A collisionless shock transfer of mass, momentum, and energy occurs from upstream to downstream. Most of the momentum and energy fluxes are carried by ions so the shock structure is affected mainly by ions. With the increase in the Mach number, the fraction of reflected ions increases and their influence on the shock structure becomes progressively more important. Here, we study the effect of the reflected ions on the overshoot strength. It is shown that directly transmitted ions are responsible for the overshoot formation and the interaction of the overshoot field with these ions alone might result in an unstable growth of the overshoot. On the contrary, reflected ions, at their second crossing of the shock, are accelerated along the shock normal and, thus, provide a stabilizing effect on the overshoot.

## I. INTRODUCTION

^{1–6}In these shocks, ion heating is due to the downstream gyration of the directly transmitted ions.

^{7,8}In collisionless plasma systems, heating is not related to irreversible processes. Let $ f ( t , r , v )$ be the distribution function of some species. The number density $n$, the hydrodynamic velocity $V$, the total pressure tensor $ p i j$, the kinetic pressure tensor $ P i j$, and the temperature $T$ are given by the following expressions:

^{9,10}

^{11}

Directly transmitted ions are those ions which cross the shock from upstream to downstream only once. Some incident ions may be unable to overcome the cross-shock potential^{12–16} and are returned to the upstream region. These are reflected ions.^{17–20} Subsequent behavior of the reflected ions depends on the shock angle, $ \theta B n$, that is, the angle between the shock normal and the upstream magnetic field vector. In quasi-perpendicular shocks, $ \theta B n > 45 \xb0$, the vast majority of the reflected ions gyrate in the upstream region come back to the shock and cross it again, thus forming a population of the reflected–transmitted ions. Only a tiny fraction of the reflected ions and only for $ \theta B n \u2009 \u2272 \u2009 50 \xb0$ do not return to the shock but escape further upstream, thus forming a population of backstreaming ions.^{21} In quasi-parallel shocks, $ \theta B n < 45 \xb0$, the number of backstreaming ions increases at the expense of reflected–transmitted ions.^{22,23} In the present study, we limit ourselves with shocks where backstreaming ions are not important.

In low Mach number shocks, there are no or almost no reflected ions. At higher Mach numbers, ion reflection becomes important^{18,24,25} and reflected–transmitted ions contribute significantly to ion heating.^{19,20} Large magnetic overshoots are present in these shocks.^{6,26–30} It was shown that ion reflection is stronger with larger overshoots.^{31} The influence of the reflected ions on the overshoot is not well understood. In analogy with the foot formation,^{24} it was assumed that overshoots appear due to the reflected ions.^{29,32} However, observations show that overshoots are present even in very low-Mach number shocks where reflected ions are absent.^{5,33,34} The overshoot is due to the fact that ion heating lags behind the deceleration of the bulk flow, so that the total, dynamic, and kinetic ion pressure drops at the transition to the level below the eventual downstream pressure, thus causing surplus of the magnetic pressure.^{8,35} Based on this principle, an approximate relation between the overshoot magnitude and the Alfvénic Mach number was proposed and used for observational analysis.^{36} Following the same reasoning as above for directly transmitted ions, we can expect that the effect of reflected–transmitted ions on the overshoot depends on whether their total pressure decreases or increases upon crossing the shock toward downstream.

## II. ION DISTRIBUTION AND OVERSHOOT: AN OBSERVED SHOCK AND GENERAL REMARKS

In what follows, we shall treat collisionless shock as a planar stationary structure in which all fields depend on the co-ordinate along the shock normal only. Low Mach number shocks are expected to be planar and stationary. It is expected that high Mach number shocks become non-planar and/or time-dependent.^{37,38} Moderate deviations from a planar shape and/or time dependence would somewhat modify the below consideration and the conclusions but would not change them drastically.^{39–42} Moreover, even rather high Mach number shocks may be planar and stationary.^{10} Rippled shocks are not planar and are time-dependent. However, the spatial scale of inhomogeneity along the shock front is substantially larger than the shock width and the temporal scale of non-stationarity is substantially larger than the time during which ions cross the shock.^{50,51} Therefore, in such shocks, Eq, (6) should be valid at least across the shock transition layer, from the beginning of the ramp to the maximum of the overshoot.^{39–42}

In order to define the framework of the present study, let the shock normal be along $x$. In a planar and stationary shock, the magnetic and electric field depend then only on $x$, and $ B x = B u \u2009 cos \u2009 \theta B n = const$. Here, subscript $u$ refers to the upstream region, $ B u$ is the upstream magnetic field magnitude, $ \theta B n$ is the angle between the shock normal, $ x \u0302$, and the upstream magnetic field vector $ B u = B u ( cos \u2009 \theta B n , 0 , \u2009 sin \u2009 \theta B n )$. The normal incidence frame (NIF) is the reference frame in which the shock does not move and the upstream flow $ ( V u , 0 , 0 )$ is along the shock normal. By the coplanarity theorem, the magnetic fields in the uniform upstream and downstream regions lie in the same plane, which will be the $ x \u2212 z$ plane in the forthcoming consideration. In NIF, $ E y = V u B u \u2009 sin \u2009 \theta B n / c$ and $ E z = 0$. The de Hoffman-Teller frame (HT) is the shock frame in which the upstream flow is along the upstream magnetic field and $ E y = 0$. HT is moving relative to NIF with the velocity $ \u2212 V u \u2009 tan \u2009 \theta B n$ in $z$-direction. The cross-shock potentials in the two frames are different, and the difference is related to the non-coplanar magnetic field inside the shock transition.^{43,44}

^{8,45}The strong decrease in the total pressure should be balanced by a strong increase in the magnetic pressure, so an overshoot arises. This can be seen in Fig. 1, which presents the reduced distribution function $ f ( x , v x )$ in NIF for the shock crossing measured by MMS1 on December 27, 2018 at 23:53:26 UTC. The shock is listed in the database of MMS shocks

^{46}as having the Alfvénic Mach number $ M = 6.7$ and the shock angle $ \theta B n = 60 \xb0$. The magnetic field and the ion distributions are obtained from the data in the burst mode,

^{47,48}which has the highest available temporal resolution. Fluctuations are not removed from the magnetic field, so that the ramp and the overshoot maximum are determined only approximately. The distribution is normalized on its maximum value and is shown on a linear scale. The velocity is transformed to NIF and normalized on the upstream plasma speed. The thin black line shows the magnetic field magnitude normalized on the upstream value. The three populations of ions (directly transmitted, reflected, and reflected–transmitted) are easily identified visually. It can be seen that the maximum velocity as a function of time decreases steadily until approximately the position of the overshoot maximum. Unfortunately, MMS does not measure properly the cold solar wind. The measured upstream electron density more than twice exceeds the measured ion density. The density compression calculated from ions, $ n i d / n i u \u2248 7.4$, exceeds the maximum allowed compression of four, while for electrons it is reasonable, $ n e d / n e u \u2248 3.3$. Therefore, any calculation of the ion pressure would not be reliable. For a long time it was thought that the overshoot is produced due to the reflected ions when they cross the shock again (reflected–transmitted ions).

^{27,29}If this were the case the pressure of these ions should have dropped upon crossing the shock. However, the figure shows that their velocities $ v x$ increase. Since the particle flux should be constant, the contribution to $ p x x$ should increase with $ v x$

^{49}(see details below in Sec. III).

## III. NON-SPECULAR REFLECTION AND TOTAL PRESSURE

^{52}This principle is in the basis of all analyses of ion reflection so far. The equations of the ion motion in the macroscopic electric and magnetic fields of the shock read as

^{4,10,44,53,54}Note that the pressure balance in the dimensionless form reads $ p x x + B 2 / 2 M 2 = const$.

If there were no magnetic field, any ion having initial velocity $ < s NIF$ would not be able to overcome the cross-shock potential and would be reflected. For a typical $ s NIF = 0.5$, the electrostatic reflection alone would reflect ions with $ v i n < 0.7$. For the above-mentioned $\beta $ and $M$, this means that only ions with $ v i n < 1 \u2212 3 v T$ are reflected. The magnetic force $ v y B z$ causes additional decrease in $ v x$ if $ v y < 0$. Thus, more ions are reflected. The magnetic contribution to the reflection is stronger for larger magnetic fields, that is, a stronger overshoot would produce more reflected ions.

## IV. NUMERICAL ANALYSIS AND VISUALIZATION

^{56}and successfully tested with observations.

^{55}The relation of $ E x$ to $ B z$ follows approximately the relation obtained from the Ohm's law with massless electrons.

^{43,57}A more sophisticated profile could include also an undershoot. Such a profile was used to perform adjustment to the shock parameters so that the pressure balance is satisfied.

^{31}Here, our objective is quite different and the part of the profile beyond the overshoot is not important. The coefficients $ k E$ and $ k B$ are determined from the conditions

^{43,44,57}

In what follows, the shock angle $ \theta = 60 \xb0$ and the ramp width $ D = 2 / M$. For the ion tracing, we have chosen: $ M = 6$, $ s NIF = 0.4$, $ s HT = 0.1$, $ B d = 3.0$, $ R add = 1$, and $ \beta = 0.5$. The parameter $R$ is obtained from (29). For the chosen $ R add$, the maximum magnetic field in the overshoot $ B m = 4.06$. The other parameters are $ W L = 0.8 D$, $ W R = 2.4 D$, and $ x L = x R = 0.4 D$. The overshoot maximum is at $ x m = 0.194$, which is near $ x L$ but not exactly there. The overshoot shape is asymmetric, with a steeper side adjacent to the ramp. The width of each side is roughly given by $ W L$ and $ W R$. Note that the normalized NIF potential at the overshoot maximum is $ s max = s NIF B m = 0.54$. We traced 320 000 ions using the Boris algorithm for solving the relativistic equations of motion for charged particles in electromagnetic fields.^{58} The particles start at the same initial position, sufficiently far upstream of the ramp, in the uniform region. The time step of $ \Delta t = 0.001$ is quite sufficient for statistics and ensures that the thinnest part, the ramp, is crossed in $ \u223c 10 2$ steps. The algorithm converges and conserves energy even for larger time steps. Slightly more than 11 000 (about 3.5%), ions were reflected. Figure 2 shows the turning points of the reflected ions. Each blue point is the first point along an ion trajectory in which $ v x$ changes sign from positive to negative. Each red point is the first later point where $ v x$ changes sign from negative to positive and the ion position is ahead of the ramp. The ramp is between the two vertical green lines on the plot. It is the thinnest part of the shock with the steepest increase in the magnetic field. All reflected ions cross the ramp and are reflected within the overshoot (the region where the magnetic field exceeds the downstream value). The overshoot maximum is marked by the vertical magenta line in the figure. The distribution of the turning points is shown in more detail in Fig. 3.

Figure 4 shows three components of the velocity as functions of $x$ for a number of reflected particles. The left panel shows $ v x$. The middle panel shows that $ v y$ is negative from the entry to the shock and up to the reflection point, that is, the magnetic force $ v y B z$ place a significant role in the ion deceleration. It becomes more significant as the ion proceeds further into the overshoot.^{59} On the second entry to the ramp $ v y$ is positive (middle panel) and $ v z$ is negative (right panel). Inside the ramp both do not change noticeably. Beyond the ramp $ v y$ begins to change due to the gyration. The component $ v x$ (left panel) increases from the start of the ramp until the maximum of the overshoot.

^{49}The reduced distribution function is the sum of these weights for all particles and all appearances in the cell.

Figure 5 presents the reduced distribution function $ f ( x , v x )$ for all ions. The scale is linear. The directly transmitted, reflected, and reflected–transmitted ions are clearly distinguished visually, except in two regions where they overlap. The region well beyond the overshoot maximum is not relevant for our study. Figure 6 shows the total pressure $ p x x ( x ) = \u222b v x 2 f ( x , v x ) d v x$ calculated for all particles together. The total pressure decreases in the region between the beginning of the ramp and the overshoot maximum and starts to increase almost immediately after the overshoot maximum. Further adjustment of the shock parameter is possible to bring the minimum of the pressure exactly to the maximum of the overshoot. We are not doing it here. The proximity of the two for our choice of parameters shows that the chosen model is quite satisfactory.

Figure 7 presents the reduced distribution function $ f ( x , v x )$ for the reflected and reflected–transmitted ions. It is easy to visually follow the local maxima of the distribution function at each $x$. The reflected–transmitted population is shown separately in Fig. 8. These are the ions which re-enter the ramp, after gyrating in the upstream region, and proceed further across the shock. The velocities $ v x$ of these ions steadily increase when they are crossing the ramp and until the overshoot maximum. Figure 9 shows the reduced distribution function $ f ( x , v y )$ for the reflected–transmitted ions only, from the point where they turn toward the shock (upstream of the ramp) to the overshoot maximum and slightly beyond. The velocity $ v y > 1$ in this region. The total pressure of the reflected–transmitted ions is shown in Fig. 10. It steadily increases from the beginning of the ramp to the overshoot maximum.

## V. CONCLUSIONS

Ions which enter the shock are affected by the electrostatic force and the magnetic force. The incident ions of the solar wind are decelerated by the cross-shock electric field directed toward upstream and due to the bending of the ion path by the increasing magnetic field of the ramp and overshoot. The larger is the magnetic field increase the larger is the magnetic contribution to the incident ion deceleration.^{31} The ions which cross the shock (the directly transmitted ions) have lower velocities $ v x$ along the shock normal than they had when entered the shock. Some ions even are stopped inside the ramp-overshoot region, and reverse their direction of motion toward upstream, thus being reflected at the shock front. These ions subsequently gyrate in the upstream region ahead of the ramp (being named reflected ions), and cross the shock again (being named reflected–transmitted ions). For the same cross-shock potential, the larger is the magnetic field increase, that is, the larger is the magnetic field in the overshoot maximum, the stronger is the drop of $ v x$ the directly transmitted ions and the larger is the fraction of ions which are reflected at the shock.

Upon gyration ahead of the ramp, the reflected ions return to the shock with large positive $ v y$ (the main magnetic field is in the $z$ direction). At their re-entry into the shock, the magnetic force $ v y B z$ is accelerating them toward downstream. This magnetic force exceeds the decelerating electrostatic force, so that their velocity $ v x$ steadily increases from the beginning of the ramp further downstream, at least up to the overshoot maximum.

We have shown above that in the collisionless plasma system, studied here, the ions which are accelerated contribute toward increasing their total pressure $ p x x = \u222b v x 2 f d 3 v$, while decelerating ions contribute toward decreasing their total pressure. Thus, during the shock crossing, the total pressure of the directly transmitted ions decreases while the total pressure of the reflected–transmitted ions increases. The ramp-overshoot region is a narrow part of the shock. Even if the shock is rippled and/or time-dependent, the spatial scale of the rippling substantially exceeds the width of the ramp-overshoot region, while the temporal scale of variations is substantially larger than the time of the shock crossing by an ion, so that the one-dimensional stationary momentum conservation (pressure balance) (6) should be a very good approximation. This expression states that the magnetic field increases at the expense of the drop of the total pressure of all ions crossing the ramp-overshoot region (the directly transmitted ions, the ions which are being reflected, and the reflected–transmitted ions together). Imagine now that the overshoot spontaneously grows a little. This would decrease the pressure of the directly transmitted ions and the ions which are being reflected. Without the reflected–transmitted ions, this would cause further increase the overshoot, as prescribed by (6). However, the increase in the overshoot would increase the number of reflected ions and, accordingly, the pressure of the reflected–transmitted ions. The latter increases from the beginning of the ramp through the overshoot, thus outplaying the decrease in the pressure of the directly transmitted ions across the same region and limiting the spontaneous overshoot growth. If the overshoot spontaneously decreases, the pressure of the directly transmitted ions goes up and the pressure of the reflected–transmitted ions goes down. Therefore, the reflected–transmitted ions play a stabilizing effect, ensuring that a balance could be achieved, where any spontaneous change of the overshoot strength would result in the corresponding change of the total pressure of all ion populations together in such a way to act against the spontaneous overshoot variation, thus making the shock structure stable.

## ACKNOWLEDGMENTS

The study was supported by the European Union's Horizon 2020 research and innovation program under Grant Agreement No. 101004131 (SHARP). The irfu-matlab software was used for the processing of the MMS data.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Michael Gedalin:** Conceptualization (equal); Formal analysis (lead); Funding acquisition (lead); Methodology (equal); Project administration (lead); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). **Prachi Sharma:** Conceptualization (equal); Formal analysis (supporting); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The MMS data are publicly accessible on the MMS Science Data Center website https://lasp.colorado.edu/mms/sdc/public/.