MHD Shocks and Collisionless Shocks

Manfred Scholer

Max-Planck-Institut für extraterrestrische PhysikGarching, Germany

The Solar/Space MHD International Summer School 2011 USTC, Hefei, China, 2011

Overview

1. Information, Nonlinearity, Dissipation

2. Shocks in the Solar System

3. MHD Rankine – Hugoniot Relations

4. de Hoffmann-Teller Frame, Coplanarity, and Shock Normal Determination

5. Resistive, 2-Fluid MHD – First Critical Mach Number

6. Specular Reflection of Ions: Quasi-Perpendicular vs Quasi-Parallel Shocks

7. Upstream Whistlers and the Whistler Critical Mach Number

8. Brief Excursion on Shock Simulation Methods

9. Quasi-Perp. Shock: Specular Reflection, Size of the Foot, Excitation of Alfven Ion Cyclotron Waves

10. Cross- Shock Potential and Electron Heating

11. Quasi-Parallel Shock: Upstream Ions, Ion-Ion Beam Instabilities, and Interface Instability

12. The Bow Shock

Electrons at the Foreshock Edge

Field-Aligned Beams

Diffuse Ions

Brief Excursion on Diffusiv Acceleration

Large-Amplitude Pulsations

Literature

D. Burgess: Collisionless Shocks, in Introduction to Space Physics, Edt. M. G. Kivelson & C. T. Russell, Cambridge University Press, 1995

W. Baumjohann & R. A. Treumann: Basic Space Plasma Physics, Imperial College Press, 1996

Object in supersonic flow – Why a shock is needed

If flow sub-sonic information about object can transmitted via sound waves against flow

Flow can respond to the information and is deflected around obstacle in a laminar fashion

If flow super-sonic signals get swept downstream and cannot inform upstream flowabout presence of object

A shock is launched which stands in upstream flow and effetcs a super- to sub-sonictransition

The sub-sonic flow behind the shock is then capable of being deflected around the object

Fluid moves with velocity v; a disturbance occurs at 0 and propagates with velocity of sound c relative to the fluid

The velocity of the disturbance relative to 0 is v + c n, where n is unit vector in any direction

(a)v<c : a disturbance from any point in a sub-sonic flow eventually reaches any point(b)v>c: a disturbance from position 0 can reach only the area within a cone given by opening angle where sin =c / v

Surface a disturbance can reach is called Mach‘s surface

Ernst Mach

Examples of a Gasdynamic Shock

‘Schlieren‘ photography

Shock attached to a bullet Shock around a blunt object:detached from the object (blunt = rounded, not sharp))

More Examples

Schematic of how a compressional wave steepens to form a shock wave(shown is the pressure profile as a function of time)

The sound speed is greater at the peak of the compressional wave where the density is higher than in front or behind of the peak. The peak will catch up with the part of the peak ahead of it, and the wave steepens. The wave steepens until the flow becomes nonadiabatic.

Viscous effects become important and a shock wave forms where steepening is balancedby viscous dissiplation.

Characteristics cross at one point at a certain time

Results in 3-valued solution

Add some physics:

Introduce viscosity in Burgers‘ equation

In MHD (in addition to sound wave) a number of new wave modes (Alfven, fast, slow)

Background magnetic field, v x B electric field

We expect considerable changes

MHD

Solar System

Solar wind speed 400 – 600 km/secAlfven speed about 40 km/sec:

There have to be shocks

Coronal Mass Ejection(SOHO-LASCO) in forbidden Fe line

Large CME observedwith SOHO coronograph

Interplanetary traveling shocks

Quasi-parallel shock

Quasi-perpendicularshock

Belcher and Davis 1971

Vsw

N

B

Corotating interaction regions and forward and reverse shock

CIR observed by Ulysses at 5 AU

70 keV

12 MeV

Decker et al. 1999

F

R

Earth‘s bow shock

Perpendicular Shock

Quasi-Parallel Shock

The Earth‘s Bow Shock

solar wind300-600 km/s

Magnetic field during various bow shock crossings

Heliospheric termination shock

Schematic of the heliosphere showing theheliospheric termination shock (at about 80 –90 AU) and the bow shock in front of the heliosphere.

Voyager 2 at the termination shock(84 AU)

Friedrichs-diagram

Rankine – Hugoniot Relations

William John Macquorn Rankine 1820 - 1872

Pierre-Henri Hugoniot 1851 - 1887

h h

F

1 2

n

t

Oblique MHD Shocks

Fast Slow

Intermediate Switch-on

Switch-off Rotational

de Hoffmann-Teller Frame (H-T frame) and Normal Incidence Frame (NIF frame)

Unit vectors

Incoming velocity

Subtract a velocity vHT perpto normal so that incomingvelocity is parallel to B

This is widely used in order to determine the shock normal from magnetic field observations

Adiabatic reflection (conservation of the magnetic moment)

Note: only predicts energy of reflected ions, not whether an ion will be reflected