## 1 Introduction

Collisionless shocks are one of the most fundamental phenomena in space plasmas and among the most efficient accelerators of charged particles. Within the magnetohydrodynamical description, a fast shock is a discontinuity on which the plasma density, magnetic field, temperature jump and the flow speed drop (De Hoffmann & Teller Reference De Hoffmann and Teller1950). Real collisionless shocks are of finite width with a rather complicated structure (Scudder *et al.* Reference Scudder, Aggson, Aggson, Mangeney, Lacombe and Harvey1986). The latter is related to absence of collisions, which results in different responses of different populations of particles to the fields in the shock front. Over the past several decades, theoretical models, numerical simulations, *in situ* observations and laboratory experiments have all been used to understand the physics of collisionless shocks (see, e.g. Bell Reference Bell1978; Kennel, Edmiston & Hada Reference Kennel, Edmiston and Hada1985; Stone & Tsurutani Reference Stone and Tsurutani1985; Sagdeev & Kennel Reference Sagdeev and Kennel1991; Bykov *et al.* Reference Bykov, Vazza, Kropotina, Levenfish and Paerels2019, and references therein). Perri *et al.* (Reference Perri, Bykov, Fahr, Fichtner and Giacalone2022) review the latest advancements in particle acceleration, with a particular focus on shock acceleration.

The only place where the shock structure may be studied with *in situ* observations is the heliosphere. The shock structure depends on the angle $\theta _{Bn}$ between the shock normal and the upstream magnetic field vector (Bale *et al.* Reference Bale, Balikhin, Horbury, Krasnoselskikh, Kucharek, Möbius, Walker, Balogh, Burgess and Lembège2005; Burgess *et al.* Reference Burgess, Lucek, Scholer, Bale, Balikhin, Balogh, Horbury, Krasnoselskikh, Kucharek and Lembège2005). Quasi-perpendicular shocks, $\theta _{Bn}>45^\circ$, have a well-developed structure of the magnetic field. Quasi-parallel shocks are believed to have a less regular shock front. However, this is not a rule and a quasi-perpendicular shock may look quasi-parallel or *vice versa*. See, e.g. the plots of the shocks in the Magnetospheric Multiscale database (Lalti *et al.* Reference Lalti, Khotyaintsev, Dimmock, Johlander, Graham and Olshevsky2022). In what follows, we address those shocks which possess a well-defined magnetic structure: a narrow ramp with the steepest increase of the magnetic field and a wider overshoot in which the magnetic field reaches its maximum value.

Most high-resolution measurements have been done at the Earth bow shock. These measurements have shown that a part of incident ions may be reflected off the shock front and return to the upstream region just ahead of the shock transition, before either crossing the shock again toward downstream or escaping into the upstream region (Gosling *et al.* Reference Gosling, Thomsen, Bame, Feldman, Paschmann and Sckopke1982; Sckopke *et al.* Reference Sckopke, Paschmann, Bame, Gosling and Russell1983, Reference Sckopke, Paschmann, Brinca, Carlson and Lühr1990; Goodrich *et al.* Reference Goodrich, Ergun, Schwartz, Wilson, Johlander, Newman, Wilder, Holmes, Burch and Torbert2019). In low-Mach-number shocks, there are no, or almost no, reflected ions and the magnetic field profile is nearly monotonically increasing or has a small overshoot (Farris, Russell & Thomsen Reference Farris, Russell and Thomsen1993; Balikhin *et al.* Reference Balikhin, Zhang, Gedalin, Ganushkina and Pope2008). The overshoot is the region where the magnetic field is larger than it is in the uniform region well downstream of the shock. The overshoot arises due to the deceleration of the ion flow and drop of the ion pressure below the uniform downstream value (Ofman *et al.* Reference Ofman, Balikhin, Russell and Gedalin2009; Gedalin, Friedman & Balikhin Reference Gedalin, Friedman and Balikhin2015; Gedalin Reference Gedalin2019*a*,Reference Gedalin*b*), while ion heating is due to the gyration of the directly transmitted ions (Gedalin Reference Gedalin1997, Reference Gedalin2021). With the increase of the Mach number the downstream magnetic field increases, which causes a stronger drop of the ion pressure and a larger overshoot (Mellott & Livesey Reference Mellott and Livesey1987; Gedalin *et al.* Reference Gedalin, Dimmock, Russell, Pogorelov and Roytershteyn2023). In supercritical shocks (the high Mach number), with an increase in the Mach number, ion reflection becomes crucial for ion heating (Chodura Reference Chodura1975; Sckopke *et al.* Reference Sckopke, Paschmann, Bame, Gosling and Russell1983; Burgess, Wilkinson & Schwartz Reference Burgess, Wilkinson and Schwartz1989; Sckopke *et al.* Reference Sckopke, Paschmann, Brinca, Carlson and Lühr1990; Zimbardo Reference Zimbardo2011; Gedalin Reference Gedalin2019*c*; Madanian *et al.* Reference Madanian, Desai, Schwartz, Wilson, Fuselier, Burch, Le Contel, Turner, Ogasawara and Brosius2021). The reflected ions are responsible for the foot formation (Woods Reference Woods1969, Reference Woods1971) and were thought being responsible for the overshoot increase (Mellott & Livesey Reference Mellott and Livesey1987). Ion reflection has been approximately described as specular for quite a while Woods (Reference Woods1969, Reference Woods1971). The specular reflection approximation assumes that ions are reflected by the cross-shock potential without entering the shock. In the upstream region the uniform magnetic field and the motional electric field ensure the drift of ions with a velocity $V_u$ along the shock. This description refers to the normal incidence frame (NIF). The NIF is the reference frame in which the shock is static and the upstream plasma flow $(V_u,0,0)$ is along the shock normal. Inside the shock transition layer, the magnetic field increases and the magnetic force is no longer balanced by the electric force due to the motional electric field. The specular reflection approximation ignores the effect of this magnetic force on the ion motion. It was found that specular reflection requires extremely high cross-shock potentials (Wilkinson & Schwartz Reference Wilkinson and Schwartz1990). It has been recently shown (Balikhin & Gedalin Reference Balikhin and Gedalin2022) that the widely used expression for the shock foot width, based on the specular reflection approximation (Gosling & Robson Reference Gosling and Robson1985), may overestimate the width by a factor of two. The significance of the magnetic field is the primary distinction between the processes of specular and non-specular reflection (Gedalin Reference Gedalin1996). The reflection process of an ion depends on the initial values of the velocity of an ion (Gedalin *et al.* Reference Gedalin, Zhou, Russell and Angelopoulos2020). The process of non-specular reflection determines the velocities of the reflected ions with which they start to gyrate in the upstream region ahead of the ramp. In quasi-perpendicular shocks most of these ions return to the shock and cross it again, constituting the population of reflected–transmitted ions. Knowledge of the distribution of the reflected ions in the upstream region and of the reflected–transmitted ions is essential for understanding of the ion heating, the effect of ions on the shock structure and possible instabilities in the foot region (Burgess *et al.* Reference Burgess, Hellinger, Gingell and Trávníček2016). So far non-specular reflection has been studied within simulations (see, e.g. Burgess *et al.* Reference Burgess, Wilkinson and Schwartz1989; Burgess & Scholer Reference Burgess and Scholer2007; Burgess *et al.* Reference Burgess, Hellinger, Gingell and Trávníček2016) or test particle analyses (Gedalin Reference Gedalin1996, Reference Gedalin2016). These are able to provide the detailed information on the reflection process and the distributions, but only in numerical form. Deeper understanding of the role of electrostatic and magnetic forces in the ion reflection requires analytical treatment. In the present paper, we propose an approximate analytical approach which allows us to determine the non-specular ion velocities and the distributions of the reflected ions ahead of the ramp and during the second crossing of the shock.

## 2 Non-specular reflection inside the shock front

In this section we consider the ion reflection process, that is, the deceleration of an ion inside the ramp and overshoot region (up to the overshoot maximum) and return to the upstream region. The forthcoming analysis is in NIF. The shock normal is along the $x$-axis. The upstream and downstream magnetic fields are in the $x$–$z$ plane. In what follows, it is convenient to use the dimensionless variables. The magnetic field is normalized on the upstream magnetic field magnitude $B_u$, the velocity is normalized on the upstream NIF plasma velocity $V_u$, the electric field is normalized on $V_uB_u/c$, the spatial coordinates are normalized on the upstream ion convective gyroradius $V_u/\varOmega _u$ and the time is normalized on the inverse upstream gyrofrequency $\varOmega _u^{-1}$, where $\varOmega _u=eB_u/mc$, $m$ being the proton mass, e and c are the proton charge and speed of light respectively . The shock is assumed planar and stationary. The normalized fields are

*a–c*)\begin{gather} B_x=\cos\theta_{Bn}, \quad B_y=0, \quad B_z=\sin\theta_{Bn} \end{gather}

*a–c*)\begin{gather}E_x=0, \quad E_y=\sin\theta_{Bn}, \quad E_z=0 \end{gather}

in the uniform upstream region ahead of the ramp, $x<0$, and

*a–c*)\begin{gather} B_x=\cos\theta_{Bn}, \quad B_y=B_y(x), \quad B_z=B_z(x) \end{gather}

*a–c*)\begin{gather}E_x=E_x(x) \quad E_y=\sin\theta_{Bn}, \quad E_z=0 \end{gather}

in the ramp-overshoot region $x>0$. Here E is the normalised cross-shock electric field. In this section we are treating the ion motion for $x>0$. The equations of motion (in the normalized form) read

We are interested in the reflected ions. An ion is reflected if, at some point $x_r< L$ inside the ramp-overshoot region, the velocity along the shock normal drops to zero, $v_x(x_r)=0$. Here, $L$ denotes the width of the region. Up to the reflection point $x=x_r$ the velocity $v_x$ does not change sign, so that it is possible to replace $({\rm d}/{\rm d}t)=v_x({\rm d}/{{\rm d}x})$ and integrate

where

is the NIF cross-shock potential which is normalized on $mV_u^2/2$. The electric field inside the ramp is highly inhomogeneous. It is determined by ion–electron decoupling, and depends on the electron temperature and peculiarities of the ion dynamics. The dependence $s(x)$ is built to be self-consistently affected by the particle motion and also affects this motion. At present, there is no good knowledge of the fine structure of the shock front. In our calculations only the total potential drop is of importance and not the details of the functional dependence $s(x)$. For the incident ions both initial $v_y$ and $v_z$ are of the order of the upstream thermal speed $v_T=\sqrt {\beta /2}/M$, normalized on $V_u$. Here, $M=V_u/v_A$ is the Alfvénic Mach number, the Alfvén speed is $v_A=B_u/\sqrt {4{\rm \pi} n_um}$, $n_u$ is the upstream ion number density, $\beta =8{\rm \pi} n_uT_u/B_u^2$ is the ratio of the upstream ion kinetic pressure to the upstream magnetic pressure and $T_u$ is the upstream ion temperature. For $\beta \lesssim 1$ and $M\sim 5$ one has $v_T\sim 0.1\ll 1$. Thus, both initial $v_{y0}$ and $v_{z0}$ are small. The non-coplanar component $B_y$ peaks inside the ramp and $\int B_y\,{\rm d}x$ is typically much smaller than $\int B_z\, {{\rm d}x}$ (Goodrich & Scudder Reference Goodrich and Scudder1984; Jones & Ellison Reference Jones and Ellison1987), so that we approximate (2.9) as follows:

Note that the effective cross-shock potential, defined by (2.14), depends on $x$ only for a given initial $v_{y0}$. The integral $A(x)$ monotonically increases. The electrostatic cross-shock potential is expected to increase with the magnetic field, up to the overshoot maximum (Goodrich & Scudder Reference Goodrich and Scudder1984; Jones & Ellison Reference Jones and Ellison1987; Schwartz *et al.* Reference Schwartz, Thomsen, Bame and Stansberry1988). An ion is reflected if $v_x=0$ at some position inside the ramp-overshoot region. Let the width of the ramp-overshoot region (up to the overshoot maximum approximately) be $L$. Without the magnetic contribution, all ions with $v_{x0}^2< s(L)$ would be reflected. For a typical $s(L)=0.5$ (Morse Reference Morse1973; Formisano Reference Formisano1982; Schwartz *et al.* Reference Schwartz, Thomsen, Bame and Stansberry1988; Dimmock *et al.* Reference Dimmock, Balikhin, Krasnoselskikh, Walker, Bale and Hobara2012; Hanson *et al.* Reference Hanson, Agapitov, Mozer, Krasnoselskikh, Bale, Avanov, Khotyaintsev and Giles2019) one has $v_{x0}\lesssim 1-3v_T$, which is in the tail of the Maxwellian distribution function (see, e.g. Sckopke *et al.* Reference Sckopke, Paschmann, Bame, Gosling and Russell1983)

so that the number of reflected ions would be small. For $v_{y0} >0$, the effective potential reduces relative to the electrostatic cross-shock potential, so that these velocities are unfavourable for ion reflection. For $v_{y0}$, the effective potential exceeds the electrostatic potential, thus enhancing the ion reflection. Therefore, the ions whose initial velocity satisfies the condition

would be reflected, while others will cross the shock without stopping inside.

Figure 1 shows the part of the velocity space of the incident Maxwellian distribution, corresponding to the reflected ions for specular and non-specular reflection, for $v_T=0.15$, $s=0.5$ and $A=0.75$. The latter is estimated as $\sim B_{\max } L$, where $L\sim \sqrt {1-s}/B_{\max }$ (Gedalin *et al.* Reference Gedalin, Golbraikh, Russell and Dimmock2022). If the reflection were specular, ${\approx }2.5\,\%$ of the total number of ions would be reflected. The non-specular reflection results in ${\approx }4\,\%$ of reflected ions. For $v_T=0.1$, the percentage of the reflected ions is 0.16 % and 0.43 %, respectively. For $v_T=0.2$ we have 7 % and 9 %, respectively.

After the ion passes through the point of reflection, it comes back to the beginning of the ramp. In the same approximation as above, the ion would have $v_{x1}=-v_{x0}$, $v_{y1}=v_{y0}$ and $v_{z1}=v_{z0}$ when it enters the upstream region again. Note that, in our approximation, the number of reflected ions and their velocities $\boldsymbol {v}_1$ do not depend on the shock angle $\theta _{Bn}$.

## 3 Gyration of the reflected ions in the upstream region

The equations of motion of the reflected ion in the upstream region are

with the initial conditions ${\boldsymbol {v}}={\boldsymbol {v}}_1$ and $x=0$ at $t=0$. The solution of the equations of motion with the specified initial conditions is

For each $x<0$, (3.5) has zero, one or two solutions $t(x)$. These solutions, when substituted into (3.6)–(3.8), give ${\boldsymbol {v}}(x)$. Equation (3.5) cannot be solved analytically. For the further analysis we shall use numerical solution of this equation. For derivation of the reduced distribution function

we have from (3.5)–(3.8) the position and velocities as functions of time for $10^6$ ions initially uniformly distributed in the range $|v_{x0}-1|<3v_T$, $|v_{y0}|<3v_T$, $|v_{z0}|<3v_T$, after selecting ions which satisfy the conditions $\sqrt {(v_{x0}-1)^2+v_{y0}^2+v_{z0}^2}<3v_T$ and $v_{x0}^2< s-Av_{y0}$. For a small time step $\Delta t=0.02$ the phase space $(x,v_x)$ is sufficiently densely filled.

We catch the reflected ions on a two-dimensional grid in the phase space $(x,v_x)$. Each ion is assigned the weight $|v_{x0}|f_u({\boldsymbol {v}}_0)$, where $v_{x0}$ is the $x$-component of the initial ion velocity, to ensure the conservation of particle flux. These weights for all the particles and all the appearances in the cell $(x,v_x)\rightarrow (x+\Delta x,v_x)\rightarrow (x+\Delta x,v_x+\Delta v_x) \rightarrow (x,v_x+\Delta v_x)\rightarrow (x,v_x)$, where the grid cell dimensions are $(\Delta x, \Delta v_x)$, are summed up to form the reduced distribution function. In what follows the basic working set of parameters chosen is follows: $M = 5$, $s = 0.5$, $A=0.75$, $\beta = 0.5$ and $\theta _{Bn} = 65^\circ$.

Figure 2 shows the one dimensional (1-D) reduced distribution function $f(x,v_x)$ for the basic set (*a*) and for $A=0.375$ (*b*). For the visualization, the distribution function is normalized so that $\sum _{ij}f(x_i,v_{xj})=1$. Here, $i$ and $j$ give the number of cells. For the smaller value of $A$, the number of reflected ions is smaller by a factor of two, and they move to a smaller distance from the shock. A part of the phase space near the ramp is free of particles, which is usually referred to as an ion phase space hole (Johlander *et al.* Reference Johlander, Vaivads, Khotyaintsev, Gingell, Schwartz, Giles, Torbert and Russell2018).

Figure 3 shows the 1-D reduced distribution function $f(x,v_x)$ for $M = 5$, $s = 0.5$, $\beta = 0.5$, $A=0.75$ and two smaller angles. For $\theta _{Bn} = 40^\circ$ there is a substantial population of reflected ions which escape further upstream and do not return to the shock, also known as backstreaming ions. Yet, a number of ions do return. They will either cross the shock and proceed further into the downstream region or are reflected once again. Analytical treatment of multiply reflected ions is beyond the scope of the present paper and the capabilities of our approximation.

## 4 Shock crossing by the reflected–transmitted ions

Upon gyration in the upstream region, in quasi-perpendicular shocks the reflected ions return to the shock and cross it again. Their dynamics inside the shock front is governed by (2.5)–(2.8). For simplicity, we analyse here the perpendicular case $\theta _{Bn}=90^\circ$, for which $B_x=B_y=0$. Accordingly, the relevant equations of motion read

with the initial conditions $v_x=v_{x2}$, $v_y=v_{y2}$ at $x=0$. The subscript $2$ refers to the velocities of the reflected ions at their re-entry to the shock. Integrating the equations, one has

where $v_x$, $v_y$, $s$ and $B_z$ are functions of $x$. Figure 4 shows the reduced distribution function $f(x,v_y)$ for the basic set of parameters. It is seen that the reflected ions re-enter the shock with large positive $v_y$. Figure 2 shows that the component $v_x>1$ at the re-entry. Since

Therefore, we may approximate $v_y\approx v_{y2}$ and

where $v_{x3}$ is the velocity upon crossing the ramp-overshoot region. Figure 5 shows the distribution function $f(x=0,v_x)$ at the re-entry to the shock (black curve) and $f(x=L,v_x)$ (red curve) of the reflected–transmitted ions upon crossing the ramp-overshoot region. Figure 6 shows the histogram of the ratio $v_{x3}/v_{x2}$. All reflected–transmitted ions acquire higher $v_x$ upon crossing the shock.

The total pressure is related to the reduced distribution function as follows:

where $p_{xx}$ is a total pressure including the dynamic and kinetic pressure. In the collisionless system the distribution function is constant along the particle trajectory in the phase space. For the reflected–transmitted ions this means

The particle flux conservation requires

so that

The immediate consequence of the increase of $v_x$ across the shock is that the contribution of the reflected–transmitted ions to $p_{xx}$ of all ions increases at the shock crossing. The momentum conservation in the planar stationary shock (pressure balance equation) reads

which means that a decrease in the plasma pressure should be balanced by the corresponding increase of the magnetic pressure. The ion pressure $p_{i,xx}$ inside the ramp-overshoot region is the sum of the decreasing pressure of the directly transmitted ions and the increasing pressure of the reflected–transmitted ions. The deceleration of the directly transmitted ions is responsible for the magnetic field increase, that is, the overshoot growth, while the reflected–transmitted ions contribute to limit the maximum of the overshoot.

## 5 Conclusions

We proposed, for the first time, an approximate analytical description of the non-specular ion reflection in the shock front, taking into account both the electrostatic deceleration and the magnetic deflection. It is shown that, in the reflection condition, the electrostatic cross-shock potential should be replaced with an effective potential proportional to the magnetic field, integrated over the reflection region, and to the initial ion velocity in the direction perpendicular to the coplanarity plane. The magnetic contribution substantially increases the number of reflected particles. The velocity distribution of the reflected ions differs significantly from that obtained within the specular reflection approximation. The reflected particles further gyrate in the upstream region and return to the shock. Using the analytical solution of the equations of ion motion in the uniform upstream region and the initial velocity space derived at the previous step, we numerically constructed the reduced distribution function of reflected ions ahead of the ramp. We have explicitly shown that a smaller magnetic contribution results in a reduction of the number of reflected particles and smaller turning distances of these reflected ions. For smaller angles, in the quasi-parallel range but still close to the boundary between the quasi-parallel and quasi-perpendicular regimes, a population of reflected ions returning to the shock coexists with a population of ions which escape further upstream (backstreaming ions). We determined the velocities of the ions at the re-entry to the shock. The component $v_x$ significantly exceeds the upstream plasma speed, the component $v_y\sim 1.5 V_u$ at this point. We have developed an analytical approximation to establish a relation between the $v_x$ of the reflected–transmitted ions at their re-entry into the shock and their $v_x$ upon crossing the ramp-overshoot region. It is shown that $v_x$ increases for all these ions, that is, they are accelerated by the magnetic force despite electrostatic deceleration. Therefore, the total pressure $p_{xx}$ of these ions increases across the shock. Thus, they contribute towards limiting the overshoot growth due to the deceleration of the directly transmitted ions.

The proposed approach is the first analytical treatment of the non-specular ion reflection and its implications. This is the lowest-order approximation and lays out a basis for more sophisticated analyses.

## Acknowledgements

*Editor Antoine C. Bret thanks the referees for their advice in evaluating this article.*

## Funding

The study was supported by the European Unions Horizon 2020 research and innovation program under grant agreement No. 101004131 (SHARP).

## Declaration of interests

The authors report no conflict of interest.