4.1. Velocity-space signature of ion energization

In figure 3(a), the ion distribution function includes both a component from the incoming beam of ions upstream, as well as the aforementioned ‘crescent’ population of reflected ions. For the $E_x$ contribution to the FPC, $C_{E_x}(v_x,v_y)$, at position $x = 22.9 d_i$, figure 3(b) shows that the incoming ion beam is being acted upon strongly by the cross-shock electric field $E_x$, but that $E_x$ has little effect on the reflected ion population at this position. On the other hand, for the $E_y$ contribution to the FPC, $C_{E_y}(v_x,v_y)$, at position $x = 22.9 d_i$, figure 3c) shows that the reflected ions principally interact with this component of the electric field, i.e. the motional electric field which supports the incoming $\boldsymbol {E} \times \boldsymbol {B}$ flow.

Figure 3. The ion distribution function $f_i(v_x,v_y)$ (a), and the $C_{E_x}$ (b) and $C_{E_y}$ (c) components of the FPC, (3.5) and (3.6), computed at $x = 22.9 d_i$ from the self-consistent Gkeyll simulation. Note that the FPC is computed in the shock–rest frame. While the bulk incoming ions are slowed down by the cross-shock electric field, $E_x$, we see the distribution of reflected ions gain energy due to the motional electric field, $E_y$, which supports the incoming supersonic $\boldsymbol {E} \times \boldsymbol {B}$ flow.

To understand this visual representation of the rate of ion energization over velocity space, recall that the FPC determines the rate of change of the phase-space energy density of a particular plasma species, $w_s(\boldsymbol {x},\boldsymbol {v},t) = m_s |\boldsymbol {v}|^2 f_s(\boldsymbol {x},\boldsymbol {v},t)/2$, due to the electric field. The phase-space energy density of the ions, $w_i$, can only change if the number of ions in that volume of phase space changes. Therefore, the nested blue and red crescents in figure 3(c) indicate that ions are accelerated by $E_y$ from the blue region to the red region. Conservation of particle number requires that the number of ions lost from the blue region is the same as the number gained in the red region, but because the red region is at higher velocity $v_y$, the net effect, obtained by integrating $C_{E_y}$ over velocity space $(v_x,v_y)$, is an increase in the ion phase-space energy density $w_i$. We also note that the observed $C_{E_y}$ signature is a larger amplitude than the observed $C_{E_x}$, such that $E_y$ dominates the energy exchange at this particular point in space. Furthermore, the FPC method computes the rate of change of energy density, so the rate of energization per ion in the low-density population of reflected ions is much higher in amplitude than the loss of energy per ion by the much more dense incoming beam.

As a first attempt to understand this signature, consider that the gradient length scale of the collisionless shock in our simulation is $L_{\text {shock}} \sim \rho _i$, where $\rho _i = v_{t_i}/\varOmega _{ci}$ is the ion Larmor, or gyro-, radius. Therefore, ions encountering this gradient in the magnetic field will not necessarily have closed orbits and smoothly transition downstream. Depending on an ion's gyrophase when it encounters this magnetic field gradient, the ion's new Larmor orbit may cause the ion to move back upstream, where the magnetic field magnitude is smaller. The increased Larmor radius of this reflected ion in the upstream region then allows the ion to gain energy along the motional electric field supporting the incoming $\boldsymbol {E} \times \boldsymbol {B}$ motion. This energization of the reflected ion population via $E_y$ is consistent with the well-known energization mechanism, shock-drift acceleration (Paschmann et al. Reference Paschmann, Sckopke, Bame and Gosling1982; Sckopke et al. Reference Sckopke, Paschmann, Bame, Gosling and Russell1983; Anagnostopoulos & Kaliabetsos Reference Anagnostopoulos and Kaliabetsos1994; Anagnostopoulos et al. Reference Anagnostopoulos, Rigas, Sarris and Krimigis1998; Ball & Melrose Reference Ball and Melrose2001; Anagnostopoulos et al. Reference Anagnostopoulos, Tenentes and Vassiliadis2009; Park et al. Reference Park, Ren, Workman and Blackman2013). However, to understand why shock-drift acceleration would produce the particular velocity-space signature observed in figure 3(c), we turn to a simplified analytic model to connect the well-known Lagrangian picture for shock-drift acceleration with the new Eulerian perspective granted by the FPC.

4.2. Shock-drift acceleration in an idealized perpendicular shock

We consider now a simplified reduction of the electromagnetic fields observed in our self-consistent simulation to a step function in the magnetic field,

(4.1)\begin{equation} B_z(x) = \left\{ \begin{array}{@{}cc} B_{u} & x \ge 0\\ B_{d} & x < 0 \end{array} \right., \end{equation}

with amplitude jump $B_{d}/B_{u}=4$. We will also continue to work exclusively in the shock–rest frame, where to a good approximation the motional electric field, $E_y$, is a constant through the entire shock. The value of the constant $E_y$, as well as the ion and electron plasma betas, are chosen so that the shock velocity is similar to the self-consistent simulation, $M_A = 4.9$ and $M_f = 3.0$. This reduced model corresponds to the limit $L_{\text {shock}}/\rho _i \ll 1$ and allows us to decompose the ion motion more easily between upstream and downstream gyro- and $\boldsymbol {E} \times \boldsymbol {B}$ motion. To mimic the geometry of the self-consistent simulation, we take $E_y < 0$ and $B_z > 0$ so that the inflow $\boldsymbol {E} \times \boldsymbol {B}$ is in the negative $x$ direction.

In figure 4, we plot (a) the trajectory of an ion in the $(x,y)$ plane and (b) its corresponding trajectory in $(v_x,v_y)$ velocity space in the shock–rest frame, where the colours indicate the corresponding segments of the trajectory. In the upstream region at $x>0$ (black), the black circle centred about the upstream $\boldsymbol {E} \times \boldsymbol {B}$ velocity (black star) corresponds to the Larmor orbit of the ion about the upstream inflow velocity in the $(v_x,v_y)$ plane. Upon first crossing the magnetic discontinuity to $x<0$, the ion changes to a Larmor gyration in the $(v_x,v_y)$ plane (blue) about the downstream $\boldsymbol {E} \times \boldsymbol {B}$ velocity (green star). In the larger amplitude downstream perpendicular magnetic field, the radius of the Larmor motion in the $(x,y)$ plane in the shock–rest frame is reduced (blue).

Figure 4. (a) Real space trajectory of an ion as it traverses the shock front and (b) velocity-space trajectory.

Depending on the ion's gyrophase when the ion crosses the magnetic discontinuity, the ion passes back upstream to $x>0$, and once again undergoes a Larmor orbit in the $(v_x,v_y)$ plane (red) about the upstream $\boldsymbol {E} \times \boldsymbol {B}$ velocity (black star). In this segment of the trajectory (red), the ion gains perpendicular energy in the shock–rest frame, graphically represented by the distance in velocity space of the ion from the origin of the $(v_x,v_y)$ plane. Finally, the ion will eventually cross back into the downstream region to $x<0$ (green), and resumes its Larmor orbit in the $(v_x,v_y)$ plane (green) about the downstream $\boldsymbol {E} \times \boldsymbol {B}$ velocity (green star). Without any additional crossings of the magnetic discontinuity, the ion will simply $\boldsymbol {E} \times \boldsymbol {B}$ drift downstream.

In the segment of the trajectory where the ion can gain energy, it is the motional electric field, $E_y$, that is doing positive work on the ion, exactly like in our self-consistent simulation. We note that this ion's dynamics—the reflection due to the magnetic gradient and energy gain from its traversal upstream and alignment with the motional electric field—is the well-known single-particle picture of shock-drift acceleration. In fact, this picture in velocity space, where a single ion gains energy via this reflection by a magnetic gradient, has been previously noted (Gedalin Reference Gedalin1996a). We wish now to connect this Lagrangian perspective on how a single ion gains energy from this reflection off a magnetic gradient to the Eulerian point-of-view we have from the FPC.

4.3. Velocity-space signature of shock-drift acceleration

To connect the single-particle picture of shock-drift acceleration with how a distribution of ions is energized, we employ the Vlasov-mapping technique (Scudder et al. Reference Scudder, Mangeney, Lacombe, Harvey, Wu and Anderson1986; Kletzing Reference Kletzing1994; Hull et al. Reference Hull, Scudder, Frank, Paterson and Kivelson1998; Hull & Scudder Reference Hull and Scudder2000; Hull et al. Reference Hull, Scudder, Larson and Lin2001; Mitchell & Schwartz Reference Mitchell and Schwartz2013, Reference Mitchell and Schwartz2014), described in Appendix D, to determine the velocity distribution function in our simplified model for the electromagnetic fields through the shock. We show in figure 5 the reconstructed ion distribution $f_i(v_x,v_y)$ (a) at $x = 0.4 d_i$ and the corresponding FPC $C_{E_y}(v_x,v_y)$ (b) computed from the motional electric field, $E_y$, and gradients of this reconstructed distribution function. In addition, we repeat figures 3(a) and (c), for reference in comparing the distribution function and generated velocity-space signature between the simplified model and self-consistent simulation.

Figure 5. Comparison of the reconstructed ion distribution function (a) and $C_{E_y}$ component of the FPC (b) computed from this reconstruction to the self-consistently produced ion distribution function (c) and $C_{E_y}$ component of the FPC (d) from the Gkeyll simulation. Using the Vlasov-mapping technique, we can connect the single-particle orbits (overplotted white (a) and black (b) lines) to the distribution function dynamics. Here, $C_{E_y}$ integrated over velocity space is net positive, which means the observed velocity-space signature corresponds to an energization process. We identify this particular velocity-space signature as the signature of shock-drift acceleration, energization of the reflected ions via the motional electric field in the upstream, via the connection between where in velocity space a single ion is energized and the specific region of velocity space where the strongest energy exchange is occurring.

In the reconstructed distribution function from the idealized model, we identify, in addition to the incoming upstream population centred at the upstream $\boldsymbol {E} \times \boldsymbol {B}$ velocity, a component of reflected particles that have returned upstream, exactly like in the self-consistent simulation. Overplotted on the ion distribution function and computed FPC from the Vlasov-mapping technique is the trajectory in $(v_x,v_y)$ for the ion analysed in figure 4, which shows that this reflected population and velocity-space signature are coincident with the red segment of the trajectory in figure 4. Integrating this field–particle correlation over velocity space simply yields the net rate of work done by $E_y$, $\int C_{E_y}(v_x,v_y) dv_x dv_y = j_yE_y$, and we find the integration to be positive. We thus identify the whole population of reflected ions as experiencing net energization, with the velocity-space signature of this energization process, shock-drift acceleration, given by figure 5(b,d).

We have now connected the Lagrangian picture of shock-drift acceleration with the Eulerian picture provided by the FPC technique, and we conclude this section noting that while shock-drift acceleration has been studied extensive theoretically and numerically (e.g. Gedalin Reference Gedalin1996a,Reference Gedalinb, Reference Gedalin1997; Gedalin, Newbury & Russell Reference Gedalin, Newbury and Russell2000; Park et al. Reference Park, Ren, Workman and Blackman2013; Guo et al. Reference Guo, Sironi and Narayan2014a,Reference Guo, Sironi and Narayanb; Park et al. Reference Park, Caprioli and Spitkovsky2015; Gedalin et al. Reference Gedalin, Zhou, Russell, Drozdov and Liu2018; Xu et al. Reference Xu, Spitkovsky and Caprioli2020), the velocity-space signature of shock-drift acceleration provides a new perspective on the energization of the ions in phase space via this process. In both cases, we understand that a portion of the distribution of ions is reflected via the magnetic field gradient and return upstream, where they can gain energy via the motional electric field. Although the single-particle trajectory in phase space guides our understanding of where we expect the ions to be gaining energy, using the FPC technique enables us to see clearly the exact region of phase space in which ions are being energized via shock-drift acceleration.

This confirmation of the velocity-space signature of shock-drift acceleration in a self-consistent simulation is a vitally important step for the comparison to measured velocity-space signatures of energy exchange using in situ spacecraft measurements; however, it is also interesting that the velocity-space signature of shock-drift acceleration is unchanged between the idealized model and a self-consistent simulation given the additional physics of the self-consistent simulation: the finite shock width and cross-shock electric field. We explore the reasons for the excellent agreement despite these two key differences between the simulation and the idealized model in Appendix E, where we find the cross-shock electric field assists in reflecting ions, which allows the ions to traverse further back upstream and gain additional energy via shock-drift acceleration. Thus, while the combination of the finite shock width and cross-shock electric field quantitatively changes the population of ions that are reflected, the qualitative signature of energization in velocity space via shock-drift acceleration remains unchanged.

5.1. Velocity-space signature of electron energization

We now examine the energization of the electrons by the simulated perpendicular collisionless shock. Similar to figure 3 for the ions, figure 6(a) shows the electron distribution function $f_e(v_x,v_y)$, figure 6(b) shows the $C_{E_x}$ and figure 6(c) shows the $C_{E_y}$ components of the FPC, (3.5) and (3.6). As shown in figure 1(e), the electron distribution broadens through the entire shock ramp, so we plot in figure 6 the results of the FPC analysis at $x_B = 21.8 d_i$, where the cross-shock electric field peaks. In figure 6(a), the centre of the distribution in the shock–rest frame is displaced away from the origin to $v_x<0$ and $v_y<0$ due to the particle drifts in the varying electric and magnetic fields through the shock ramp. Because the thermal width of the electron velocity distribution is much larger than the net drift of the distribution in the shock-rest frame, computing the $C_{E_x}$ and $C_{E_y}$ correlations using (3.5) and (3.6) leads to the qualitative ‘two-lobed’ velocity-space signatures observed in figure 6(b,c). The small drifts—i.e. $|v_x/v_{te}| \ll 1$—lead to a slight asymmetry of the two-lobed structure, so we overplot contours of constant $C_{E_{x,y}}$ to make these slight asymmetries more visually apparent. Although the gain (red) and loss (blue) of electron energy largely cancels out upon integration over velocity space $(v_x,v_y)$, asymmetries in the two lobes lead to a non-zero net energization.

Figure 6. The electron distribution function $f_e(v_x,v_y)$ (a), and the $C_{E_x}$ (b) and $C_{E_y}$ (c) components of the FPC, (3.5) and (3.6), computed at $x_B = 21.8 d_i$ from the self-consistent Gkeyll simulation. Note that the FPC is computed in the shock–rest frame. The contours on the FPC plots, which are the same in both $C_{E_x}$ and $C_{E_y}$, make clear that $E_y$ leads to a net loss of electron energy, whereas $E_x$ yields a net increase of electron energy.

In figure 6(b), the asymmetry of the velocity-space signature leads to a net energization of the electrons by the cross-shock component of the electric field $E_x$. Additionally, in contrast to the shock-drift acceleration of the ions by the motional electric field $E_y$ seen in figure 3(c), we see in figure 6(c) that the electrons experience a net loss of energy due to the $E_y$ component of the electric field. As with the ion analysis, we now turn to an idealized model for the electron dynamics through the shock layer to understand these velocity-space signatures for the electron energization.

5.2. Adiabatic heating in an idealized perpendicular shock

Unlike the ions for which $L_{\text {shock}} \lesssim \rho _i$, the electron gyroradius is much smaller than the gradient length scale of the shock, $L_{\text {shock}} \gg \rho _e$, and we thus expect the electrons to stay well magnetized through the shock. Therefore, we adopt a simplified model for the electron dynamics through the shock by taking a shock ramp with a linearly increasing magnetic field over a length $L= 2 d_i$ and mass ratio $m_i/m_e=1836$, which satisfy the limit $L_{\text {shock}} \gg \rho _e$. The other parameters are the same as the simple shock model used for the ion analysis in § 4.2: a magnetic field increase of $B_{d}/B_{u}=4$, a constant and uniform motional electric field $E_y<0$ in the shock–rest frame, with the same ion and electron plasma betas such that the shock velocity is comparable to the self-consistent simulation $M_A = 4.9$ and $M_f = 3.0$. Note that this idealized model for the electrons has no cross-shock component of the electric field, $E_x=0$; the implications of this choice are discussed at length in the upcoming subsections.

In this model, the increase in the magnetic field magnitude through the ramp leads to a steady decrease in the $\boldsymbol {E} \times \boldsymbol {B}$ velocity as the plasma flows through the shock transition. In addition, the gradient of the magnetic field magnitude in the shock-normal direction induces a $\boldsymbol {\nabla } B$ drift in the $+y$ direction. In figure 7(a), we plot the profile of the perpendicular magnetic field $B_z(x)$ (blue) and the motional electric field $E_y(x)$ (red) along the shock normal direction, and figure 7(b) shows the trajectory of an electron in the $(x,y)$ plane as it flows through the shock ramp over $0 \le x/d_i \le 2$. The trajectory plot shows clearly the $\boldsymbol {\nabla } B$ drift in the $+y$ direction. As the electrons flow through the shock ramp over $0 \le x/d_i \le 2$ and undergo a $\boldsymbol {\nabla } B$ drift in the $+y$ direction, figure 7(c) shows that the net energization for a distribution of electrons $j_y E_y$ is positive.

Figure 7. (a) Profiles along the shock normal direction of the perpendicular magnetic field $B_z$ (blue) and the motional electric field $E_y$ (red), (b) trajectory in the $(x,y)$ plane of an electron as it traverses the finite-width ramp in the magnetic field and (c) the rate of work done by the electric field on the distribution of particles $j_y E_y$.

In the region where the perpendicular magnetic field changes magnitude, $0 \le x/d_i \le 2$, the $\boldsymbol {\nabla } B$ drift is anti-aligned with the motional electric field, thus leading to net energization of electrons. In fact, the rate of energization of the electrons by the $\boldsymbol {\nabla } B$ drift in the motional electric field is precisely that needed to conserve the first adiabatic invariant of the electron, i.e. the magnetic moment $\mu = m_e v_\perp ^2/2 B_z$. This energization via conservation of the electron's adiabatic invariant is thus often referred to as adiabatic heating.

We can show the relationship between the energization via the $\boldsymbol {\nabla } B$ drift and the conservation of the electron's magnetic moment by considering the change in the perpendicular kinetic energy of the electrons,

(5.1)\begin{equation} \frac{\textrm{d} m_e v_\perp^2/2}{\textrm{d} t}=q_e u_{\boldsymbol{\nabla} B} E_y,\end{equation}

where the magnitude of the $\boldsymbol {\nabla } B$ drift in the $+y$ direction is given by

(5.2)\begin{equation} u_{\boldsymbol{\nabla} B}= \frac{m_e v_\perp^2}{2 q_e B_z} \left(\frac{1}{B_z} \frac{\partial B_z}{\partial x} \right).\end{equation}

For the static fields in this model, the total time derivative is dominated by the $\boldsymbol {E} \times \boldsymbol {B}$ velocity, $\textrm {d} / \textrm {d} t = \partial /\partial t + u_x \partial /\partial x = u_{E \times B} \partial /\partial x$. Substituting $u_{E \times B}=E_y/B_z$, we can manipulate (5.1) to obtain

(5.3)\begin{equation} \frac{\partial }{\partial x} \frac{m_e v_\perp^2}{2B_z}= \frac{\partial \mu }{\partial x}=0,\end{equation}

proving that the electron's magnetic moment $\mu$ is conserved. As before, we now wish to connect this single-particle, Lagrangian picture of adiabatic heating with the Eulerian picture provided by the FPC technique.

5.3. Cross-shock electric field impact on velocity-space signature of adiabatic heating

We again employ the Vlasov-mapping technique, as in § 4.3, to reconstruct the electron distribution function through the idealized shock model and the resulting FPC velocity-space signatures. Figure 8(a) presents the resulting electron distribution function $f_e(v_x,v_y)$ and figure 8(b) shows the $C_{E_y}$ correlation at position $x_A = 1.8 d_i$ in the model, where the velocity-integrated energy transfer rate $j_y E_y$ is positive, as seen in figure 7(c). Note that the small shift (relative to $v_{te}$) of the distribution to $v_y>0$, due to the $\boldsymbol {\nabla } B$ drift, leads to an asymmetry in the velocity-space signature because of the $v_y^2$ weighting in (3.6) for $C_{E_y}$.Footnote ⁷ Although a large part of the energy transfer rate represented by this two-lobed velocity-space signature cancels out upon integration over $v_y$, the slight asymmetry leads to a net positive energization of the electrons, which yields $j_y E_y>0$, as plotted in figure 7(c).

Figure 8. Comparison of the reconstructed electron distribution function (a) and $C_{E_y}$ component of the FPC (b) computed from this reconstruction to the self-consistently produced electron distribution function (c) and $C_{E_y}$ component of the FPC (d) from the Gkeyll simulation. Unlike with the ion comparison presented in figure 5, the model $C_{E_y}$ displays the opposite asymmetry from the $C_{E_y}$ computed from the self-consistent simulation. Even though the signatures are qualitatively similar, this first computation of $C_{E_y}$ suggests that $E_y$ is responsible for a net loss of energy through the shock.

We compare the predicted velocity-space signature of electron adiabatic heating from the idealized shock model in figure 8(b) to the Gkeyll simulation results, where we plot in figure 8(c) the $f_e(v_x,v_y)$ and figure 8(d) the $C_{E_y}$ from the simulation at position $x_B=21.8 d_i$. While the correlation $C_{E_y}$ from the simulation has the same qualitative, two-lobed structure indicative of drift energization of the electrons, the net drift of the electron velocity distribution has $v_y<0$, and therefore the resulting asymmetry in $C_{E_y}$ has the opposite overall sign, which leads to a net loss of energy for the electrons due to $E_y$. How can we reconcile these apparently contradictory results for the electron energization by the motional electric field $E_y$, particularly given that we have shown in (5.3) that the $\boldsymbol {\nabla } B$ drift in the $+y$ direction leads to adiabatic heating of the electrons?

We can check if the adiabatic invariant of a distribution of electrons,

(5.4)\begin{equation} \mu_e= \frac{1}{\displaystyle \int f_e \, \textrm{d} \boldsymbol{v}} \left\{ \int \frac{m_e [ (v_x - u_x)^2 + (v_y - u_y)^2 ]}{2 B_z} f_e \, \textrm{d} \boldsymbol{v} \right\},\end{equation}

is constant through the shock, and whether our intuition about how the electron temperature should increase through a magnetic field gradient has merit. Note that in the 1D-2V geometry of the simulated perpendicular shock where both velocity coordinates are perpendicular to the magnetic field, the adiabatic invariant of the distribution of electrons can be simplified to

(5.5)\begin{equation} \mu_e = \frac{T_{{\perp},e}}{B_z}, \end{equation}

where $T_{\perp ,e}$ is the temperature of the electrons perpendicular to the magnetic field. Indeed, as shown in figure 9, the adiabatic invariant for the distribution of electrons, $\mu _e$, is well conserved through the shock transition.

Figure 9. The electron adiabatic invariant, $\mu _e = T_\perp /B_z$ (blue solid), the electron temperature (red solid) and the magnetic field (black dashed) normalized to their value upstream and plotted through the shock. The electron temperature rises commensurate with the compression of the magnetic field such that the electron $\mu _e$ is well conserved through the shock.

This result suggests that the cross-shock electric field is complicating the electron dynamics and energy exchange through the shock. Here we can leverage the Vlasov-mapping model to explore the effect of the cross-shock electric field on the energetics of the electrons. Figure 10(b) presents a comparison of the electron trajectories, figure 10(c) shows the rate of energization of the electrons by the electric field $\boldsymbol {j}_e \boldsymbol {\cdot } \boldsymbol {E}$ and figure 10(d) presents the cumulative electron energization integrated from upstream $\int _{x_{\textrm {up}}}^x \,\textrm {d} \boldsymbol {j}_e \boldsymbol {\cdot } \boldsymbol {E}$ for two models: (i) a ‘full model’ (solid) which integrates electron trajectories in the self-consistently produced electromagnetic fields in figure 10(a); and (ii) a ‘zero $E_x$ model’ (dashed), in which we artificially set the cross-shock electric field to zero.

Figure 10. (a) Electromagnetic fields approximated from the self-consistent Gkeyll simulation. (b) Example electron trajectories for full model (black) and zero $E_x$ model (blue), which show qualitatively different drifts in the $y$ direction. (c) Rate of work done by the components of the electric field, $j_yE_y$ (red) and $j_xE_x$ (blue) for the full model (solid) and zero $E_x$ model (dashed), along with total $\boldsymbol {j} \boldsymbol {\cdot } \boldsymbol {E}$ (black). Note that the total energization (black, solid and dashed) is the same for both cases. (d) Cumulative work done integrated from upstream $\int _x \boldsymbol {j} \boldsymbol {\cdot } \boldsymbol {E}$. The electrons experience adiabatic heating in both cases, although the detailed mechanisms of energization involve qualitatively different drifts.

The trajectories in figure 10(b) show a clear qualitative difference between the zero $E_x$ model and the full model: in the zero $E_x$ model (blue), the transverse drift through the shock ramp is relatively weak and in the $+y$ direction, whereas the full model (black) yields a much larger amplitude drift in the opposite direction. This qualitative difference in the transverse drift direction explains the stark differences in $C_{E_y}$ between the idealized model in figure 8(b) and the simulation in figure 8(d).

Looking at the rate of electron energization by the electric field in figure 10(c), we indeed see that $j_{ye} E_y$ (red dashed) is positive for the zero $E_x$ model (and is the only means of energization of the electrons because $E_x=0$), but $j_{ye} E_y$ is negative for the full model (red solid). However, when the energization by $j_{xe} E_x$ (blue solid) is combined with $j_{ye} E_y$ (red solid), the net rate of energization $\boldsymbol {j}_e \boldsymbol {\cdot } \boldsymbol {E}$ of the two models is exactly the same (black solid and red dashed overlap). Therefore, although the dynamics of the electrons differ qualitatively in the presence or absence of the cross-shock electric field, their energization is the same in either case. To explain this puzzling finding, we exploit the limit $L_{\text {shock}} \gg \rho _e$ to execute a guiding-centre drift analysis of the electron energization.

5.4. Guiding-centre drift analysis of electron energization

In the idealized model presented in § 5.2, there are only two drifts: an $\boldsymbol {E} \times \boldsymbol {B}$ drift in the $x$ direction, due to a constant $E_y$ through the magnetic field ramp, and a $\boldsymbol {\nabla } B$ drift in the $y$ direction, due to the linearly increasing magnetic field. The introduction of a cross-shock electric field, along with the transition from a single-particle picture to a distribution of particles, adds several new drifts to the full list of potentially dynamically important drifts. We now not only have an $\boldsymbol {E} \times \boldsymbol {B}$ drift that has a component in the $x$ direction, due to the motional electric field $E_y$ supporting the incoming supersonic flow, but also a new $y$ component, due to the cross-shock electric field $E_x$. For a distribution of electrons the $\boldsymbol {\nabla } B$ drift in the $y$ direction is modified from its single particle form,

(5.6)\begin{equation} u_{\boldsymbol{\nabla} B}^{SP} = \frac{m_e v_{{\perp},e}^2}{2 q_e B^2_z} \frac{\partial B_z}{\partial x} \hat{\boldsymbol{y}}, \end{equation}

to

(5.7)\begin{equation} u_{\boldsymbol{\nabla} B} = \frac{1}{q_e n_e} \frac{p_{{\perp},e}}{B^2_z} \frac{\partial B_z}{\partial x} \hat{\boldsymbol{y}}; \end{equation}

where $p_{\perp ,e}$ is the electron perpendicular pressure,

(5.8)\begin{equation} p_{{\perp},e} = \frac{1}{2} m_e \int (\boldsymbol{v}_\perp{-} \boldsymbol{u}_{e \perp})^2 f_e \, \textrm{d} \boldsymbol{v}. \end{equation}

We now must also consider a polarization drift in the $x$ direction, and we note that the total time derivative as the electrons flow through the shock is dominated by the convective contribution $\textrm {d} / \textrm {d} t = \partial /\partial t + \boldsymbol {U} \boldsymbol {\cdot } \boldsymbol {\nabla } \simeq U_x \partial /\partial x$, which gives

(5.9)\begin{equation} u_{\textrm{d} E/\textrm{d} t} = \frac{1}{\varOmega_{cs} B_z} \frac{\textrm{d} \boldsymbol{E}}{\textrm{d} t} \sim \frac{1}{\varOmega_{cs} B_z} U_x \frac{\partial E_x}{\partial x} \hat{\boldsymbol{x}}. \end{equation}

Finally, we have the magnetization drift, $\boldsymbol {\nabla } \times \boldsymbol {M}$, a bulk drift in the $y$ direction due to the increasing density through the shock ramp,

(5.10)\begin{equation} u_{\boldsymbol{\nabla} \times \boldsymbol{M}} = \frac{\boldsymbol{\nabla} \times \boldsymbol{M}}{q_e n_e} =\frac{1}{q_e n_e} \boldsymbol{\nabla} \times \left(- \frac{p_{{\perp}, e}}{B_z} \right) = \frac{1}{q_e n_e} \frac{\partial }{\partial x} \left(\frac{p_{{\perp},e}}{B_z} \right)\hat{\boldsymbol{y}}, \end{equation}

where the magnetization vector $\boldsymbol {M}$ (Hazeltine & Waelbroeck Reference Hazeltine and Waelbroeck1998) is given by

(5.11)\begin{equation} \boldsymbol{M} ={-}p_\perp \frac{\boldsymbol{B}}{|\boldsymbol{B}|^2}. \end{equation}

Although there can, in principle, be a polarization drift in the $y$ direction due to the variation in $E_y$ through the shock, in the shock rest frame $E_y$ changes very little, so this drift is negligible. We note that the $\boldsymbol {\nabla } B$ drift and magnetization drift can be combined to form the diamagnetic drift,

(5.12)\begin{align} u_{\mbox{diamag}} & = u_{\boldsymbol{\nabla} B} + u_{\boldsymbol{\nabla} \times \boldsymbol{M}} = \frac{1}{q_e n_e} \left[ \frac{p_{e \perp}}{B^2_z} \frac{\partial B_z}{\partial x} + \frac{\partial }{\partial x} \left(\frac{p_{e \perp}}{B_z} \right) \right], \nonumber\\ & = \frac{1}{q_e n_e} \frac{1}{B_z} \frac{\partial p_{e \perp}}{\partial x} ={-}\frac{1}{q_e n_e} \frac{\boldsymbol{\nabla} p_{e \perp} \times \boldsymbol{B}}{|\boldsymbol{B}|^2}. \end{align}

This generalization from the single-particle picture to a distribution of particles is somewhat subtle and often dubbed Spitzer's paradox from the early work on plasma equilibria in a fusion context (Spitzer Reference Spitzer1952, Reference Spitzer1962; Qin et al. Reference Qin, Tang, Rewoldt and Lee2000). While the concept of pressure is ill-defined for a single particle, magnetic field gradients must inevitably be balanced by pressure gradients in equilibrium because the $\boldsymbol {\nabla } B$ drift depends on the particles’ velocity, and thus different parts of the distribution of particles will experience different $\boldsymbol {\nabla } B$ drifts. As a consequence, the pressure of the plasma must change through the magnetic field gradient, presuming of course that the plasma is magnetized. To understand this generalization from the single-particle picture to a distribution of particles, a detailed derivation of these drifts from the first moment of the Vlasov equation can be found in Appendix F.

We plot these drifts in the shock-rest frame in figure 11(a) as a function of the distance through the shock, which shows that there is a clear ordering of the magnitude of these drifts: both components of the $\boldsymbol {E} \times \boldsymbol {B}$ drift are dominant, the $\boldsymbol {\nabla } B$ and magnetization $\boldsymbol {\nabla } \times \boldsymbol {M}$ drifts provide equal contributions at a smaller amplitude, and the polarization drift is yet smaller. To demonstrate that the sum of these drifts indeed completely describes the bulk flow of the plasma in the shock–rest frame, we show in figure 11(b) that the first moment of the electron distribution agrees well with the sum of the drifts for both $x$ and $y$ components.

Figure 11. (a) A comparison of the major drifts through the shock evaluated in the shock–rest frame of reference: (i) $\boldsymbol {E} \times \boldsymbol {B}$ drift in $x$ (black) and $y$ (green dashed), (ii) the $\boldsymbol {\nabla } B$ drift in $y$ (blue), (iii) the magnetization drift in $y$ (red dashed), and (iv) the polarization drift in $x$ (magenta dashed–dotted). We check that these drifts sum to the total first moment computed from the electron distribution function (b) as well as determine how each of these drifts contributes to the overall energy exchange, $\boldsymbol {j}_e \boldsymbol {\cdot } \boldsymbol {E}$ (c) and compare the $\boldsymbol {j}_e \boldsymbol {\cdot } \boldsymbol {E}$ computed from these drifts to the total $\boldsymbol {j}_e \boldsymbol {\cdot } \boldsymbol {E}$ computed from moments of the electron distribution function. Note that in comparing how each of these drifts contributes to $\boldsymbol {j}_e \boldsymbol {\cdot } \boldsymbol {E}$, we plot the polarization drift multiplied by $E_x$ (green dashed–dotted), the $\boldsymbol {\nabla } B$ and magnetization drifts multiplied by $E_y$ in the shock-rest frame (blue and red dashed, respectively), and the total $\boldsymbol {j}_e \boldsymbol {\cdot } \boldsymbol {E}$ arising from both components of the $\boldsymbol {E} \times \boldsymbol {B}$ flow (black), as we expect the total energization due to the $\boldsymbol {E} \times \boldsymbol {B}$ flow to be zero.

The rate of energization of the electrons in the drift approximation is equal to the rate of work done by the electric field on the drifting electrons, given by

(5.13)\begin{equation} \boldsymbol{j}_{d,e} \boldsymbol{\cdot} \boldsymbol{E} = q_e n_e \boldsymbol{U}_d \boldsymbol{\cdot} \boldsymbol{E}, \end{equation}

where $\boldsymbol {U}_d$ is the total drift motion, which can be decomposed into the contributions by the individual drifts identified above in, e.g. (5.7)–(5.11). This analysis follows in the vein of studies of electron energization in reconnection using the guiding centre approximation (Dahlin, Drake & Swisdak Reference Dahlin, Drake and Swisdak2014, Reference Dahlin, Drake and Swisdak2015, Reference Dahlin, Drake and Swisdak2016, Reference Dahlin, Drake and Swisdak2017; Drake et al. Reference Drake, Arnold, Swisdak and Dahlin2019). However, we emphasize again that we have generalized from a single-particle picture to a distribution function picture as part of our goal to understand the energization of the electrons in phase space, and thus our equivalent guiding centre energization equation has additional terms, such as the energization of the plasma via the magnetization drift (5.10), which do not appear in the guiding centre energy equation of a single particle, e.g. (1) in Dahlin et al. (Reference Dahlin, Drake and Swisdak2014) and Dahlin et al. (Reference Dahlin, Drake and Swisdak2016).Footnote ⁸ We reiterate that the perspective provided by the Lagrangian point-of-view, wherein one considers the energization of individual particles, has significant merit, but that the Eulerian perspective has its own advantages, and to obtain the Eulerian point-of-view we must generalize from single particles to distributions of particles.

In figure 11(c), we plot the rate of electron energization by each of these four drifts in the shock–rest frame. The first key takeaway from this figure is that, although the two components of the $\boldsymbol {E} \times \boldsymbol {B}$ drift dominate the total drift motion, there is no net work done by the $\boldsymbol {E} \times \boldsymbol {B}$ drift as we expect because $(\boldsymbol {E} \times \boldsymbol {B}) \boldsymbol {\cdot } \boldsymbol {E}=0$. So, while the energization arising from one component of the $\boldsymbol {E} \times \boldsymbol {B}$ drift may appear large, summed over all components the energization must be zero, as shown in figure 11(c). In this regard, separately plotting the contributions to the rate of electron energization $\boldsymbol {j}_e \boldsymbol {\cdot } \boldsymbol {E}$ by the different components of the $\boldsymbol {E} \times \boldsymbol {B}$ drift can be somewhat misleading, as both components of the $\boldsymbol {E} \times \boldsymbol {B}$ flow are much larger than the other drifts.

That the full $\boldsymbol {E} \times \boldsymbol {B}$ drift leads to zero net energization of the particles also explains the puzzling finding in our single-particle modelling of the electron energization shown in figure 10(c). Although the $j_{e_x} E_x$ and $j_{e_y} E_y$ from the full model were larger than and significantly different from the zero $E_x$ model, when summed they yielded the same net rate of energization of the electrons as the zero $E_x$ model. This cancellation is exactly the result of the two components of ‘energization’ from the $\boldsymbol {E} \times \boldsymbol {B}$ flow cancelling, as we know they must. The remaining net energization is then solely from the other drifts and their alignment with the motional electric field $E_y$.

In figure 11(d), we check that the total rate of energization of the electrons by the electric field in the shock–rest frame, $\boldsymbol {j}_e \boldsymbol {\cdot } \boldsymbol {E}$, agrees with the sum of the energization by the $\boldsymbol {\nabla } B$, magnetization and polarization drifts, finding good agreement. Note that for this comparison, we are computing the electron current in the shock–rest frame from the electron distribution function, i.e.

(5.14)\begin{equation} \boldsymbol{j}_e = \int q_e (\boldsymbol{v}' - U_{\text{shock}} \hat{\boldsymbol{x}}) f_e \, \textrm{d}\boldsymbol{v}'. \end{equation}

After the comparison between the full model and zero-$E_x$ model in figure 10(c,d) revealed that the total $\boldsymbol {j}_e \boldsymbol {\cdot } \boldsymbol {E}$ was roughly equivalent between the two models, when the zero-$E_x$ model only had energy exchange, due to the alignment of the $\boldsymbol {\nabla } B$ drift with the motional electric field, we might have anticipated that the only energy gain arose from this same adiabatic heating process from the idealized model in § 5.2. Importantly though, we see from the drift analysis in figure 11(c) that the model in § 5.2 must be generalized to the case when a distribution of particles is drifting.

While the energy gain by a single electron $\boldsymbol {\nabla } B$ drifting in the model fields is exactly the energy gain required for that single electron's magnetic moment $\mu$ to be conserved, in the self-consistent simulation it is not only the $\boldsymbol {\nabla } B$ drift that ensures the electron distribution's adiabatic invariant $\mu _e$ is well conserved in the shock. We also have an equal contribution to the energization from the magnetization drift. Together, as shown by (5.12), the $\boldsymbol {\nabla } B$ and magnetization drifts are equivalent to the diamagnetic drift, $u_{\mbox {diamag}} = u_{\boldsymbol {\nabla } B} + u_{\boldsymbol {\nabla } \times \boldsymbol {M}}$, so another perspective on the electron energization via adiabatic heating is the adiabatic invariant of a distribution of electrons, $\mu _e$ in (5.4), is conserved due to the alignment of the diamagnetic drift and the motional electric field, $E_y$.

Two important questions remain: (i) what is responsible for the slight disagreement between the energy gain arising from the $\boldsymbol {\nabla } B$ and magnetization drifts and $\boldsymbol {j}_e \boldsymbol {\cdot } \boldsymbol {E}$ from velocity moments of the electron distribution function in figure 11(d)?; and (ii) even if the physics of adiabatic heating is the same with the electrons gaining energy via the alignment of drifts with the motional electric field, is the velocity-space signature of adiabatic heating the same as that predicted by the Vlasov-mapping model in figure 10(b)? To answer the first question, we have performed a more realistic mass ratio simulation, $m_i/m_e = 400$, in Appendix G where we find better agreement between the energy gain arising from the $\boldsymbol {\nabla } B$ and magnetization drifts and $\,\boldsymbol {j}_e \boldsymbol {\cdot } \boldsymbol {E}$ from velocity moments of the electron distribution function. Thus, the small disagreement between these methods of measuring the electron's energy gain in the $m_i/m_e = 100$ simulation is simply a result of the fact that the electron gyroradius is not asymptotically smaller than the shock's extent. To answer the second question, we seek a means of eliminating the $\boldsymbol {E} \times \boldsymbol {B}$ component of the energy exchange in the FPC.

5.5. Velocity-space signature of adiabatic electron heating

If the large $\boldsymbol {E} \times \boldsymbol {B}$ flows are polluting the analysis of the overall exchange of the energy, when fundamentally the electron heating principally arises from the alignment of the $\boldsymbol {\nabla } B$ and magnetization drifts with the motional electric field, we return to the velocity-space signature plotted in figures 6(c) and 8(d) to determine how we might remove the contribution from the large $\boldsymbol {E} \times \boldsymbol {B}$ flows to the FPC signal. Guided by the fact that the transverse electric field component $E_y$ governs the adiabatic heating of the electrons, we seek to eliminate the large contribution to the rate of energization associated with the $y$-component of the $\boldsymbol {E} \times \boldsymbol {B}$ drift (which is ultimately cancelled by energization associated with its $x$-component, as we show in Appendix B). Therefore, we transform to a frame of reference moving in the transverse direction at the same velocity as the $y$-component of the $\boldsymbol {E} \times \boldsymbol {B}$ drift, ${\boldsymbol {U}}_{td}= -E_x/B_z {\hat {\boldsymbol {y}}}$. We define the transverse drift frame of reference: (i) in the $x$ direction, or shock-normal direction, the shock is at rest; (ii) in the $y$ direction, or transverse direction, the frame moves at a velocity equal to the $y$-component of the local $\boldsymbol {E} \times \boldsymbol {B}$ drift in the shock–rest frame.

Critically, the electric field transforms from the shock–rest frame ($sf$, unprimed) to the transverse drift frame ($td$, double primed), as

(5.15)\begin{equation} \boldsymbol{E}''= \boldsymbol{E} + \boldsymbol{U}_{td} \times \boldsymbol{B} = \boldsymbol{E} + \left(-\frac{E_x}{B_z}{\hat{\boldsymbol{y}}}\right) \times B_z {\hat{\boldsymbol{z}}} = \boldsymbol{E} -E_x {\hat{\boldsymbol{x}}} = E_y {\hat{\boldsymbol{y}}}, \end{equation}

where we have assumed $E_z=0$ in this 1D-2V perpendicular shock. Therefore, in the transverse drift frame, the cross-shock electric field is zero, $E_x''=0$, and the motional electric field is unchanged from the shock–rest frame, $E_y''=E_y$. The $y$-component of the velocity coordinate in the transverse drift frame is

(5.16)\begin{equation} v_y''= v_y - U_{td,y} = v_y + \frac{E_x}{B_z}. \end{equation}

For completeness, the $x$ velocity coordinate and magnetic field are unchanged from the shock–rest frame, $v_x''=v_x$ and $\boldsymbol {B}''=\boldsymbol {B}$.

Although the transverse drift frame changes with position through the shock as the cross-shock electric field changes along the normal direction, one can determine this frame of reference at any position from a local, single-point measurement of the electric field. In contrast, other drifts generally depend on gradients, and therefore cannot be uniquely specified using only single-point measurements. The benefit of the transverse drift frame of reference is not only that the rate of energization associated with the total $\boldsymbol {E} \times \boldsymbol {B}$ drift is equal to zero, which is true in any frame of reference, but also that the rates of energization associated with each component of the $\boldsymbol {E} \times \boldsymbol {B}$ drift are separately zero.Footnote ⁹

Because $E_x''=0$ in the transverse drift frame, we need only examine the $y$ correlation, $C_{E_y''}(v_x'',v_y'')$, to explore the energization of the electrons. We plot in figure 12(a) the correlation $C_{E_y''}(v_x'',v_y'')$ in the transverse drift frame at position $x_B=21.8 d_i$, and compare it with the corresponding correlation $C_{E_y}(v_x,v_y)$ in the shock–rest frame shown in figure 12(b) (a repeat of figure 6c).

Figure 12. A comparison of the FPC from $E_y$, where $v_x$ and $v_y$ are shifted to the shock-rest frame and local $\boldsymbol {E} \times \boldsymbol {B}$ frame respectively, i.e. the transverse drift frame, $C_{E''_y}(v''_x, v''_y)$ (a) to our previous computation of the FPC from $E_y$ using only a frame transformation in $v_x$ to the shock–rest frame, $C_{E_y}(v_x,v_y)$ (b). Note that panel (b) is a repeat of figure 6(c) and panel (d) is a repeat of figure 8. While the previous correlation, $C_{E_y}$, suggested the electrons were losing energy in this degree of freedom, the newly transformed correlation, $C_{E''_y}$, demonstrates that the electrons are in fact gaining energy once the $\boldsymbol {E} \times \boldsymbol {B}$ motion in this degree of freedom is subtracted. This energization mechanism, caused by an alignment of the $\boldsymbol {\nabla } B$ and magnetization drifts with the motional electric field, $E_y$, is the same mechanism responsible for energizing the electrons in the idealized model, and the velocity-space signature for this adiabatic heating now exactly matches the results of the idealized model presented in §§ 5.2 and 5.3.

We note two things immediately from figure 12. First, transformation to the transverse drift frame has switched the sign of the net rate of electron energization (from integrating the correlation over velocity space) relative to the case in the shock–rest frame, made clear by counting the overplotted equally spaced contours in the blue-red, two-lobe structure of each case. Second, the correlation $C_{E_y''}(v_x'',v_y'')$ in the transverse drift frame is strikingly similar to the correlation found in figure 8(b) computed from the reconstructed distribution function from the idealized model. Both points are perhaps unsurprising, as this transformation has removed the energy exchange in the $v_y$ degree of freedom due to the $\boldsymbol {E} \times \boldsymbol {B}$ flow, and we are left with a similar energization mechanism found in the idealized model: alignment of the $\boldsymbol {\nabla } B$ and magnetization drifts with the motional electric field, $E_y$. The alignment of these two combined drifts (constituting the diamagnetic drift) with the motional electric field is exactly what is required for the electron distribution's adiabatic invariant $\mu _e$ to be well-conserved and for the electrons to gain energy through the increasing magnetic field of the perpendicular shock. Figure 12(a) contains the second key result of this paper, the velocity-space signature of adiabatic electron heating.

It is worth emphasizing that transformation to the transverse drift frame not only revealed the same velocity-space signature for adiabatic heating as in the idealized model analysed in §§ 5.2 and 5.3, but also allowed the extraction of an energization signature which was buried in the large background energy exchange from the $\boldsymbol {E} \times \boldsymbol {B}$ flow. Ultimately, the adiabatic heating via alignment of the $\boldsymbol {\nabla } B$ and magnetization drifts with the motional electric field was masked by the large oscillation of energy between the $v_x$ and $v_y$ degrees of freedom due to the two components of the $\boldsymbol {E} \times \boldsymbol {B}$ motion. The need to carefully consider the frame of reference in which the energization analysis is performed, especially due to the impact the frame of reference choice has on the cross-shock electric field, has been pointed out in previous studies (Goodrich & Scudder Reference Goodrich and Scudder1984).

One must therefore exercise extreme caution in attributing energization to a particular drift when separately considering the work done by the different components of the electric field. In figure 10(c), for example, it would be easy to attribute erroneously the energization of the electrons to the cross-shock electric field $E_x$. Rather, the energization arising from the $x$-component of the $\boldsymbol {E} \times \boldsymbol {B}$ drift and $E_x$ is exactly cancelled by a loss of energy arising from the $y$-component of the $\boldsymbol {E} \times \boldsymbol {B}$ drift and $E_y$. The net result is a much smaller positive rate of energization due to the motional electric field $E_y$ and the remaining drifts in the transverse direction.

That the addition of the magnetization drift did not qualitatively alter the velocity-space signature of adiabatic electron heating can be understood by considering the general qualitative features of drift energization. In the drift approximation for a warm electron distribution relevant to most heliospheric plasmas, it is often true that the magnitude of the drift (not including the dominant $\boldsymbol {E} \times \boldsymbol {B}$ drift, which cannot energize particles) is much less than the thermal velocity of the electrons, $U_{d,e} \ll v_{te}$. In this case, the centre of the drifting velocity distribution is offset from the origin of velocity space, but this offset will be much smaller than the thermal width of the distribution. When the correlation is computed by taking the velocity derivatives of $f_e$ and multiplying by the appropriate component of the electric field weighted by $|\boldsymbol {v}|^2$, the result will generally produce a two-lobed velocity-space signature, qualitatively similar to that shown in figure 12(a).

A careful consideration of the drifts is essential to interpret properly the mechanism responsible for the adiabatic heating. However, much of the interest in collisionless shocks is focused not on adiabatic heating and instead on identifying mechanisms of non-adiabatic heating. In fact, because the observed temperature increase in the electrons is entirely in the perpendicular temperature to conserve the electron distribution's adiabatic invariant, even if the electron response is initially adiabatic, this anisotropy will itself be a source of instabilities, such as the whistler anisotropy instability (Gary Reference Gary1993; Gary, Liu & Winske Reference Gary, Liu and Winske2011; Wilson et al. Reference Wilson III, Koval, Szabo, Breneman, Cattell, Goetz, Kellogg, Kersten, Kasper, Maruca and Pulupa2013b, Reference Wilson, Chen, Wang, Schwartz, Turner, Stevens, Kasper, Osmane, Caprioli, Bale, Pulupa, Salem and Goodrich2020), which will further complicate the energy exchange.

Likewise, the transverse drift frame relies on the electrons being magnetized so that their $\boldsymbol {E} \times \boldsymbol {B}$ motion is well-defined. While the transformation is sensible here because $L_{\text {shock}} \gg \rho _e$, there are many observations of collisionless shocks where the shock ramp is not asymptotically larger than the electron gyroradius (Hobara et al. Reference Hobara, Balikhin, Krasnoselskikh, Gedalin and Yamagishi2010) and thus would warrant caution in the application of the transverse drift frame to reveal any masked energization signatures such as the velocity-space signature of adiabatic heating in figure 12(a). In addition, the transverse-drift frame is a non-inertial frame because the $\boldsymbol {E} \times \boldsymbol {B}$ velocity is changing through the shock. While the transformation to the instantaneous transverse-drift frame at a single point in configuration space in a simulation is a perfectly reasonable frame transformation, care will be required in applying this technique to the analysis of spacecraft data, which inevitably average over a small volume of configuration space.

Nevertheless, the velocity-space signatures of the mechanisms that govern non-adiabatic heating are likely to be entirely distinct from the simple two-lobed appearance of adiabatic heating, and so will be easy to distinguish using a field–particle correlation analysis. In addition, it is useful to first characterize the ‘background’ signature of adiabatic heating, as represented by the typical velocity-space signature shown in figure 12(a). In cases where the adiabatic response produces a temperature anisotropy that drives instabilities, the energetic response, and thus the electron's velocity-space signature through the shock, is likely to be characterized by a combination of both adiabatic and non-adiabatic signatures.