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.     

Incompressible MHD Turbulence

The inertial range of incompressible MHD turbulence is most conveniently described in terms of counter propagating waves. Shear Alfvén waves control the cascade dynamics. Slow waves play a passive role and adopt the spectrum set by the shear Alfvén waves. Cascades composed entirely of shear Alfvén waves do not generate a significant measure of slow waves. MHD turbulence is anisotropic with energy cascading more rapidly along k _⊥ than along k _∥. Anisotropy increases with k _⊥ such that the excited modes are confined inside a cone bounded by k _∥∝ k _perp ^2/3. The opening angle of the cone, θ(k _⊥)∝ k _⊥ ^-1/3, defines the scale dependent anisotropy. MHD turbulence is generically strong in the sense that the waves which comprise it are critically damped. Nevertheless, deep inside the inertial range, turbulent fluctuations are small. Their energy density is less than that of the background field by a factor θ²(k _⊥)≪. MHD cascades are best understood geometrically. Wave packets suffer distortions as they move along magnetic field lines perturbed by counter propagating wave packets. Field lines perturbed by unidirectional waves map planes perpendicular to the local field into each other. Shear Alfvén waves are responsible for the mapping's shear and slow waves for its dilatation. The former exceeds the latter by θ^-1(k _⊥)≫ 1 which accounts for dominance of the shear Alfvén waves in controlling the cascade dynamics. This revised version was published online in July 2006 with corrections to the Cover Date.     

Hydrodynamic and turbulent motions in the galactic disk

Statistics in absorption 21-cm data show two main types of clouds at low galactic latitudes: dense small clouds, many of them with molecular cores, with dispersions σ≈1.5 km s⁻¹ and large clouds forming the fine features of the spiral arms (the shingle like features) with a dispersion range α≈3–4 km s⁻¹. Sizes and dispersions of both types of clouds are compatible with the Kolmogorov law of turbulence: σ∞d ^1/3. The large clouds forming the shingle-like features can be considered as the largest clouds of a Kolmogorov spectrum (the initial vortices), or as the hydrodynamic features with minimum sizes in the Galaxy. In order to define hydrodynamic motions in the same sense as given by Ogrodnikov (1965) we use here the tensorial form of the Helmholtz theorem to obtain an approximation for the hydrodynamic motions depending on distances and seen from the local standard of rest:V _r ∞r. The intermediate range of sizes between turbulent motions and hydrodynamic motions is 100<d<300 pc which is also the range of sizes of the large clouds forming the fine features of the spiral arms. A classification on of motions in the Galaxy is postulated: (a) a basic rotation motion given by an smooth unperturbed curveΘ _b (R) associated to the old disk population. (b) Systematic motions of the spiral arms. (c) Systematic motions in the fine structure of the arms. For scale sizes smaller than these fine features one has turbulent motions according to the Kolmogorov law. The densities and sizes of the turbulent clouds behave asn _H ∝d ⁻² in a range of sizes 7 pc<d<300 pc. The obtained gas densities of the clouds are confirmed with the dust densities from photometric studies. The conditions for gravitational binding of the clouds are analyzed. Factors as the geometry and the magnetic field within the clouds increases the critic densities for gravitational binding. When we consider these factors we find that the wide component clouds have densities below such a critical value. The narrow component clouds have densities similar or above the critical value; but the real fraction of collapsing clouds remains unknown as far as the factor of geometry and the inner magnetic field of each cloud are not determinated.     

Combination Scattering by Anisotropic Langmuir Turbulence with Application to Solar Radar Experiments

We develop a theory for radar signal scattering by anisotropic Langmuir turbulence in the solar corona due to a t+l ⇄ t process. Langmuir turbulence is assumed to be generated within a cone by a narrow type III burst electron beam. Using wave-kinetic theory we obtain expressions for the frequency shift, scattering cross-section of the turbulence, coefficient of absorption (due to scattering) and optical depth. On the basis of those expressions we give some estimates for an echo spectrum. We show that the minimum radar echo frequency shift is determined by the minimal phase velocity of the Langmuir waves, the maximum shift is determined by the electron beam velocity, but in any case it can not exceed −wt/2 (decay) and wt (coalescence), where wt is the frequency of a radar signal. The angular characteristics of the scattered signal differ dramatically for the cases of coalescence and decay. The signal is scattered into a narrow cone high above the specular reflection point (wp ≪ wt), but in the vicinity of wp ∼ wt/2 the red-shifted echo is scattered isotropically, while the blue-shifted echo is scattered into a even narrower cone. We show that absorption (due to scattering) increases with increasing radar frequency. The dependence of the absorption on the local plasma frequency is strongly determined by the Langmuir turbulence spectrum. Our theory shows that the role of the nonlinear scattering process t+l ⇄ t is essential and that such process can be used for radar studies of the spectral energy density of anisotropic Langmuir turbulence.     

Variable synchrotron emission from BL Lacertae objects. II. Optical and X-ray flares in HBL source 1ES 1959+650

This paper presents the results of the optical R band and 1.5–12 keV band X-ray monitoring of the high-energy peaked BL Lacertae source 1ES 1959+650 performed during 2002–2007 with the 70 cm Meniscus Telescope of Abastumani Astrophysical Observatory (Georgia) and the All-Sky Monitor on board the Rossi X-ray Time Explorer, respectively. The observed long- and short-term outbursts are fitted with the lightcurves obtained by means of the modeling of synchrotron flares that are assumed to be the result of a propagation of the relativistic shock waves through the jet of 1ES 1959+650, pointed to the observer. Different values of the input parameters (shock velocity, particles’ spectral index, sizes of emission region, minimum and maximum Lorentz factors of the particles etc.) are used in order to fit the simulated lightcurves whose constructed by means of observational data. This investigation shows that both shock velocity and physical conditions in the jet of 1ES 1959+650 should be variable from flare to flare. The shocks are found to be mildly relativistic with the apparent speeds β=0.46–0.85, expressed in the units of c. Spectral index of the particle energy distribution varied from 2.10 to 2.17 for the long-term flares while it is higher in the case of short-term outbursts: s=2.32–2.45 that is suggested to be a result of the deceleration of shock front during its passage through the shell situated downstream the Mach disc. The average strength of a turbulent magnetic field ranged from 0.025 gauss to 0.04 gauss for different long-term flares while the values of 0.07–0.14 gauss were adopted for the different short-term outbursts. The lengths of variable jet area found to be of 0.13–0.47 pc with the transverse extents of (0.5–1.0)×10¹⁷ cm in the case of long-term flares. The same characteristics for short-term outbursts were (2.74–5.5)×10¹⁶ cm and (0.2–04)×10¹⁷ cm, respectively. We conclude that both shock velocity and properties of pre-shocked plasma were not the same in 1ES 1959+650 for the different flaring epochs.     

Properties of the negative effective magnetic pressure instability

As was demonstrated in earlier studies, turbulence can result in a negative contribution to the effective mean magnetic pressure, which, in turn, can cause a large‐scale instability. In this study, hydromagnetic mean‐field modelling is performed for an isothermally stratified layer in the presence of a horizontal magnetic field. The negative effective magnetic pressure instability (NEMPI) is comprehensively investigated. It is shown that, if the effect of turbulence on the mean magnetic tension force vanishes, which is consistent with results from direct numerical simulations of forced turbulence, the fastest growing eigenmodes of NEMPI are two‐dimensional. The growth rate is found to depend on a parameter β_* characterizing the turbulent contribution of the effective mean magnetic pressure for moderately strong mean magnetic fields. A fit formula is proposed that gives the growth rate as a function of turbulent kinematic viscosity, turbulent magnetic diffusivity, the density scale height, and the parameter β_*. The strength of the imposed magnetic field does not explicitly enter provided the location of the vertical boundaries are chosen such that the maximum of the eigenmode of NEMPI fits into the domain. The formation of sunspots and solar active regions is discussed as possible applications of NEMPI (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solar Flare Phenomena as Phase Transition Caused by Frustration of Current Percolation

We present analysis of flare process as "phase transition" phenomena caused by frustration of current percolation in turbulent current sheet. We show that numerous plasma instabilities in the sheet will form random resistors network with "bad resistors"-turbulent domains and "good resistors"-normal plasma domains. We show that current percolation in random inhomogeneous turbulent current sheet like to another percolated systems is able to produce phase transition with drastic change of global properties of system as whole (conductivity, heat-conductivity, elasticity,) on the threshold value of critical density of "bad" elements (p= p _c ). Another property of solar flares, what may be understood on the base of percolation approach is observed universal power dependence of frequency of flares and microflares (elementary events-spikes) on their amplitude: N _W ∝ W ^−k . It may be explained as natural sequence of universal power dependence of clusters' masses in percolated systems on their sizes. The slope of resulted spectra is determined by the fractal dimension of clusters and depends on feedback between current propagation and turbulence generation. We show that percolation approach allow to explain phenomena of preflare bursts-precursors observed in radio and hard X-ray. It may be understood as results of pre-catastrophic lose of elasticity of system to small disturbance on the percolation threshold, with formation of short life nuclear of "new phase". This revised version was published online in July 2006 with corrections to the Cover Date.     

A Consistent one-Dimensional Model for the Turbulent Tachocline

The first consistent model for the turbulent tachocline is presented, with the turbulent diffusivity computed within the model instead of being specified arbitrarily. For the origin of the 3D turbulence a new mechanism is proposed. Owing to the strongly stable stratification, the mean radial shear is stable, while the horizontal shear is expected to drive predominantly horizontal, quasi-2D motions in thin slabs. Here I suggest that a major source of 3D overturning turbulent motions in the tachocline is the secondary shear instability due to the strong, random vertical shear arising between the uncorrelated horizontal flows in neighboring slabs. A formula for the vertical diffusivity due to this turbulence, Equation (9), is derived and applied in a simplified 1D model of the tachocline. It is found that Maxwell stresses due to an oscillatory poloidal magnetic field of a few hundred gauss are able to confine the tachocline to a thickness less than 5 Mm. The integral scale of the 3D overturning turbulence is the buoyancy scale, on the order of 10 km, and its velocity amplitude is a few m s^–1, yielding a vertical turbulent diffusivity on the order of 10⁸ cm² s^–1.     

Giant Radio Halos in Galaxy Clusters as Probes of Particle Acceleration in Turbulent Regions

Giant radio halos in galaxy clusters probe mechanisms of particle acceleration connected with cluster merger events. Shocks and turbulence are driven in the inter-galactic medium (IGM) during clusters mergers and may have a deep impact on the non-thermal properties of galaxy clusters. Models of turbulent (re)acceleration of relativistic particles allow good correspondence with present observations, from radio halos to γ-ray upper limits, although several aspects of this complex scenario still remain poorly understood.     

Mass-Stripping Analysis of an Interstellar Cloud by a Supernova Shock

The interaction of supernova shocks and interstellar clouds is an important astrophysical phenomenon since it can result in stellar and planetary formation. Our experiments attempt to simulate this mass-loading as it occurs when a shock passes through interstellar clouds. We drive a strong shock using the Omega laser (∼5kJ)into a foam-filled cylinder with an embedded Al sphere(diameterD=120 μm) simulating an interstellar cloud. The density ratio between Al and foamis∼9. We have previously reported on the interaction between shock and cloud, the ensuing Kelvin-Helmholtz and Widnall instabilities, and the rapid stripping of all mass from the cloud. We now present a theory that explains the rapid mass-stripping. The theory combines (1) the integral momentum equations for a viscous boundary layer, (2) the equations for a potential flow past a sphere, (3) Spalding's law of the wall for turbulent boundary layers, and (4) the skin friction coefficient for a turbulent boundary layer on a flat plate. The theory gives as its final result the mass stripped from a sphere in a turbulent high Reynolds number flow, and it agrees very well with our experimental observations.     

Diamagnetic pumping near the base of a stellar convection zone

The property of inhomogeneous turbulence in conducting fluids to expel large‐scale magnetic fields in the direction of decreasing turbulence intensity is shown as important for the magnetic field dynamics near the base of a stellar convection zone. The downward diamagnetic pumping confines a fossil internal magnetic field in the radiative core so that the field geometry is appropriate for formation of the solar tachocline. For the stars of solar age, the diamagnetic confinement is efficient only if the ratio of turbulent magnetic diffusivity η_T of the convection zone to the (microscopic or turbulent) diffusivity η_in of the radiative interior is η_T/η_in ≥ 10⁵. Confinement in younger stars requires larger η_T/η_in. The observation of persistent magnetic structures on young solar‐type stars can thus provide evidence for the nonexistence of tachoclines in stellar interiors and on the level of turbulence in radiative cores. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Variable synchrotron emission from BL Lacertae objects. I. Shock-in-jet scenario

This paper presents a modeling of the variable synchrotron emission in the BL Lacertae sources (BLLs). Flux variability is assumed to be a result of the interaction between a relativistic shock wave with a magnetized jet material. Long-term flares (of months to years durations) are modeled via the propagation of a plane relativistic shock wave though the emission zone of a cylindrical form with the radius R and length H. As for short-term bursts (lasting from days to weeks), they may result from shock passage through the jet inhomogeneities such as a shell of enhanced density downstream to a Mach disc, originated due to pressure imbalance between the jet and its ambient medium. Emitting particles (electrons) gain the energies, sufficient to produce synchrotron photons at optical—X-ray frequencies, via the first-order Fermi mechanism. Observation’s frequency is the main parameter determining a rate of the increase/ decay of the emission via the characteristic decay time of emitting electrons. The magnetic field, assumed to be turbulent with an average field constant throughout the entire emission zone, is another key parameter determining the slope of a lightcurve corresponding to the flare—the higher strength the magnetic field has, the steeper the lightcurve is. The rest input parameters (shock speed, jet viewing angle, maximum/minimum energies of the electrons, particles’ density etc.), as well the strength of average magnetic field, influence the energy output from a flare.     

Observations of a 2→3 Type Interplanetary Intermediate Shock

An intermediate shock-like event was observed by Voyager 2 on 9 January 1979. The discontinuity structure is well identified to be a 2→3 type intermediate shock by fitting the Rankine – Hugoniot relations. The shock satisfies the following conditions: i) the plasma density increases from the upstream region to the downstream region, ii) The normal Alfvén Mach number (M _AN) is greater than unity in the preshock state and less than unity in the postshock state, iii) The fast-mode Mach numbers in the upstream and downstream regions are less than unity and both the slow-mode Mach numbers are greater than unity, iv) The tangential component of the magnetic field changes sign across the shock front.     

Geoeffectiveness of interplanetary shocks during solar minimum (1995–1996) and solar maximum (2000)

Plasma and magnetic field parameter variations across fast forward interplanetary shocks are analyzed during the last solar cycle minimum (1995–1996, 15 shocks), and maximum year 2000 (50 shocks). It was observed that the solar wind velocity and magnetic field strength variation across the shocks were the parameters better correlated with Dst. Superposed epoch analysis centered on the shock showed that, during solar minimum, B _z profiles had a southward, long-duration variation superposed with fluctuations, whereas in solar maximum the B _z profile presented 2 peaks. The first peak occurred 4 hr after the shock, and seems to be associated with the magnetic field disturbed by the shock in the sheath region. The second peak occurred 19 hr after the shock, and seems to be associated with the ejecta fields. The difference in shape and peak in solar maximum (Dst peak =−50 nT, moderate activity) and minimum (Dst peak =−30 nT, weak activity) in average Dst profiles after shocks are, probably, a consequence of the energy injection in the magnetosphere being driven by different interplanetary southward magnetic structures. A statistical distribution of geomagnetic activity levels following interplanetary shocks was also obtained. It was observed that during solar maximum, 36% of interplanetary shocks were followed by intense (Dst≤−100 nT) and 28% by moderate (−50≤Dst <−100 nT) geomagnetic activity. During solar minimum, 13% and 33% of the shocks were followed by intense and moderate geomagnetic activity, respectively. Thus, during solar maximum a higher relative number of interplanetary shocks might be followed by intense geomagnetic activity than during solar minimum. One can extrapolate, for forecasting goals, that during a whole solar cycle a shock has a probability of around 50–60% to be followed by intense/moderate geomagnetic activity.     

Tachocline Confinement by an Oscillatory Magnetic Field

Helioseismic measurements indicate that the solar tachocline is very thin, its full thickness not exceeding 4% of the solar radius. The mechanism that inhibits differential rotation to propagate from the convective zone to deeper into the radiative zone is not known, though several propositions have been made. In this paper we demonstrate by numerical models and analytic estimates that the tachocline can be confined to its observed thickness by a poloidal magnetic field B _p of about one kilogauss, penetrating below the convective zone and oscillating with a period of 22 years, if the tachocline region is turbulent with a diffusivity of η∼10¹⁰ cm² s⁻¹ (for a turbulent magnetic Prandtl number of unity). We also show that a similar confinement may be produced for other pairs of the parameter values (B _p, η). The assumption of the dynamo field penetrating into the tachocline is consistent whenever η≳10⁹ cm² s⁻¹. Supplementary material to this paper is available in electronic form at http://dx.doi.org/10.1023/A:1013389631585     

Turbulence in fast-mode shocks as a triggering mechanism in a solar flare

If a solar flare originates from the dissipation of magnetic energy, available in abundance in a larger region, this dissipation must take place very rapidly. A local topological change in the magnetic field structure may be sufficient to start the dissipation process. Such a change in topology might be obtained by fast reconnection in a smaller region, such as e.g. in the Sweet-Parker model, as a result of current-driven microinstabilities.Among the candidates satisfying the requirements to obtain large enough currents, such as magnetically neutral or current sheets and MHD shocks, the latter are shown to be most probable. In a fast MHD shock the (thermal) results of turbulence do in fact destroy the conditions for turbulence. However, in this work we show numerically that the nonlinear steepening mechanism of such a shock is able to restore the driving current for a large range of parameters and over a long time. This is still true if the most difficult threshold for turbulence, being that for Langmuir turbulence, is to be achieved. The critical parameter, not only for the occurrence of turbulence but also for the restoration of the driving current, is the shock thickness.     

Propagation of the Bastille Day 2000 CME Shock in the Outer Heliosphere

The Bastille Day (14 July) 2000 CME is a fast, halo coronal mass ejection event headed earthward. The ejection reached Earth on 15 July 2000 and produced a very significant magnetic storm and widespread aurora. At 1 AU the Wind spacecraft recorded a strong forward shock with a speed jump from ∼ 600 to over 1000 km s⁻¹. About 6 months later, this CME-driven shock arrived at Voyager 2 (∼ 63 AU) on 12 January 2001 with a speed jump of ∼ 60 km s⁻¹. This provides a good opportunity to study the shock propagation in the outer heliosphere. In this study, we employ a 2.5-D MHD numerical model, which takes the interaction of solar wind protons and interstellar neutrals into account, to investigate the shock propagation in detail and compare the model predictions with the Voyager 2 observations. The Bastille Day CME shock undergoes a dramatic change in character from the inner to outer heliosphere. Its strength and propagation speed decay significantly with distance. The model results at the location of Voyager 2 are in good agreement with in-situ observations. Supplementary material to this paper is available in electronic form at http://dx.doi.org/10.1023/A:1014293527951     

The characteristics of ion acoustic shock waves in non-Maxwellian plasmas with (G′/G)-expansion method

Korteweg-de-Vries-Burger (K-dVB) equation is derived for ion acoustic shock waves in electron-positron-ion plasmas. Electrons and positrons are considered superthermal and are effectively modeled by a kappa distribution in which ions are as cold fluid. The analytical traveling wave solutions of the K-dVB equation investigated, through the (G′/G)-expansion method. These traveling wave solutions are expressed by hyperbolic function, trigonometric functions are rational functions. When the parameters are taken special values, the shock waves are derived from the traveling waves. It is observed that the amplitude ion acoustic shock waves increase as spectral index κ and kinematic viscosity η _i,0 increases in which with increasing positron density β and electron temperature σ the shock amplitude decreases. Also, numerically the effect different parameters on the nonlinearity A and dispersive B terms and wave velocity V investigated.     

The Effect of Abnormal Granulation on Acoustic Wave Travel Times and Mode Frequencies

Observations indicate that in plage areas (i.e. in active regions outside sunspots) acoustic waves travel faster than in the quiet Sun, leading to shortened travel times and higher p-mode frequencies. Coupled with the 11-year variation of solar activity, this may also explain the solar cycle variation of oscillation frequencies. While it is clear that the ultimate cause of any difference between the quiet Sun and plage is the presence of magnetic fields of order 100 G in the latter, the mechanism by which the magnetic field exerts its influence has not yet been conclusively identified. One possible such mechanism is suggested by the observation that granular motions in plage areas tend to be slightly “abnormal”, dampened compared to the quiet Sun. In this paper we consider the effect that abnormal granulation observed in active regions should have on the propagation of acoustic waves. Any such effect is found to be limited to a shallow surface layer where sound waves propagate nearly vertically. The magnetically suppressed turbulence implies higher sound speeds, leading to shorter travel times. This time shift Δ τ is independent of the travel distance, while it shows a characteristic dependence on the assumed plage field strength. As a consequence of the variation of the acoustic cutoff with height, Δ τ is expected to be significantly higher for higher frequency waves within the observed regime of 3 – 5 mHz. The lower group velocity near the upper reflection point further leads to an increased envelope time shift, as compared to the phase shift. p-mode frequencies in plage areas are increased by a corresponding amount, Δ ν/ν=ν Δ τ. These characteristics of the time and frequency shifts are in accordance with observations. The calculated overall amplitudes of the time and frequency shifts are comparable to, but still significantly less than (by a factor of 2 to 5), those suggested by measurements.     

The Plasma and Magnetic Field Characteristics of a Double Discontinuity in Interplanetary Space

A double discontinuity is a rarely observed compound structure composed of a slow shock layer and an adjoining rotational discontinuity layer in the downstream region. In this paper, we report the observations of a double discontinuity detected by Wind on May 15, 1997. This double discontinuity is found to be the front boundary of a magnetic cloud boundary layer. We strictly identify the shock layer and the rotational discontinuity layer by using the high-resolution plasma and magnetic field data from Wind. The observed jump conditions of the upstream and downstream region of the slow shock layer are in good agreement with the Rankine – Hugoniot relations. The flow speeds in the shock frame U _n <V _Acos θ _Bn on both sides of the slow shock layer. In the upstream region, the slow Mach number M _s1=U _n1/V _s1 is 1.95 (above unity), and in the downstream region, the slow Mach number M _s2=U _n2/V _s2 is 0.31 (below unity). Here V _A and V _s represent the Alfvén speed and the local slow magnetosonic speed, respectively, and θ _Bn is the angle between the direction of the magnetic field and the shock normal. The magnetic cloud boundary layer observed by Wind was also detected by Geotail 48 min later when the spacecraft was located outside the bow shock of the magnetosphere. However, Geotail observations showed that its front boundary was no longer a double discontinuity and the rotational discontinuity layer disappeared, indicating that this double discontinuity was unstable when propagating from Wind to Geotail.