2. Observational illustration

In order to illustrate what we are supposed to be able to explain, figure 1 shows the magnetic field profile and the reduced ion distribution

(2.1)\begin{equation} f(t,v_n)=\int F(\boldsymbol{v},t) \, {\rm d} v_{t_1}\, {\rm d} v_{t_2}. \end{equation}

Here, $n$ denotes the direction along the shock normal, $t_1,t_2$ are two perpendicular directions along the shock front, and $F(\boldsymbol {v},t)$ is the three-dimensional distribution as a function of time, measured in the spacecraft frame. The shock crossing was observed by MMS 1 on 07 October 2015 at 12:07:10 and documented in the database https://zenodo.org/record/6343989${\tt \#}$.Yxxbbi0Rokw (Lalti et al. Reference Lalti, Khotyaintsev, Dimmock, Johlander, Graham and Olshevsky2022). The model shock normal is used (Farris & Russell Reference Farris and Russell1994). The listed shock parameters are: the Alfvénic Mach number $M\approx 5$, the angle between the shock normal and the upstream magnetic field $\theta \approx 70^\circ$ and $\beta _u=0.4$. In the figure the magnetic field magnitude is normalized on the upstream magnetic field $B_u$, and the normal component of the ion velocity is shifted into NIF and normalized on the upstream plasma speed $V_u$. For the visualization the distribution is normalized on $\max f(t,v_n)$ in the whole range of the figure and plotted in the log scale. The magnetic compression $B_d/B_u\approx 3$. Here, and in what follows, $u$ denotes upstream and $d$ denotes downstream. Since the measurements of the cold solar wind are not very precise, the values of $v_n$ are very approximate and the plot is for illustrative purposes only. The arrows in the figure show the main features which are expected to affect the ion motion, the overshoot and the undershoot and the population of reflected ion which is expected to be affected. One blue arrow points to the position where the dynamic pressure drops while the kinetic pressure is not large yet. This is the position of the overshoot, that is, the point where the magnetic pressure is maximum. The second blue arrow points to the position where the kinetic pressure is maximum, this point corresponds to the undershoot. These conclusions follow from the approximate constancy of the total pressure across the shock (Gedalin Reference Gedalin2019a,Reference Gedalinb).

Figure 1. The magnetic field magnitude, normalized to the upstream magnetic field magnitude (black curve) and the reduced ion distribution function. See details in text.

3. Overshoot and ion reflection

Ion reflection in the shock front occurs due to the combined influence of the electric and magnetic fields. In order to elucidate the effect of both let us consider the motion of an ion entering the ramp. The most part of the magnetic field jump occurs from the beginning of the ramp to the maximum of the overshoot. This is also the region where the cross-shock electric field acts. Let the shock normal be in the $x$ direction and the upstream magnetic field reside in the $x-z$ plane. The equations of motion for the ion inside the ramp-overshoot region are

(3.1)\begin{gather} \frac{{\rm d} v_x}{{\rm d} t}=\frac{e}{m_i}E_x+\frac{e}{m_ic} (v_yB_z-v_zB_y), \end{gather}

(3.2)\begin{gather}\frac{{\rm d} v_y}{{\rm d} t}=\frac{e}{m_i}E_y+\frac{e}{m_ic} (v_zB_x-v_xB_z), \end{gather}

(3.3)\begin{gather}\frac{{\rm d} v_z}{{\rm d} t}=\frac{e}{m_ic} (v_xB_y-v_yB_x). \end{gather}

Here, we assume, for simplicity of the analysis, that the shock is planar and stationary. Then in NIF $E_z=0$, $E_y=V_uB_u\sin \theta /c=\text {const.}$ and $E_x=-({\rm d}\phi /{{\rm d} x})$, where $\phi$ is the cross-shock potential depending on the coordinate along the shock normal. The subscript $u$ denotes upstream and $\theta$ is the angle between the shock normal and the upstream magnetic field. At the entry to the ramp $v_{x0}-V_u$, $v_{y0}$ and $v_{z0}$ are of the order of the upstream thermal speed $v_T$. According to the observations (see, e.g. Sckopke et al. Reference Sckopke, Paschmann, Bame, Gosling and Russell1983), the incident ion distribution is usually approximated by an isotropic Maxwellian distribution. For the present analytical consideration it is not essential, $v_T$ is just the measure of the spread in the velocity space. Upon crossing the shock the ion distribution becomes strongly non-gyrotropic and gradually gyrotropizes and isotropizes. The ratio of the thermal speed to the upstream plasma speed $v_T/V_u=\sqrt {\beta _{ui}/2}/M$ and decreases with the increase of the Mach number. An ion is reflected if at some $x$ the ion velocity $v_x=0$. Up to this point $v_x>0$ and we may replace $({\rm d} / {\rm d} t)=v_x({\rm d}/{{\rm d} x})$ and integrate (3.1)–(3.3) from the entry to the ramp, $x_0$, to the reflection point $x_r$, as follows:

(3.4)\begin{gather} -\frac{1}{2}v_{x0}^2=\frac{e}{m_i}\int_{x_0}^{x_r} E_x\, {\rm d} x+\frac{e}{m_ic_i} \int_{x_0}^{x_r} (v_yB_z-v_zB_y){{\rm d} x}, \end{gather}

(3.5)\begin{gather}v_y-v_{y0}=\frac{e}{m_ic}\int_{x_0}^{x_r}\left(\frac{B_u\sin\theta V_u}{v_x}-B_z\right){{\rm d} x} + \frac{e}{m_ic}\int_{x_0}^{x_r}\left(\frac{v_zB_x}{v_x}\right){{\rm d} x}, \end{gather}

(3.6)\begin{gather}v_z-v_{z0}=\frac{e}{m_ic}\int_{x_0}^{x_r}\left(B_y- \frac{v_yB_x}{v_x}\right){{\rm d} x}, \end{gather}

or

(3.7)\begin{equation} \frac{1}{2}v_{x0}^2=\frac{e}{m_i}\phi(x_r)-\frac{e}{m_ic} \int_{x_0}^{x_r} (v_yB_z-v_zB_y)\,{{\rm d} x}.\end{equation}

In the right-hand side of this expression $v_y,v_z,B_y,B_z$ depend on $x$, so that the integral cannot be calculated exactly but can be estimated. The width of the ramp itself is of the order of the ion inertial length or smaller. The distance from the beginning of the ramp to the overshoot maximum is substantially smaller than the ion convective gyroradius $V_u/\varOmega _u$. Thus, the narrow transition approximation would be appropriate. In the expressions for $v_y$ and $v_z$ the component $v_x$ is of the order of $V_u$ throughout, so that the variations of $v_y$ and $v_z$ are proportional to the width and may be neglected in (3.7). Since the ion starts at the point where $B_y=0$, the lowest-order estimate would be

(3.8)\begin{gather} \frac{m_i}{2}v_{x}^2=\frac{m_i}{2}v_{x0}^2-e\phi_{\rm eff}, \end{gather}

(3.9)\begin{gather}\phi_{\rm eff}=\phi(x_r)-\frac{1}{c}v_{y0} \int_{x_0}^{x_r}B_z\, {\rm d} x. \end{gather}

At the reflection point $v_x=0$ and therefore

(3.10a,b)\begin{equation} \frac{m_iv_{x0}^2}{2}=e\phi_{\rm eff}, \quad v_{x0}= \sqrt{\frac{2e\phi_{\rm eff}}{m_i}},\end{equation}

which gives

(3.11)\begin{equation} \frac{V_u-v_{x0}}{v_T}=\frac{M\left(1- \sqrt{\dfrac{2e\phi_{\rm eff}}{m_iV_u^2}} \right)}{\sqrt{\beta_{ui}/2}}.\end{equation}

If the magnetic breaking is not taken into account $\phi _{\rm eff}=\phi \sim 0.5(m_iV_u^2/2e)$ (Morse Reference Morse1973; Schwartz et al. Reference Schwartz, Thomsen, Bame and Stansberry1988; Dimmock et al. Reference Dimmock, Balikhin, Krasnoselskikh, Walker, Bale and Hobara2012). For typical $\beta \sim 1$ (Farris et al. Reference Farris, Russell and Thomsen1993) and $M=4$ the ratio (3.11) is $>1.6$. That is, the reflected ions come from the tail of the incident distribution. The number of these reflected ions should be small, but even a modest increase of the effective potential may result in a significant increase of the number of reflected ions. The addition to the effective potential (3.9) due to the magnetic braking (the second term on the right-hand side) can be estimated as follows:

(3.12)\begin{equation} v_{y0} \int_{x_0}^{x_r}B_z\, {\rm d} x \sim v_TB_{\max} L,\end{equation}

where $B_{\max }$ is the maximum magnetic field in the ramp-overshoot region and $L$ is the distance from the entry to the ramp to this maximum. Without an overshoot, $B_{\max } =B_d$, and $L$ is the ramp width, which is of the order of the ion inertial length or smaller (Newbury, Russell & Gedalin Reference Newbury, Russell and Gedalin1998; Hobara et al. Reference Hobara, Balikhin, Krasnoselskikh, Gedalin and Yamagishi2010). With the overshoot $B_{\max }$ may be substantially larger than $B_d$, and $L$ may be of the order of the upstream ion convective gyroradius (Bale et al. Reference Bale, Bale, Mozer, Mozer, Horbury and Horbury2003, Reference Bale, Balikhin, Horbury, Krasnoselskikh, Kucharek, Möbius, Walker, Balogh, Burgess, Lembège, Lucek, Scholer, Schwartz and Thomsen2005; Krasnoselskikh et al. Reference Krasnoselskikh, Balikhin, Walker, Schwartz, Sundkvist, Lobzin, Gedalin, Bale, Mozer, Soucek, Hobara and Comişel2013). Thus, the contribution of the overshoot to the effective potential is significant and its presence should enhance ion reflection.

4. Model

The objective of this study is to understand what the effects of the overshoot and undershoot are. Since there is no hope of solving the equations of motion analytically, we will perform a test particle analysis in a model shock profile. We are not going to reproduce the observed profile and distributions, especially when some important parameters, like the shock width and the cross-shock potential, are either not known or cannot be determined with the precision required for quantitative comparison. We therefore choose a shock model, that is, the electric and magnetic fields, and trace ions in these fields. Tracing is done in NIF. Although at the Mach number of $M\approx 5$ the shock can be expected to be rippled, we ignore here possible deviations from planarity and time dependence. In what follows all fields depend only on one coordinate, $x$, which is the direction of the shock normal, pointing toward downstream, by definition. The non-coplanarity direction is $y$. In the analysis we use the normalized variables $\boldsymbol {B}/B_u$, $\boldsymbol {v}/V_u$, $c\boldsymbol {E}/V_uB_u$, $\varOmega _ux/V_u$ and $\varOmega _u t$ without changing the notation. The basic profile is given by

(4.1a–c)\begin{gather} B_x=\cos\theta, \quad B_z=\frac{R+1}{2}+\frac{R-1}{2}\tanh \frac{3x}{D}, \quad B_y=C_B\frac{{\rm d} B_z}{{\rm d} x}, \end{gather}

(4.2a–c)\begin{gather}E_x={-}C_E\frac{{\rm d} B_z}{{\rm d} x}, \quad E_y=\sin\theta, \quad E_z=0. \end{gather}

Here, $D$ is the ramp width in terms of the upstream convective gyroradius $V_u/\varOmega _u$, $\varOmega _u=eB_u/m_ic$ is the upstream ion (proton) gyrofrequency, $V_u$ is the upstream plasma speed in NIF and $\theta$ is the angle between the upstream magnetic field vector and the shock normal. The magnetic compression is $B_d/B_u=\sqrt {R^2\sin ^2\theta +\cos ^2\theta }$. The coefficients $C_B$ and $C_E$ are determined from the conditions

(4.3)\begin{gather} -\int_{-\infty}^\infty E_x\, {\rm d} x=s_{\rm NIF}, \end{gather}

(4.4)\begin{gather}s_{\rm NIF}-s_{\rm HT}=\tan\theta\int_{-\infty}^\infty B_y\, {\rm d} x, \end{gather}

where $s=2e\phi /m_iV_u^2$ is the normalized cross-shock potential. In order to avoid any temptation to compare with the observed shock we use slightly different parameters. The chosen shock angle is $\theta =60^\circ$ and $\beta _i=8{\rm \pi} n_uT_{iu}/B_u^2=0.5$, where $T_{iu}$ is the upstream ion temperature. The chosen Mach number is $M=4$. At these shock parameters rippling is expected to be less pronounced (Ofman & Gedalin Reference Ofman and Gedalin2013), so that the planar stationary model may be expected to be a satisfactory approximation. The magnetic compression $B_d/B_u=2.8$ is obtained from the Rankine–Hugoniot relations (see, e.g. Kennel Reference Kennel1988). The chosen ramp width is taken as one ion inertial length $D=1/M$ (Newbury et al. Reference Newbury, Russell and Gedalin1998; Hobara et al. Reference Hobara, Balikhin, Krasnoselskikh, Gedalin and Yamagishi2010), and the chosen cross-shock potentials are $s_{\rm HT}=0.1$ and $s_{\rm NIF}=0.35$ (Morse Reference Morse1973; Schwartz et al. Reference Schwartz, Thomsen, Bame and Stansberry1988; Dimmock et al. Reference Dimmock, Balikhin, Krasnoselskikh, Walker, Bale and Hobara2012; Cohen et al. Reference Cohen2019; Hanson et al. Reference Hanson, Agapitov, Mozer, Krasnoselskikh, Bale, Avanov, Khotyaintsev and Giles2019; Pope et al. Reference Pope, Gedalin and Balikhin2019; Schwartz et al. Reference Schwartz, Ergun, Kucharek, Wilson, Chen, Goodrich, Turner, Gingell, Madanian, Gershman and Strangeway2021).

Overshoot is added using

(4.5)\begin{equation} \delta B_z=B_1\left(1+\tanh\frac{3(x-x_L)}{W_L}\right) \left(1-\tanh\frac{3(x+x_R)}{W_R}\right),\end{equation}

with the additions to $B_y$ and $E_x$ using the same relations as above. Note that the width of the shock including the ramp and the overshoot is of the order of the upstream ion convective gyroradius (Bale et al. Reference Bale, Bale, Mozer, Mozer, Horbury and Horbury2003, Reference Bale, Balikhin, Horbury, Krasnoselskikh, Kucharek, Möbius, Walker, Balogh, Burgess, Lembège, Lucek, Scholer, Schwartz and Thomsen2005; Krasnoselskikh et al. Reference Krasnoselskikh, Balikhin, Walker, Schwartz, Sundkvist, Lobzin, Gedalin, Bale, Mozer, Soucek, Hobara and Comişel2013). Initially, Maxwellian distributed 40 000 ions were traced across the shock and the corresponding distribution functions $f(x,\boldsymbol {v})$, projections, and moments were numerically derived using the staying time method (Gedalin Reference Gedalin2016; Gedalin, Pogorelov & Roytershteyn Reference Gedalin, Pogorelov and Roytershteyn2021). We shall use the one-dimensional reduced distribution

(4.6)\begin{equation} f(x,v_x)=\int_{-\infty}^\infty \, {\rm d} v_y \int_{-\infty}^\infty \, {\rm d} v_z f(x,\boldsymbol{v}), \end{equation}

and the two-dimensional reduced distribution integrated over a slab $x_1< x< x_2$

(4.7)\begin{equation} f(v_x,v_y)=\int_{x_1}^{x_2} \,{{\rm d} x} \int_{-\infty}^\infty \, {\rm d}v_z f(x,\boldsymbol{v}). \end{equation}

Mean quantities are defined far downstream from the shock

(4.8)\begin{equation} \langle F(\boldsymbol{v})\rangle=\frac{\displaystyle \int F(\boldsymbol{v}) f(x\rightarrow \infty, \boldsymbol{v})\, {\rm d}^3\boldsymbol{v}}{\displaystyle \int f(x\rightarrow \infty,\boldsymbol{v})\, {\rm d}^3\boldsymbol{v}}. \end{equation}

We will be interested in the normalized parallel and perpendicular temperatures defined as follows:

(4.9)\begin{equation} T_\parallel{=} \langle((\boldsymbol{v}-\langle\boldsymbol{v}\rangle) \boldsymbol{\cdot}\hat{\boldsymbol{b}})^2\rangle, \quad T_\perp{=} \tfrac{1}{2} \langle((\boldsymbol{v}-\langle\boldsymbol{v}\rangle) \times\hat{\boldsymbol{b}})^2\rangle, \end{equation}

where $\hat {\boldsymbol {b}}$ is the direction of the downstream magnetic field. The temperatures are normalized on $m_iV_u^2$.

5. Ion distributions

We start with the basic profile without an overshoot. Figure 2 shows the one-dimensional (1-D) reduced distribution $f(x,v_x)$. There is weak ion reflection. The gyration of the downstream distribution as a whole is clearly seen, as well as the gradual gyrophase mixing. Ion phase space holes are quite pronounced. Figure 3 shows the 2-D reduced distribution integrated over the slab $1.5< x<1.6$. The halo of the ions which were reflected and crossed the shock again to the downstream region (hereafter ‘reflected-transmitted’ ions) is weak and their contribution to the downstream temperature is small. The core of the directly transmitted ions is clearly non-gyrotropic. One arrow on the plot points to the downstream drift velocity, which is approximately $V_d\approx 1/B_d$. The other one points to the maximum of the distribution (‘centre’). The distance between the two in the velocity space determines the speed of gyration of the distribution as a whole and, ultimately, the amount of perpendicular heating. The parallel heating is negligible $T_{d,\parallel }/T_{u,\parallel }=1$. The perpendicular downstream temperature is $T_{d,\perp }/m_iV_u^2\approx 0.14$, $T_{d,\perp }/T_{u,\perp }\approx 9$ and the anisotropy is $T_{d,\parallel }/T_{d,\perp }\approx 0.11$.

Figure 2. Reduced 1-D distribution $f(x,v_x)$, together with the magnetic profile.

Figure 3. Two-dimensional reduced distribution $f(v_x,v_y)$ integrated over the slab $1.5< x<1.6$.

We add an overshoot using (4.5) with $W_L=D$, $W_R=3D$, $X_L=0.5D$, $X_R=0.5D$ and $B_1=1$. The maximum magnetic field is $B_m\approx 3.86$. Since the potential closely follows the magnetic profile the maximum potential at the overshoot is $s_m\approx s_{\rm NIF}(B_m/B_d)\approx 0.48$. Figure 4 shows the 1-D reduced distribution for the profile with an overshoot. There is a substantial population of reflected ions. Ion phase space holes are clearly seen. Figure 5 shows a halo of reflected-transmitted ions which contribute substantially to the downstream temperature. The core of directly transmitted ions is clearly non-gyrotropic. Ion reflection causes parallel heating $T_{d,\parallel }/T_{u,\parallel }\approx 3.87$. Perpendicular heating dominates, $T_{d,\perp }/m_iV_u^2\approx 0.18$, $T_{d,\perp }/T_{u,\perp }\approx 11.6$. The anisotropy is weaker than in the case without overshoot, $T_{d,\parallel }/T_{d,\perp }\approx 0.33$. The higher magnetic field of the overshoot enhances ion reflection. There are more ions which are turned back before reaching the maximum of the overshoot, due to the magnetic deflection adding to the electrostatic deceleration. There are more ions which turn back after crossing the maximum of the overshoot, since the drift speed is smaller and the ratio of the gyration speed to the drift speed increases. On the other hand, the higher cross-shock potential at the ramp overshoot reduces the ion velocity just after the overshoot, which reduces the gyration speed of the bulk of the directly transmitted ions and thus reduces heating of the core. This elevated potential also reduces reflection behind the overshoot.

Figure 4. Reduced 1-D distribution $f(x,v_x)$, together with the magnetic profile.

Figure 5. Two-dimensional reduced distribution $f(v_x,v_y)$ integrated over the slab $1.5< x<1.6$.

In order to separate the effect of the potential from the effect of the magnetic overshoot, we reduce the downstream cross-shock potential to $s_{\rm NIF}=0.25$ so that the potential at the overshoot is now $s_m\approx 0.344$. Figure 6 shows the 1-D reduced distribution for the profile with an overshoot and reduced cross-shock potential. Ion reflection is weaker but still contributes substantially to the downstream heating. Figure 7 show the corresponding 2-D reduced integrated distribution. The distance from the centre to the downstream drift speed is smaller than for the case without the overshoot but larger than for the case with overshoot and original potential. Accordingly, the core heating is larger and the relative contribution of the reflected ions is smaller. The parallel heating is noticeable, $T_{d,\parallel }/T_{u,\parallel }\approx 2.43$, because of the reflected ions. Perpendicular heating dominates, $T_{d,\perp }/m_iV_u^2\approx 0.17$, $T_{d,\perp }/T_{u,\perp }\approx 11.18$, as in both previous cases. It is slightly weaker because of the reduced contribution of reflected ions. The anisotropy is weaker than in the case without overshoot but stronger than in the case with unreduced potential, $T_{d,\parallel }/T_{d,\perp }\approx 0.22$.

Figure 6. Reduced 1-D distribution $f(x,v_x)$, together with the magnetic profile.

Figure 7. Two-dimensional reduced distribution $f(v_x,v_y)$ integrated over the slab $1.5< x<1.6$.

The effects of the overshoot and the potential at the overshoot are shown in figure 8. The distribution $f_d(v_\parallel, v_\perp )$ is calculated by switching to HT sufficiently far from the shock in the downstream uniform region. Here, $v_\parallel =\boldsymbol {v}\cdot \hat {\boldsymbol {b}}$ and $v_\perp =\sqrt {v^2-v_\parallel ^2}$. The heating of the core in the perpendicular direction is the largest without overshoot (left) and smallest with the overshoot and unreduced potential (middle). The heating of the core in the parallel direction is nearly the same in all cases. The overall heating, which is just the measure of the spread in the velocity space, is determined by the contribution of the reflected ions, which is the largest with the overshoot and unreduced potential and somewhat smaller for reduced potential. Without overshoot the contribution of the reflected ions is negligible.

Figure 8. Distributions calculated in HT. (a) No overshoot. (b) Overshoot with unreduced potential. (c) Overshoot with reduced potential.

Figure 9 shows the total off-diagonal pressure

(5.1)\begin{equation} P_{xy}(x)=\int v_xv_yf(x,\boldsymbol{v})\, {\rm d}^3\boldsymbol{v}, \end{equation}

as a measure of non-gyrotropy. Upon the shock crossing the distribution becomes non-gyrotropic and gradually gyrotropizes: the oscillations of $P_{xy}$ damp with the distance from the shock. The gyrotropization is the slowest without overshoot (blue curve) and fastest with overshoot and unreduced potential (black curve).

Figure 9. The off-diagonal pressure $P_{xy}$ as a measure of non-gyrotropy. Blue: no overshoot. Black: overshoot and unreduced potential. Red: overshoot and reduced potential.