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

(2.1a–c)\begin{gather} B_x=\cos\theta_{Bn}, \quad B_y=0, \quad B_z=\sin\theta_{Bn} \end{gather}

(2.2a–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

(2.3a–c)\begin{gather} B_x=\cos\theta_{Bn}, \quad B_y=B_y(x), \quad B_z=B_z(x) \end{gather}

(2.4a–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

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

(2.6)\begin{gather}\frac{{\rm d}v_y}{{\rm d}t}=\sin\theta_{Bn}+ v_z\cos\theta_{Bn}-v_xB_z, \end{gather}

(2.7)\begin{gather}\frac{{\rm d}v_z}{{\rm d}t}=v_xB_y-v_y\cos\theta_{Bn}, \end{gather}

(2.8)\begin{gather}\frac{{\rm d}x}{{\rm d}t}=v_x. \end{gather}

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

(2.9)\begin{gather} v_x^2-v_{x0}^2={-}s(x) + 2\int_0^x (v_yB_z-v_zB_y)\,{{\rm d}x}', \end{gather}

(2.10)\begin{gather}v_y=v_{y0} + \int_0^x\left(\frac{\sin\theta_{Bn}+ v_z\cos\theta_{Bn}}{v_x}-B_z\right){{\rm d}x}', \end{gather}

(2.11)\begin{gather}v_z=v_{z0}+\int_0^x\left(B_y-\frac{v_y\cos\theta_{Bn}}{v_x}\right) {{\rm d} x}', \end{gather}

where

(2.12)\begin{equation} s(x)={-}2\int_0^x E_x(x')\,{{\rm d}x}', \end{equation}

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:

(2.13)\begin{gather} v_x^2-v_{x0}^2={-}s_{{\rm eff}}(x), \end{gather}

(2.14)\begin{gather}s_{{\rm eff}}(x)=s(x) - v_{y0}A(x), \end{gather}

(2.15)\begin{gather}A(x)=2\int_0^xB_z(x ')\,{{\rm d} x}'. \end{gather}

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)

(2.16)\begin{equation} f_u(v_x,v_y)=\frac{n_u}{(2{\rm \pi})^{3/2} v_T^3}\exp\left(-\frac{(v_x-1)^2}{2v_T^2}-\frac{v_y^2}{2v_T^2}\right), \end{equation}

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

(2.17)\begin{gather} v_{x0}^2< s_{{\rm eff}}(L) \end{gather}

(2.18)\begin{gather}s_{{\rm eff}}(L)=s(L)-v_{y0}A(L), \end{gather}

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.

Figure 1. Initial Maxwellian distribution of ions with $v_T=0.15$. The ions to the left of the red line would be reflected if the reflection process were specular. The ions to the left of the blue line are non-specularly reflected. The contours for $|\boldsymbol {v}-\boldsymbol {V}_u|=v_T$, $2v_T$ and $3v_T$ are shown.

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

(3.1)\begin{gather} \frac{{\rm d}v_x}{{\rm d}t}= v_y\sin\theta_{Bn} \end{gather}

(3.2)\begin{gather}\frac{{\rm d}v_y}{{\rm d}t}=\sin\theta_{Bn}+v_z\cos\theta_{Bn}-v_x\sin\theta_{Bn} \end{gather}

(3.3)\begin{gather}\frac{{\rm d}v_z}{{\rm d}t}={-}v_y\cos\theta_{Bn} \end{gather}

(3.4)\begin{gather}\frac{{\rm d} x}{{\rm d}t}=v_x, \end{gather}

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

(3.5)\begin{align} x& =(v_{x1}\cos^2\theta_{Bn}+v_{z1}\sin\theta_{Bn}\cos\theta_{Bn}+ \sin^2\theta_{Bn})t\nonumber\\ & \quad+ (v_{x1}\sin\theta_{Bn} -v_{z1}\cos\theta_{Bn}-\sin\theta_{Bn})\sin\theta_{Bn}\sin t\nonumber\\ & \quad- v_{y1}\sin\theta_{Bn} \cos t+v_{y1}\sin\theta_{Bn} \end{align}

(3.6)\begin{align} v_x& =(v_{x1}\cos\theta_{Bn}+v_{z1}\sin\theta_{Bn})\cos\theta_{Bn} + \sin^2\theta_{Bn} \nonumber\\ & \quad+(v_{x1}\sin\theta_{Bn} -v_{z1}\cos\theta_{Bn}-\sin\theta_{Bn})\sin\theta_{Bn}\cos t + v_{y1}\sin\theta_{Bn}\sin t \end{align}

(3.7)\begin{align} v_y& =v_{y1}\cos t_{Bn}-(v_{x1}\sin\theta_{Bn} -v_{z1}\cos\theta_{Bn}-\sin\theta_{Bn})\sin t\end{align}

(3.8)\begin{align} v_z& = (v_{x1}\cos\theta_{Bn}+v_{z1}\sin\theta_{Bn}) \sin\theta_{Bn}-\sin\theta_{Bn}\cos\theta_{Bn}\nonumber\\ & \quad-(v_{x1}\sin\theta_{Bn} -v_{z1}\cos\theta_{Bn}-\sin\theta_{Bn})\cos\theta_{Bn}\cos t -v_{y1}\cos\theta_{Bn}\sin t . \end{align}

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

(3.9)\begin{equation} f(x,v_x)=\int f(x,{\boldsymbol{v}})\,{\rm d}v_y\,{\rm d}v_z \end{equation}

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 2. Normalized 1-D reduced distribution function $f(x,v_x)$ for non-specularly reflected ions only with $M = 5$, $s = 0.5$, $\beta = 0.5$ and $\theta _{Bn} = 65^\circ$; (a) $A=0.75$, (b) $A=0.375$. Ions which have positive $v_x$, return to the shock and cross it again.

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.

Figure 3. Normalized 1-D reduced distribution function $f(x,v_x)$ for non-specularly reflected and backstreaming ions both with $M = 5$, $s = 0.5$, $\beta = 0.5$ and $A=0.75$; (a) $\theta _{Bn} = 45^\circ$, (b) $\theta _{Bn} = 40^\circ$. There is a substantial population of backstreaming ions for the smaller angle case. In the right panel, it is clearly visible that ions which have $v_x\approx 0$ mainly do not cross the shock again and escape further upstream and appear as the backstreaming ions.

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

(4.1)\begin{gather} v_x\frac{{\rm d}v_x}{{\rm d}x}=E_x+v_yB_z \end{gather}

(4.2)\begin{gather}v_x\frac{{\rm d}v_y}{{\rm d}x}=1-v_xB_z, \end{gather}

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

(4.3)\begin{gather} v_x^{2}=v_{x2}^2-s+2\int_0^xv_y B_z \,{{\rm d}x}', \end{gather}

(4.4)\begin{gather}v_y=v_{y2}+\int_0^x\left(\frac{1}{v_x}-B_z\right){{\rm d}x}', \end{gather}

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

(4.5)\begin{equation} \left|\int_0^L\left(\frac{1}{v_x}-B_z\right){{\rm d}x}' \right|< \left|\int_0^L\left(1-B_z\right){{\rm d}x}' \right|< A. \end{equation}

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

(4.6)\begin{equation} v_{x3}^{2}\approx v_{x2}^2-s+2Av_{y2}, \end{equation}

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.

Figure 4. Normalized 1-D reduced distribution function $f(x,v_y)$ for non-specularly reflected ions only with $M = 5$, $s = 0.5$, $\beta = 0.5$, $\theta _{Bn} = 65^\circ$ and $A=0.75$. Ions which havelarge positive $v_y$, return to the shock and cross it again.

Figure 5. 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–transmittedions upon crossing the ramp-overshoot region. The distribution functions are normalized so that $\int f\,{\rm d}v_x=1$.

Figure 6. The histogram of the ratio $v_{x3}/v_{x2}$. All reflected–transmitted ions are accelerated across the shock.

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

(4.7)\begin{equation} p_{xx}=m\int v_x^2 f(x,{\boldsymbol{v}}) \,{\rm d}^3{\boldsymbol{v}}, \end{equation}

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

(4.8)\begin{equation} f(x_2,{\boldsymbol{v}}_2)=f(x_3,{\boldsymbol{v}}_3). \end{equation}

The particle flux conservation requires

(4.9)\begin{equation} v_{x2}f(x_2,{\boldsymbol{v}}_2)\,{\rm d}^3{\boldsymbol{v}}_2=v_{x3}f(x_3,{\boldsymbol{v}}_3)\,{\rm d}^3{\boldsymbol{v}}_3, \end{equation}

so that

(4.10)\begin{equation} p_{3,xx}=\int v_{x3}^2f(x_3,{\boldsymbol{v}}_3)\,{\rm d}^3{\boldsymbol{v}}_3=\int v_{x3} v_{x2} f(x_2,{\boldsymbol{v}}_2)\,{\rm d}^3{\boldsymbol{v}}_2>p_{2,xx}. \end{equation}

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

(4.11)\begin{equation} p_{{\rm tot},xx}+\frac{B^2}{8{\rm \pi}}={\rm const}., \end{equation}

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.