1. Collisionless Shocks Manfred Scholer
Max-Planck-Institut für extraterrestrische Physik
Garching, Germany
The Solar/Space MHD International Summer School 2011
USTC, Hefei, China, 2011

2. Tom Gold, 1953: Solar flare plasma injection creates a thin collisionless shock

4. Criticality
Above first critical Mach number resistivity (by whatever mechanism, e.g. ion sound anomalous resistivity) cannot provide all the dissipation required by the Rankine-Hugoniot conditions.
Conclusion: additional dissipation needed.
Question: what is this additional dissipation?

5. Critical Mach Number
(Leroy, Phys. Fluids 1983)
2-Fluid (one-dimensional) resistive MHD equations:
Momentum equation for ions, momentum equation for massless electrons,
energy equation for electrons
Solve electron momentum equation for the electric field and insert into
ion momentum equation
Use Maxwells equation for curl B to substitute ion velocity for the electron
velocity
Integrate ion momentum equation to obtain ion velocity as function of
magnetic field and electron pressure

6. Critical Mach Number -II
Assume that ions remain cold through the shock and eletrons are heated by Ohmic friction
Obtain from the integrated momentum equation an equation for the x derivative of the electron pressure
Insert this derivative into the energy equation of the electrons and obtain an equation for dv/dx
This equation exhibits a singularity at the critical Mach number

12. Low beta, almost
perpendicular
Edmiston & Kennel et al. 1984

13. Important Parameters:
Oblique Shocks: Quasi-Parallel and Quasi-Perpendicular Shocks
Shock normal angle QBn
Trajectories of
specularly reflected ions
Important Parameters:
Mach number MA
Ion/electron beta

15. This is why 45 degrees between shock normal and
magnetic field is the dividing line between quasi-parallel
and quasi-perpendicular shocks

16. The Whistler Critical Mach number

18. Quasi-perpendicular bow shock
ion inertial length km
Magnetic field data in shock normal coordinates
versus distance from the shock in km
Horbury et al., 2001

19. Magnetic Field the Quasi-Parallel Bow Shock
Greenstadt et al., 1993

20. Cluster measurements of large amplitude
Pulsations (also called SLAMSs)
Lucek et al

21. Classification of Computer Simulation Models of Plasmas
Kinetic Description
Fluid Description
Full particle
codes
PIC
Vlasov
Codes
Hybrid Code
MHD Codes
Vlasov hybrid
code
Two-fluid code

22. Simulation Methods 1. Hybrid Method Ions are (macro) particles
Electrons are represented as a charge-neutralizing fluid
Electric field is determined from the momentum equation
of the electron fluid
Assume massless electrons and solve for electric field
is determined from the electrical current via
where is the bulk velocity of the ions

23. Simulation Methods 2. Particle-In-Cell (PIC) Method
Both species, ions and electrons, are represented as particles
Poisson‘s equation has to be solved
Spatial and temporal scales of the electrons
(gyration, Debye length) have to be resolved
Disadventage:
Needs huge computational resources
Adventage:
Gives information about processes on electron scales
Describes self-consistently electron heating and acceleration

24. Hybrid Simulation of 1-D or 2-D Planar Quasi-Parallel Collisionless Shocks
Inject a thermal distribution from the left hand side of a numerical box
Let these ions reflect at the right hand side
The (collective) interaction of the incident and reflected ions results
eventually in a shock which travels to the left
Ion phase space vx - x
(velocity in units of Mach number)
Diffuse ions
Transverse magnetic field component
Large amplitude waves
dB/B ~ 1

25. Quasi-Perpendicular Collisionless Shocks
1. Specular reflection of ions
2. Size of foot
3. Downstream exciation of the ion cyclotron instability
4. Electron heating

26. Schematic of a quasi-perpendicular supercritical shock

27. Schematic of Ion Reflection and Downstream Thermalization
Upstream
Esw B
Downstream
Shock
vsw
Core
Foot
Ramp

28. Specular reflection in HT frame: guiding center motion
is directed into downstream
About 30% of incoming solar wind is specularly reflected

29. Specularly reflected ions in the foot of the quasi-perpendicular bow shock –
in situ observations
Sckopke et al. 1983
Sun
Specularly reflected
ions
Solar wind
Ion velocity space distributions for an inbound bow shock crossing.
Phase space density is shown in the ecliptic plane with sunward
flow to the left.
Sckopke et al

30. Size of the foot

33. Reflection of particles and upstreamacceleration leads to
increase of the kinetic temperature.
Is this the downstream thermalization?
No! Process is time reversible. For dissipation we must have
an irreversible process; entropy must increase!
Scatter the resulting distribution!

34. Downstream Thermalization and Wave Excitation
Yong Liu et al
Downstream Thermalization and Wave Excitation
(Low Alfven Mach Number Case)
Predicted magnetic field fluctuation
power spectra obtained from the resonance
condition
Scattering leads to wave generation and
to a bi-spherical distribution

35. Downstream thermalization : Alfven ion cyclotron instability
Winske and Quest 1988
Oblique propagating Alfven Ion Cyclotron waves produced by the perpendicular/parallel temperature anisotropy
(Davidson & Ogden, Phys. Fluids, 1975)

36. Situation in the foot region of a perpendicular shockB
Ion and electron
distributions in the foot
Ions: unmagnetized
Electrons: magnetized

37. Possible microinstabilities in the foot
Wave type Necessary condition
Buneman inst Upper hybrid Du >> vte
(Langmuir)
Ion acoustic inst Ion acoustic Te >> Ti
Bernstein inst Cyclotron harmonics Du > vte
Modified two-stream inst. Oblique whistler Du/cosq > vte

38. Instabilities in the Foot and Shock Re-Formation
Instability between incoming ions and incoming electrons leads to perpendicular ion trapping
Reflected ions not effected

39. Shock Ripples Electron acceleration (test particle electrons
Burgess 2006
Shock Ripples
Electron acceleration (test particle electrons
in hybrid code shock)
Ripples are surface waves on shock front
Move along shock surface with Alfven
velocity given by magnetic field in
overshoot
Shocks with no ripples
Shock with ripples

40. Electron Heating
Electron heating at heliospheric shocks is small. Ratio of downstream to upstream
temperature about Downstream temperature usually amounts to an average of
about 12% of the upstream flow energy (for a wide range of shock parameters,
Including Mach number). Laboratory shock studied in the 70s had downstream to upstream
electron temperature ratios of up to 70!
Macrosacopic scales larger than electron gyroradius – electrons are magnetized
whereas ions are de- or only partially magnetized
Decoupling of ions and electrons at the shock and different thermalization histories

41. Electron distributiuon function through the shock
Feldman et al

42. Cross-shock potential and electron heating
Offset peak produced by
shock potential drop in the HT frame
Filled in by scattering

46. Quasi-Parallel Collisionless Shocks
Parker (1961): Collisionless parallel shock is due to firehose instability when upstream
plasma penetrates into downstream plasma
Golden et al. (1973) Group standing ion cyclotron mode excited by interpenetrating
beam produces turbulence of parallel shock waves
Early papers did not recognize importance of backstreaming ions
1. Excitation of upsteam waves and downstream convection
2. Upstream vs downstream directed group velocity
3. Mode conversion of waves at shock
4. Interface instability
5. Short Large Amplitude Magnetic Pulsations

47. Diffuse upstream ions
Paschmann et al

48. excites ion - ion beam instabilities in the upstream region
Free energy due to relative streaming of diffuse upstrem ions and solar wind
excites ion - ion beam instabilities in the upstream region

49. Electromagnetic Ion/Ion Instabilities
Gary, 1993
Ion/ion right hand resonant
(cold beam)
propagates in direction of beam
resonance with beam ions
right hand polarized
fast magnetosonic mode branch
Ion/ion nonresonant
(large relative velocity, large beam
density)
Firehose-like instability
propagates in direction opposite
to beam
Ion/ion left hand resonant
(hot beam)
resonance with hot ions flowing
antiparallel to beam
left hand polarized
on Alfven ion cyclotron branch
Ion distribution functions and associated
cyclotron resonance speed.

50. Upstream Waves: Resonant Ion/Ion Beam Instability
Backstreaming ions excite upstream
propagating waves by a resonant
ion/ion beam instability
Cyclotron resonance condition for
beam ions
dispersion relation
assume beam ions are specularly
reflected
( in units of , in units of )
Wavelength (resonance) increase with increasing Mach number

51. Dispersion Relatiosn of Magnetosonic Waves

52. Doppler-shift of dispersion relation from upstream plasma frame into shock frame
Negative w : phase velocity toward shock

53. Dispersion relation of upstream propagating whistler in shock frame:
Dispersion curve is shifted below zero line

54. At low Mach number waves (with large k) have upstream directed
Dopplershift into Shock Frame
(positive :phase velocity directed upstream)
Downstream directed group
velocity
Group standing
Phase standing
At low Mach number waves (with large k) have upstream directed
group velocity; they are phase-standing or have downstream directed phase velocity.
At higher Mach number the group velocity is reduced until it points back toward shock

55. of three different Mach numbers
Upstream wave spectra (2-D (x-t space) Fourier analysis) for simulated shocks
of three different Mach numbers
Krauss-Varban and Omidi 1991
Upstream waves are close
to phase-standing. Group
velocity directed upstream
Upstream waves are close
to group standing.
Group and phase velocity
directed towards shock
Shock periodically reforms itself when group velocity directed downstream

56. Winske et al. 1990 Interface Instability
In the region of overlap between cold solar
wind and heated downstream plasma waves
are produced by a right hand resonant
instability (solar wind is background, hot plasma
is beam).
Wave damping
Medium Mach number shock:
decomposition in positive and
negative helicity
Scholer, Kucharek, Jayanti 1997

57. Medium Mach Number Shock (2.5<MA<7)
Krauss-Varban and Omidi 1991
Interface waves have small
wavelength and are heavily
damped
Far downstream only upstream
generated F/MS waves survive
F/MS waves are mode converted
into AIC waves
Right: wavelet analysis of magnetic
field of a MA=3.5 shock ). Two
different wavelet components.

58. Interface Instability – High Mach Number Shocks
In high Mach number shocks the right hand resonant and right hand nonresonant
instability are excited. The downstream turbulence is dominated by these
large wavelength interface waves
(back to Parker and Golden et al.)
Scholer et al

59. Energetic Particles at

60. Electron Acceleration at the Quasi-Perpendicular Bow Shock
Upstream edge of the foreshock
Anderson et al

61. Foreshock velocity dispersion
Most energetic electrons at
foreshock edge
Lower energies deeper in the
foreshock

62. Electron acceleration at the foreshock edge
(quasi-perpendicular shock)

64. Reflected electrons: Ring beam with sharp edges
Portion reflected by magnetic mirroring
modified by effect of cross-shock potential
Larger loss cone due to
cross-shock potential

65. Magnetic mirror criterion
One contour of incident
electron distribution

66. Field –aligned beams (FABs) upstream of the quasi-perpendicular shock
Solar wind speed
Paschmann et al
Kuchaek et al

67. Specular reflection in HT frame and subsequent pitch angle scattering
Scattering in the de HT frame
Specular reflection in HT frame and subsequent pitch angle scattering

68. Test particles in hybrid simulations of a quasi-perpendicular shock
Burgess 1987
QBn
40o
Simulation beams in vx-vz phase space as the
angle QBn is increased. The line is the direction
of the upstream magnetic field.
45o
50o

69. Trajectory of a directly reflected particle
plotted in the shock frame. Top left:
typical magnetic field trace. Right panels:
time history of position and component forces.
Simulation beam density as a function of
QBn for various values of upstream ion b
(ratio of incident particle flux to backstreaming
beam flux). .

70. Quysi-Perpendicular Shock Ripples – Cluster Observations
Low Pass
Band Pass
High Pass
Low Pass + Band Pass B Field and density
at 4 Cluster spacecraft
Wavelength of ripples about 30 ion inertial lenghts
(2000 – 3000 km)
Moullard et al

71. Diffuse ions upstream of Earh‘s quasi-parallel bow shock
Paschmann et al

72. Diffuse Ion Spectra Upstream of Earth‘s Bow Shock
Log(flux) versus log(energy/charge)
Log(flux) versus energy/charge (linear)
Spectra are exponentials in energy/charge with the same
e-folding energy for all species
Ipavich et al. 1981

73. Cluster observations of diffuse upstream ions
Particle density at 24 – 32 keV at two
spacecraft, SC1 (black) and SC3 (green).
SC3 was about 1.5 Re closer to the bow
shock .
Kis et al

74. The gradient of upstream ion density in the energy range 24-32 keV as a function
of distance from the bow shock (Cluster) in a lin vs log representation. The gradient
(and thus the density itself) falls off exponentially.
Strong indication for diffusive transport in the upstream region:
Downstream convection is balanced by upstream diffusion

75. Diffusive Shock Acceleration
The Parker transport equation for energetic particles
1. Diffusion
2. Convection
3. Adiabatic decelation

77. Consider a one-dimensional flow Ux(x) as shown.
(the shock ratio U1 / U2 < 4)

79. Solution of time-dependent Parker equation results in
characteristic acceleration time for acceleration to to velocity v
(Laplace transform the Parker equation)

80. (stationary, planar shock):
Earth‘s bow shock
Between L, diffusion coefficient , k and solar wind velocity vsw the following relation holds
(stationary, planar shock):
Time scale for diffusive shock acceleration:
e-folding distance
With L = 3 RE at 30 keV and vsw = 600 km/s one obtains:
t acc = 120 s

82. Short Large Amplitude Magnetic Structures SLAMSs
and Shock Reformation
Oservations of a pulsationss at Earth‘s
bow shock. Top: temporal profile of magnetic
field magnitude; bottom: hodogram in one pulsation.
Pulsations mostly LH polarized in plasma frame.
Attached is a RH (in plasma frame) whistler.
Hybrid simulation of a quasi-
parallel shock showing shock
reformation.
Burgess 1989; Scholer and Teasawa 1989
Schwartz et al. 1992

83. Large amplitude magnetic pulsations comprise the quasi-parallel shock
Upstream waves – interacion with diffuse ions – Pulsations – shock structure
A collisionless quasi-parallel shock as
due to formation, convection, growth,
deceleration and merging of short large
amplitude magnetic structures.
Pulsations have a finite transverse extent.
Thus the shock is patchy when viewed,
e.g., over the shock surface.
The downstream state is divided into
plasma within Pulsation and in
inter-Pulsation region.
Schwartz and Burgess 1991

84. The Problem of Particle Injection at Quasi-Parallel Shocks
Into a Diffusive Acceleration Mechanism

85. Nonlinear phase trapping in large amplitude monochromatic wave
Sugiyama and Terasawa, 1999
Ions move non-resonantly in large amplitude upstream and
downstream monochomatic waves
Equation of motion in shock frame:
where Q = phase angle between V^ and wave field,
and Ww proportional to wave amplitude
Depending on phase angle a solar wind ion may get turned
around and moves upstream

86. Note: this is a non-resonant mechanism
Nonlinear phase trapping in large amplitude monochromatic wave
Ion is trapped between upstream and downstream wave train and gains energy
Note: this is a non-resonant mechanism