wind and Heliosphere, Part 2 CTR
1/20 W Physics of Plasmas RJS PS2
1/22 F MHD including Waves RJS
1/25 M Solar Wind Interactions: Magnetized Planets YM PS3
1/27 W Solar Wind Interactions: Unmagnetized Planets YM
1/29 F Collisionless Shocks CTR
2/1 M Mid-Term PS4
2/3 W Solar Wind Magnetosphere Coupling I CTR
2/5 F Solar Wind Magnetosphere Coupling II;
The Inner Magnetosphere I CTR
2/8 M The Inner Magnetosphere II CTR PS5
2/10 W Planetary Magnetospheres CTR
2/12 F The Auroral Ionosphere RJS
2/17 W Waves in Plasmas 1 RJS PS6
2/19 F Waves in Plasmas 2 RJS
2/26 F Review CTR/RJS PS7
2/29 M Final

3. Computer Exercises: X SOLAR
X SOLAR is designed to assist you in understanding the solar magnetic field, the solar wind and charged particle motion.
It makes calculations based on your input. It gives quantitative results that often can be used to verify or establish physical relationships.

4. X SOLAR: Potential Field
You can construct dipole, quadrupole and octupole magnetic fields and examine them from different viewpoints.
You can examine the effect of the so-called source surface where field lines become radial.
You can examine realistic solar field line distributions.
You can examine maps of the field lines back to the surface of the Sun.

5. X SOLAR: Solar Wind
You can study the Parker spiral magnetic field for different solar wind speeds in the equatorial plane and in 3D.
You can examine the configuration of the heliospheric current sheet.

6. X SOLAR: Particle Motion
You can examine how charged particles move in magnetic and electric fields of different geometry
Uniform B
Cross electric and magnetic field
Curved magnetic fields
Mirror geometry magnetic fields
Harris current sheet
Dipole magnetic field

7. Discontinuities in the Plasma
Plasmas in space are often cellular, consisting of large regions of nearly uniform plasma separated by a thin boundary across which the properties of the plasma change rapidly.
Tangential discontinuities have no flow or magnetic field across them and the total pressure plasma thermal plus magnetic is constant.
Examples: Venus ionopause; Earth’s magnetopause at times; solar wind discontinuities usually
Rotational discontinuities have a component of flow and magnetic field across them. The magnetic field strength is usually constant across them.
Examples: Earth’s magnetopause at times
Collisionless shocks have flow and field changes across the boundary including compression of the magnetic field and density and heating. Processes are non-linear.
Examples: Shocks in steepening stream interactions; bow shocks
Stream interactions where the solar wind increases by more than the speed of the fast magnetosonic speed can generate shocks. Here is a pair of shocks bounding a stream interaction.

8. Shocks in Different Frames of Reference
Shocks can form when an obstacle moves with respect to the unshocked gas.
Examples: ICMEs in solar wind
Shocks can form when a gas encounters an obstacle.
Examples: Planetary bow shocks in the solar wind

9. Dissipation in Collisionless Shocks
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 compression of the gas and a change in speed.
The low-entropy upstream side has high velocity at a bow shock and the high-entropy downstream side has smaller velocity.
An ICME shock has just the opposite velocity change in the observer’s frame because the shock is moving.
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 to couple the plasma. A collective force.
MHD can be used to show how the bulk parameters change across the shock.
Shock Front
Upstream
(low entropy)
Downstream
(high entropy) vu vd

10. Magnetic Structure of a Collisionless Shock
A collisionless shock is a thin structure that makes an irreversible change in the plasma, increasing entropy.
There is a thin ramp where the field strength increases, the plasma density increases, and the plasma slows. This region has the scale of an ion-gyroradius or less and cannot be treated with MHD. MHD is applicable on both sides of the shock, however.
Ignoring waves for the moment, the field along the shock normal is constant.
The field component along the shock plane is zero except in the shock ramp. This is the non-coplanarity component.
The jump is all in the BL component.
Waves appear upstream (2 types) and down.

11. Importance of Collisionless Shocks
Shocks are responsible for the generation of very energetic particles at the bow shock, in interplanetary space, in interstellar space, in intergalactic space.
Shocks convert dynamic pressure into kinetic pressure that in turn alters the plasma conditions in planetary magnetosheaths. Thus they can control the rate of reconnection at the dayside magnetopause and control the coupling of the solar wind plasma to the magnetosphere.
Shocks are very macroscopic in size, but depend for their dissipation on non-linear kinetic processes. Thus this study pushes our understanding of the physics of plasmas.
Marginally critical quasi-perpendicular shock

12. How Shocks Strengthen
If a shock moves into a region in which the fast mode wave speed is decreasing, its Mach number increases. Shocks moving outward in the solar wind do this.
This occurs at stream interactions as they move out in the solar wind.
This occurs at ICMEs as they move outward from the Sun.
Here we see one shock overtaking another. Soon they will be one shock.
The high-resolution panels to the right show the magnetic field across the shocks.
The oscillations around the field jumps are characteristic of shocks and aid in the dissipation required in such shocks.

13. The Shock Normal
It is important to know the direction in which the shock is traveling, i.e. the direction of its normal.
If a shock passes four spacecraft then the locations of the spacecraft and the times of arrival can be used to determine n,
where i = 1,2,3 labels the vectors and time difference between one spacecraft and the other three.
The rate of transport of magnetic flux across the shock is the tangential component of the electric field. The magnetic flux does not pile up in the shock so that tangential E is conserved consistent with Maxwell’s equations. This implies that the upstream and downstream magnetic field and shock normal are coplanar.
Thus, the cross product of the upstream and downstream magnetic fields must be coplanar
Since the magnetic field is divergenceless,
(magnetic coplanarity)
Avoid using the minimum variance technique at the shock. There is no basis for using this in general.
Conservation of momentum gives the shock velocity

14. Rankine-Hugoniot Equations
The Rankine-Hugoniot equations are MHD conservation equations along the shock normal
Continuity
Tangential momentum
Normal momentum
Energy
Normal magnetic field
Faraday’s law
These equations can be solved to obtain the jump in plasma conditions across the shock.
We can parameterize these jumps with four dimensionless parameters
Plasma beta
Fast magnetosonic Mach number
Angle between upstream B and normal
Ratio of specific heats
MHD does not tell us the source of the dissipation.
Contour plots of jump in density and field
as a function of θBN, MMS for 3 betas

15. 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 magnetic pressures in the normal direction.

16. Conservation of energy
Conservation of transverse momentum: The subscript t refers to components that are transverse to the shock (i.e. parallel to the shock surface).
Conservation of energy
where we have used [Adiabatic equation of state]
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

17. Types of Discontinuities in Ideal MHD
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 are conservation laws and a multitude of other discontinuities can also be described by these equations.
Types of Discontinuities in Ideal MHD
Contact 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.

18. 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.
Perpendicular Shock
P and B increase at 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.

19. Configuration of magnetic field lines for fast and slow shocks
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.

20. Rankine-Hugoniot Solutions: Density and Temperature
When beta is large (bottom), the magnetic field exhibits little control on the plasma. The density increase is dependent only on Mach number and not on the direction of the field, but when the field is strong (top) the behavior varies with θBN.
At 90°, θBN the density jump is similar independent of beta as is the field jump.
At 0° θBN the field does not change when beta is large, but it does change when beta is very small.
This is a switch-on shock.

21. Rankine-Hugoniot: Btan and Temperature
Here we show the jump in the component of the field in the shock plane and the temperature jump as a function of Mach number and θBN.
For a high beta plasma, the magnetic field has little effect on the jump across the shock.
We note that while the jump in B, Btan, and N cannot be bigger than 4 if γ is 5/3, the temperature is unbounded. At a very strong high beta shock, all the dynamic pressure will become thermal pressure downstream.
At low beta and low Mach number, we see the effect of the switch-on shock on Btan.

22. Dependence on Mach Number: Quasi-perpendicular Shocks
Low Mach number, low beta shocks are termed laminar shocks and have simple structure and generally regular periodic waves.
Shocks are also classified as
Perpendicular θBN > 89°
Quasi-perpendicular 70° < θBN <89 °
Oblique 45° < θBN < 70°
Quasi-parallel 15° < θBN < 45°
Parallel θBN < 15°
Laminar shocks are believed to provide dissipation through wave damping.
Whistler waves can phase stand in the flow while the wave energy moves upstream and damps.
Five low Mach number shocks ranging from subcritical to slightly supercritical

23. Overshoots and Ion Reflection
Low beta, supercritical shock
quasi-perpendicular shock
Ion distributions at low beta,
Super critical quasi-perpendicular shock
When Mach number increases above some limit, there is a peak field strength behind the shock and irregular waves downstream
At low Mach numbers, all the ions can penetrate the shock potential and the ramp magnetic field that oppose the ions’ motion.
At supercritical Mach numbers, some ions cannot get through the magnetic ramp and return upstream only to be carried through the shock by the solar wind convection electric field with a large gyro-radius and a high perpendicular temperature.
If the magnetic field is within 39° of the shock normal, the parallel motion of the reflected ion takes it upstream and forms an ion beam. This produces upstream waves.

24. 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.

25. Turbulence Upstream of Interplanetary Shocks
BL component across 4 interplanetary
shocks showing upstream whistler
BM component upstream
of 5 interplanetary shocks
Power spectra of the waves upstream
of the 5 shocks to the left
Interplanetary shocks are associated with the acceleration of energetic particles.
The acceleration is attributed to Fermi acceleration in the turbulence overtaken by the interplanetary shock.

26. High Mach Number Quasi Parallel Shocks
Shock reformation at the quasi-parallel shock
Hybrid simulation of shock reformation
Waves excited in front of the quasi-parallel shock are swept backward toward the shock.
When they get close to the shock they amplify and reform the shock.
This is also seen in hybrid simulations.

27. High Mach Number Quasi-Perpendicular Shocks
Two supercritical nearly
perpendicular shocks
Quasi-perpendicular shock with
downstream ion-cyclotron waves
Quasi-perpendicular shock with
downstream mirror-mode waves
The presence of an overshoot and an undershoot with ion gyroradius scale sizes and the reflection of ions at the shock indicate that ion trajectories are important in determining shock structure.
Classical plasma instabilities do exist downstream of the shock. Depending on Mach number, beta and ion composition, ion-cyclotron or mirror-mode waves can grow.
The shock seldom assumes electron scale lengths but can do so if the shock normal is close to 90° to the upstream field.

28. Electron Acceleration: Flat-top Distributions
Electrons downstream of the bow shock develop flat-topped distributions.
This has been attributed to their adiabatic behavior and their interaction with the shock potential.
Simulations suggest that ion-driven instabilities such as the modified two stream instability are operative.
Fully kinetic simulation of counter-streaming ion beams
(Winske and Yin, 2005)
Electron distributions parallel to B

29. Electron Foreshock: Fast Fermi Acceleration
Production of an electron beam
in the de Hoffman-Teller frame
Location of the electron and ion foreshocks
When an interplanetary field line approaches the bow shock, it produces an electron beam parallel to the field line beginning at the point of tangency.
The electrons that have a large V┴ and a V‖ comparable to or less than the velocity of the deHoffman-Teller frame cannot get through the ramp while at the same time they experience a mirror force away from the tangent point.
The energy of the beam becomes very large near 90° but the flux goes to zero.

30. Simulations I
Hybrid simulations (Omidi et al, 2005)
MHD
Great macroscopic description, but no insight into dissipation processes
Hybrid
2D planar gives ion dissipation but no electron physics.
2D, 3D global gives geometry’s effects on wave processes but only ion physics handled properly

31. PIC simulation of a quasi-perpendicular shock
Simulations II
Fully Kinetic, Particle-in-Cell
Periodic boundary conditions may affect physics
Resolution poor and runs time consuming
Qualitatively instructive
PIC simulation of a quasi-perpendicular shock
showing effects of MT-SI (Scholer and Matsukiyo, 2005)