Overshoot dependence on the cross-shock potential

. Coherent downstream oscillations of the magnetic ﬁeld in shocks are produced due to the coherent ion gyration and quasi-periodic variations of the ion pressure. The amplitude and the positions of the pressure maxima and minima depend on the cross-shock potential and upstream ion temperature. Two critical cross-shock potentials are deﬁned: the critical gyration potential (CGP) which separates the cases of increase or decrease of the component of the velocity of the distribution center along the shock normal, and the critical reﬂection potential (CRP) above which ion reﬂection becomes signiﬁcant. In weak very 5 low upstream kinetic-to-magnetic pressure ratio, β , shocks CRP exceeds CGP. For potentials below CGP the ﬁrst downstream maximum of the magnetic ﬁeld is shifted farther downstream and is larger than the second one. For higher potentials the ﬁrst maximum occurs just behind the ramp and is lower than the second one. With the increase of the upstream temperature CGP exceeds the CRP. For potentials below CRP the effects of ion reﬂection are negligible and the shock proﬁle is similar to that of very low β shocks. If the potential exceeds CRP ion reﬂection is signiﬁcant, the magnetic ﬁeld increase toward the overshoot 10 becomes steeper, and the largest peak occurs at the downstream edge of the ramp.

understanding of the shock structure substantially improved due to these high quality observations and also due to numerical simulations. The frontier of the observational shock studies has shifted recently towards the processes occurring within few ion convective gyroradii in both directions from the ramp along the shock normal (Dimmock et al., 2012;Wilson et al., 2012Wilson et al., , 2014Johlander et al., 2016;Burgess et al., 2016;Eselevich et al., 2017;Wilson III et al., 2017;Gingell et al., 2017).
Magnetic profiles of collisionless shocks are rarely monotonic, even for low-Mach numbers (Greenstadt et al., 1975;Green-5 stadt et al., 1980;Russell et al., 1982a;Mellott and Greenstadt, 1984;Farris et al., 1993;Balikhin et al., 2008;Russell et al., 2009;Kajdič et al., 2012). Since the peak value of the downstream oscillations increases with the increase of the Mach number, for a long time is was believed that overshoots are produced by ion reflection in super-critical shocks (Livesey et al., 1982;Russell et al., 1982b;Sckopke et al., 1983;Scudder et al., 1986;Mellott and Livesey, 1987). Super-critical shocks are the shocks with the Mach number exceeding the critical Mach number (Edmiston and Kennel, 1984;Kennel, 1987), so that 10 resistivity (Edmiston and Kennel, 1984) and thermal conductivity (Kennel, 1987) 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 B d /B u = 1.3. The oscillating trail behind the ramp exhibited all features expected for a supercritical shocks, like the largest first peak, spatially periodical peaks, and gradual decrease of the peak amplitude. Such oscillations, albeit often less ordered, were found to be common in low-Mach number 15 shocks Kajdič et al., 2012). They were successfully explained as a result of coherent ion gyration upon crossing the shock ramp and subsequent collisionless relaxation due to gyrophase mixing Ofman et al., 2009;Ofman and Gedalin, 2013;Gedalin, 2015;Gedalin et al., 2015Gedalin et al., , 2018. It has been shown that the largest peak amplitude is determined mainly by the magnetic compression and cross-shock potential, while the damping rate of the oscillations is related to the upstream thermal-to-fluid speed ratio (Gedalin, 2015). Shapes of the downstream profile, like relative peaks of 20 the first oscillations and steepness of the magnetic field increase up to the first peak, vary considerably among observed shocks, even subcritical ones. Sufficient attention has not been devoted so far 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. 25

Weak low-β shocks
In what follows B u is the upstream magnetic field magnitude, T u is the upstream ion temperature, n u is the upstream ion number density, v T = T u /m is the upstream ion thermal speed, m is the ion mass, and β = 8πn u T u /B 2 u 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 30 shock. We treat the shock as a jump in the magnetic field from B u to B d = RB u occurring within a narrow ramp. Accordingly, the fluid drift speeds upstream and downstream are V u and V d = V u /R. We shall also neglect the electron contribution in the plasma pressure and treat ions as a monoenergetic beam entering the shock with the velocity V u along the shock normal. The analysis is done in the normal incidence frame, where x is along the shock normal (toward downstream) and z is along the magnetic field. The equations of motion for ions inside the ramp arė We integrate the equations of motion across the ramp assuming |v y | ∼ v T V u , where v T is the thermal speed of upstream 5 ions. In this approximation we get Here u denotes the ion velocity at the downstream edge of the ramp while v(x) denotes the ion velocity at the position x inside the ramp. The second term in (3) is a small correction for ramp width (c/ω pi ) and v T /V u 1. Here (c/ω pi ) is the ion 10 inertial length. This small correction can be neglected for our purposes. In (4) the only term is small but nonzero. Thus, if the cross-shock potential is φ = s(mV 2 u /2e), the ion velocity just after crossing the jump is The ion motion is then described as a drift along the shock normal with the velocity V u /R and gyration around the magnetic field: where Ω d = eB d /m i c is the downstream ion gyrofrequency. For a cold beam all ions move together and the coordinate along the shock normal is given by 20 In general, it is not possible to derive an analytical expression for v x (x). For our purposes it is sufficient to restrict ourselves is a single-valued function. Let us define the critical gyration potential (CGP) s cr = 1 − 1/R 2 . For s < s cr the initial gyrophase ϕ ≈ 0, so that dv x /dx < 0 at the downstream edge of the ramp. For s > s cr the initial gyrophase is ϕ ≈ π, so that dv x /dx > 0 at the downstream edge of the 25 ramp.
The total (dynamic and kinetic) ion pressure is given by where we have used the mass conservation nv x = n u V u . Pressure balance requires p i,xx +B 2 /8π = const, so that the magnetic field has maxima at the minima of the ion pressure. The latter occur at the minima of v x . For s < s cr the velocity decreases inside the ramp and keeps decreasing down to v x,min = V d − v ⊥ at Ω d t + ϕ = π which approximately corresponds to x l = πV d /Ω d for ϕ ≈ 0. Thus, the first maximum of the magnetic field occurs at x l at the pressure With the increase of s the relative contribution of u y in v ⊥ increases which moves the position of the first pressure minimum closer 5 to the ramp. For s > s cr 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 one has p h > p l which means that the first peak will be lower than the subsequent ones corresponding to the pressure minima For a cold ion beam the amplitude of further pressure oscillations does not change. Finite temperature leads to the divergence 10 of the ion trajectories and gradual gyrophase mixing. The divergence occurs already at the shock crossing since the downstream x,u − 2eφ/m, and the spread in v x,u results in a more substantial spread in v x,d . Moreover, there is nonzero v y which affects the gyration speed v ⊥ and ϕ, which are now different for different particles: The downstream ion pressure including finite temperature is obtained as an integral over the distribution It has been shown (Gedalin et al., 2015;Gedalin, 2016a) that finite temperature results in the collisionless relaxation during which the downstream ion distribution gyrotropizes and the pressure oscillations damp out. The relaxation is faster for larger v T /V u . In oblique shocks the mechanism of the generation of downstream oscillations is the same. Relaxation is faster for 20 lower angles θ between the shock normal and the upstream magnetic field (Gedalin, 2015;Gedalin et al., 2015).
With the increase of the magnetic compression CGP rapidly increases. At R = 2 this critical value is s cr = 0.75. Although such high cross-shock potentials cannot be completely excluded, they are not observed often (Dimmock et al., 2012). Thus, we expect that in most shocks the potential is below 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 valid, strictly speaking, only 25 for sufficiently low-β = 8πn u T u /B 2 u shocks since the number of quasi-reflected and/or reflected ions rapidly increases with the increase of v T /V u , where v T = T u /m is the upstream thermal speed of ions (Gedalin, 2016b). In the narrow shock approximation all ions having initially mv 2 x /2 < eφ cannot cross the ramp. This mode of reflection is efficient when 1 − Deceleration of quasi-reflected ions inside the ramp can be expected to result in faster reduction of the ion pressure with the distance from the upstream edge of the ramp, that is, steeper increase of the magnetic field.

Advanced test particle analysis vs observations
The principles of the advanced test particle analysis have been described in detail by Gedalin and Dröge (2013). In brief, a model magnetic field profile is chosen, supplemented with a model electric field shape. The basic upstream plasma parameters, that is, ion and electron β and the angle between the shock normal and the upstream magnetic field θ are chosen and remain fixed during the analysis. Choosing a magnetic compression ratio R, the rest of the significant parameters are varied. With 5 each set of the parameters ions are numerically traced across the shock, the ion pressure is determined, and the corresponding magnetic field is derived from the pressure balance. The parameters are varied until reasonable agreement is achieved with the adopted model profile: the asymptotic values of the magnetic field should be equal and the fluctuations as small as possible. It has been found that the most influential parameters are the Alfvenic Mach number M and the normalized cross-shock potential s . There is also weak dependence on the shock width D. The magnetic profile chosen for the analysis is taken in the following 10 form: with B x = B u cos θ, B y ∝ dB z /dx, and E x ∝ dB z /dx. The coefficients of proportionality are constrained by the chosen values of the normal incidence frame cross-shock potential s N IF and the de Hoffman-Teller potential s HT (Goodrich and Scudder, 1984;Scudder et al., 1986;Schwartz et al., 1988). The latter was found to almost not affect the ion motion and was 15 kept s HT = 0.1 in the subsequent analysis. The post-tracing magnetic field was derived from the condition where the ion pressure was determined numerically and for the electron pressure the polytropic equation of state p e /n 5/3 was used, together with the quasineutrality. convenience. The coordinate is measured in r g = V u /Ω u . It is clearly seen that for the low potential the first peak is shifted farther downstream from the ramp and its amplitude is higher than that of the second peak. In the case of the higher potential the 25 first peak occurs at the downstream edge of the ramp and its amplitude is lower than that of the second one. Figure 2 illustrates the difference in the behavior of the normal component of the ion velocity, v x , in both cases. In the low potential case this component continues to decrease well beyond the ramp. Subsequent dips become more and more shallow with the distance from the ramp. In the high potential case v x starts to increase upon crossing the ramp. The second dip is deeper because lower v x are achieved, as explained above.  With the increase of the magnetic compression CGP rapidly increases. For B d /B u = 2 CGP is rather high: c cr = 0.75. In most shocks the cross-shock potential is expected to be below this value (Dimmock et al., 2012). In low-β i plasmas all ions are directly transmitted across the shock without reflection and the above findings can be summarized as follows: a) below CGP 5 the first peak is the strongest, b) with the increase of the potential toward CGP the first peak moves closer to the ramp, c) upon crossing CGP the first peak stands at the downstream edge of the ramp and is no longer the strongest.

Effects of ion reflection
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 10 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 toward upstream, but turns around again inside the ramp moving toward downstream. In-ramp reflection occurs when an ion changes its direction of motion inside the ramp and starts moving toward upstream. In both modes reflection occurs due to the combined effects of the electric and magnetic forces. Since 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 15 most strongly with the increase of the magnetic compression B d /B u . It also increases with the increase of the ratio v T /V u = β i /2/M and with the decrease of the cross-shock potential s (Gedalin, 1996). The inverse dependence on the cross-shock potential is related to the fact that chances of a downstream gyrating ion to return to the ramp are higher if the gyration speed is higher, while the cross-shock potential takes energy from an ion upon crossing the ramp. Efficiency of in-ramp reflection increases with the increase of the ratio v T /V u and the cross-shock potential s Gedalin, 2016b). 20 It can be most simply explained in the approximation of specular reflection which ignores magnetic deflection. A particles THEMIS-c i 2011-03-30/08:09:40±360s  with initial v x is reflected within the ramp if m i v 2 x /2 < qφ. For an initial Maxwellian distribution, 5% of incident ions are reflected if m i (V u − 2v T ) 2 /2 = qφ which allows us to define the critical reflection potential (CRP) s 5% = (1 − v T /V u ) 2 . In this approximation in-ramp reflection does not depend either on the magnetic compression or shock angle and is stronger for lower Mach numbers for given β i 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 4 illustrates the difference between ion populations and the terminology proposed by Gedalin (2016b) 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 these ions acquire energy in NIF so that they should be clearly distinguished from the directly transmitted ions inside the ramp and behind it, both in a distribution plot or in a spectrogram. In both cases there should be a noticeable gap between the two.

15
In low-β i and small B d /B u both modes of reflection should be suppressed. In high Mach number shocks B d /B u is large while v T /V u = β i /2/M is small unless β i is large. In such shocks post-ramp reflection should dominate. In marginally critical and weakly supercritical shocks in-ramp reflection should dominate unless β i is too small. One can expect that in-ramp reflection would cause a sharper drop of the ion pressure and therefore a steeper increase of the magnetic field. A more detailed analysis can be done numerically where the cross-shock potential s and ion β i are fully controlled. Figure 5 shows the results of the test-particle adjustment for a shock with β i = 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 s cr = 0.7 but above the corresponding CRP s 5% = 0.49. The profile (left panel) shows a steeper increase toward the overshoot with the first peak exceeding the subsequent peaks. The same panel shows the ion orbits and the right panel shows a slice of the ion distribution which covers a half of the ramp adjacent to the upstream. Both clearly show 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 6 shows the results of the test-particle adjustment for the same compression ratio and cross-shock potential but lower 10 β i = 0.05. In this case there are very few quasi-reflected ions and the shock profile follows the low-β prescription shown in the top panel of Figure 2. The magnetic field increase toward the overshoot is less steep and the first peak is shifted further downstream. Figure 7 shows the results of the test particle adjustment for the same compression ratio and β i = 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 15 shown in the top panel of Figure 2. The magnetic field increase toward the overshoot is less steep and the first peak is shifted further downstream.   Figure 8 shows the results of the test particle adjustment for the same compression ratio and β i = 0.4 but lower cross-shock potential s = 0.4. This value is slightly above the value of s 5% , 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.

Observations
A detailed example of a pair of very low-Mach number shocks with the magnetic compression of B d /B u ≈ 1.2 and β i ≈ 0.08 5 is given by Pope et al. (2019), Figure 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 ones, while the shock with a higher potential has the second peak higher than the first one. The first peak follows a steep magnetic field increase and is the largest. Thus, we expect that s 5% < s < s cr , which is in a good agreement with the adjusted value of s = 0.65. Figure 10 shows the corresponding gap for the analyzed shock in Figure 8. The spectrogram is made in the reference frame ("spacecraft") moving with the velocity 1.5V u along the shock normal. Figure 11 shows a similar gap in 2011/11/28 THEMIS C measured shock spectrogram in which reflected ions are detected. It is not possible to compare directly the gap for the 5 analyzed shock with observations since the analysis is done in the normal incidence frame while the observed spectrograms are produced in the spacecraft frame. Figure 12 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 β i ≈ 0.2. The angle is large θ = 86 • while the Mach number is lower M ≈ 1.65.
The corresponding CGP is s cr ≈ 0.5 and CRP is s 5% ≈ 0.38. The absence of ions reflected inside the ramp indicates insufficient 10 potential, so that we expect that s < 0.38. Adjustment using the advanced test particle analysis results in s ≈ 0.35.

Discussion and conclusions
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 ones. Therefore, any cross-check of less reliable measurements on the basis of better ones is important. In particular, 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-5 shock potential. It appears that certain limitations can be placed on the potential using knowledge of the Mach number, magnetic compression, β i , 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 c cr = 1 − (B u /B d ) 2 and CRP c 5% = (1 − 2v T /V u ) 2 are approximations which do 10 not take into account properly the ramp width and the shock angle. Yet, they provide certain limits on possible cross-shock potentials consistent with the measured Mach number, β i , 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 s cr < s < s 5% the first downstream peak is at the downstream edge of the ramp and is weaker than the 15 second one. For s < s cr < s 5% and for s < s 5% < s cr the first downstream peak is shifted farther downstream and it is the strongest. For s 5% < s < s cr reflected ions are seen, the rise toward the overshoot is substantially steeper, 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, marginallycritical and weakly supercritical shocks. Higher super-criticality will require separate study, including also post-ramp reflected ions.