Quasi-perpendicular supercritical shocks are characterized by the presence of
a magnetic foot due to the accumulation of a fraction of the incoming ions
that is reflected by the shock front. There, three different plasma
populations coexist (incoming ion core, reflected ion beam, electrons) and
can excite various two-stream instabilities (TSIs) owing to their relative
drifts. These instabilities represent local sources of turbulence with a wide
frequency range extending from the lower hybrid to the electron cyclotron.
Their linear features are analyzed by means of both a dispersion study and
numerical PIC simulations. Three main types of TSI and correspondingly
excited waves are identified:

Oblique whistlers due to the (so-called
“fast”) relative drift between reflected ions/electrons; the waves
propagate toward upstream away from the shock front at a strongly oblique
angle (

Quasi-perpendicular whistlers due to the (so-called “slow”) relative
drift between incoming ions/electrons; the waves propagate toward the shock
ramp at an angle

Extended Bernstein waves which also propagate in the quasi-perpendicular domain, yet are due to the (so-called “fast”) relative drift between reflected ions/electrons; the instability is an extension of the electron cyclotron drift instability (normally strictly perpendicular and electrostatic) and produces waves with a magnetic component which have frequencies close to the electron cyclotron as well as wavelengths close to the electron gyroradius and which propagate toward upstream.

A hallmark of supercritical shocks in collisionless plasmas is the presence
of a sizable ion population that is reflected off of the steep shock front.
These ions carry a substantial amount of energy: they are the source of
microturbulence within the shock front and are fundamental to the
transformation of directed bulk flow energy into thermal energy, a tenet of
shock physics. For quasi-perpendicular geometries, the reflected ions'
velocity, as seen in the normal incidence frame, is in large part directed at
90

Whistler waves are an attribute of collisionless fast-mode shocks. They have
been observed in association with shocks in space for a very long time

In this article, we present a synthetic view of the plasma microinstabilities
which can occur in the foot of supercritical quasi-perpendicular shocks as the
result of the relative drifts between incoming ions, reflected ions, and
electrons. Figure 1 illustrates the relations between the three plasma
populations in the shock's foot. The resulting instabilities cover
wavelengths from the ion inertia length to the electron gyroradius and
frequencies from the lower-hybrid to the electron cyclotron. The study can be
viewed as an extension of our previous work, which was focussed on 90

Model of ion and electron populations in the foot region of a
supercritical perpendicular shock extracted at a given time from a 1-D PIC
simulation.

Our notations are as follows:

Results of linear dispersion analysis are presented in Sect. 2 for a stable
situation without beam. We first address the cold plasma model in Sect. 2.1.
In Sect. 2.2 we show that the electrons are in a kinetic regime and that
thermal effects are very important, unless an extremely small

In the cold plasma model, the mode which can propagate in the frequency range
above the ion cyclotron frequency is the right-handed wave. It is often
referred to as the R-X mode because it becomes the extraordinary wave in
perpendicular propagation

Orientation of the wavevector

Figure 3 shows the dispersion relation in a log–log representation for
different angles

Solutions of the cold dispersion relation (without ion beam) above
the ion cyclotron frequency

Plasma parameters in normalized units.

For drawing Fig. 3, we choose parameter values such as mass ratio and
magnetization that are the same as those used in the simulations to be shown
later, namely

Ion temperature effects are negligible. Indeed, since

By contrast, the electrons' temperature has significant effects on the
dispersion properties. We first examine the impact on the real part of the
frequency, which is shown in Fig. 4 versus
wavenumber

Solutions of the dispersion relation (without ion beam) in a linear,
2-D

What about the discrepancy in the shaded area where the wavenumbers are
large? Unlike the ions, the electrons are in a kinetic regime where they can
resonate with the waves and this causes damping. We have numerically solved
the full dispersion relation for three electron temperatures and show the
results including the damping in Fig. 5. As for Fig. 4, contours describe the
real part of the frequency

Solutions of the full kinetic dispersion relation (without ion beam)
including thermal effects for various

Figure 5b and c demonstrate how the damping progressively weakens when the
electrons become colder. When

Composite figure illustrating the damping role of the resonant
velocities on the electron distribution (for

The terms in the full dispersion tensor are numerous and reflect different
contributions. As shown in Appendix A, since the electrons are magnetized,
their contribution to the tensor's elements is made of combinations of Bessel
functions and derivatives with the plasma dispersion function

We point out that the two resonant velocities

We will see in Sect. 3 that the two-stream instabilities occur at large angles (

Similar to Fig. 3 with the addition of beam modes (colored dashed
lines) described by Eq. (

Consider now the situation where an ion population is drifting at speed

We revert to Fig. 3 for the three angles considered (

When whistlers propagate obliquely (for example here

We now derive an equation that describes the locations of all WH
intersections in a

Let us now suppose that we want to destabilize waves propagating at very
large angles, close to

Species characteristics in normalized units.

Section 2.2 has stressed that a population of warm electrons substantially
damps the obliquely propagating whistlers at most wavenumbers and angles. The
damping rate even becomes very strong for large wavenumbers (see Fig. 5a). In
order to have an effective instability, this damping rate needs to be
overcome by a sufficient growth rate driven by the ion drift. We investigate
the question by numerically solving the full electromagnetic dispersion
relation for varying parameters. In the spirit of the lower-hybrid regime,
the ions are taken as unmagnetized, whereas the electrons are magnetized. We
work in the frame where the total momentum density vanishes, sometimes called
the “proper frame”. We emphasize that the drift and density values of the
reflected ion beam and the incoming ion core cannot be arbitrarily chosen yet
must satisfy the zero current condition for an application to the shock's
foot, namely

Let us first examine the case where an ion population drifts fast with
respect to the electrons, a situation represented by the reflected beam in
Fig. 1b. The cold model then predicts two zones of instability for oblique
whistlers at fixed

Effect of the electron temperature on the growth rate. Solution of
the full dispersion relation for oblique whistlers (

Numerical solutions of the dispersion relation in

Numerical solutions of the dispersion relation in

Section 3.1 has demonstrated that the relatively slow drift of the incoming
ion core/electrons enables destabilization of quasi-perpendicularly propagating
whistlers with frequencies around

Zoomed-in view similar to Fig. 9c for a slow drift case such as
the ion core with

Reverting to the schematic plot in Fig. 7a, which treats the case where an
ion population drifts fast with respect to the electrons, one sees that for
nearly perpendicular propagation (here

For the present work, the important feature to note is the quasi-absence of
damping for waves with

Waves with much larger wavenumbers and frequencies than those in
Fig. 9c can also be destabilized by the same ion beam (with

We employ a 1-D electromagnetic PIC code with periodic boundary conditions and initially load
a plasma made of the three components depicted in Fig. 1b. The simulation is
carried out in the same frame as used for the dispersion analysis, often
known as the proper frame, where the total momentum vanishes. The components,
namely electrons, ion beam, and ion core, are given identical characteristics
(defined in Table 2) to those used for the dispersion computations of
Sects. 3.2 and 4. In the spirit of lower-hybrid instabilities, the simulation
treats the ions as unmagnetized, whereas the electrons are magnetized. Since
we are restricted to a 1-D 3V code, the direction of the unstable wavevector

Two series of simulations are performed – one along an oblique direction

The simulation results presented here are preliminary due to the use of a 1-D code. Our objectives are twofold. First, we want to confirm by actual PIC results the dispersion analysis which makes up the bulk of this paper. Second, the aim is to “calibrate” the various instabilities and prepare the stage for 2-D simulations to come. In this light the simulation runs will be limited to the “linear” stage. The nonlinear stage has to be addressed with 2-D simulations since coupling between waves propagating at various angles can be expected in this regime.

Here, we set

Whistler instability for oblique propagation,

Figure 13 further documents the characteristics of the waves excited in the
simulation. In order to check whether the frequencies are those expected from
the dispersion analysis, we set up probes at fixed positions and register the
local magnetic field versus time. A sample of such a record is shown in
Fig. 13a, which exhibits oscillations with a period on the order of

Whistler instability for oblique propagation,

Quasi-perpendicular instability for

Another feature to analyze is the polarization of the excited waves. To this
end, we construct hodograms of the magnetic and transverse electric fields by
means of the following procedure. The components

Here, we set

Let us now examine the power spectra of the fields at time

Figure 14c shows the spectra of

We now investigate the frequencies involved in the two types of waves in
order to further the link between the simulation run and the dispersion
analysis presented in Sects. 3 and 4. Probes at fixed positions record the
local magnetic and electric fields versus time. Figure 15 shows a sample of
such recordings. The magnetic signal, which is illustrated by

Quasi-perpendicular instability for

The quasi-perpendicular whistlers are the result of an instability that has
been known for a long time and is generally referred to as modified
two-stream instability (MTSI)

Because a quasi-perpendicular whistler has a wavevector at a slight angle off
90

The other, parallel fluctuating current

We now further discuss the relation of the two-stream instabilities studied in this paper with work published previously. Our aim is to provide a unified context while remaining in a “linear” regime. We believe that nonlinear stages need be addressed with 2-D simulations due to the potential coupling between waves that propagate at various angles once they reach large amplitudes.

For perpendicular and quasi-perpendicular propagation, two different
instabilities can arise. By quasi-perpendicular, we mean here a few degrees
off 90

The second instability occurs around the lower-hybrid frequency and is due to
the slow drift of the ion core versus the electrons. The wavevectors are thus
directed toward the shock ramp. As shown with Eq. (

The third instability generates whistlers that propagate obliquely with
respect to

Our present study is at the crossroads of the works of

Finally, we should mention that the MMS mission, with its four spacecraft, has
made measurements of whistlers in the Earth's bowshock which can establish
wavevector characteristics together with plasma populations down to the sub-ion inertia scales. While a typical lower-hybrid period is

In this study we have examined the wave activity that can possibly develop in
the foot of quasi-perpendicular shocks, as it arises from the relative drifts
across the background magnetic field

Generalized Bernstein waves with wavelengths close to the electron gyroradius
which propagate toward upstream at angles within a few degrees off
90

Quasi-perpendicular whistlers with wavelength covering several times the electron inertia length
(such that

Oblique whistlers with wavelengths close to the ion inertia length which
propagate toward upstream at angles about 50

Access to the raw data may be provided upon reasonable request to the authors.

The geometry used is displayed in Fig. 2: the background magnetic field

The dielectric tensor

When the plasma is made of various populations, it is convenient to write the
dielectric tensor as

In our case, we are interested in waves with frequencies ranging from the lower hybrid up to the electron cyclotron. Therefore, the ions will be unmagnetized but the electrons magnetized.

For the ions' contribution we have four integrals of the following type to
evaluate:

The electron distribution is modeled by an isotropic Maxwellian without
drift. However, the electrons being magnetized, their contributions to the
dielectric tensor are made of the usual sums combining modified Bessel
functions and plasma dispersion function. The Bessel functions

Our purpose here is to assess how the parallel component of the electric
field

The wave Eq. (

In order to extract further information from Eq. (

The authors declare that they have no conflict of interest.

Laurent Muschietti gratefully acknowledges beneficial
conversations with Art Hull and also thanks the LATMOS, where
he has worked at in France, for the hospitality and the access to computing
facilities while some of this study was done. The PIC simulations were
performed on ADA, a machine of the supercomputer center “IDRIS”, institute
of the CNRS (