Collisionless shocks (CSs) are one of the most efficient accelerators of charged particles in the universe. They are present in virtually all plasma environments at scales from ∼1 cm in the terrestrial labs to ∼1 Mpc in galaxy clusters. A CS is a multi-scale object, where the highest energies are achieved at the largest scales within the diffusive process due to scattering at fluctuations far upstream and far downstream and multiple crossings of the shock. The latter occur within the scatter-free region; thus, ion dynamics in the shock front is intimately related to the large-scale acceleration: while the diffusive acceleration occurs at scales much larger than the shock width, the spectrum of the accelerated particles is essentially determined by conservation laws at the scatter-free shock transition. The fields in the shock front are responsible for ion heating, generation of backstreaming ion beams (; ; ; ; ), acceleration of pickup ions , and injection into the diffusive mechanism . Thus, the structure of the shock front is the central problem of the shock physics. The shock structure can be studied with in situ measurements only at heliospheric shocks. Qualitative understanding of the shock structure has substantially improved due to these high quality observations and also due to numerical simulations. The frontier of observational shock studies has recently shifted towards the processes occurring within a few ion convective gyroradii in both directions from the ramp along the shock normal (; ; ; ; ; ; ).

Magnetic profiles of collisionless shocks are rarely monotonic, even for low-Mach numbers (; ; ; ; ; ; ; ). As the peak value of the downstream oscillations increases with the increase in the Mach number, for a long time it was believed that overshoots were produced by ion reflection in super-critical shocks  (; ; ; ; ). Super-critical shocks are the shocks with the Mach number exceeding the critical Mach number ; thus, the resistivity and thermal conductivity alone cannot provide necessary dissipation to sustain a shock. Eventually coherent downstream oscillations were observed at a very low-Mach number shock with the Alfvenic Mach number of M=1.3 and magnetic compression of Bd/Bu=1.3. The oscillating trail behind the ramp exhibited all of the features expected for a supercritical shocks, such as the largest first peak, spatially periodical peaks, and the gradual decrease in the peak amplitude. Such oscillations, albeit often less ordered, were found to be common in low-Mach number shocks . They were successfully explained as a result of coherent ion gyration upon crossing the shock ramp and subsequent collisionless relaxation due to gyrophase mixing (; ; ; ; ). It has been shown that the largest peak amplitude is mainly determined by the magnetic compression and cross-shock potential, whereas the damping rate of the oscillations is related to the upstream thermal-to-fluid speed ratio . Shapes of the downstream profile, like relative peaks of the first oscillations and steepness of the magnetic field increase up to the first peak, and vary considerably among observed shocks, even subcritical shocks. Thus far, sufficient attention has not been devoted to the relation of the details of the magnetic oscillation pattern to the shock parameters and ion kinetics in the shock front. In particular, amplitudes and positions of the first peaks, which are not yet distorted by gyrophase mixing, may provide information about the cross-shock potential as well about the ion transmission and reflection.

In what follows, Bu is the upstream magnetic field magnitude, Tu is the upstream ion temperature, nu is the upstream ion number density, vT=Tu/m is the upstream ion thermal speed, m is the ion mass, and β=8πnuTu/Bu2 is the upstream kinetic-to-magnetic pressure ratio. The corresponding parameters for electrons are denoted by adding index e. In order to explain the basic mechanism of producing the downstream oscillations, let us consider a simplified model of a perpendicular shock. We treat the shock as a jump in the magnetic field from Bu to Bd=RBu occurring within a narrow ramp. Accordingly, the fluid drift speeds upstream and downstream are Vu and Vd=Vu/R, respectively. We shall also neglect the electron contribution in the plasma pressure and treat ions as a monoenergetic beam entering the shock with the velocity Vu along the shock normal. The analysis is carried out in the normal incidence frame, where x is along the shock normal (in the downstream direction) and z is along the magnetic field. The equations of motion for ions inside the ramp are 1v˙x=qmEx+qmcvyBz2v˙y=qmc(VuBu-vxBz). We integrate the equations of motion across the ramp assuming |vy|∼vT≪Vu, where vT is the thermal speed of upstream ions. In this approximation we get 3ux22-Vu22=-qmϕ+∫qmcvyBzdx4uy=∫qBumcVuvx-BzBudx. Here u denotes the ion velocity at the downstream edge of the ramp, whereas v(x) denotes the ion velocity at the position x inside the ramp. The second term in Eq. () is a small correction for ramp width ≲(c/ωpi) and vT/Vu≪1. Here (c/ωpi) is the ion inertial length. This small correction can be neglected for our purposes. In Eq. () the only term is small but nonzero. Thus, if the cross-shock potential is ϕ=s(mVu2/2e), the ion velocity just after crossing the jump is 5vx(x=0)=Vu1-s,vy(x=0)=uy. The ion motion is then described as a drift along the shock normal with the velocity Vu/R and gyration around the magnetic field: 6vx(t)=Vd+v⟂cos⁡(Ωdt+φ)7vy(t)=v⟂sin⁡(Ωdt+φ)8v⟂2=(Vu1-s-R)2+uy2, where Ωd=eBd/mic is the downstream ion gyrofrequency. For a cold beam, all ions move together and the coordinate along the shock normal is given by 9x(t)=Vdt+v⟂Ωsin⁡(Ωdt+φ)-sin⁡φ. In general, it is not possible to derive an analytical expression for vx(x). For our purposes it is sufficient to restrict ourselves to weak gyration, v⟂<Vd, so that dx(t)/dt=vx>0 and x(t) is invertible; thus, t(x) is a single-valued function. Let us define the critical gyration potential (CGP) scr=1-1/R2. For s<scr, the initial gyrophase φ≈0, so that dvx/dx<0 at the downstream edge of the ramp. For s>scr, the initial gyrophase is φ≈π, so that dvx/dx>0 at the downstream edge of the ramp.

The total (dynamic and kinetic) ion pressure is given by 10pi,xx=minvx2=minuVuvx, where we have used the mass conservation nvx=nuVu. Pressure balance requires pi,xx+B2/8π=const, so that the magnetic field has maxima at the minima of the ion pressure. The latter occur at the minima of vx. For s<scr, the velocity decreases inside the ramp and keeps decreasing down to vx,min=Vd-v⟂ at Ωdt+φ=π which approximately corresponds to xl=πVd/Ωd for φ≈0. Thus, the first maximum of the magnetic field occurs at xl at the pressure pl=minuVu(Vd-v⟂). With the increase in s, the relative contribution of uy in v⟂ increases which moves the position of the first pressure minimum closer to the ramp. For s>scr, the velocity decreases inside the ramp but starts to increase just behind it. Thus, the first maximum of the magnetic field occurs at x=0 (the downstream edge of the ramp) at the pressure ph=minuVu(Vd-ux). As ux<v⟂, one has ph>pl which means that the first peak will be lower than the subsequent peaks corresponding to the pressure minima pl.

For a cold ion beam the amplitude of further pressure oscillations does not change. Finite temperature leads to the divergence of the ion trajectories and gradual gyrophase mixing. The divergence already occurs at the shock crossing, as the downstream ion velocity vx,d=vx,u2-2eϕ/m, and the spread in vx,u results in a more substantial spread in vx,d. Moreover, there is nonzero vy that affects the gyration speed v⟂ and φ, which are now different for different particles: 11v⟂2=vx,u2-2eϕ/m-Vd2+vy,u212cos⁡φ=vx,u2-2eϕ/m-Vdv⟂. The downstream ion pressure including finite temperature is obtained as an integral over the distribution 13pi,xx=mi∫vx2f(v)d3v. It has been shown () that finite temperature results in collisionless relaxation during which the downstream ion distribution gyrotropizes and the pressure oscillations damp out. The relaxation is faster for larger vT/Vu. In oblique shocks the mechanism of the generation of downstream oscillations is the same. Relaxation is faster for lower angles θ between the shock normal and the upstream magnetic field .

With the increase in the magnetic compression, the CGP rapidly increases. At R=2 this critical value is scr=0.75. Although such high cross-shock potentials cannot be completely excluded, they are not observed often . Thus, we expect that in most shocks the potential is below the CGP. Yet, in many shocks the first magnetic peak occurs right at the downstream edge of the ramp. In many cases, it is also the largest peak. The above analysis is only valid, strictly speaking, for sufficiently low-β=8πnuTu/Bu2 shocks, as the number of quasi-reflected and/or reflected ions rapidly increases with the increase in vT/Vu, where vT=Tu/m is the upstream thermal speed of ions . In the narrow shock approximation, all ions that initially have mvx2/2<eϕ cannot cross the ramp. This mode of reflection is efficient when 1-s∼vT/Vu. Deceleration of quasi-reflected ions inside the ramp can be expected to result in the faster reduction of the ion pressure with the distance from the upstream edge of the ramp – that is, a steeper increase in the magnetic field.

Ion reflection occurs in supercritical and marginally critical shocks. Ion reflection is a kinetic process and the fate of an ion entering a shock front depends on the initial velocity of the ion. There are two major modes of ion reflection: post-ramp and in-ramp reflection. Post-ramp reflection occurs when an ion crosses the ramp, gyrates behind it, and returns back to the ramp to cross it in the upstream direction, but turns around again inside the ramp moving in the downstream direction. In-ramp reflection occurs when an ion changes its direction of motion inside the ramp and starts moving in the upstream direction. In both modes reflection occurs due to the combined effects of the electric and magnetic forces. As the transition from upstream to ramp and further downstream is continuous, there is no strict separation between the two modes. Efficiency of the post-ramp reflection increases most strongly with the increase in the magnetic compression Bd/Bu. It also increases with the increase of the ratio vT/Vu=βi/2/M and with the decrease in the cross-shock potential s . The inverse dependence on the cross-shock potential is related to the fact that the chances of a downstream gyrating ion returning to the ramp are higher if the gyration speed is higher, whereas the cross-shock potential takes energy from an ion upon crossing the ramp. The efficiency of in-ramp reflection increases with the increase in the vT/Vu ratio and the cross-shock potential s . It can be most simply explained in the approximation of specular reflection, which ignores magnetic deflection. A particles with initial vx is reflected within the ramp if mivx2/2<qϕ. For an initial Maxwellian distribution, 5 % of incident ions are reflected if mi(Vu-2vT)2/2=qϕ which allows us to define the critical reflection potential (CRP) s5%=(1-vT/Vu)2. In this approximation, in-ramp reflection does not depend on the magnetic compression nor the shock angle and is stronger for lower Mach numbers for given βi and s. In reality, magnetic deflection enhances the reflection, which is never specular. In what follows, we distinguish between reflected and quasi-reflected ions. Figure  illustrates the difference between ion populations and the terminology proposed by .

The first turning point is the first point along the ion trajectory where the sign of vx changes from positive (in the downstream direction) to negative (in the upstream direction). The second turning point is the first point at the ion trajectory where vx changes its sign from negative to positive. A directly transmitted ion may have no turning points at all and may have turning points behind the ramp. Figure a shows the trajectory of a directly transmitted ion which does have turning points. In this case, the second turning point, marked using a blue arrow, occurs behind the ramp. The ion trajectory shown in Fig. 4b belongs to a quasi-reflected ion. In this case, the second turning point occurs inside the ramp. For a reflected ion (Fig. 4c), the second turning point is in the upstream region ahead of the ramp. Figure 4d shows, for completeness, a trajectory of a backstreaming ion which has several turning points in the ramp vicinity and eventually escapes in the upstream direction. Quasi-reflected and reflected ions have similar energies and similar gyrating distributions. The difference is that quasi-reflected ions do not appear in the upstream region and do not contribute to foot formation. Each reflected or quasi-reflected ion makes a loop and moves along the shock front. As a result, all of these ions acquire energy in NIF; therefore, they should be clearly distinguished from the directly transmitted ions inside the ramp and behind it, both in a distribution plot and in a spectrogram. In both cases there should be a noticeable gap between the two.

In low-βi and small Bd/Bu shocks, both modes of reflection should be suppressed. In high Mach number shocks, Bd/Bu is large, whereas vT/Vu=βi/2/M is small unless βi is large. In such shocks, post-ramp reflection should dominate. In marginally critical and weakly supercritical shocks, in-ramp reflection should dominate unless βi is too small. One can expect that in-ramp reflection would cause a sharper drop of the ion pressure and, therefore, a steeper increase in the magnetic field. A more detailed analysis can be carried out numerically where the cross-shock potential s and ion βi are fully controlled.

Figure  shows the results of the test-particle adjustment for a shock with βi=0.2 and magnetic compression R=1.85. The adjustment of the downstream magnetic field predicted by the test-particle analysis to the initial model field is achieved with the cross-shock potential s=0.65, which is below the corresponding CGP scr=0.7 but above the corresponding CRP s5%=0.49. Figure 5a shows a steeper increase toward the overshoot with the first peak exceeding the subsequent peaks. The same panel also shows the ion orbits and Fig. 5b shows a slice of the ion distribution which covers a half of the ramp adjacent to the upstream. Both clearly display the presence of a non-gyrotropic distribution of quasi-reflected ions. The incident and quasi-reflected populations are clearly separated in the velocity space and in energies.

Figure  shows the results of the test-particle adjustment for the same compression ratio and cross-shock potential but lower βi=0.05.

In this case there are very few quasi-reflected ions and the shock profile follows the low-β prescription shown in Fig. a. The magnetic field increase toward the overshoot is less steep and the first peak is shifted further downstream.

Figure  shows the results of the test-particle adjustment for the same compression ratio and βi=0.2 but lower cross-shock potential s=0.4.

In this case there are also very few quasi-reflected ions and the shock profile follows the low-β prescription shown in Fig. a. The magnetic field increase toward the overshoot is less steep and the first peak is shifted further downstream.

Figure  shows the results of the test-particle adjustment for the same compression ratio and βi=0.4 but lower cross-shock potential s=0.4.

This value is slightly above the value of s5%, so that the number of reflected ions is noticeable. Yet, the first maximum is shifted to downstream and the magnetic field increase toward the overshoot is not steep.

A detailed example of a pair of very low-Mach number shocks with the magnetic compression of Bd/Bu≈1.2 and βi≈0.08 is given by , Fig. 4, where the cross-shock potentials are also calculated from observations and shown to agree well with the theoretical findings above. Namely, the shock with a lower potential has the first peak higher than the successive peaks, whereas the shock with a higher potential has the second peak higher than the first peak.

Figure  shows the magnetic profile of a of a subcritical shock observed by THEMIS-B plotted over the ion spectrogram. The shock crossing occurred at 22 January 2012, 06:01:47. The estimated shock parameters are similar to those of Fig. : Bd/Bu=1.85, θ=65∘, βi≈0.14, and M=2.6. The corresponding CGP is scr≈0.71 and the CRP is s5%≈0.64. The spectrogram shows that a number of ions are quasi-reflected at the ramp. This is seen as a gap in the ion distribution inside the ramp. This gap cannot be seen using the standard “tplot” procedure of SPEDAS, as the resolution is low. The IDL function “contour” carries out an interpolation, similar to what is undertaken when calculating distribution functions from a discrete set of measurements in a number of energy channels and angle detectors. With this interpolation, the gap becomes visible. Such quasi-reflection requires a sufficiently high cross-shock potential, capable of stopping slow ions inside the ramp (marked using a red arrow in Fig. 9). The first peak follows a steep magnetic field increase and is the largest. Thus, we expect that s5%<s<scr, which is in a good agreement with the adjusted value of s=0.65.

Figure  shows the corresponding gap for the analyzed shock in Fig. . The spectrogram is made in the reference frame (“spacecraft”) moving with the velocity 1.5Vu along the shock normal. Figure  shows a similar gap in 28 November 2011 THEMIS-C-measured shock spectrogram in which reflected ions are detected. It is not possible to directly compare the gap for the analyzed shock with observations, as the analysis is carried out in the normal incidence frame whereas the observed spectrograms are produced in the spacecraft frame.

Figure  shows the magnetic profile of a THEMIS-C-observed shock. This shock is also subcritical. It has a lower magnetic compression R=1.4 with a slightly higher βi≈0.2. The angle is large θ=86∘, whereas the Mach number is lower M≈1.65. The corresponding CGP is scr≈0.5 and the CRP is s5%≈0.38. The absence of ions reflected inside the ramp indicates insufficient potential; thus, we expect that s<0.38. Adjustment using the advanced test-particle analysis results in s≈0.35.

Magnetic field measurements at heliospheric shocks are by far the best quality measurements with regard to both precision and resolution. The resolution of particle measurements is much worse: their precision is limited by geometric factors and the finite number of detectors. Measurements of electric field are typically the most difficult. Therefore, any cross-check of less reliable measurements on the basis of better measurements is important. This is particular true if measurements of the magnetic field would enable us to fill gaps in particle and cross-shock measurements that would substantially improve our ability to compare observations and theory.

In the present paper, we examine the implications of the shape of the downstream magnetic oscillation trail for the cross-shock potential. It appears that certain limitations can be placed on the potential using knowledge of the Mach number, magnetic compression, βi, and first peaks of the downstream magnetic field. The two critical kinetic phenomena are the gyration of the center of the incident distribution upon crossing the shock and the onset of ion reflection within the ramp. These two features are related to the two critical values of the cross-shock potential that have been defined in the simplified case of a narrow perpendicular shock. The derived CGP ccr=1-(Bu/Bd)2 and the CRP c5%=(1-2vT/Vu)2 are approximations which do not properly take the ramp width and the shock angle into account. Yet, they provide certain limits on possible cross-shock potentials consistent with the measured Mach number, βi, and magnetic compression. Numerical test-particle analyses have shown that these limits are in good agreement with the parameters obtained by adjustment of the predicted profile to the required downstream asymptotic value.

It is found that for scr<s<s5% the first downstream peak is at the downstream edge of the ramp and is weaker than the second peak. For s<scr<s5% and for s<s5%<scr, the first downstream peak is shifted farther downstream and it is the strongest. For s5%<s<scr, reflected ions are seen, the rise toward the overshoot is substantially steeper, and the first downstream peak is at the downstream edge of the ramp and is the strongest. Thus, observations of the downstream magnetic oscillations may be used to place restrictions on the cross-shock potential. At this stage, the analysis is limited to subcritical, marginally critical, and weakly supercritical shocks. Higher supercriticality will require separate study, also including post-ramp reflected ions.