Non-locality of ion reflection at the shock front: dependence on the shock angle

Abstract

In typical heliospheric collisionless shocks most of the mass, momentum and energy are carried by ions. Therefore, the shock structure should be most affected by ions. With the increase of the Mach number, ion reflection becomes more and more important, and reflected ions participate in shaping the shock profile. Ion reflection at the collisionless shock is a non-local process: the reflected–transmitted ions re-enter the shock front far from the reflection point. The direction and the magnitude of this shift depend on the shock angle. The distance between the reflection point and the re-entry point is of the order of the upstream ion convective gyroradius and exceeds the shock width. The non-locality of ion reflection may have implications for shock rippling since reflected ions may carry perturbations along the shock front.

1. Introduction

In typical heliospheric collisionless shocks most of the mass, momentum and energy are carried by ions. Therefore, the shock structure should be most affected by ions. In sufficiently low-Mach-number shocks, all or almost all ions cross the shock from the upstream region to the downstream region and proceed further without return (directly transmitted ions). In these shocks ion heating is due to the gyration of directly transmitted ions (Gedalin 1997, 2021) and the downstream magnetic structure, including moderate overshoot, is due to the non-gyrotropy of the downstream ion distributions. This non-gyrotropy causes spatially quasi-periodic variations of the total, dynamic and kinetic, pressure of ions, and corresponding variations of the magnetic pressure in the opposite phase (Balikhin et al. 2008; Gedalin, Friedman & Balikhin 2015). With the increase of the Mach number, ion reflection becomes progressively more and more important, and the contribution of reflected–transmitted ions to ion heating is substantial (Sckopke et al. 1983; Scudder et al. 1986; Sckopke et al. 1990). For a long time, ion reflection was described in the convenient but physically not feasible approximation of specular reflection (Gosling et al. 1982). Specular reflection assumes that an ion with the bulk flow velocity is instantly reflected by a potential wall in such a way that the normal component of the ion velocity flips while the others do not change. The above description implies working in the normal incidence frame (NIF), in which the incident plasma flow velocity (upstream velocity) is along the shock normal. The specular reflection approximation has been used for years and is still often used for observational estimate of the shock speed and conversion of time in the spacecraft frame to a spatial coordinate system (Gosling & Thomsen 1985). In reality, ions should enter the shock transition where the magnetic field increases and the cross-shock potential builds up (Morse 1973; Formisano 1982; Schwartz et al. 1987, 1988; Dimmock et al. 2012; Cohen et al. 2019) before the normal component of their velocity drops to zero and changes sign. It was shown, theoretically and observationally (Sckopke et al. 1983; Gedalin 1996, 2016) that ion reflection is essentially non-specular. This has been shown already in early simulations (Burgess, Wilkinson & Schwartz 1989) but remained unnoticed. It was shown recently (Balikhin & Gedalin 2022) that the widely used expression of Gosling & Thomsen (1985) overestimates the actual excursion of the reflected ions along the shock normal by a factor of two. This has to be taken into account when converting spacecraft measurements of the magnetic field as a function of time to the spatial dependence of the magnetic field.

Present focus in the studies of ion reflection by shocks is mainly on the energies the ions can achieve (see, e.g. Trattner et al. 2023), on the ion phase space holes (see, e.g. Wang et al. 2022), on the observations of multiple ion phase space holes as a rippling signature (Johlander et al. 2018) and on the role of reflected ions in rippling formation (Lowe & Burgess 2003; Moullard et al. 2006; Burgess & Scholer 2007; Johlander et al. 2016; Gingell et al. 2017; Omidi et al. 2021). The motion of the reflected ions is not studied in detail, which is understandable, since neither observations nor simulations follow individual ion trajectories but measure ions appearing at a certain position and/or time, without precise knowledge of where they came from.

Shock structure depends, in general, on the angle $\theta _{Bn}$ between the shock normal and the upstream magnetic field vector. Most quasi-perpendicular shocks, $\theta _{Bn}>45^\circ$, have a clear structure where the steepest magnetic field increase, the ramp, is followed by an overshoot, undershoot and possibly more coherent magnetic oscillations (Bale et al. 2005). Quasi-parallel shocks, $\theta _{Bn}<45^\circ$, often have a less regular and more extended transition with no clear ramp and overshoot (Burgess et al. 2005). This is, however, not a rule. In many cases, a quasi-parallel shock has a clear magnetic profile with the same structural elements as in quasi-perpendicular shocks. An example is shown in figure 1. This shock crossing is taken from the database of the Magnetospheric MultiScale (MMS) shock crossings (Lalti et al. 2022). The shock angle is determined in the database as $\theta _{Bn}\approx 32^\circ$, well within the quasi-parallel range. The black line shows the magnetic field magnitude measured in the fast mode, 16 vectors per sec. The high temporal resolution of measurements reveals plenty of fluctuations/small-scale structure. The thin red line shows the magnetic profile with the small-scale fluctuations and structure removed. The removal is done by applying wavelet transform with the Daubechies 10 wavelet, removing the 7 finest levels of the total 13 levels and applying the inverse transform. The denoised profile exhibits the basic elements of the shock profile of a typical quasi-perpendicular shock. Note that the wavelet transform smooths the transition from the foot to the ramp. In what follows, our discussion refers to the shocks with a well-defined structure (after removal of small-scale fluctuations/structure), like the one shown in figure 1.

Figure 1. A quasi-parallel shock measured by MMS 1. The shock angle is $\theta _{Bn}\approx 32^\circ$. The black line shows the magnetic field magnitude measured in the fast mode, 16 vectors per sec. The high temporal resolution of measurements reveals plenty of fluctuations/small-scale structure. The thick red line shows the magnetic profile with the small-scale fluctuations and structure removed. The removal is done by applying wavelet transform with the Daubechies 10 wavelet, removing the 7 finest levels of the total 13 levels and applying the inverse transform.

After being reflected, ions gyrate in the upstream magnetic and electric field ahead of the shock transition before coming to the shock again and crossing it toward downstream, thus becoming a reflected–transmitted ion population. The distance between the reflection point and the re-entry point may well exceed the width of the shock transition. Thus, the whole process is non-local in space. This non-locality is known and was used for explanation of the foot formation (Woods 1971; Leroy et al. 1982) as well as for the mechanism of shock drift acceleration by multiple reflection (see, e.g. Zank et al. 1996). In both cases only the reflected ion motion in the direction of the NIF motional electric field is of interest. The motional electric field is in the direction perpendicular to the shock normal and to the upstream magnetic field vector. However, the reflected ions move also in the direction of the magnetic field. If the shock is planar and stationary, there seem to be no other implications of the non-locality, because of the translational invariance along the shock front. However, at large Mach numbers, shocks are expected to become rippled (Lowe & Burgess 2003; Moullard et al. 2006; Burgess & Scholer 2007; Lobzin et al. 2008; Ofman & Gedalin 2013; Burgess et al. 2016; Johlander et al. 2016; Gingell et al. 2017; Johlander et al. 2018; Omidi et al. 2021). If this is the case, an ion which is reflected at some position at the shock front re-enters the shock transition, which may have different parameters. On the other hand, if a perturbation occurs at some shock position, ions, which are reflected at this position, convey this perturbation to a quite another position on the shock front. In the absence of translational invariance on a rippled surface, this may cause a stabilizing effect or instability. Thus, knowledge of the dependence of the shift of a reflected ion between the reflection and re-entry points is important to understanding of rippling development and for interpretation of observations. The latest hybrid simulations (Burgess et al. 2016; Omidi et al. 2021) suggest that rippling is generated by the instability between the reflected ions and the background plasma, and that the direction and speed of ripple propagation is determined by the reflected ion velocity in the turnaround point. The direction of the shift between the reflection point and the re-entry point is consistent with this velocity. In the present paper, we use test particle analysis in a model shock to study the non-locality of ion reflection and corresponding ion distributions within the shock transition. Such test particle analyses are actually an extension of the theory on systems where the equations of motion cannot be solved analytically. Modelling of the shock profile is a necessary part of any theoretical investigation and allows us also to extract information from observations (Gedalin et al. 2022). Although test particle analysis is sometimes considered inferior compared with sophisticated self-particle simulations, it has the advantage of controlling each parameter separately, which allows us to determine multi-parameter dependencies. The spatial resolution is not limited by any grid cell size and each particle trajectory can be easily followed. The method has already demonstrated its efficiency. As an example, it has been used to explain the downstream magnetic oscillations as a response to non-gyrotropy of gyrating downstream ion distributions, which was later confirmed by hybrid simulations and detailed observations (Balikhin et al. 2008; Ofman et al. 2009; Pope, Gedalin & Balikhin 2019). As another example, the influence of $\alpha$-particles on the shock structure was first elucidated using the test particle analysis and later supported by simulations (Gedalin 2017, 2019; Ofman et al. 2019). Test particle analysis has been used recently for studies of pickup ion acceleration at the shock front (Zirnstein et al. 2021). In the present study, we focus on the dependence of the direction and magnitude of the shift of a reflected particle along the shock front, without taking into account possible non-planarity or time dependence of the shock.

2. The model

We adopt a simple model including overshoot

(2.1)\begin{gather} B_{{\rm bas}}=\frac{R_{{\rm bas}}-1}{2}+\frac{R_{{\rm bas}}+1}{2}\tanh\frac{3x}{D}, \end{gather}

(2.2)\begin{gather}B_{{\rm add}}=R_{{\rm add}}\left(1-\tanh\frac{3(x-c_r)}{w_r}\right)\left(1+\tanh\frac{3(x-c_l)}{w_l}\right), \end{gather}

(2.3)\begin{gather}B=B_{{\rm bas}}+B_{{\rm add}}, \end{gather}

(2.4)\begin{gather}B_x=\cos\theta_{Bn}, \end{gather}

(2.5)\begin{gather}B_z=B\sin\theta_{Bn}, \end{gather}

(2.6)\begin{gather}B_y=k_B\left(\frac{{\rm d}B}{{\rm d} x}\right), \end{gather}

(2.7)\begin{gather}E_x=-k_E\left(\frac{{\rm d}B}{{\rm d} x}\right), \end{gather}

(2.8)\begin{gather}E_y=\sin\theta_{Bn}. \end{gather}

(2.9)\begin{gather}E_z=0. \end{gather}

Here, the magnetic field is normalized on the upstream magnetic field $B_u$. All coordinates and lengths are normalized on the upstream convective gyroradius, $V_u/\varOmega _u$, where $V_u$ is the plasma speed in NIF, and $\varOmega _u=eB_u/m_pc$ is the upstream proton gyrofrequency, where e is the proton charge and m_p is the proton mass. The related upstream ion inertial length is $c/\omega _{pi}$, where $\omega _{pi}=\sqrt {4{\rm \pi} e^2n_u/m_p}$ is the upstream proton plasma frequency, and $n_u$ is the upstream proton number density. The Alfvénic Mach number is defined as usual, $M_A=V_u/v_A$, where the Alfvén speed is $v_A=B_u/\sqrt {4{\rm \pi} n_um_p}$. In the above expressions the angle $\theta _{Bn}$ between the shock normal and the upstream magnetic field is used. The expression (2.1) describes a monotonic increase of the main magnetic field, $B_z$, across the shock front, while (2.2) adds an overshoot. The magnetic compression is given by

(2.10)\begin{equation} \frac{B_d}{B_u}=\sqrt{R^2\sin^2\theta_{Bn}+\cos^2\theta_{Bn}}. \end{equation}

The relation of the non-coplanar component $B_y$ approximately follows the theoretical analysis (Gedalin 1998) and was successfully applied to observational analysis (Gedalin et al. 2022). The electric field is normalized on $V_uB_u/c$. The coefficients $k_E$ and $k_B$ are found from the condition on the cross-shock potential

(2.11)\begin{gather} -\int E_x\,{\rm d}x=\phi_{{\rm NIF}}\equiv s_{{\rm NIF}}(m_pV_u^2/2e), \end{gather}

(2.12)\begin{gather}-\int (E_x+V_u\tan\theta_{Bn} B_y/c)\,{{\rm d}x}=\phi_{{\rm HT}}\equiv s_{{\rm HT}}(m_pV_u^2/2e). \end{gather}

The subscript ${\rm HT}$ means de Hoffman–Teller frame, which is the shock frame where the upstream plasma flow is along the upstream magnetic field. The parameters $\theta _{Bn}$, $B_d/B_u$, $R_{{\rm add}}$, $s_{{\rm NIF}}$, $s_{{\rm HT}}$, $D$, $c_l$, $c_r$, $w_l$ and $w_r$ completely define the planar stationary shock profile used in the analysis. In the present study we fix the following parameters: $M_A=6$, $B_d/B_u=3$, $R_{{\rm add}}=1$, $s_{{\rm NIF}}=0.4$, $s_{{\rm HT}}=0.1$, $D=2/M$, $w_l=D$, $w_r=3D$, $c_l=0.5D$, $c_r=0.5D$. The incident ion distribution is Maxwellian with $\beta _i=8{\rm \pi} n_uT_{iu}/B_u^2=1$. The shock angle is varied. Figure 2 illustrates the magnetic field given by (2.1)–(2.6), for the same $B_d/B_u$, $R_{{\rm add}}$ and $D$, for two different shock angles, $\theta _{Bn}=60^\circ$ (a quasi-perpendicular shock) and $\theta _{Bn}=40^\circ$ (a quasi-parallel shock). In the quasi-parallel shock the non-coplanar component $B_y$ is significant and even becomes larger than $B_z$ in a small region, which results in the appearance of a small peak inside the ramp on the profile of the magnetic field magnitude. Ions are not sensitive to such small details of the shock profile.

Figure 2. The components of the magnetic field and the magnetic field magnitude, according to the model: $B_z$ (blue), $B_y$ (red) and $|\boldsymbol {B}|$ (black). The component $B_x$ is constant and not shown.

3. Results

We trace 320 000 initially Maxwellian distributed ions across the shock by solving the equations of motion in the model fields. Figure 3 shows $x$ vs $v_x$ for reflected ions only, together with the magnetic field magnitude, for $\theta _{Bn}=75^\circ$. The red line marks the coordinate $x=0$ at which the distributions shown below are derived. Note that the ions are reflected after the ramp but before the maximum of the overshoot. This means that the reflection is due to the combined effect of the deceleration by the cross-shock electric field and deflection by the increasing magnetic field (Gedalin et al. 2023).

Figure 3. Black: $x$ vs $v_x$ for the reflected ions only. Blue: the magnetic field magnitude. The shock angle is $\theta _{Bn}=75^\circ$. Red line marks the position for which the cuts are presented.

Figure 4 shows two-dimensional cuts, $v_x-v_y$ and $v_x-v_z$, of the three-dimensional distribution function of all ions crossing the position $x=0$ in any direction. The black arrow marks the population of directly transmitted ions. The blue arrow marks the ions which are reflected, that is, move toward upstream at this position. The red arrow marks the reflected–transmitted ions, that is, the ions which were reflected, gyrated in the upstream and are crossing again the shock toward downstream. In the $v_x-v_y$ cut the three populations are clearly distinct, in the $v_x-v_z$ cut there is substantial overlap of directly transmitted and reflected–transmitted populations.

Figure 4. Two-dimensional cuts of the ion distribution at the red line position for $\theta _{Bn}=75^\circ$. (a) The reduced two-dimensional distribution function $f(v_x,v_y,x=0)$. (b) The reduced two-dimensional distribution function $f(v_x,v_z,x=0)$. The black arrow points to the directly transmitted ions. The blue arrow indicates the ions which are being reflected. The red arrow shows the reflected–transmitted ions.

Figure 5 shows the $y$ and $z$ positions of the ions crossing the shock at $x=0$. The reflected–transmitted ion population is significantly shifted along $y$, by one ion convective gyroradius and more, with a substantial dispersion of their $z$-coordinates.

Figure 5. The $y$ and $z$-coordinates of the ions crossing the red line position for $\theta _{Bn}=75^\circ$. The red arrow points to the population of the reflected–transmitted ions. The directly transmitted and reflected ions overlap on this plot.

Figure 6 shows $x$ vs $v_x$ for reflected ions only, together with the magnetic field magnitude, for $\theta _{Bn}=60^\circ$, and it is rather similar to figure 3. Note that the magnetic field in the overshoot maximum is slightly smaller in this case. The left panel of figure 7 is similar to the left panel of figure 4. The right panel shows that reflected–transmitted ions acquire larger negative $v_z$ for $\theta _{Bn}=60^\circ$ than for $\theta _{Bn}=75^\circ$. The three panels of figure 8 show $y$ and $z$ coordinates of ions crossing the plane $x=0$, similarly to figure 5, but for three different values of the cross-shock potential, $s=0.3, 0.4, 0.5$. The value of the cross-shock potential affects the dispersion of the positions but the position of the maximum is affected only weakly. This maximum is at $\Delta y\approx 1.2$, $\Delta z\approx -0.5$ for $s=0.4$, so that the ions are moving at the angle $\approx 65^\circ$ to the main magnetic field $B_z$. For larger $s$ the maximum is slightly farther from the coordinate origin, for smaller $s$ it is slightly closer. The $z$-coordinates of the reflected–transmitted ions experience a systematic shift toward negative values.

Figure 6. Black: $x$ vs $v_x$ for the reflected ions only. Blue: the magnetic field magnitude. The shock angle is $\theta _{Bn}=60^\circ$. Red line marks the position for which the cuts are presented.

Figure 7. Two-dimensional cuts of the ion distribution at the red line position for $\theta _{Bn}=60^\circ$. (a) The reduced two-dimensional distribution function $f(v_x,v_y,x=0)$. (b) The reduced two-dimensional distribution function $f(v_x,v_z,x=0)$. The black arrow points to the directly transmitted ions. The blue arrow indicates the ions which are being reflected. The red arrow shows the reflected–transmitted ions.

Figure 8. The $y$ and $z$-coordinates of the ions crossing the red line position for $\theta _{Bn}=60^\circ$ and three different values of $s=0.3, 0.4, 0.5$. The value of the cross-shock potential affects the dispersion of the positions but only weakly affects the position of the maximum.

Figure 9 illustrates the effect of the magnetic compression and overshoot strength. For the lower magnetic field ion reflection is substantially weaker but the effect of the maximum of the position is not noticeable.

Figure 9. The $y$ and $z$-coordinates of the ions crossing the red line position for $\theta _{Bn}=60^\circ$. Panel (a) corresponds to the basic set of parameters, $B_d/B_u=3$ and $R_{{\rm add}}=1$. Panel (b) corresponds to the reduced magnetic field, $B_d/B_u=2.4$ and $R_{{\rm add}}=0.8$. There are far fewer reflected ions for the lower magnetic field but the position of the maximum almost does not change.

In what follows we present the results of ion tracing in the model shock front for two smaller angles (Figures 10–17). The tracing procedure and the analysis are the same as for the two previous cases. The set of figures for $\theta _{Bn}=50^\circ$ is similar to that for $\theta _{Bn}=75^\circ$, while for $\theta _{Bn}=40^\circ$ we show also the effects of $s$ and $B_d/B_u$ and $R_{{\rm add}}$, as is done above for $\theta _{Bn}=60^\circ$. Variation of these parameters affects mainly the dispersion in the positions of the ions but only weakly affects the position of the maximum. There is further shift of the re-entry point in the $z$-direction. The major difference from the cases of larger angles is the appearance of backstreaming ions. These ions are produced by multiple reflection, as shown in figure 11. In all cuts these backstreaming ions are indicated with a green arrow. The number of backstreaming ions seems to increase with the decrease of the shock angle, at the expense of the reflected–transmitted ions.

Figure 10. Black: $x$ vs $v_x$ for the reflected ions only. Blue: the magnetic field magnitude. The shock angle is $\theta _{Bn}=50^\circ$. Red line marks the position for which the cuts are presented.

Figure 11. Black: $x-v_x$ for the multiply reflected ions only. Blue: the magnetic field magnitude. The shock angle is $\theta _{Bn}=50^\circ$. Red line marks the position for which the cuts are presented.

Figure 12. Two-dimensional cuts of the ion distribution at the red line position for $\theta _{Bn}=50^\circ$. (a) The reduced two-dimensional distribution function $f(v_x,v_y,x=0)$. (b) The reduced two-dimensional distribution function $f(v_x,v_z,x=0)$. The black arrow points to the directly transmitted ions. The blue arrow indicates the ions which are being reflected. The red arrow shows the reflected–transmitted ions. The green arrow points to the backstreaming ions.

Figure 13. The $y$ and $z$-coordinates of the ions crossing the red line position for $\theta _{Bn}=50^\circ$. The red arrow points to the reflected–transmitted ions. The green arrow points to the multiply reflected ions.

Figure 14. Black: $x$ vs $v_x$ for the reflected ions only. Blue: the magnetic field magnitude. The shock angle is $\theta _{Bn}=40^\circ$. Red line marks the position for which the cuts are presented.

Figure 15. Two-dimensional cuts of the ion distribution at the red line position for $\theta _{Bn}=40^\circ$. (a) The reduced two-dimensional distribution function $f(v_x,v_y,x=0)$. (b) The reduced two-dimensional distribution function $f(v_x,v_z,x=0)$. The black arrow points to the directly transmitted ions. The blue arrow indicates the ions which are being reflected. The red arrow shows the reflected–transmitted ions. The green arrow points to the backstreaming ions.

Figure 16. The $y$ and $z$-coordinates of the ions crossing the red line position for $\theta _{Bn}=40^\circ$ and three different values of $s=0.3, 0.4, 0.5$. The value of the cross-shock potential affects the dispersion of the positions but only weakly affects the position of the maximum. The red arrow points to the reflected–transmitted ions for which $v_x>0$. The green arrow points to the multiply reflected ions for which $v_x<0$.

Figure 17. The $y$ and $z$-coordinates of the ions crossing the red line position for $\theta _{Bn}=40^\circ$. Panel (a) corresponds to the basic set of parameters, $B_d/B_u=3$ and $R_{{\rm add}}=1$. Panel (b) corresponds to the reduced magnetic field, $B_d/B_u=2.4$ and $R_{{\rm add}}=0.8$. There are far fewer reflected ions for the lower magnetic field but the position of the maximum almost does not change. The red arrow points to the reflected–transmitted ions for which $v_x>0$. The green arrow points to the multiply reflected ions for which $v_x<0$.

Reduction of the magnetic field does not affect the position of the two maxima for $\theta _{Bn}=40^\circ$. For $s=0.4$ these maxima are at $\Delta y\approx 1.6$ and $\Delta z\approx -1.2$ for reflected–transmitted ions, and $\Delta y\approx 1.2$ and $\Delta z\approx -2.5$ for multiply reflected (backstreaming) ions. The maxima move closer to the coordinate origin for smaller $s$ and farther from it for larger $s$. The angle between $\Delta \boldsymbol {r}=(\Delta y,\Delta z)$ and $B_z$ is $\approx 53^\circ$.

4. Conclusions

Ion reflection at a collisionless shock front is a manifestly non-local process: the distance between the reflection point of an ion and the re-entry point of this ion into the shock transition is substantial and exceeds the upstream ion convective gyroradius. For quasi-perpendicular shocks the shift between the two points is mainly in the direction perpendicular to the magnetic field. For quasi-parallel shocks the shift in the perpendicular direction is similar to the shift in quasi-perpendicular shocks. It is accompanied by the shift in the direction opposite to the magnetic field direction. The latter parallel shift may significantly exceed the perpendicular shift. In a planar stationary shock this non-locality may seem of no importance because of the translational symmetry along the shock front. This is, however, not true. Increase of $B_z$ ahead of the shock, the magnetic foot, is related to the current in the $y$-direction produced by the motion of reflected ions in this direction. For smaller shock angles the motion of reflected ions in the $z$-direction may become more substantial than in the $y$-direction, so that a current in the $z$-component is produced which should cause appearance of nonzero $B_y$ in the foot.

Non-locality should be taken into account when interpreting observations. In multi-spacecraft missions different probes often observed different magnetic profiles. Distributions of reflected ions measured by a probe may have been produced at a shock crossing with parameters different from those at the measurement position.

Non-locality may be related to the development of rippling since a perturbation at one site is conveyed along the shock front by the reflected ions. If, for example, an overshoot increases at some site, ion reflection is enhanced at this site (Gedalin et al. 2023). A larger number of reflected ions enter the shock at a certain distance from the perturbed site. Higher ion pressure there would cause decrease of the overshoot, because of the momentum conservation, that is, the pressure balance. At this stage we are unable to quantify this possible instability. However, if it occurs, we can estimate the velocity of ripples as $\boldsymbol {V}_{{\rm ripple}}\sim \varOmega \Delta \boldsymbol {r}/{\rm \pi}$, where $\Delta \boldsymbol {r}$ is the vector from the reflection point to the re-entry point of the reflected–transmitted ions and we estimate that this motion occurs roughly during half of the upstream gyroperiod, ${\rm \pi} /\varOmega _u$. The latter gives an estimate of the surface wave (rippling) period. Accordingly, the wavelength may be estimated as $\sim | \Delta \boldsymbol {r}|$. For our $\theta _{Bn}=60^\circ$ case this estimate gives $V_{{\rm ripple}} \approx 0.4 V_u$ at $\approx 65^\circ$ to $B_z$. For the $\theta _{Bn}=40^\circ$ case we have $V_{{\rm ripple}} \approx 0.65 V_u$ at $\approx 53^\circ$ to $B_z$.

Johlander et al. (2016) report MMS observations of a rippled shock with $\theta _{Bn}=83^\circ$ and $M=4.2$. The rippling wavelength was estimated as $\approx V_u/\varOmega _u$ (one in our dimensionless units) and the speed was estimated as $\sim 0.3 V_u$. The propagation angle could not be determined. Burgess et al. (2016) performed three-dimensional hybrid simulations of a rippled shock. Although their analysis was devoted to the waves in the foot, the figures there show that the wavelength of ripples is $\sim V_u/\varOmega _u$. The speed was estimated as $\sim 0.5 V_u$. Our estimates of the speed and wavelength are in better agreement with the values found in observations (Johlander et al. 2016) and three-dimensional hybrid simulations (Burgess et al. 2016) than could be hoped for by the planar stationary model. Further comparison requires development of a more sophisticated model which would include rippling (Gedalin & Ganushkina 2022), and more three-dimensional simulations. Since the direction of the reflected ion motion is neither perpendicular nor parallel to the upstream magnetic field, two-dimensional simulations with the field in-plane or out-of-plane may suppress rippling or misestimate the rippling parameters. This has been shown in the comparison of two- and three-dimensional hybrid similations (Burgess et al. 2016), where it is found that fluctuations carried by the reflected ions are coupled to fluctuations propagating along the magnetic field. Non-locality of the reflection process may be directly related to the cross-field component of the rippling. The coupling with the other component may arise due to the developing rippling which causes variation of the shock angle along the shock front. Even initially perpendicular shocks become locally oblique (Burgess et al. 2016), so that the shift acquires component along the magnetic field. This issue is beyond the scope of this study and will be investigated separately.

In the cases $\theta _{Bn}=50^\circ$ and $\theta _{Bn}=40^\circ$ there appears a population of multiply reflected ions which eventually escape upstream. As long as the number of these ions is small relative to the number of the reflected–transmitted ions, the effect of the multiply reflected ions can be expected to be minor. With further decrease of the shock angle they may become progressively more important. If rippling is indeed related to the vector of the shift of reflected ions from the reflection point to the point of the re-entry into the shock, these ions may affect the rippling pattern by adding another wavelength or making the surface wave a wavepacket rather than a quasi-monochromatic wave. However, the main effect of the backstreaming ions may be causing instabilities ahead of the shock. Generated waves may drift toward the shock and interact with it, thus providing another mechanism of rippling generation (Hao et al. 2016). Shocks, which are strongly affected by waves excited due to backstreaming ions, are not within the scope of the present study.