2. Observations

At present, the MMS is widely regarded as the state of the art since MMS measures particles also at high resolution. The DC magnetic field and magnetic fluctuations are measured separately by different instruments. The DC component is measured by the FluxGate Magnetometer at a rate of 128 samples s$^{-1}$ in the burst mode (Russell et al. Reference Russell, Anderson, Baumjohann, Bromund, Dearborn, Fischer, Le, Leinweber, Leneman and Magnes2016). The Search-Coil Magnetometer (SCM) measures magnetic waveforms which are provided at a rate of 8192 s s$^{-1}$ (Le Contel et al. Reference Le Contel, Leroy, Roux, Coillot, Alison, Bouabdellah, Mirioni, Meslier, Galic and Vassal2016). The Spin-plane Double Probes and Axial Double Probes provide the electric field vector (EDP) at a rate of 8192 s s$^{-1}$ with nominal precision of ${\sim }1$ mV m$^{-1}$ (Ergun et al. Reference Ergun, Tucker, Westfall, Goodrich, Malaspina, Summers, Wallace, Karlsson, Mack and Brennan2016; Lindqvist et al. Reference Lindqvist, Olsson, Torbert, King, Granoff, Rau, Needell, Turco, Dors and Beckman2016; Torbert et al. Reference Torbert, Russell, Magnes, Ergun, Lindqvist, Le Contel, Vaith, Macri, Myers and Rau2016). These electric field measurements include the DC component. Level 2 data are provided at the MMS Science Data Center https://lasp.colorado.edu/mms/sdc/public/. Electron distributions are measured by the Fast Plasma Instrument (FPI) at 30 ms cadence, in the burst mode (Pollock et al. Reference Pollock, Moore, Jacques, Burch, Gliese, Omoto, Avanov, Barrie, Coffey and Dorelli2016). Figure 1 illustrates the available measurements by MMS. Figure 1(a) shows the magnetic field and electron temperature in a moderate-Mach-number, $M_A$, quasi-perpendicular, $\theta _{Bn}=65^\circ$, terrestrial bow shock crossing by MMS1, chosen rather arbitrarily from the database developed by the SHARP collaboration (Lalti et al. Reference Lalti, Khotyaintsev, Dimmock, Johlander, Graham and Olshevsky2022). The crossing occurred on 2017-11-02 at 08:29:17 UTC. Figure 1(b–d) belongs to the same crossing. Figure 1(e) is for the shock on 2020-04-22 at 16:44:00 UTC with $M_A\approx 16$ and $\theta _{Bn}=55^\circ$. The main shock transition, according to the DC magnetic field, lasts 5 s, $0< t<5$, and the increase of the electron temperature from the upstream value to the maximum occurs there (figure 1a), which means that electron heating is prompt and is determined by the fields there. Figure 1(c) shows that the whole shock transition is covered by a wavetrain of the AC magnetic field with a period of 1 s. Figure 1(d) shows that the electric field bursts are intermittent. A most abundant burst has the shape of a wavepacket of a duration of 20–30 ms and a period of 1 ms. Other typical electric waveforms are a unipolar/solitary spike or a bipolar spike, both of 1 ms duration (Goodrich et al. Reference Goodrich, Ergun, Schwartz, Wilson, Newman, Wilder, Holmes, Johlander, Burch and Torbert2018; Vasko et al. Reference Vasko, Mozer, Krasnoselskikh, Artemyev, Agapitov, Bale, Avanov, Ergun, Giles and Lindqvist2018, Reference Vasko, Wang, Mozer, Bale and Artemyev2020; Wang et al. Reference Wang, Vasko, Mozer, Bale, Artemyev, Bonnell, Ergun, Giles, Lindqvist and Russell2020, Reference Wang, Vasko, Mozer, Bale, Kuzichev, Artemyev, Steinvall, Ergun, Giles and Khotyaintsev2021; Vasko et al. Reference Vasko, Mozer, Bale and Artemyev2022). It has to be noted that conversion of the duration into spatial dependence requires thorough analysis of the wave properties. The magnetic and electric activity presented in figure 1 has been found in quasi-perpendicular and quasi-parallel shocks as well, from the lowest- to highest-Mach-number shocks in the SHARP database.

Figure 1. (a) The magnitude of the DC magnetic field and the electron temperature calculated from FPI burst mode distributions, at 30 ms cadence. (b) The reduced distribution function $f(x,v_x)$ obtained from the same FPI data. (c) One component of the SCM magnetic field measured at a rate of 8192 s s$^{-1}$. (d) One component of the EDP electric field measured at a rate of 8192 s s$^{-1}$. (e) The most abundant shape of the small-scale electric field.

3. Interaction with a single spike

The focus of this study is the effect of a single spike shape on the distribution of electrons. We numerically solve the equations of motion of electrons:

(3.1)\begin{gather} \dot{\boldsymbol{v}}={-}\frac{e}{m_e} \left(\boldsymbol{E}+\frac{1}{c}\boldsymbol{v}\times\boldsymbol{B}\right), \end{gather}

(3.2a,b)\begin{gather}\dot{\boldsymbol{r}}=\boldsymbol{v},\quad \dot{X}\equiv \frac{{\rm d}\kern0.7pt X}{{\rm d} t}, \end{gather}

where the magnetic field $\boldsymbol {B}$ is a background field and the electric field includes both the background field and the spike field. The typical scale of variation of the background fields is $\gtrsim c/\omega _{pi}$, where $\omega _{pi}=\sqrt {4{\rm \pi} ne^2/m_p}$, while the spatial scale of a spike is substantially smaller than $c/\omega _{pe}=(m_e/m_p)^{1/2}(c/\omega _{pi})$. Therefore, the background fields are taken as constant. In particular, the background electric field is zero, so that initially, electrons flow along the local magnetic field $\boldsymbol {B}=B_0(\cos \theta, 0, \sin \theta )$. Note that $\theta$ is not necessarily the same as $\theta _{Bn}$, that is, the coordinate $x$ is not necessarily along the shock normal. We consider a spike as a planar stationary structure depending only on one coordinate $E_x=E_x(x)$. In this approach, the direction of the spike propagation with respect to the local magnetic field is $\theta$; the direction of propagation with respect to the shock normal is not relevant. Three types of the small-scale electrostatic field $E_x$ are modelled.

1. A unipolar spike is modelled as a Gaussian:

(3.3)\begin{equation} E_x={-}\frac{\phi}{\sqrt{2{\rm \pi}} L}\exp\left(-\frac{(x-x_c)^2}{2L^2}\right)\end{equation}

of width $L$ and the centre at $x_c$. The potential across the shape is $\phi$. Unipolar spikes were not found in the above observations but, in principle, the whole cross-shock potential might be made of a series of such spikes, and it is at least of theoretical interest to analyse the electron interaction of such a spike.

2. A bipolar spike is modelled as a difference of Gaussians given by (3.3):

(3.4)\begin{equation} E_x={-}\frac{\phi_1}{\sqrt{2{\rm \pi}} L_1}\exp\left(-\frac{(x-x_1)^2}{2L_1^2}\right)- \frac{\phi_2}{\sqrt{2{\rm \pi}} L_2}\exp\left(-\frac{(x-x_2)^2}{2L_2^2}\right). \end{equation}

The centres of the Gaussians, $x_1$ and $x_2$, are shifted and the amplitudes $\phi _1$ and $\phi _2$ are of opposite signs. The total cross-spike potential is $\phi _1+\phi _2$, while the maximum potential is $\approx \max (|\phi _1|,|\phi _2|)$.

3. A wavepacket is modelled using the expression

(3.5)\begin{equation} E_x=E_0\cos(2{\rm \pi} x/W)\exp\left(-\frac{x^2}{2L^2}\right), \end{equation}

where $W$ is the wavelength of the wavetrain and $L$ is the width of the Gaussian envelope. The spatial scale of a single spike is much smaller than the spatial scale of the magnetic field variation and substantially smaller than the electron inertial length.

The initial electron distribution is taken as bi-Maxwellian:

(3.6)\begin{equation} f(v_\parallel,v_\perp)=\frac{1}{(2{\rm \pi})^{3/2}v_T^3} \exp\left(-\frac{(v_\parallel{-}V_d)^2}{2Av_T^2}-\frac{v_\perp^2}{2v_T^2}\right), \end{equation}

where $v_\parallel =\boldsymbol {v}\boldsymbol {\cdot }\boldsymbol {B}/|\boldsymbol {B}$, $v_\perp ^2=v^2-v_\parallel ^2$, $V_d$ is the flow velocity along the magnetic field, $v_T^2=T_\perp /m_e$ is the perpendicular thermal speed, $T_\perp$ is the perpendicular temperature and $A=T_\parallel /T_\perp$ is the anisotropy ratio, that is, the ratio of the parallel and perpendicular temperatures. The perpendicular electron thermal gyroradius is

(3.7)\begin{equation} \rho_e=\frac{v_{Te}}{\varOmega_e}=\frac{m_e}{m_i} \sqrt{\frac{m_i}{m_e}} \frac{v_{Ti}}{\varOmega_i} =\sqrt{\frac{m_e}{m_i}}\sqrt{\frac{\beta}{2}} \frac{v_A}{\varOmega_i}=\frac{c}{\omega_{pe}}\sqrt{\frac{\beta_e}{2}}, \end{equation}

where $v_{Te}=\sqrt {T_e/m_e}$ is the electron thermal speed, $\varOmega _e=eB/m_ec$ is the electron gyrofrequency, $\omega _{pe}=\sqrt {4{\rm \pi} n_ee^2/m_e}$ is the electron plasma frequency, $B$ and $n$ are the local magnetic field magnitude and electron number density, respectively, and $\beta _e=8{\rm \pi} nT_e/B^2$. For anisotropic distributions $T_e=T_\perp$. We also define $V_d=V_u/\cos \theta$, that is, the incident electron flow velocity along the $x$ direction is $V_u$, and $M=V_u/v_A$ is the Alfvénic Mach number of this flow. Note that all parameters defined above are local and not the shock parameters, e.g. the Mach number of the flow is not the shock Mach number. The parameter $V_d$ is the drift velocity of the electrons measured in the spike frame and not in the shock frame.

It is convenient to normalize:

(3.8a–c)\begin{gather} \boldsymbol{v}/V_u\rightarrow \boldsymbol{v}, \quad \boldsymbol{B}/B_u\rightarrow\boldsymbol{B}, \quad \boldsymbol{E}/(V_uB/c)\rightarrow \boldsymbol{E}, \end{gather}

(3.9a,b)\begin{gather}\varOmega_et\rightarrow t, \quad \omega_{pe}\boldsymbol{r}/c\rightarrow \boldsymbol{r}, \end{gather}

which allows rewriting the equations of motion in the form

(3.10)\begin{gather} \dot{v}_x={-}E_x+v_y\sin\theta , \end{gather}

(3.11)\begin{gather}\dot{v}_y=v_z\cos\theta-v_x\sin\theta , \end{gather}

(3.12)\begin{gather}\dot{v}_z={-}v_y\sin\theta , \end{gather}

(3.13)\begin{gather}\dot{\boldsymbol{r}}=M\left(\frac{m_e}{m_p}\right)^{1/2}\boldsymbol{v} . \end{gather}

In all these spikes electrons are demagnetized and their motion is non-adiabatic. We start tracing with incident bi-Maxwellian distribution drifting towards the spike in the positive direction of the $x$ axis. We follow electrons until they appear well behind the spike (downstream) or well ahead of the spike (backstreaming), and the distributions $f(v_\parallel, f_\perp)$ are derived in both regions. These distributions do not depend on what position the electrons are registered at. Since the fields depend only on $x$ and do not depend on time, the electron number flux in the $x$ direction should be conserved throughout, which is achieved by assigning the weight $|v_{0x}|$ to each electron. Here $\boldsymbol {v}_0$ is the initial velocity of an electron.

3.1. A unipolar spike

We start with a solitary unipolar spike with a small positive cross-spike potential. Figure 2 shows the electrostatic field and the cross-spike potential for the chosen spike. The spike parameters are: the width $L=0.01(c/\omega _{pe})$ and the amplitude corresponding to $2.4\times 10^{-3}(m_pV_u^2/2)$. Typical potentials within spikes are $\phi \lesssim 0.1 T_e$ (Kamaletdinov et al. Reference Kamaletdinov, Vasko, Artemyev, Wang and Mozer2022). For typical $T_e\sim 10$ eV and $m_pV_u^2\sim 1$ keV, one has $\phi /(m_pV_u^2/2)\sim 2\times 10^{-3}$.

Figure 2. (a) The electrostatic field of the unipolar spike used for electron tracing. (b) The corresponding cross-spike potential.

Figure 3 shows the electron distributions. There were 80 000 electrons in the initial bi-Maxwellian with $\beta _\perp =0.3$ and $A=3$. The tracing was done for $\theta _{Bn}=60^\circ$ and $M=6$. The high initial $\beta _\parallel =0.9$ results in $v_{T\parallel }/V_d=\sqrt {A\beta _\perp /2}\cos \theta (m_p/m_e)^{1/2}/M\approx 0.4$ which means that a significant fraction of the electrons in the initial bi-Maxwellian distribution move initially in the negative direction of the $x$ axis and do not encounter the spike. These electrons are excluded from the analysis. The bi-Maxwellian distribution $f_0(v_\parallel,v_\perp )$ is shown in figure 3(a), while the distribution of the forward (towards the spike) moving electrons $f_i(v_\parallel,v_\perp )$ is shown in figure 3(b). Figure 3(c) shows the distribution $f_d(v_\parallel,v_\perp )$ of the electrons that crossed the spike and proceeded further downstream without return. Figure 3(d) (empty in this case) is for the distribution $f_b(v_\parallel,v_\perp )$ of the ions that were reflected off the spike back to the upstream region and proceed further upstream without return. Of the total initial 80 000 particles 66 % encountered the spike and crossed it. We characterize the effect of the spike using the following parameters:

(3.14a–c)\begin{equation} v_{l}=\left\langle \frac{v_{{\parallel}}}{V_u}\right\rangle,\quad T_l=\left\langle \frac{(v_{{\parallel}}-v_l)^2}{V_u^2}\right\rangle,\quad T_\perp{=}\frac{1}{2} \left\langle \frac{v_\perp^2}{V_u^2}\right\rangle . \end{equation}

The averaging $\langle \cdots \rangle$ is calculated with each of the four distribution functions mentioned above. For the chosen parameters $\theta$, $M$, $\beta$ and $A$ one has $v_{0l}=1/\cos \theta \approx 1.97$, $T_{0l}\approx 22.7$, $T_{0p}=T_{0l}/3\approx 7.6$, $v_{il}\approx 4.6$, $T_{il}\approx 10.5$ and $T_{ip}\approx 7.6$. For the downstream distribution $v_{dl}\approx 5.25$, $T_{dl}\approx 8.4$ and $T_{dp}\approx 7.75$. Thus, the electrons are accelerated across the spike, cooled in the parallel direction and slightly heated in the perpendicular direction. This means that the anisotropy ratio $T_l/T_p$ decreases.

Figure 3. Interaction with the unipolar spike. (a) The initial bi-Maxwellian. (b) Only the incident electrons moving towards the spike. (c) The electrons crossing the spike and proceeding further downstream. (d) No backstreaming electrons reflected off the spike into the upstream region.

Next, we consider a bipolar spike which consists of the previous unipolar spike and its opposite with a shift of $0.02(c/\omega _{pe})$ between the centres. Figure 4 shows the electrostatic field and the cross-spike potential for the chosen bipolar spike. The potential reaches its maximum between the Gaussians but the overall potential is zero. The distributions are shown in figure 5. In this case, about 2 % of the incident electrons are returned to the upstream region and stream backward. The corresponding parameters are $v_{dl}\approx 4.7$, $T_{dl}\approx 10$, $T_{dp}\approx 7.6$, $v_{bl}\approx -0.3$, $T_{bl}\approx 0.05$ and $T_{bp}\approx 7.9$. No acceleration of the downstream electrons occurs, and the temperatures are almost the same as for the forward-moving incident electron distribution.

Figure 4. (a) The electrostatic field of the bipolar spike used for electron tracing. (b) The corresponding cross-spike potential.

Figure 5. Interaction with the bipolar spike. (a) The initial bi-Maxwellian. (b) Only the incident electrons moving towards the spike. (c) The electrons crossing the spike and proceeding further downstream. (d) The backstreaming electrons reflected off the spike into the upstream region.

Finally, we analyse the interaction of the same incident electron distribution with the wavepacket shown in figure 6. The wavelength is $0.04(c/\omega _{pe})$ and the width of the Gaussian envelope is $0.08(c/\omega _{pe})$. The maximum potential is $1.2\times 10^{-3} (m_pV_u^2/2)$. About 12 % of the incident electrons are reflected. The corresponding distributions are shown in figure 7. The parameters of the downstream and backstreaming electrons are $v_{dl}\approx 5.2$, $T_{dl}\approx 9.3$, $T_{dp}\approx 7.8$, $v_{bl}\approx -0.7$, $T_{bl}\approx 0.22$ and $T_{bp}\approx 6.5$.

Figure 6. (a) The electrostatic field of the wavepacket used for electron tracing. (b) The corresponding cross-spike potential.

Figure 7. Interaction with the wavepacket. (a) The initial bi-Maxwellian. (b) Only the incident electrons moving towards the spike. (c) The electrons crossing the spike and proceeding further downstream. (d) The backstreaming electrons reflected off the spike and proceeding into the upstream region.

In all three types of spikes, the maximal potentials are similar. All three are capable of reflecting electrons, the wavepacket being the most efficient. The backstreaming electrons have perpendicular temperatures substantially higher than parallel temperatures.

To understand the dependence on the shock angle, electron tracing was performed for the above bipolar structure and wavepacket and $\theta _{Bn}=30^\circ$. Figure 8 shows the downstream and backstreaming electron distributions for $\theta =30^\circ$. The main effect is the increase of the fraction of backstreaming ions: 3 % for the bipolar structure and 15 % for the wavepacket.

Figure 8. The downstream and backstreaming electron distributions for $\theta =30^\circ$. (a) The interaction with the bipolar structure. (b) The interaction with the wavepacket.

4. Interaction with multiple spikes

The shock front is filled with a large number of spikes, and the electron distribution is eventually formed due to interaction with many small-scale structures, as well as the mean electric field and the inhomogeneous magnetic field. The roles of the large-scale and small-scale electric fields have been elucidated in a simple model (Gedalin Reference Gedalin2020) showing that the total energetics is related to the overall cross-shock potential while the small-scale fields shape the electron distribution. The details of the effect of the spikes should be the subject of a separate study. Here we analyse the interaction of electrons with several spikes in a row, at the spatial scale where the magnetic field variations may be neglected. Figure 9 shows the electric field and the potential of 10 consecutive identical bipolar spikes and the downstream and backstreaming electron distributions for $\theta =60^\circ$ and the same $M=6$, $\beta _\perp =0.3$ and $A=3$ as above.

Figure 9. (a) The electric field and the potential of 10 consecutive bipolar spikes. (b) The downstream and backstreaming electron distributions for $\theta =60^\circ$.

Figure 10 shows the electric field and the potential of 10 consecutive identical wavepackets, as well as the downstream and backstreaming electron distributions for $\theta =60^\circ$ and the same $M=6$, $\beta _\perp =0.3$ and $A=3$ as above. The main effect of multiple encounters with spikes is the enhancement of reflection: about 7 % of backstreaming ions for the multiple bipolar structures and about 22 % for the multiple wavepackets.

Figure 10. (a) The electric field and the potential of 10 consecutive wavepackets. (b) The downstream and backstreaming electron distributions for $\theta =60^\circ$.

It is of interest to check what would happen if a mean electric field were present also. Figure 11 shows the electric field and the potential of 10 consecutive identical wavepackets and a superimposed large-scale electric field, together with the downstream and backstreaming electron distributions for $\theta =60^\circ$ and the same $M=6$, $\beta _\perp =0.3$ and $A=3$ as above. The large-scale field is modelled as follows:

(4.1)\begin{equation} E_x=E_0\exp\left(-\frac{(x-x_0)^2}{2w_0^2}\right), \end{equation}

where $E_0=-0.5(V_uB/c)$, $w_0=c/\omega _{pe}$ and $x_0$ is in the middle of the region filled with the spikes. As would be expected, the mean electric field, which accelerates electrons across the spikes downstream, reduces the fraction of backstreaming electrons from 22 % without the field to 14 % with the field.

Figure 11. (a) The electric field and the potential of 10 consecutive wavepackets and a superimposed large-scale electric field (only a part is shown). (b) The downstream and backstreaming electron distributions.

Finally, figure 12 shows the effect of the fields in figure 11 on the electron dynamics when the initial distribution is anisotropic with $A=0.3$. There is no significant difference compared with the case of $A=3$. In particular, the fraction of backstreaming electrons only slightly increases to 24 %.

Figure 12. (a) The initial bi-Maxwellian. (b) Only the incident electrons moving towards the spike. (c) The electrons crossing the spike and proceeding further downstream. (d) The backstreaming electrons reflected off the spike into the upstream region.