There is an increasing amount of observational evidence in space plasmas
for the breakdown of inertial-range spectra
of magnetohydrodynamic (MHD) turbulence
on spatial scales smaller than the ion-inertial length.
Magnetic energy spectra often exhibit a steepening,
which is reminiscent of dissipation of turbulence energy,
for example in wave–particle interactions.
Electric energy spectra, on the other hand,
tend to be flatter than those of MHD turbulence,
which is indicative of a dispersive process
converting magnetic into electric energy
in electromagnetic wave excitation.
Here we develop a model of the scaling laws and the power spectra
for the Hall inertial range in plasma turbulence.
In the present paper we consider a two-dimensional geometry with no wave vector component
parallel to the magnetic field as is appropriate in Hall MHD.
A phenomenological approach is taken.
The Hall electric field attains an electrostatic
component when the wave vectors are perpendicular to the mean magnetic field.
The power spectra of Hall turbulence are steep for the magnetic field
with a slope of

The recent availability of multi-spacecraft missions
such as Cluster

Physically speaking, in a medium of density

This ion-scale Hall turbulence is, for example in the solar wind,
two-dimensional with both wave vectors and fluctuating magnetic fields
confined to the plane perpendicular with respect to
the mean field

Dispersion analyses performed on the fluctuations showed the absence of any
clear spectral eigenmodes (in linear Vlasov theory)
which would result from dispersion relations in the presumable
ion-scale wavenumber-frequency domain.
At the best, there were rather weak indications found only
of otherwise expected kinetic Alfvén,
whistler, and ion Bernstein modes

Based on these observations, we consider in the following a phenomenological
turbulence model of stationary inertial-range spectra evolving
in ion-scale turbulence.
We will show that, qualitatively, such a model
reproduces ion-inertial-range spectra measured by the MMS spacecraft
in the vicinity of the magnetopause

Limitations of Hall MHD have been discussed,
for example, by

The results of our endeavor can be summarized as follows:
The Hall electric field attains the electrostatic
component when the wave vectors are perpendicular
to the mean magnetic field.
Scaling laws are derived for the magnetic field
and electric field in a power-law form.
In the case of the compressible magnetic field fluctuations
(with the parallel fluctuations of the magnetic field),
the energy spectra have a slope of

Separation of ion and electron motion in a streaming magnetized plasma generates
a Hall current

The magnetic field fluctuations

It follows from Eq. (

What concerns the incompressible part (Eq. 6), so its phase speed becomes
a function of the transverse magnetic Hall field

Ions are non-magnetic. So, since the Hall field is electrostatic and
the wavenumber and electric field are aligned, the ions respond to the
presence of an electric Hall field via Poisson's equation to generate
an electric-fluctuation-related density fluctuation:

In order to proceed quantitatively, we need to construct wavenumber
scaling laws for the spectra of the various field fluctuations.
Subsequently we intend to determine the ratio

To this end we turn to the application of a phenomenological turbulence
model

In the two-dimensional compressible turbulence configuration,
the electron flow velocity is confined to the plane
perpendicular to the mean magnetic field,
but the magnetic field fluctuation

The energy spectrum for the Hall electric field fluctuation follows
from the relation

Since for the turbulent Hall velocity fluctuations we have

Finally coming to the spectrum of density fluctuations,
we invoke Eq. (

It is interesting to compare the density spectrum with

The Hall magnetic energy spectrum is steeper than
the Kolmogorov-type one with wavenumber ratio

In this section we briefly turn to the incompressible Hall spectra.
As we had already noted, they play a lesser role in Hall turbulence
for the quadratic dependence on the incompressible Hall magnetic field
fluctuation component

The scaling law in incompressible magnetic field fluctuations
is determined by the estimate of the flow velocity
in the perpendicular plane for the

Note that purely two-dimensional turbulence,
in which the gradients, wave vectors, and
field fluctuations are confined to the perpendicular plane,
is rather improbable in electron magnetohydrodynamics (EMHD) because the

Figure

Panels

Since the total turbulence spectra in the ion-inertial range are composed of the superposition of Hall and non-Hall contributions, it becomes fairly clear that the ion-inertial-range spectra must deviate quite strongly from the inertial-range spectra of hydrodynamic turbulence (Richardson–Kolmogorov) or that of hydromagnetic turbulence (Iroshnikov–Kraichnan).

The present communication dealt exclusively with the effect of the generation
of Hall current turbulence in collisionless stationary homogeneous and
isotropic inertial-range magnetohydrodynamic turbulence on ion-inertial
scales

The first interesting result of this endeavor was that in stationary homogeneous turbulence the Hall contribution can be separated into compressive and non-compressive parts. It turned out that the compressive contribution to Hall turbulence dominates as it is 1st order in the turbulent magnetic field perturbation the Hall effect introduces. It was also found that the compressive Hall turbulence corresponds to kind of a zero-frequency ion wave whose complex phase speed is given by the ratio of electric and magnetic fluctuations. This phase speed increases with shrinking scale across the ion-inertial range being linearly proportional to the turbulent wavenumber.

Knowing the relations between the turbulent field fluctuations under the conditions when the Hall effect has to be taken into account in collisionless stationary and homogeneous turbulence, we considered the turbulent inertial Hall state. Turning to a dimensional analysis we were able to obtain the relevant scaling laws for the power spectral densities with respect to wavenumber holding in inertial-range Hall turbulent power law spectra.

Transition to phenomenological electron magnetohydrodynamics enabled
the construction of the Hall inertial-range turbulent scaling laws on
ion-inertia scales, an important and to our knowledge new finding which
possibly enables the identification of the ion-inertial range
from observation of magnetic, kinetic, and density turbulent power
law shapes. For instance, the Hall turbulence model qualitatively
explains the Hall-range energy spectra of the Kelvin–Helmholtz-type
turbulence at the magnetopause

Compressive inertial-range Hall magnetic power spectra scale
like

While the presented model is qualitatively similar to previous observations in
that the magnetic energy spectra become steeper in the kinetic range,
observed slopes are often steeper than

The turbulent Hall electric power spectra directly map the turbulent velocity power spectra, the most important kinetic power spectra in any turbulence. Since these at short scales are very difficult to measure, the observation of Hall turbulence should give a direct clue to their identification.

Hall turbulence quite strongly affects the inertial-range turbulent density spectra on ion-inertial scales,
as recently suggested

The data-analysis-motivated model of

The Hall electric field attains the electrostatic component when the wave vectors are perpendicular or nearly perpendicular to the magnetic field. This applies to both the compressible and incompressible cases of magnetic fluctuations. The energy spectrum of the Hall electric field has a flatter spectral slope than that of the magnetic field.

Care must be exercised when analyzing
electric field data and estimating the
phase speed by reference to the

Some observations

The parallel fluctuating component dominates
if both compressive and incompressible fluctuations are excited
by the electron flow. The normalized perpendicular component of the
magnetic field is smaller than the parallel component according to

The relative contribution between the parallel and perpendicular
components of the magnetic field depends on the length scales.
Using Eq. (

The increasing sense of the smaller-scale (or higher-frequency)
density spectrum is indeed found using the Spektr-R spacecraft
data in the solar wind

In summary, we believe that the detailed analysis of the particular
properties of the Hall inertial-range turbulence
contributes to the clarification of the behavior of
the plasma and electromagnetic field
on the ion-inertia scales

In the observational studies of space plasma turbulence, various spectral observations have been performed in the past two decades, and there is an increasing amount of evidence that the magnetic energy spectrum exhibits a dissipative sense (steeper sense) of the spectral curve. Occasionally, it has even been called the dissipation (or ion dissipation) range. Excitation of ion-kinetic electromagnetic waves (such as highly oblique whistler mode, kinetic Alfvén mode, and ion Bernstein mode) is another possible scenario (which leads to the notion of dispersive range instead of dissipation range). Our model for the Hall turbulence serves as a likely candidate to explain the steepening of the magnetic energy spectra neither as dissipation range nor as dispersive range but as Hall inertial range.

In the theoretical studies, clarification of the spectral shapes in the Hall inertial range should provide a useful background for the distinction among the inertial-range behavior and dissipation of turbulence. Our Hall turbulence model shows that the inertial range can have a transition from fluid scales (which is for MHD) to ion scales (which is for Hall MHD) in a dissipationless manner. The dissipation of turbulent fluctuations in collisionless plasmas remains poorly understood. The difference in the spectral shapes from Hall inertial range would be interpreted as a sign of the onset of dissipation.

No data sets were used in this article.

All the authors worked equally on the idea development, calculation, and manuscript writing.

The authors declare that they have no conflict of interest.

This research has been supported by the Austrian Research Promotion Agency (FFG) (Austrian Space Applications Programme (ASAP, grant no. 853994)).

This paper was edited by Minna Palmroth and reviewed by two anonymous referees.