Collisionless Shocks Manfred Scholer
Max-Planck-Institut fr extraterrestrische Physik Garching, Germany The Solar/Space MHD International Summer School 2011 USTC, Hefei,China, 2011 Tom Gold, 1953: Solar flare plasma injection creates a thin collisionless shock 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? Critical Mach Number (Leroy, Phys. Fluids1983) 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 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 Low beta, almost perpendicular Edmiston & Kennel et al. 1984 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 This is why 45 degrees between shock normal and
magnetic field isthe dividing line between quasi-parallel and quasi-perpendicular shocks The Whistler Critical Mach number 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 Magnetic Field the Quasi-Parallel Bow Shock
Greenstadt et al., 1993 Cluster measurements of large amplitude
Pulsations (also called SLAMSs) Lucek et al Classification of Computer Simulation Models of Plasmas
Kinetic Description Fluid Description Full particle codes PIC Vlasov Codes Hybrid Code MHDCodes Vlasov hybrid code Two-fluid code 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 Simulation Methods 2. Particle-In-Cell (PIC) Method
Both species, ions and electrons, are represented as particles Poissons 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 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 Quasi-Perpendicular Collisionless Shocks
1.Specular reflection of ions 2. Size of foot 3.Downstream exciation of the ion cyclotron instability 4.Electron heating Schematic of a quasi-perpendicular supercritical shock Schematic of Ion Reflection and Downstream Thermalization
Upstream Esw B Downstream Shock vsw Core Foot Ramp Specular reflection in HT frame: guiding center motion
is directed into downstream About 30% of incoming solar wind is specularly reflected Specularly reflected ions in the foot ofthe 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 Size of the foot 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! Downstream Thermalization and Wave Excitation
Yong Liu et al Downstream Thermalization and Wave Excitation (Low Alfven Mach Number Case) Predictedmagnetic field fluctuation power spectra obtained from the resonance condition Scattering leads to wave generation and to a bi-spherical distribution Downstream thermalization : Alfven ion cyclotron instability
Winske and Quest1988 Oblique propagating Alfven Ion Cyclotron waves produced by the perpendicular/parallel temperature anisotropy (Davidson & Ogden, Phys. Fluids, 1975) Situation in the foot region of a perpendicular shock
B Ion and electron distributions in the foot Ions:unmagnetized Electrons:magnetized 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 Instabilities in the Foot and Shock Re-Formation
Instability between incoming ions and incoming electrons leads to perpendicular ion trapping Reflected ions not effected Shock Ripples Electron acceleration (test particle electrons
Burgess2006 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 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 Electron distributiuon function through the shock
Feldman et al Cross-shock potential and electron heating
Offset peak produced by shock potential drop in the HT frame Filled in by scattering 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 Diffuse upstream ions Paschmann et al excites ion - ion beam instabilities in the upstream region
Free energy due to relative streaming of diffuse upstremions and solar wind excites ion - ion beam instabilities in the upstream region 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. 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 Dispersion Relatiosn of Magnetosonic Waves Doppler-shift of dispersion relation from upstream plasma frame into shock frame
Negative w : phase velocity toward shock Dispersion relation of upstream propagating whistler in shock frame:
Dispersion curve is shifted below zero line 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 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 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, Jayanti1997 Medium Mach Number Shock (2.5