2 Approach: numerical ion tracing

Since we are interested in separating the effects of the distribution from all other effects, it is reasonable to control all other shock parameters. It is most efficiently done by tracing ions as test particles across a model shock. A slightly supercritical shock is expected to be a nearly planar and stationary low-Mach number shock, with a weak overshoot. Let subscripts $u$ and $d$ denote upstream and downstream, respectively. We choose the coordinates so that $x$ is along the shock normal, pointing towards downstream, and $x$–$z$ is the coplanarity plane. The magnetic field profile is described as follows:

(2.1)\begin{equation} B_z=B_u\sin\theta\left(\frac{R-1}{2}+\frac{R+1}{2}\tanh\frac{3x}{D}\right), \end{equation}

where $B_u$ is the upstream magnetic field magnitude, $\theta$ is the angle between the shock normal and the upstream magnetic field vector, and the magnetic compression is

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

The parameter $D$ gives the shock width. We do not include any overshoot in this analysis, since its influence is expected to be minor. The upstream proton gyrofrequency is $\varOmega _u=eB_u/m_pc$. The upstream proton plasma frequency is $\omega _{p}=\sqrt {4{\rm \pi} e^2n_u/m_p}$, where $n_u$ is the upstream proton number density. The ion inertial length is $c/\omega _{p}$ and the Alfvén speed is $v_A=c\varOmega _u/\omega _{p}$. Let the upstream plasma velocity along the shock normal in the shock frame be $V_u$, then the Alfvénic Mach number is $M=V_u/v_A$. The upstream convective proton gyroradius is $\rho _p=V_u/\varOmega _{u}=M(c/\omega _{p})$. In what follows we shall use dimensional parameters and variables as follows:

(2.3a–d)\begin{equation} \frac{x}{\rho_p}\rightarrow x,\quad \varOmega_u t\rightarrow t, \quad \frac{\boldsymbol{v}}{V_u}\rightarrow \boldsymbol{v}, \quad \frac{\boldsymbol{B}}{B_u}\rightarrow \boldsymbol{B}. \end{equation}

The two most convenient shock frames and the de Hoffman–Teller frame (HT), in which the upstream and downstream plasma velocities are along the upstream and downstream magnetic field vectors, respectively, and the normal incidence frame (NIF), in which the upstream plasma velocity is along the shock normal. In HT the motional electric field $E_y{({\rm HT})}=0$ vanishes identically, while in NIF it is constant $E_y^{({\rm NIF})}=V_uB_u\sin \theta /c$ throughout the shock. In both frames $E_z=0$. The shape of the cross-shock electric field $E_x^{({\rm HT})}$ is chosen as follows:

(2.4)\begin{equation} E_x^{({\rm HT})}={-}K_E\frac{{\rm d}B_z}{{\rm d} x}, \end{equation}

where the coefficient $K_E$ is determined from the cross-shock potential

(2.5)\begin{equation} \phi_{{\rm HT}}={-}\int_{-\infty}^\infty E_x\,{\rm d}x\end{equation}

and is one of the model parameters. In the dimensionless form we have

(2.6)\begin{equation} s_{{\rm HT}}=e\phi_{{\rm HT}}/(m_pV_u^2/2). \end{equation}

The non-coplanar magnetic field is chosen in a similar form,

(2.7)\begin{equation} B_y=K_B\frac{{\rm d}B_z}{{\rm d} x}, \end{equation}

while

(2.8)\begin{equation} \phi_{{\rm NIF}}-\phi_{{\rm HT}}=\frac{V_u\tan\theta}{c}\int_{-\infty}^\infty B_y\,{\rm d}x. \end{equation}

For $V_u/\cos \theta \ll c$ the transformation between the two frames is non-relativistic which means that the magnetic field can be considered the same. The parameter

(2.9)\begin{equation} s_{{\rm NIF}}=e\phi_{{\rm NIF}}/(m_pV_u^2/2) \end{equation}

is also the model parameter which determines the coefficient $K_B$. In the present analysis $s_{{\rm NIF}}=0.4$, $s_{{\rm HT}}=0.1$ and $D=(c/\omega _{p})$. The chosen magnetic compression is $B_d/B_u=2.6$ and $\theta =70^\circ$. The Mach number $M=2.9$ is derived from isotropic Rankine–Hugoniot relations (Kennel Reference Kennel1988) with $\beta _p=8{\rm \pi} n_uT_u/B_u^2=0.2$. Here $T_u$ is the upstream ion temperature.

The initial upstream distributions of ions are taken either Maxwellian

(2.10)\begin{equation} f^{(M)}_u(\boldsymbol{v})=\frac{n_u}{(2{\rm \pi})^{3/2}v_T^3} \exp \left(-\frac{(\boldsymbol{v}-\boldsymbol{V})^2}{2v_T^2}\right)\end{equation}

or a $\kappa$ distribution

(2.11)\begin{equation} f^{(K)}_u(\boldsymbol{v})=\frac{n_u}{{\rm \pi}^{3/2}v_T^3} \left(\kappa-\frac{3}{2}\right)^{{-}3/2}\frac{\varGamma(\kappa+1)}{\varGamma(\kappa-1/2)} \left(1+\frac{(\boldsymbol{v}-\boldsymbol{V})^2}{(2\kappa-3)v_T^2}\right)^{-\kappa-1}, \end{equation}

where in both cases

(2.12)\begin{gather} n_u\boldsymbol{V}=\int f_u(\boldsymbol{v})\,{\rm d}^3\boldsymbol{v}, \end{gather}

(2.13)\begin{gather}n_uv_T^2=\int (\boldsymbol{v}-\boldsymbol{V})^2f_u(\boldsymbol{v})\,{\rm d}^3\boldsymbol{v} \end{gather}

and $\varGamma$ is the $\varGamma$-function. In the present analysis $\kappa =4$ (Smith et al. Reference Smith, Renfroe, Pogorelov, Zhang, Gedalin and Kim2022).

3 Downstream distributions and heating parameters

Figure 1 shows the distributions functions $f_u(v_\parallel,v_\perp )$ and $f_d(v_\parallel,v_\perp )$ for the Maxwellian distributed incident ions. Here

(3.1)\begin{gather} v_\parallel{=}\frac{\boldsymbol{v}\boldsymbol{\cdot} \boldsymbol{B}}{|\boldsymbol{B}|}, \end{gather}

(3.2)\begin{gather}v_\perp{=}\frac{|(\boldsymbol{v}-\boldsymbol{V})\times \boldsymbol{B}|}{|\boldsymbol{B}|}, \end{gather}

where $\boldsymbol {B}$ is the local magnetic field. The upstream distributions are gyrotropic by choice. The downstream distributions become gyrotropic well beyond the shock transition. The distributions are in HT. The maximum of the downstream distribution is not at $v_\perp =0$ which shows that initially the whole distribution gyrates around the magnetic field. After gyrotropization, this gyration is the main contribution to the velocity dispersion, that is, to the temperature. There is a distinct population of reflected ions. Their density is low and there is a clear gap between the directly transmitted ions and reflected ions. The figure uses a logarithmic scale with $f_{\min }/f_{\max }>10^{-6}$. The upstream temperature of the ions is $T_u/m_pV_u^2=0.0118$. The downstream parallel temperature is $T_{d,l}=T_u$, while the downstream perpendicular temperature is $T_{d,p}/m_pV_u^2=0.0973$.

Figure 1. The upstream (left-hand side) and downstream (right-hand side) gyrotropic distributions for initially Maxwellian distributed ions, on a log scale.

Figure 2 shows the distributions functions $f_u(v_\parallel,v_\perp )$ and $f_d(v_\parallel,v_\perp )$in the same format as in figure 1. The population of the reflected ions is substantially larger, and there is no gap between the directly transmitted and reflected ions. The upstream temperature of the ions is $T_u/m_pV_u^2=0.0118$, as for the Maxwellian ions. The downstream parallel temperature is $T_{d,l}/m_pV_u^2=0.0126$, while the downstream perpendicular temperature is $T_{d,p}/m_pV_u^2= 0.1004$. The 10 % difference in the downstream perpendicular temperature is due to the reflected ions. The difference in the downstream density is negligible.

Figure 2. The upstream (left-hand side) and downstream (right-hand side) gyrotropic distributions for initially $\kappa$-distributed ions, on a log scale.

Figure 3 shows the downstream reduced distribution

(3.3)\begin{equation} f_d(v_\perp)=\int_{-\infty}^\infty f_d(v_\parallel,v_\perp)\,{\rm d}v_\parallel, \end{equation}

for the initially Maxwellian ions (blue) and initially $\kappa$ ions (red). The left-hand vertical black line stands at the maximum of the distribution of the directly transmitted population. The right-hand line marks the maximum of the reflected ions. The middle line is for the minimum of the distribution. There are no noticeable differences in the positions of the three points in both cases. For $\kappa$ ions the maximum of the reflected ions and the integral under the curve part, corresponding to these ions, are substantially larger. In the minimum the distribution function does not drop as strongly as for the Maxwellian ions.

Figure 3. The downstream reduced distribution for the initially Maxwellian ions (blue) and initially $\kappa$ ions (red). The left-hand vertical black line stands at the maximum of the distribution of the directly transmitted population. The right-hand line marks the maximum of the reflected ions. The middle line is for the minimum of the distribution.

Figure 4 shows the reduced distribution function

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

throughout the shock for initially Maxwellian ions. The gyration and relaxation of directly transmitted ions is clearly seen, as well as the gyrating reflected ions. There is a phase space hole separating the two populations. Reflected ions are seen in the upstream region up to the distance $\approx 0.5(V_u/\varOmega _u)$.

Figure 4. The reduced distribution function $f(x,v_x)$ throughout the shock for initially Maxwellian ions.

Figure 5 throughout the shock for initially $\kappa$ ions. Now the two populations, albeit clearly seen, are not separated by phase space holes. Reflected ions are seen in the upstream region up to the distance $\approx (V_u/\varOmega _u)$. However, their density there is hardly detectable in reality. The sharp drop of the distribution at $0.5(V_u/\varOmega _u)$ for Maxwellian ions versus the smooth decrease over $-(V_u/\varOmega _u)< x<-0.5(V_u/\varOmega _u)$ is because of the much faster drop of the tail for Maxwellian ions. The slower decreasing tail of $\kappa$ is also responsible for filling the phase space holes.

Figure 5. The reduced distribution function $f(x,v_x)$ throughout the shock for initially $\kappa$ ions.