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Collisionless Shocks Manfred Scholer Max-Planck-Institut für 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. 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 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 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 is the dividing line between quasi-parallel and quasi-perpendicular shocks The Whistler Critical Mach number Quasi-perpendicular bow shock ion inertial length i 60 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. 2008 Classification of Computer Simulation Models of Plasmas Kinetic Description Vlasov Codes Full particle codes PIC Vlasov hybrid code Fluid Description Hybrid Code MHD Codes 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 nm e dV e Ve B en(E ) p e dt c Assume massless electrons and solve for electric field 1 1 pe E x (Ve B) x c en x Ve is determined from the electrical current via Ve Vi j ne where Vi is the bulk velocity of the ions 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 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 Shock Upstream Downstream Esw vsw B B 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 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 Sckopke et al. 1983 flow to the left. 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! Yong Liu et al. 2005 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 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) 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 (Langmuir) Du >> vte Ion acoustic inst. Ion acoustic Te >> Ti Cyclotron harmonics Du > vte Bernstein inst. Modified two-stream inst. Oblique whistler Du/cosq > vte Instabilities in the Foot and Shock Re-Formation i e 0.05 Instability between incoming ions and incoming electrons leads to perpendicular ion trapping Reflected ions not effected Burgess 2006 Shock Ripples Ripples are surface waves on shock front Move along shock surface with Alfven velocity given by magnetic field in overshoot Electron acceleration (test particle electrons in hybrid code shock) Shock with ripples Shocks with no ripples Electron Heating Electron heating at heliospheric shocks is small. Ratio of downstream to upstream temperature about 3 - 4. 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. 1982 Cross-shock potential and electron heating Filled in by scattering Offset peak produced by shock potential drop in the HT frame 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. 1981 Free energy due to relative streaming of diffuse upstrem ions 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 distribution functions and associated cyclotron resonance speed. Ion/ion left hand resonant (hot beam) propagates in direction of beam resonance with hot ions flowing antiparallel to beam left hand polarized on Alfven ion cyclotron branch 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 k r vb c dispersion relation kv A assume beam ions are specularly reflected vb 2vsw ( in units of c , k in units of c / v A) r k r 1 /( 2 M A 1) 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 : phase velocity toward shock Dispersion relation of upstream propagating whistler in shock frame: Dispersion curve is shifted below zero line Dopplershift into Shock Frame (positive Downstream directed group velocity :phase velocity directed upstream) 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 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 Interface Instability Winske et al. 1990 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 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. 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. 1997 Energetic Particles at Electron Acceleration at the Quasi-Perpendicular Bow Shock Upstream edge of the foreshock Anderson et al. 1979 Foreshock velocity dispersion Most energetic electrons at foreshock edge Lower energies deeper in the foreshock Electron acceleration at the foreshock edge (quasi-perpendicular shock) Reflected electrons: Ring beam with sharp edges Larger loss cone due to cross-shock potential Portion reflected by magnetic mirroring modified by effect of cross-shock potential Magnetic mirror criterion One contour of incident electron distribution Field –aligned beams (FABs) upstream of the quasi-perpendicular shock Solar wind speed Paschmann et al. 1981 Kuchaek et al. 2004 Scattering in the de HT frame Specular reflection in HT frame and subsequent pitch angle scattering 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 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 (ratio of incident particle flux to backstreaming beam flux). . 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. 2006 Diffuse ions upstream of Earh‘s quasi-parallel bow shock Paschmann et al. 1981 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 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. 2004 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 Diffusive Shock Acceleration The Parker transport equation for energetic particles 1. Diffusion 2. Convection 3. Adiabatic decelation Consider a one-dimensional flow Ux(x) as shown. (the shock ratio U1 / U2 < 4) Solution of time-dependent Parker equation results in characteristic acceleration time for acceleration to to velocity v (Laplace transform the Parker equation) v acc xx1 xx2 3 dv U1 U 2 U1 U2 Earth‘s bow shock Between L, diffusion coefficient , and solar wind velocity vsw the following relation holds (stationary, planar shock): L v sw Time scale for diffusive shock acceleration: t acc 3|| 2 vsw 3L vsw e-folding distance With L = 3 RE at 30 keV and vsw = 600 km/s one obtains: t acc = 120 s 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 quasiparallel shock showing shock reformation. Burgess 1989; Scholer and Teasawa 1989 Schwartz et al. 1992 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 The Problem of Particle Injection at Quasi-Parallel Shocks Into a Diffusive Acceleration Mechanism 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: dVx w V sin Q dt where Q = phase angle between V and wave field, and w proportional to wave amplitude Depending on phase angle a solar wind ion may get turned around and moves upstream 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