On the linear analysis of unstable radiative shocks

We study the stability properties of strong hydrodynamic shocks and their associated radiative cooling layers. We explore a range of conditions which covers both molecular and atomic gas impacting against a rigid wall. Through a linear analysis employing a cooling function of the form  Λ∝ρ^β T ^α  and a specific heat ratio of γ, we determine the overstability regime in the parameter space consisting of  α, β  and γ. In general, if α is sufficiently low, the fundamental mode leads to long-wavelength growing oscillations. For the fundamental mode, we find that values of γ corresponding to molecular hydrodynamics lead to a significantly restricted instability range for α in comparison with the shocks in a monatomic medium. The conditions for the growth of higher-order modes, however, are relatively unchanged. This predicts that certain molecular shocks are prone to displaying signatures of small-scale rapid variability. Dissociative shocks, however, can be subject to a large-scale overstability if subsequent molecule formation in the cooling layer abruptly increases the cooling rate. In contrast to the dynamical rippling overstability, the cooling overstability is suppressed for a sufficiently low specific heat ratio.     

Jump relations across a shock in non-ideal gas flow

Generalized forms of jump relations are obtained for one dimensional shock waves propagating in a non-ideal gas which reduce to Rankine-Hugoniot conditions for shocks in idea gas when non-idealness parameter becomes zero. The equation of state for non-ideal gas is considered as given by Landau and Lifshitz. The jump relations for pressure, density, temperature, particle velocity, and change in entropy across the shock are derived in terms of upstream Mach number. Finally, the useful forms of the shock jump relations for weak and strong shocks, respectively, are obtained in terms of the non-idealness parameter. It is observed that the shock waves may arise in flow of real fluids where upstream Mach number is less than unity.     

Instability of isothermal stellar wind bowshocks

We present hydrodynamical simulations illustrating the instability of stellar wind bowshocks in the limit of an isothermal equation of state. In this limit, the bowshock is characterized by a thin dense shell bounded on both sides by shocks. In a time-averaged sense the shape of this bowshock shell roughly matches the steady state solution of Wilkin (1996)[ApJ, 459, L31], although the apex of the bowshock can deviate in or out by a factor of two or more. The shape of the bowshock is distorted by large amplitude kinks with a characteristic wavelength of order the standoff distance from the star. The instability is driven by a strong shear flow within the shock-bounded shell, suggesting an origin related to the nonlinear thin-shell instability. This instability occurs when both the forward bowshock and the reverse wind shock are effectively isothermal and the star is moving through the interstellar medium with a Mach number greater than a few. This work therefore suggests that ragged, clumpy bowshocks should be expected to surround stars with a slow, dense wind (which leads to rapid cooling behind the reverse wind shock), whose velocity with respect to the surrounding interstellar medium is of order 60 km s⁻¹ (leading both to rapid cooling behind the forward bowshock and sufficiently high Mach numbers to drive the instability).     

Relaxation of protostellar accretion shocks using the smoothed particle hydrodynamics

It is believed that protostellar accretion disks are formed from nearly ballistic infall of molecular matter in rotating core collapse. Collisions of this infalling matter leads to formation of strong supersonic shocks, which if they cool rapidly, result in accumulation of that material in a thin structure in the equatorial plane. Here, we investigate the relaxation time of the protostellar accretion post‐shock gas using the smoothed particle hydrodynamics (SPH). For this purpose, a one‐dimensional head‐on collision of two molecular sheets is considered, and the time evolution of the temperature and density of the post‐shock region simulated. The results show that in strong supersonic shocks, the temperature of the post‐shock gas quickly increases proportional to square of the Mach number, and then gradually decreases according to the cooling processes. Using a suitable cooling function shows that in appropriate time‐scale, the center of the collision, which is at the equatorial plane of the core, is converted to a thin dense molecular disk, together with atomic and ionized gases around it. This structure for accretion disks may justify the suitable conditions for grain growth and formation of proto‐planetary entities (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Deeply Penetrating Banded Zonal Flows in the Solar Convection Zone

We discuss the spectrum arising from synchrotron emission by fast cooling (FC) electrons, when fresh electrons are continually accelerated by a strong blast wave, into a power-law distribution of energies. The FC spectrum has so far been described by four power-law segments divided by three break frequencies nusa相似文献   

The dependence of the diffusion coefficient on the mixing length in Kato's overstable zone

The treatment of the overstable zone by Langer, Sugimoto &38; Fricke is extended so that the effects of radiative heat loss from convective elements and of the mixing length are included. As the overstability investigated by Kato arises because of the radiative heat loss from the convective element, this loss may play a role in the heat flux. In the formulation presented in this work, the diffusion coefficient depends on the mixing length when the mixing length is very small. In calculations with the method of Langer et al., the efficiency factor of the diffusion coefficient of chemical species is sometimes much smaller than 1. This exceedingly small efficiency factor may be explained by the dependence of the diffusion coefficient on the mixing length. We find that the neutral condition of Kato's overstable zone is ∇ _rad = ∇ _ad.     

On the manifestation of Richtmyer-Meshkov instability in an inhomogeneous interstellar medium with radiative cooling

We numerically simulate the evolution of the plane two-dimensional deformations of a contact discontinuity that is impulsively accelerated by a shock wave. We take into account the effects of radiative cooling and perturbation scale lengths on the dynamics and shape of the forming density inhomogeneities. For moderately intense shocks in a stellar wind and for strong shocks from a supernova, we show that the radiative cooling processes do not affect significantly the growth rate of the initial perturbations and the total mass of the forming condensations. However, the density of the matter compressed by the transmitted shock wave increases dramatically. At the same time, the contribution from long-wavelength perturbations to the deformation of the contact surface decreases significantly. In the case of shock propagation from a supernova, the initial conditions have been found to be a factor that can affect the morphology of the shocked interstellar medium.     

Generation of magnetic fluctuations near a shock front in a partially ionized medium

We investigate the generation mechanism of long-wavelength Alfvénic disturbances near the front of a collisionless shock that propagates in a partially ionized plasma. The wave generation and dissipation rates are calculated in the linear approximation. The instability is attributable to a current of energetic particles upstream of the shock front. The generation of long-wavelength magnetic fluctuations is most pronounced for strong shocks, but the effect is retained for shocks with a moderate particle acceleration efficiency without any noticeable modification of the shock structure by the pressure of accelerated particles. The mode generation time for supernova remnants in a partially ionized interstellar medium is shown to be shorter than their age. Long-wavelength magnetic disturbances determine the limiting energies of the particles accelerated at a shock by the Fermimechanism. We discuss the application of the mechanism under consideration to explaining the observed properties of the SN 1006 remnant.     

Particle acceleration by ultrarelativistic shocks: theory and simulations

We consider the acceleration of charged particles near ultrarelativistic shocks, with Lorentz factor     . We present simulations of the acceleration process and compare these with results from semi-analytical calculations. We show that the spectrum that results from acceleration near ultrarelativistic shocks is a power law,     , with a nearly universal value     for the slope of this power law.
We confirm that the ultrarelativistic equivalent of the Fermi acceleration at a shock differs from its non-relativistic counterpart by the occurrence of large anisotropies in the distribution of the accelerated particles near the shock. In the rest frame of the upstream fluid, particles can only outrun the shock when their direction of motion lies within a small loss cone of opening angle     around the shock normal.
We also show that all physically plausible deflection or scattering mechanisms can change the upstream flight direction of relativistic particles originating from downstream by only a small amount:     . This limits the energy change per shock crossing cycle to     , except for the first cycle where particles originate upstream. In that case the upstream energy is boosted by a factor     for those particles that are scattered back across the shock into the upstream region.     

Dissipative accretion flows around a rotating black hole

We study the dynamical structure of a cooling dominated rotating accretion flow around a spinning black hole. We show that non-linear phenomena such as shock waves can be studied in terms of only three flow parameters, namely the specific energy     , the specific angular momentum (λ) and the accretion rate     of the flow. We present all possible accretion solutions. We find that a significant region of the parameter space in the     plane allows global accretion shock solutions. The effective area of the parameter space for which the Rankine–Hugoniot shocks are possible is maximum when the flow is dissipation-free. It decreases with the increase of cooling effects and finally disappears when the cooling is high enough. We show that shock forms further away when the black hole is rotating compared to the solution around a Schwarzschild black hole with identical flow parameters at a large distance. However, in a normalized sense, the flow parameters for which the shocks form around the rotating black holes are produced shocks closer to the black hole. The location of the shock is also dictated by the cooling efficiency in that higher the accretion rate     , the closer is the shock location. We believe that some of the high-frequency quasi-periodic oscillations may be due to the flows with higher accretion rate around the rotating black holes.     

Synchrotron emission in the fast cooling regime: which spectra can be explained?

We consider the synchrotron emission from relativistic shocks assuming that the radiating electrons cool rapidly (either through synchrotron or any other radiation mechanism). It is shown that the theory of synchrotron emission in the fast cooling regime can account for a wide range of spectral shapes. In particular, the magnetic field, which decays behind the shock front, brings enough flexibility to the theory to explain the majority of gamma-ray burst spectra even in the parameter-free fast cooling regime. Also, we discuss whether location of the peak in observed spectral energy distributions of gamma-ray bursts and active galactic nuclei can be made consistent with predictions of diffusive shock acceleration theory, and find that the answer is negative. This result is a strong indication that a particle injection mechanism, other than the standard shock acceleration, works in relativistic shocks.     

Viscous overstability and eccentricity evolution in three-dimensional gaseous discs

We investigate the growth or decay rate of the fundamental mode of even symmetry in a viscous accretion disc. This mode occurs in eccentric discs and is known to be potentially overstable. We determine the vertical structure of the disc and its modes, treating radiative energy transport in the diffusion approximation. In the limit of very long radial wavelength, an analytical criterion for viscous overstability is obtained, which involves the effective shear and bulk viscosity, the adiabatic exponent, and the opacity law of the disc. This differs from the prediction of a two-dimensional model. On shorter wavelengths (a few times the disc thickness), the criterion for overstability is more difficult to satisfy because of the different vertical structure of the mode. In a low-viscosity disc a third regime of intermediate wavelengths appears, in which the overstability is suppressed as the horizontal velocity perturbations develop significant vertical shear. We suggest that this effect determines the damping rate of eccentricity in protoplanetary discs, for which the long-wavelength analysis is inapplicable and overstability is unlikely to occur on any scale. In thinner accretion discs and in decretion discs around Be stars overstability may occur only on the longest wavelengths, leading to the preferential excitation of global eccentric modes.     

Oscillatory instability of radiative shocks with multiple cooling processes

The stand-off shock formed in the accretion flow on to a stationary wall, such as the surface of a white dwarf, may be thermally unstable, depending on the cooling processes which dominate the post-shock flow. Some processes lead to instability, while others tend to stabilize the shock. We consider competition between the destabilizing influence of thermal bremsstrahlung cooling, and a stabilizing process which is a power law in density and temperature. Cyclotron cooling and processes which are of order 1, 3/2 and 2 in density are considered. The relative efficiency and power-law indices of the second mechanism are varied, and particular effects on the stability properties and frequencies of oscillation modes are examined.     

Analytical structure of steady radiative shocks in magnetic cataclysmic variables

In this paper we study specific classes of radiating shocks which are widely spread in astrophysical environments. We present new analytical solutions available for any exponents of typical power-law cooling function which generalize the five special cases (corresponding to specific values of these exponents) of radiating shocks structure and proceed to the analytical determination of physical quantities. Then an application of the results for the accretion shock in polar is realized. A discussion of the reproducibility of cooling layer in laboratory is proposed using scaling laws.     

Theoretical and Experimental Studies of Radiative Shocks

This paper deals with the radiative shock from both theoretical and numerical points of view. It is based on the whole experimental results obtained at Laboratoire d'Utilisation des Lasers Intenses (LULI, école Polytechnique). Radiative shocks are high-Mach number shocks with a strong coupling between radiation and hydrodynamics which leads to a structure governed by a radiative precursor. These shocks are involved in various astrophysical systems: stellar accretion shocks, pulsating stars, interaction between supernovae and the interstellar medium. In laboratory, these radiative shocks are generated using high power lasers. New diagnostics have been implemented to study the geometrical shape of the shock and the front shock density. Data were obtained varying initial conditions for different laser intensities and temperature. The modeling of these phenomena is mainly performed through numerical simulations (1D and 2D) and analytical studies. We exhibit results obtained from several radiative hydrodynamics codes. As a result, it is possible to discuss about the influence of the geometry and physical parameters introduced in the 1D and 2D models.     

The weak shock in the core of the Perseus cluster

The dissipation of energy from sound waves and weak shocks is one of the most promising mechanisms for coupling active galactic nucleus (AGN) activity to the surrounding intracluster medium, and so offsetting cooling in cluster cores. We present a detailed analysis of the weak shock found in deep Chandra observations of the Perseus cluster core. A comparison of the spectra either side of the shock front shows that they are very similar. By performing a deprojection analysis of a sector containing the shock, we produce temperature and density profiles across the shock front. These show no evidence for a temperature jump coincident with the density jump. To understand this result, we model the shock formation using 1D hydrodynamic simulations including models with thermal conduction and  γ < 5/3  gas. These models do not agree well with the data, suggesting that further physics is needed to explain the shock structure. We suggest that an interaction between the shock and the Hα filaments could have a significant effect on cooling the post-shock gas.
We also calculate the thermal energy liberated by the weak shock. The total energy in the shocked region is about 3.5 times the work needed to inflate the bubbles adiabatically, and the power of the shock is around  6 × 10⁴⁴ erg s⁻¹  per bubble, just over  10⁴⁵ erg s⁻¹  in total.     

Fundamentals of collisionless shocks for astrophysical application, 2. Relativistic shocks

In this concise review of the recent developments in relativistic shock theory in the Universe we restrict ourselves to shocks that do not exhibit quantum effects. On the other hand, emphasis is given to the formation of shocks under both non-magnetised and magnetised conditions. We only briefly discuss particle acceleration in relativistic shocks where much of the results are still preliminary. Analytical theory is rather limited in predicting the real shock structure. Kinetic instability theory is briefed including its predictions and limitations. A recent self-similar relativistic shock theory is described which predicts the average long-term shock behaviour to be magnetised and to cause reasonable power-law distributions for energetic particles. The main focus in this review is on numerical experiments on highly relativistic shocks in (i) pair and (ii) electron-nucleon plasmas and their limitations. These simulations do not validate all predictions of analytic and self-similar theory and so far they do not solve the injection problem and the self-modification by self-generated cosmic rays. The main results of the numerical experiments discussed in this review are: (i) a confirmation of shock evolution in non-magnetised relativistic plasma in 3D due to either the lepton-Weibel instability (in pair plasmas) or to the ion-Weibel instability; (ii) the sensitive dependence of shock formation on upstream magnetisation which causes suppression of Weibel modes for large upstream magnetisation ratios σ>10⁻³; (iii) the sensitive dependence of particle dynamics on the upstream magnetic inclination angle θ _Bn , where particles of θ _Bn >34° cannot escape upstream, leading to the distinction between ‘subluminal’ and ‘superluminal’ shocks; (iv) particles in ultra-relativistic shocks can hardly overturn the shock and escape to upstream; they may oscillate around the shock ramp for a long time, so to speak ‘surfing it’ and thereby becoming accelerated by a kind of SDA; (v) these particles form a power-law tail on the downstream distribution; their limitations are pointed out; (vi) recently developed methods permit the calculation of the radiation spectra emitted by the downstream high-energy particles; (vii) the Weibel-generated downstream magnetic fields form large-amplitude vortices which could be advected by the downstream flow to large distances from the shock and possibly contribute to an extended strong field region; (viii) if cosmic rays are included, Bell-like modes can generate upstream magnetic turbulence at short and, by diffusive re-coupling, also long wavelengths in nearly parallel magnetic field shocks; (ix) advection of such large-amplitude waves should cause periodic reformation of the quasi-parallel shock and eject large-amplitude magnetic field vortices downstream where they contribute to turbulence and to maintaining an extended region of large magnetic fields.     

Convective instability and overstability in the sunspot umbra

Convective instability and overstability of a plane-parallel medium in the presence of a uniform vertical magnetic field is studied. We take into account both the effect of penetration of disturbances into sub-adiabatically stratified semi-infinite upper and lower layers, and the effect of thermal damping of disturbances introduced by the compressibility of the medium. If the degree of super-adiabaticity in temperature stratification in the middle layer is relatively large compared with that of sub-adiabaticity in the upper and lower layers, the range of physical conditions in which overstable convection occurs is very narrow. However, if it is not the case the range is rather wide. In the normal sunspot umbra, only disturbances with selected sizes in vertical dimension will be able to become overstable. The overstable convection might be a cause of some dynamical features (such as dots) in the sunspot umbra, but will not be the main contributor for transporting mechanical energy in the sunspot region.     

In-shock cooling in numerical simulations

We model a one-dimensional shock-tube using smoothed particle hydrodynamics and investigate the consequences of having finite shock-width in numerical simulations caused by finite resolution of the codes. We investigate the cooling of gas during passage through the shock for three different cooling regimes.
For a theoretical shock temperature of 10⁵ K, the maximum temperature of the gas is much reduced. When the ratio of the cooling time to shock-crossing time was 8, we found a reduction of 25 per cent in the maximum temperature reached by the gas. When the ratio was reduced to 1.2, the maximum temperature reached dropped to 50 per cent of the theoretical value. In both cases the cooling time was reduced by a factor of 2.
At lower temperatures, we are especially interested in the production of molecular hydrogen, and so we follow the ionization level and H₂ abundance across the shock. The effect of in-shock cooling is substantial: the maximum temperature the gas reaches compared with the theoretical temperature is found to vary between 0.15 and 0.81, depending upon the shock strength and mass resolution. The downstream ionization level is reduced from the theoretical level by a factor of between 2.4 and 12.5, and the resulting H₂ abundance by a factor of 1.35 to 2.22.
At temperatures above 10⁵ K, radiative shocks are unstable and will oscillate. We find that the shock jump temperature varies by a factor of 20 because of these oscillations.
We conclude that extreme caution must be exercised when interpreting the results of simulations of galaxy formation.     

The excitation, propagation and dissipation of waves in accretion discs: the non-linear axisymmetric case

We analyse the non-linear propagation and dissipation of axisymmetric waves in accretion discs using the ZEUS-2D hydrodynamics code. The waves are numerically resolved in the vertical and radial directions. Both vertically isothermal and thermally stratified accretion discs are considered. The waves are generated by means of resonant forcing, and several forms of forcing are considered. Compressional motions are taken to be locally adiabatic  ( γ =5/3)  . Prior to non-linear dissipation, the numerical results are in excellent agreement with the linear theory of wave channelling in predicting the types of modes that are excited, the energy flux by carried by each mode, and the vertical wave energy distribution as a function of radius. In all cases, waves are excited that propagate on both sides of the resonance (inwards and outwards). For vertically isothermal discs, non-linear dissipation occurs primarily through shocks that result from the classical steepening of acoustic waves. For discs that are substantially thermally stratified, wave channelling is the primary mechanism for shock generation. Wave channelling boosts the Mach number of the wave by vertically confining the wave to a small cool region at the base of the disc atmosphere. In general, outwardly propagating waves with Mach numbers near resonance  ℳ_r≳0.01  undergo shocks within a distance of order the resonance radius.