The concept of the de Hoffmann–Teller frame is revisited for a high Mach-number quasi-perpendicular collisionless shock wave. Particle-in-cell simulation shows that the local magnetic field oscillations in the shock layer introduce a residual motional electric field in the de Hoffmann–Teller frame, which is misleading in that one may interpret that electrons were not accelerated but decelerated in the shock layer. We propose the concept of the adaptive de Hoffmann–Teller (AHT) frame in which the residual convective field is canceled by modulating the sliding velocity of the de Hoffmann–Teller frame. The electrostatic potential evaluated by Liouville mapping supports the potential profile obtained by electric field in this adaptive frame, offering a wide variety of applications in shock wave studies.

Understanding collisionless shock waves remains one of the challenges in
space and astrophysical plasmas. The shock dissipation mechanism converts the
kinetic energy of the incoming flow partly into thermal energy and partly
into the energy of supra-thermal particles

The Liouville theorem formulating the phase-space density conservation has
successfully been applied to map the electron velocity distribution function
in the upstream region onto that in the shock transition layer and further in
the downstream region. This procedure, referred to as Liouville mapping,
provides the cross-shock potential that can explain the origin of the thermal
and supra-thermal populations of electrons

On the other hand, the collisionless shock is known to become more dynamic at
a sufficiently high Mach number in that the structure of the transition layer
becomes non-stationary, exhibiting various kinds of wave–particle
interactions. If strong gradients occur in electric and magnetic fields at
small scales in the shock front, one may expect that adiabatic heating of
electrons can no longer function. For example,

The demagnetization of electrons in the shock ramp is expected also for small-scale large-amplitude oscillations of the magnetic field direction. Furthermore, the de Hoffmann–Teller frame is no longer able to provide the electric field unique to the shock transition layer, since both the residual component of the motional electric field and the electrostatic field are measured simultaneously. In other words, the application of the constant sliding velocity for the de Hoffmann–Teller frame is no more valid to study the electric field nature of the shock transition layer properly. The breakdown of the validity of the de Hoffmann–Teller frame weakens the accurate measurement of the electric field for the cross-shock potential. However, the method of Liouville mapping has successfully been applied to determine the shock potential that is responsible for the electron heating. A question arises naturally: in which reference frame can we construct the shock potential profile using the electric field data?

Here, we present a numerical simulation study of the non-stationary shock. We obtain the answer that the motional electric field needs to be locally canceled within the shock transition layer by modulating the sliding velocity for the de Hoffmann–Teller frame. By doing so, it is possible to measure the electrostatic field and the shock potential for the electron heating properly as Liouville mapping does. We refer to the modulated frame as the Adaptive de Hoffmann–Teller frame (AHT). Without this correction, one may be misled to the conclusion that the electric field was acting to decelerate incoming electrons due to the dominance of residual component of the motional field in the de Hoffmann–Teller frame. We perform the one-dimensional particle-in-cell (PIC) simulation of a high Mach number, low-beta, quasi-perpendicular collisionless shock. Earlier PIC simulations have already shown that the shock wave becomes highly non-stationary under such a condition with various kinds of instabilities developing in the shock foot and the ramp regions. We track the spatial evolution of the electric field, the magnetic field, and the electrons through the shock when its transition is the steepest, and obtain the shock potential in two different ways: from the electric field measurement in the AHT frame and from Liouville mapping.

The shock wave is produced numerically using the “em1D” code

The simulation box consists of a mesh with 40 000 cells. Each cell has an
equal size, the Debye length

Figure

Snapshot of the magnetic field (the

The cross-shock potential is evaluated in two different ways: first by
integrating the electric field in the adaptive de Hoffmann–Teller frame and
second by Liouville mapping. The electric field in the adaptive de
Hoffmann–Teller frame is constructed as follows.

The shock potential is obtained by the integration of the electric field over
the spatial coordinate along the shock normal in the de Hoffmann–Teller
frame (

Cross-shock potential obtained by the three different methods: electric field integration in the global de Hoffmann Teller frame (solid curve in black in the lower panel); electric field integration in the adaptive de Hoffmann Teller frame (solid curve in black in the upper panel); and electrostatic potential obtained by Liouville mapping (in gray).

We evaluate the shock potential using the Liouville mapping of the electron
distribution function at various distances from the upstream region (the
simulation border on the left-hand side) to the shock transition layer. Liouville mapping is an alternative procedure to find the electrostatic
potential by fitting the two distribution functions using the least square
method on the assumption that the magnetic moments of individual particles
are conserved. Frame transformation into AHT system was not applied to Liouville mapping. We used Liouville mapping

The distribution function in Fig.

Additionally, the validity of Liouville mapping is examined by comparing the distributions functions using two different methods: one is the direct measurement by counting the number of particles with various velocities, and the other one is the exact mapping (without assuming energy conservation nor magnetic moment conservation) of the phase-space density associated with the individual tracked particles. We find that the two distribution functions agree with each other for nearly adiabatic particles within the accuracy of magnetic moment conservation by greater than 90 % between the initial and final stage of the mapping.

Example of the electron distribution function at the time shown in
Fig.

In the case of stationary shock, one may safely construct the de Hoffmann Teller frame and apply the method of Liouville mapping to determine the cross-shock potential. In the case of non-stationary shock, the existence and the uniqueness of the de Hoffmann Teller frame are no longer guaranteed. Nevertheless, the use of Liouville mapping is a valid approach. We find in this study that the construction of an adequate reference frame (the adaptive de Hoffmann–Teller frame) is possible for the study of electric field to validate the potential obtained by Liouville mapping by modulating the sliding velocity and correcting locally for the spatially oscillating magnetic field. Without this correction, one obtains the cross-shock potential in the de Hoffmann–Teller frame with the opposite sign, and may be misled to the conclusion that the shock potential were not accelerating but decelerating the electrons.

In the first-order picture, the high-frequency electrostatic waves should not
influence the determination of the de Hoffmann–Teller frame because the
impact of waves on the electromagnetic component

The cross-shock potential was analyzed using in situ measurements, e.g., in
the NIF frame, by

Various methods have been alternatively devised to evaluate the cross-shock
potential

These procedures are based on specific assumptions that are mutually dependent. Moreover, the irreversibility problem is not yet solved. Above all, we find that Liouville mapping is robust in that the method can be applied even to a non-stationary shock. In this way, the time evolution of the shock potential can be tracked throughout the shock reformation process. However, Liouville mapping relies on the assumption that the electrons are adiabatic, and the validity of this assumption needs to be examined by other means. In the adaptive de Hoffmann–Teller frame, the electric field can directly be associated with the electrostatic potential. This frame would be a convenient choice in order to track the evolution of the electron distribution function and that of the electric field through the shock wave.

This work was financially supported by ECSTRA/PECS and TUNED/STAR of the Romanian Space Agency under contract 11/2012, Collaborative Research Center 963 Astrophysical Flows, Instabilities and Turbulence by the German Science Foundation. This work was also supported by the Austria–Romania bilateral project ULF-MAG (Austrian Agency for International Cooperation in Education and Research, Centre for International Cooperation & Mobility under contract RO12/2014 and 741/2014). The research leading to these results has received funding from the European Community's Seventh Framework Programme under the grant agreement 313038/STORM. Topical Editor L. Blomberg thanks one anonymous referee for their help in evaluating this paper.