ANGEOAnnales GeophysicaeANGEOAnn. Geophys.1432-0576Copernicus GmbHGöttingen, Germany10.5194/angeo-33-345-2015Adaptation of the de Hoffmann–Teller frame for quasi-perpendicular collisionless shocksComişelH.h.comisel@tu-bs.dehttps://orcid.org/0000-0002-5028-8482NaritaY.https://orcid.org/0000-0002-5332-8881MotschmannU.Institut für Theoretische Physik,
Technische Universität Braunschweig, Mendelssohnstr. 3, 38016 Brunswick, GermanyInstitute for Space Sciences, Atomiştilor 409,
P.O. Box MG-23, Bucharest-Măgurele 077125, RomaniaSpace Research Institute, Austrian Academy of Sciences,
Schmiedlstr. 6, 8042 Graz, AustriaInstitut für Geophysik und extraterrestrische Physik, Technische
Universität Braunschweig, Mendelssohnstr. 3, 38106 Brunswick, GermanyDeutsches Zentrum für Luft- und Raumfahrt,
Institut für Planetenforschung, Rutherfordstr. 2, 12489 Berlin, GermanyH. Comişel (h.comisel@tu-bs.de)17March20153333453505December201426February201527February2015This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://angeo.copernicus.org/articles/33/345/2015/angeo-33-345-2015.htmlThe full text article is available as a PDF file from https://angeo.copernicus.org/articles/33/345/2015/angeo-33-345-2015.pdf
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.
Space plasma physics (shock waves)Introduction
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 . There are
different approaches to explain electron acceleration and heating processes
in the collisionless shocks: (1) through the large-scale quasi-stationary
electrostatic potential (called hereafter the cross-shock potential) and
(2) through turbulent heating of the shock ramp
.
In this study, we limit the study to the understanding the heating process in a
quasi-static electrostatic field using a numerical simulation in order to obtain a
stationary picture of collisionless shock. Of course,
fluctuating electromagnetic and electrostatic waves may contribute to heating
,
which is beyond the scope of our current study.
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 e.g.,. The cross-shock potential is subject to the choice of the frame
. For example, it can be evaluated in the normal incident
frame (NIF) in which the flow is aligned with the shock normal direction or
in the de Hoffmann–Teller (HT) frame in which the flow is aligned with the
upstream magnetic field on the both upstream and downstream sides. These two
frames are related to each other by the Galilean transform using a constant
frame velocity tangential to the shock front (called the sliding velocity).
The utility of the de Hoffmann–Teller frame lies in that the motional (or
convective) electric field is canceled such that one can study the electric
field that arises from the electrostatic potential without being confused by
the motional field. Furthermore, the use of the de Hoffmann–Teller frame can
naturally be extended to the entire shock region so long as the electron flow
velocity and the local magnetic field remain nearly parallel to each
other within the shock transition layer .
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, have shown that
such short-scale large-amplitude structures of the electric field can switch
the adiabatic heating regime into a non-adiabatic one, in accordance with
previous theoretical studies of , and in situ
measurements, e.g., and .
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.
Particle-in-cell simulation
The shock wave is produced numerically using the “em1D” code
. Using this code, an
electron–proton plasma is injected in the one-dimensional simulation box from
the left-hand side. The plasma streams toward the right-hand side (the
positive x direction). The simulation box is set under a uniform magnetic
field at an angle of 81∘ from the shock normal (which points in the
negative x direction). The upstream magnetic field has two components,
Bx and Bz. At the boundary on the right-hand side of the simulation
box, the ions and the electrons are reflected by the “wall”. The shock wave
is formed and propagates in the negative x direction. After a sufficiently
long time, the shock wave reaches the boundary on the left-hand side and the
whole simulation box becomes the downstream region. The shock wave in the
simulation box is related to that in NIF in that the shock wave is not at
rest but propagates in the negative x direction.
The simulation box consists of a mesh with 40 000 cells. Each cell has an
equal size, the Debye length λD. Time, length, and particle
velocity are normalized to the inverse proton cyclotron frequency
(Ωi), the electron inertial length (λe),
and the speed of light in vacuum (c), respectively. The magnetic field and
electron density are scaled to their respective upstream values. The
electrostatic potential is given in units of the product of the upstream
magnetic field B0 and the Debye length. Unless noted, we use the Gaussian
units elsewhere. Five hundred particles for electrons and ions are set in each
cell. Ions are assumed to be protons, but the ion-to-electron mass ratio is
set to 1000 for efficient computation. The upstream plasma flow obtains
Alfvén Mach number 8. When the shock wave develops, the Alfvén Mach
number reaches a value of about 10. The value of plasma beta is 0.2 in the
both species with a Maxwellian initial incident particle velocity
distribution. The ratio of the electron plasma frequency to the electron
gyrofrequency is 8. Note that the values of beta and the frequency ratio are
constrained to the maximum computational load, and not set to reproducing the
collisionless shock in space such as Earth's bow shock (which is by far too
demanding compared to the computation capacity available to date). During the
simulation run, the shock wave forms and propagates, while it is highly
non-stationary in that the shock transition layer exhibits the re-formation
process.
Figure displays the snapshot (or the spatial profile) of
the magnetic field (the z or tangential component to the shock surface),
the electron density, and the electron phase-space density at the time about
4.5 ion gyroperiods. No smoothing is applied here. It is interesting to note
that the wave activity is the smallest in the maximum phase of the
reformation shock in our simulation. Highly oblique shocks may exhibit
turbulent ramp regions if the detectors have high enough time resolution in
the spacecraft observations
.
The shock re-formation reaches its maximum and the transition layer is the
narrowest with the largest transition amplitude in the magnetic field data.
The transition is clearest and sharpest from upstream to downstream in the
magnetic field and the density profile at this time. Wave activity is present
throughout the foot region (x≃1000λe), the ramp
region (x≃1020λe), and the overshoot region (x≃1040λe). In contrast, at the time one half-cycle earlier
in re-formation (at the time about 3.5 ion gyroperiods), the foot region is
extended over a larger spatial scales with higher wave amplitudes.
Snapshot of the magnetic field (the z component), the electron
density, and the subset of electron phase-space density at the time around
4.5 ion gyroperiods. The magnetic field and the electron density are
normalized to the respective background values.
Cross-shock potential
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.
E(AHT)=E(NIF)+U×B+δU×B+U×δB,
where the symbol E(NIF) denotes the electric field in the
NIF frame obtained by correcting for the shock propagating speed, U
and δU are the sliding velocity for the de Hoffmann–Teller frame
and its modulation for the adaptive frame, and B and δB
are the asymptotic upstream magnetic field (far from the shock transition) and
the spatial oscillation of the magnetic field within the transition layer. To
obtain the electric field in the adaptive frame, the sliding velocity is
modulated in the third term on the right-hand side to compensate for the
residual motional electric field originating in the
magnetic field fluctuation (the fourth term on the right-hand side). The
electric field is obtained first in the NIF frame E(NIF),
and then transformed into the de Hoffmann–Teller frame
E(HT) (using the first and the second terms) and the
adaptive de Hoffmann–Teller frame E(AHT) (including the
third and the fourth terms). We set the direction of the shock normal to the
negative x axis in our one-dimensional shock simulation in the adaptive de
Hoffmann–Teller frame, and the local residual motional electric field is
compensated with respect to the flow in the shock normal direction. It is
worth mentioning that local compensation for the motional electric field in
three dimensions is also possible.
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 (Φ(HT)) and the adaptive frame (Φ(AHT)).
The potential is set to zero in the limit of far upstream.
Figure displays the electrostatic potentials
Φ(HT) and Φ(AHT) as a function of the spatial
coordinate around the shock transition layer at the time of shock
re-formation maximum (4.5 ion gyroperiods after the simulation kickoff).
Here, we applied smoothing in the HT and AHT potential profiles because of
numerical noise. The x axis range in Fig. is extended to the border of simulation box (the wall). The electric potential should ideally be close to zero far from the shock
but remains finite because of the relatively small box setup. It is interesting to note that
the potentials have different signs. The potential has mostly negative values
in the de Hoffmann–Teller frame. This is the effect of the residual
component of the motional electric field (the third term in the equation)
which originates in the spatially oscillating magnetic field. The
representation of the potential in the de Hoffmann–Teller frame is
misleading, since one might interpret it to mean that the shock potential was
decelerating electrons. On the other hand, when the correction is undertaken
for the oscillating magnetic field (by adding the third and the fourth
terms), the shock potential is represented with the positive sign and the
association of the potential with the electron acceleration is justified.
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 in the
de Hoffmann–Teller frame as a reference potential because Liouville
mapping is a coordinate-free method. Figure displays an example
of the fitting procedure in Liouville mapping at the time 4.5 ion
gyroperiods and the spatial coordinate x=1020λe and for
the pitch angle 65∘. The electron distribution function obtained by
the PIC simulation is shown by the solid curve in black, and that obtained by
Liouville mapping is in dotted symbols in gray. The profile of the
cross-shock potential is obtained by Liouville mapping along the normal
direction from the upstream region to the shock. This potential is
over-plotted in Fig. in gray. While the maximum potential
value is different from the potential in the adaptive de Hoffmann–Teller
frame, the sign and the asymptotic behavior of the potential agreements between
Liouville mapping and the adaptive frame.
The distribution function in Fig. is obtained in the de
Hoffmann–Teller frame for accelerated and heated electrons at the shock
ramp. The sliding velocity vdHT of this frame is rather high
(vdHt≃0.2c), since the shock has a high Mach number and a
large angle from the upstream magnetic field. Incident particles with higher
perpendicular energies are most likely reflected by the magnetic mirroring at
the shock ramp, while particles with lower perpendicular energies can be
trapped in the potential well of the existing overshoot .
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 de Hoffmann–Teller frame, the spatial
coordinate x=1020λe (at the shock ramp), and the pitch
angle 65∘ obtained by the PIC simulation (solid line in black) and
that obtained by Liouville mapping (in gray diamonds).
Summary and discussion
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.
have proved that short-scale large-amplitude electric field
structures within the electric field profile lead to incoherent
heating of the electrons. In our study, we observe large-amplitude
oscillations of the magnetic field direction that result in a residual motional
electric field in the de Hoffmann–Teller frame. The local demagnetization of
the adiabatic electrons should alter the quality of Liouville mapping, but
overall, the electron thermalization remains primarily controlled by the
coherent heating.
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 ve×B is small, see e.g.,
, , and . In the second-order
picture, however, there might be a possibility that the high-frequency
electric field oscillation affects the electron bulk motion.
The cross-shock potential was analyzed using in situ measurements, e.g., in
the NIF frame, by , , and
, but determination of the cross-shock potential from
direct spacecraft observations is still far too inaccurate in the de
Hoffmann–Teller frame because the electric field is too small.
Various methods have been alternatively devised to evaluate the cross-shock
potential : evaluation using the electron fluid momentum
equation, that using the electron fluid energy equation, that using the
electron polytrope assumption, and that using Liouville mapping.
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.
Acknowledgements
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.
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