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 -7/3 for compressible magnetic turbulence;
they are flatter for the Hall electric field with a slope of -1/3.
Our model for the Hall turbulence gives a possible explanation
for the steepening of the magnetic energy spectra in the solar wind
as an indication of neither the dissipation range
nor the dispersive range but as the Hall inertial range.
Our model also reproduces
the shape of energy spectra in Kelvin–Helmholtz turbulence
observed at the Earth's magnetopause.
Introduction
The recent availability of multi-spacecraft missions
such as Cluster , THEMIS ,
and MMS
together with substantial advances in their instrumentation
and the subsequent data analysis opened up the door to a more detailed
study of space plasma turbulence on ion scales when ion inertia comes into
play. On those scales ions demagnetize and ultimately decouple magnetically
from the electron motion. The dropout of ions from magnetic dynamics
necessarily leaves its signature in the turbulent power spectra,
possibly causing deviations from the conventionally accepted
inertial-range slopes of turbulence.
Typical ion-inertial-range scale lengths in the solar wind range from
100 to 1000 km, which correspond to the turbulent wavenumber interval
10-6≲(2π)-1k≲10-5 m-1. Multiplying, for instance,
with a nominal solar wind speed of ∼500 km s-1
this interval maps to the frequency range
0.5≲(2π)-1ω≲5 Hz.
Physically speaking, in a medium of density n moving at velocity V the main
signature of the ion-inertial range,
sometimes also called the ion-kinetic regime or ion-dissipation range,
is the presence of Hall currents. These are pure electron currents
jH=-enE×B/B2 flowing
perpendicular to the magnetic B and convection electric
E=-V×B fields.
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 B0. Ion-kinetic-scale power-law spectra
(though limited to the frequency domain)
were observationally obtained separately
for the magnetic
and electric fields. They were found to be
reminiscent of a turbulent inertial range typical for scale-invariant
turbulence spectra of Richardson–Kolmogorov or Iroshnikov–Kraichnan type.
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
in this range.
The breakdown of linear-mode theory thus indicates that the
frequencies deviate from simple
Doppler-shifted linear modes by random sweeping which would be due to the
large-scale variations of the flow
such as eddies or Alfvénic fluctuations,
sideband formation caused in both,
a weakly turbulent kinetic wave–wave coupling, or
steepening in the course of nonlinear evolution.
Also solitary-structure formation resulting from phase coherence
seems to be absent.
The study by indicates the existence of
the kinetic Alfvén mode in the magnetosheath region as obtained
from the wave analysis for the fluctuations in the MMS data using
the Alfvén ratio. No dispersion analysis is performed.
On the other hand, the study by exhibits
a frequency scattering in the observationally determined
dispersion relation with an indication of a kinetic-drift mirror mode.
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 ,
whenever the Kelvin–Helmholtz instability is excited and decays
into smaller-scale Kelvin–Helmholtz vortices until a spectrum
of low-frequency small-scale turbulence is produced.
We consider a two-dimensional geometry which has no parallel wave vector component.
The full expression for the Hall electric field contains also parallel wave vector components
which in Hall MHD are neglected.
Limitations of Hall MHD have been discussed,
for example, by .
The concept of Hall turbulence is valid in the limit where
the electron temperature is much greater
than the ion temperature and when the inverse of the linear transit time
for an ion is much smaller than the turbulent frequency and
the inverse of the linear transit time for an electron, respectively.
Thus, in the instance where the temperature of the ions is finite, phase-mixing
and damping of modes ought to be taken into account. This causes
deviations from Hall MHD.
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 -7/3 and -1/3
for the magnetic field and the Hall electric field, respectively.
The amplitude ratio of the Hall electric field to
the fluctuating magnetic field (hereafter, the E–B ratio) has
a linear dependence on the wavenumber. The density power
spectrum has a positive spectral slope with an index of +5/3.
In the incompressible case with the perpendicular
fluctuations of the magnetic field,
the energy spectra have a slope of -2 and -1
for the magnetic and electric fields, respectively.
The E–B ratio has a dependence of the wavenumber with a power of 1/2.
An important lesson from the model construction is
that the Hall electric field is dependent on
the wavenumber and the E–B ratio also shows
the wavenumber dependence.
The Hall fluctuation fieldsThe Hall electric fields
Separation of ion and electron motion in a streaming magnetized plasma generates
a Hall current jH. Referring to the magnetized electron
equation of motion, which is the
generalized collisionless Ohm's law for the electric field, the Hall current produces
its specific Hall electric field:
EH=-1enjH×B.
The Hall current jH is perpendicular to the magnetic and electric fields.
In turbulence theory one is interested in the fluctuations of the fields given
in the form δF=F-F0,
with F referring to any relevant turbulent field,
magnetic, electric, flow velocity, density, and so on,
denoting fluctuations with prefix δ and in the following suppressing the index 0
on all mean fields. Since the electrons remain magnetic and continue their mean flow,
the mean Hall electric field is just the mean convection electric field of
the flow EH0=E0=-V0×B0; hence, the mean Hall current
is jH0=-en0EH0, which is of no interest. Due to their inertia the ions
continue to participate in the flow also on those scales in the collisionless plasma.
Thus the zeroth-order Hall current
can be neglected when considering fluctuations in turbulence.
The fluctuation of the Hall current taken in the moving frame
V0=E0=0 is obtained as
δjH=-eδ[nE×BB2]H≃-enδEHB×BB.
Electric field variations
contribute primarily to the linear Hall current fluctuations,
with fluctuating density and magnetic field contributions
being of higher order. As expected the turbulent Hall current
lies in the plane perpendicular to the mean field B and is perpendicular
to the fluctuation in the Hall electric field. In the stationary observer's frame
there would be a number of other terms which, however, disappear in the moving
frame, the case which we are interested in here. There would also be
higher-order Hall current terms when folding with the
1st-order magnetic and electric field fluctuations
which we neglect to lowest order here.
Relations between the electric and magnetic fields
The magnetic field fluctuations δBH in Hall turbulence
have two components: one compressive component
δB‖ parallel to the mean field B and the other perpendicular component δB⟂.
It is convenient to introduce an orthogonal coordinate system
with base vectors e1 and e2 perpendicular to the mean field,
and e‖ along the mean field.
Moreover, we are free to choose the direction of the perpendicular wave vector,
letting e1 refer to k⟂.
The Hall magnetic field has no divergence, so it must be perpendicular
to k. This yields
δB=(0,δB⟂,δB‖).
The fluctuation of the Hall electric field is given by
δEH=1enδjH×B-δnnE.
The last term on the right containing the fluctuations in density
and their contribution to δEH is important only in the
stationary frame where E≠0.
Using Ampère's law μ0δj=∇×δB
(from here on suppressing the index H on the fluctuations when
dealing exclusively with Hall fluctuations in Hall MHD) yields
δE=1enμ0B×(∇×δB).
It follows from Eq. ()
that in both cases the Hall electric field is along the perpendicular Hall
wave vector, i.e., along e1. This shows that the fluctuation part
of the Hall electric field is purely electrostatic, a property of which
we can make use below. Switching to the Fourier representation with
∇→ik we obtain
5δE⟂(c)=ienμ0k⟂δB‖B6δE⟂(i)=ienμ0k⟂|δB⟂|2
for the compressive δE(c) and incompressible
δE(i) components of the wavenumber-parallel
(i.e., longitudinal fluctuation sense)
electric field fluctuations, respectively.
This shows that the incompressible electric field is 2nd order
in the magnetic fluctuation and will in principle be small and negligible.
The ratio of the two components
δE⟂(i)δE⟂(c)=|δB⟂|2BδB‖
depends on the sign of the compressive component of
the magnetic field δB‖. Since its right-hand side
only contains magnetic components, it can be used in spacecraft observations
to estimate the ratio on the left. With this information the dominant
turbulent Hall electric field is given by the compressive magnetic Hall field
component, and the phase speed of the compressive Hall fluctuations is found to be
δE⟂(c)δB‖=ik⟂Benμ0=iVAk⟂cωi
with VA the Alfvén speed and ωi the ion plasma frequency.
Squaring this, we obtain
(δE⟂(c)VAδB‖)2=k⟂2c2ωi2.
The right-hand side of this expression is the rudiment of either a very
low-frequency ion wave or ion whistler dispersion relation which for ion
waves can be written as k⟂2c2/ωi2=ω2/ωi2-1 and which is
reproduced for ω=0. Thus the compressive part of Hall turbulence
can be understood as the zero-frequency fluctuations of transverse ion waves
k⟂2ṼA2Ωi2=1-ω2Ωi2VA2U2
in the limit ω→0, where U=δE⟂(c)/δB‖ is the complex phase
speed, and ṼA=VA2/U is a modified Alfvén speed, which shows
that these waves are essentially ion whistlers or modified zero-frequency Alfvén waves. Resolving for the fictitious frequency ω
one obtains
ωΩi=±UVA1+k⟂2ṼA2Ωi2∼±ik⟂cωiη
as can be shown by using the above expressions for U. Here η≈0
is a very small number. Nevertheless it is seen that, in principle,
these waves would have linear dispersion ω∼k⟂ if attaining
any however small frequency. In addition, they would be damped.
What concerns the incompressible part (Eq. 6), so its phase speed becomes
a function of the transverse magnetic Hall field δB⟂.
We should note that the ratio of electric-to-magnetic fields is
widely used in spacecraft observations in various plasma domains
as an estimator of the plasma convective motion
and the phase speed of the electromagnetic
wave .
Hall current-related density fluctuations
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:
δnn=iϵ0enk⟂δE⟂(c)=k⟂2VA2ωi2δB‖B.
Here, only the compressive field component contributes because of
its linearity. It shows that the relative density fluctuations are
completely determined by the compressive Hall magnetic field fluctuations
δB‖. This is an important conclusion as it shows that the turbulent
density spectrum caused by the Hall effect is proportional to the
turbulent compressive magnetic Hall spectrum whose wavenumber dependence
is raised by the power of k⟂4, an effect which should become observable
in the density spectrum in the scale range where ion inertia becomes susceptible.
There the Hall modification of the density spectrum adds to the non-Hall
deformation of the density spectrum derived in our former publication .
We will briefly return to this item below after having constructed
the power spectrum of the magnetic fluctuations in the Hall field case.
Ion-scale inertial-range spectra
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 δE/δB as well as
the energy spectra in an attempt to obtain a scaled model of
the ion-inertial-scale field fluctuations as the necessary step to derive
the turbulent inertial-range power spectra of the fields on ion-inertial-scale lengths 1≲kc/ωi<kc/ωe (where ωe denotes the
electron plasma frequency).
To this end we turn to the application of a phenomenological turbulence
model in two-dimensional electron magnetohydrodynamics
which is appropriate in our case. We have already made use of two-fluid
plasma theory above when referring to the presence of the Hall effect
in the generalized Ohm's law.
Let us introduce the following scaling
δB∝ℓαm∝k-αm
for the turbulent magnetic field. We, moreover, normalize all relevant
fields and scales to the Alfvén speed VA,
ion cyclotron frequency Ωi, and mean magnetic field B as follows:
14δB→δB̃=δBB15δV→ṽ=δVVA16δE→δẼ=δEVAB17δn→δñ=δnn18k→k̃=kVAΩi.
Compressible magnetic turbulence
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 B‖ is compressible.
The effect of the Hall effect on the fluctuation spectrum
implies that the magnetic field becomes
increasingly extended and compressed
like an elastic spring. The flow velocity is determined by the gradient
of the stream function as
ṽ=(e‖×∇̃)δB̃‖,
with the parallel fluctuation component δB̃‖ of the
magnetic field playing the role of a stream function.
The eddy interaction time in units of the ion gyro-period Ωi-1 becomes
τ̃∝(k̃ṽ)-1∝k̃-2δB̃‖-1.
For the energy transfer rate, which is assumed to be constant
over the entire turbulent inertial range, we have
ϵ̃∝|δB̃‖|2τ̃∝|δB̃‖|3k̃⟂2.
This leads to the magnetic field scaling
δB̃‖∼cmϵ̃1/3k̃⟂-2/3,
where a proper scaling coefficient cm has been introduced.
With these expressions, the magnetic energy spectrum becomes
Emag=|δB̃‖|2Δk̃⟂∼cm2ϵ̃2/3k̃⟂-7/3,
where the wavenumber interval is
scaled to the wavenumber itself as Δk̃⟂∼k̃⟂, i.e., assuming
an equidistant grid on the logarithmic scale.
This k-7/3 scaling of the magnetic energy spectrum is
intriguing in view of the same scaling
which had been obtained in numerical simulations for
isotropic Hall magnetohydrodynamic (Hall MHD) turbulence
, the exact case which
underlies our endeavor.
The energy spectrum for the Hall electric field fluctuation follows
from the relation δẼ=k̃⟂δB̃‖ together with
Eq. () as
Eelec=k̃⟂2|δB̃‖2Δk̃⟂∼cm2ϵ̃2/3k̃⟂-1/3.
These two expressions can be used to calculate the ratio δE(c)/δB
of the fluctuation amplitudes
δẼ(c)δB̃‖=EelecEmag∼k̃⟂,
which scales as the first power of the normalized wavenumber, indicating
that the normalized fluctuation phase speed Ũ referred to
linear increases above
with decreasing scale in the ion-inertial range.
Thus, this dependence remains unchanged and is in fact confirmed by the model.
Since for the turbulent Hall velocity fluctuations we have ṽ=δẼ,
the kinetic energy spectrum Ekin
scales like the electric power spectrum:
Ekin=Eelec.
As expected, the electric fluctuation spectrum in the Hall effect
maps the kinetic fluctuation spectrum.
Finally coming to the spectrum of density fluctuations,
we invoke Eq. (), which in its rescaled form reads
δñ=iVAc2k̃⟂δẼ(c).
It yields the turbulent Hall density power spectrum as
Edens=|δñ|2Δk̃⟂∼VAc4cm2ϵ̃2/3k̃⟂5/3.
Most interestingly, this spectrum is of an inverse Kolmogorov type.
Because of the relation between the Hall fluctuations in density and electric–magnetic fields, one of course expects that the presence of the Hall
effect in the ion-inertial range affects the shape of the density power spectrum.
This is indeed the case. In the ion-inertial-scale range the Hall effect
seems to practically compensate for the general spectral Kolmogorov slope
of the density power spectrum, causing it to flatten substantially.
Scaling-wise speaking,
this is quite a strong effect, the degree of whose signature
in observed density power spectra does, however, depend on the various scaling
constants in the spectral contributions. One may, however, speculate that the
notoriously frequently observed k̃-1 slope in the density power spectra
in the solar wind around the presumable ion-inertial-scale range,
for example in ,
may result from the contribution of the Hall effect to the inertial-range
spectrum of ion-inertial-scale turbulence.
It is interesting to compare the density spectrum with k⟂5/3
for the Hall scaling (Eq. )
with the Kolmogorov–Poisson density spectrum with the k⟂1/3 scaling
obtained earlier Eq. 24 for non-Hall turbulence. The ratio
of the two expressions is
EdensHEdensK∼VAc2cm2cKk⟂4/3.
It still depends on the unknown constant of proportionality cm
which must be determined otherwise. However, the deformation of the spectral scaling
caused by the Hall turbulence is stronger than in the non-Hall case. Its contribution
might thus become important, even though numerically its contribution to the density variation
is smaller than that of the Kolmogorov–Poisson spectrum, because
VA≪c. The difference in the spectral slopes of k⟂4/3 indicates that
the Hall density spectrum becomes increasingly more effective at larger wavenumbers.
The Hall magnetic energy spectrum is steeper than
the Kolmogorov-type one with wavenumber ratio
EmagHEmagK∼k⟂-2/3.
Finally, the ratio of the kinetic power spectra yields a flatter Hall kinetic energy spectrum than Kolmogorov:
EkinHEkinK∼k⟂4/3.
Incompressible magnetic turbulence
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 δB⟂ which would enable us to neglect it completely.
However, for the sake of completeness we provide the corresponding
expressions here below.
The scaling law in incompressible magnetic field fluctuations
is determined by the estimate of the flow velocity
in the perpendicular plane for the E×B drift
motion of the electron fluid,
ṽ⟂=δẼ=k̃⟂|δB̃⟂|2.
The timescale for the interaction is
τ̃∝ℓ̃⟂ṽ⟂∝k̃⟂-2δB̃⟂-2.
The energy transfer rate, again assumed to be
constant in the inertial range, is
ϵ̃∝|δB̃⟂|2τ̃∝k̃⟂2|δB̃⟂|4.
The scaling law for the magnetic field follows from Eq. () as
δB̃⟂∼cmϵ̃1/4k̃⟂-1/2,
and the magnetic energy spectrum reads
Ẽmag=|δB̃⟂|2Δk̃⟂∼cm2ϵ̃1/2k̃⟂-2.
The energy spectrum for the Hall electric field
and that for the kinetic energy have
the same spectral forms because again ũ⟂=δẼ,
yielding
Ẽelec=Ẽkin∼cm4ϵ̃k̃⟂-1.
The ratio δE(i)/δB follows as
δẼ(i)δB̃⟂∼cmϵ̃1/4k̃⟂1/2.
Even though the magnetic fluctuation field is incompressible,
the density varies because of the electrostatic nature of the Hall field.
The density spectrum for the Hall electric field is
Edens∼VAc4cmϵ̃k̃⟂.
Compared to the compressible case it increases at a lesser power
with decreasing scale though also acting to compensate for the Kolmogorov slope.
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
E×B drift motion causes the electric
current in the perpendicular plane. It
generates a parallel magnetic field fluctuation δB‖.
This has been taken care of in the previous section which included
the dominant Hall effect on the spectra.
Figure shows the schematic shapes of the Hall field spectra
in the ion-inertial range for the two cases of compressible and
incompressible Hall turbulence. Their absolute contribution to
the observed turbulence depends on the proportionality factor cm
which enters all above expressions. It is, however, seen that
the kinetic energy in the Hall range dominates
the magnetic energy. This is of course reasonable because the Hall effect
is the result of the turbulence in the flow velocity. The dominant
effect is provided by the compressive part of Hall turbulence.
The compressive Hall magnetic spectrum decays slightly more steeply than
the Kolmogorov spectrum. One therefore expects that observed magnetic
spectra in stationary Hall turbulence in the ion-inertial-scale range
will obey inertial-range spectral indices k-7/3.
On the contrary, however, the Hall density power spectra should exhibit a spectral
increase in the ion-inertial-scale Hall range which is due to
the reaction of the density to the
Hall electric turbulence. Naturally this effect is
more strongly expressed in the compressible case.
Panels (a) and (b) show
the energy spectra for the magnetic field (Emag),
the Hall electric field (Eelec), and
the flow velocity (Ekin), and
the density fluctuation (Edens)
for compressible magnetic turbulence (a)
and incompressible turbulence (b).
Panels (c) and (d) are the E–B ratio
for compressible magnetic turbulence (c)
and incompressible turbulence (d).
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).
Conclusion and discussions
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 1≲kc/ωi<kc/ωe where ion inertia takes over to determine
the dynamics and ions de-magnetize. This magnetohydrodynamic range also refers
to Hall magnetohydrodynamics or electron magnetohydrodynamics.
We first discussed in detail the appearance and properties of the Hall
effect under conditions of interest in turbulence. We then switched and referred to
a phenomenological scaling model. We derived the wavenumber scalings
of the turbulent fluctuations and turbulent power spectral
densities under Hall conditions.
These investigations refer to the stationary frame of turbulence.
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.
Spectral shapes
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 in that
(1) the (electron) flow velocity and the electric field exhibit
the same spectral curve perpendicular to
the mean magnetic field; and (2) the magnetic energy spectrum is
markedly steeper than that of the kinetic energy
and the electric field energy.
Compressive inertial-range Hall magnetic power spectra scale
like ∼k⟂-7/3, steeper than Richardson–Kolmogorov
and Iroshnikov–Kraichnan while being, in some cases, in agreement
with numerical simulations. This suggests that observed gradually
increasing slopes in turbulent magnetic power spectra and
becoming steeper at shorter scales than Kolmogorov may indicate
that Hall turbulence on those scales takes over, and that the inertial-range
turbulence enters the ion-inertia scales.
If this happens, no reference is required to any sophisticated
kind of hidden dissipation mechanism. Rather,
this changing slope is quite a clear indication of the ion-inertial scale
coming into play, clearer than the recalculation
of scales via Taylor's hypothesis.
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 -7/3, for example, as in
, , and .
It should be noted that the theory predicts the energy spectra
in the wave vector domain and the observations often have access to
the spectra in the frequency domain.
Possible reasons for the difference in the spectral slope
between the theory and the observations include
the presence of dispersive waves and
the non-Gaussian frequency broadening in the random sweeping effect.
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 .
Hall density power spectra increase in their most important compressive and thus
dominant section as k⟂+5/3, which is an inverse Kolmogorov increase!
They contribute to the earlier found deviation from
inertial-range slope.
Observations should distinguish its absolute
contribution. This cannot be determined from phenomenological scaling theory.
The obtained steep spectral increase, when overlaid on ordinary spectra, might
contribute to the occasionally observed and still mysterious k-1 spectral slopes.
The data-analysis-motivated model of
introduces an ad hoc measure α
of the compression distinguishing between the incompressible
(α=0) and isotropic compressible (|α|=1) cases.
It maps the spectral slope of the magnetic field
energy from k-7/3 in the incompressible case
to (-7+6α)/3 in the compressible case. Our
physically motivated Hall MHD model differs
from that of in that the slope -7/3
is obtained for the compressible field fluctuations.
Electrostatic nature
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 E–B ratio, in particular,
in the ion-kinetic range. In the compressible case the E–B ratio
depends linearly on the wavenumber k⟂
as considered earlier in Cluster data analysis and
hybrid plasma simulations ,
while the incompressible case exhibits a k⟂ dependence,
The character of the electric field needs to be evaluated when performing
the E–B ratio analysis. It should be determined whether the electric field
is of electromagnetic nature, representing a dispersive wave,
or it is electrostatic, in which case it results from
Hall turbulence.
Parallel vs. perpendicular components of the magnetic field
Some observations
indicate the dominance of
the perpendicular magnetic field component
in the kinetic range.
Our scaling laws are derived separately for the
parallel one. It predicts that the Hall electric field
associated with the parallel component of the magnetic field
should dominate the electric spectrum (Eqs. –).
The magnetic energy spectrum can be dominated by
either parallel or perpendicular fluctuations. However note
that the scaling contains the undetermined numerical
constant cm, which determines the absolute value.
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
δB̃‖∼|δB̃⟂|2.
This follows from the electron flow velocity
ṽ⟂∼k̃⟂δB̃‖
and the association to the perpendicular component
ṽ⟂∼k̃⟂|δB̃⟂|2.
In Hall MHD the flow velocity is E×B
passive, being subject to
the magnetic and electric fields.
The relative contribution between the parallel and perpendicular
components of the magnetic field depends on the length scales.
Using Eq. () and
Eq. () yields
δB‖δB⟂∝k̃-1/6.
Therefore, the contribution of the parallel component
of the magnetic field is reduced with increasing wavenumber.
Density spectrum
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 .
provide a theoretical explanation
of the density spectrum bump using the convected fluid model which
the present theory
extends to the inclusion of Hall dynamics.
In the magnetosheath, to date no such increase has been observed in the
electron density spectrum based on the spacecraft data.
Figure 3 in shows a flattening
of the density spectrum at spacecraft-frame frequencies of 10 Hz or higher,
but this flattening is more likely associated
with the Poisson noise in the particle measurements, indicating that
clean, proper density spectrum measurements
will be an important future task in the observational study
of the Hall domain physics.
their Fig. 4, bottom panel shows that the density
has about the same fluctuation power as the magnetic field
at lower frequencies, indicating similar density and magnetic
spectral slopes, with density spectrum estimated for electrons
and inert ions.
Theoretically, information about the density spectrum can also be obtained
making use of either the continuity equation or
the quasi-static approximation
.
Gyro-kinetic treatment
provide a detailed description of
ion-scale turbulence for weakly collisional plasmas through in a gyro-kinetic treatment.
Gyro-kinetic theory is a reduced anisotropic limit of Hall MHD with comparable results
to that of the authors. However, the gyrokinetic theory, unlike Hall MHD, incorporates
phase mixing due to Landau damping (not cyclotron resonance).
In weak turbulence of energy-cascading kinetic Alfvén waves,
gyro-kinetic theory predicts inertial-range energy spectra
(in the perpendicular wavenumber domain) with spectral slopes
k⟂-1/3 for the electric and
k⟂-7/3 the magnetic fields,
and spectral density slopes k⟂-7/3. These are identical to
the compressive magnetic field fluctuations obtained here.
Concluding remarks
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 k⟂c/ωi>1,
length scales shorter than that for the fluid or
magnetohydrodynamic picture of turbulence.
The wavenumber scaling laws and the corresponding power spectra
are derived for Hall turbulent magnetic, electric,
velocity, and density field in the phenomenological approach.
The Hall inertial range is of great interest for many reasons
in the both observational and theoretical sense.
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.
Data availability
No data sets were used in this article.
Author contributions
All the authors worked equally on the idea development, calculation, and manuscript writing.
Competing interests
The authors declare that they have no conflict of interest.
Financial support
This research has been supported by the Austrian Research Promotion Agency (FFG) (Austrian Space Applications Programme (ASAP, grant no. 853994)).
Review statement
This paper was edited by Minna Palmroth and reviewed by two anonymous referees.
ReferencesAlexandrova, O., Carbone, V., Veltri, P., and Sorriso-Valvo, L.:
Small-scale energy cascade of the solar wind turbulence,
Astrophys. J., 674, 1153–1157,
10.1086/524056, 2008.Alexandrova, O., Saur, J., Lacombe, C., Mangeney, A.,
Mitchell, J., Schwartz, S. J., and Robert, P.:
Universality of solar-wind turbulent spectrum
from MHD to electron scales,
Phys. Rev. Lett., 103, 165003,
10.1103/PhysRevLett.103.165003, 2009.Angelopoulos, V.: The THEMIS Mission, Space Sci. Rev., 141, 5–34,
10.1007/s11214-008-9336-1, 2008.Bale, S. D., Kellogg, P. J., Mozer, F. S., Horbury, T. S., and Reme, H.:
Measurement of the electric fluctuation spectrum of
magnetohydrodynamic turbulence,
Phys. Rev. Lett., 94, 215002,
10.1103/PhysRevLett.94.215002, 2005.
Biskamp, D., Schwarz, E., and Drake, J. F.:
Two-dimensional electron magnetohydrodynamic turbulence,
Phys. Rev. Lett., 76, 1264–1267, 1996.Burch, J. L., Moore, T. E., Torbert, R. B., and Giles, B. L.:
Magnetospheric multiscale overview and science objectives,
Space Sci. Rev., 199, 5–21, 10.1007/s11214-015-0164-9, 2016.Breuillard, H., Matteini, L., Argall, M. R., Sahraoui, F., Andriopoulou, M.,
Le Contel, O., Retinò, A., Mirioni, L., Huang, S. Y., Gershman, D. J.,
Ergun, R. E., Wilder, F. D., Goodrich, K. A., Ahmadi, N., Yordanova, E., Vaivads, A.,
Turner, D. L., Khotyaintsev, Yu. V., Graham, D. B., Lindqvist, P.-A., Chasapis, A.,
Burch, J. L., Torbert, R. B., Russell, C. T., Magnes, W., Strangeway, R. J.,
Plaschke, F., Moore, T. E., Giles, B. L., Paterson, W. R., Pollock, C. J.,
Lavraud, B., Fuselier, S. A., and Cohen, I. J.:
New insights into the nature of turbulence in the Earth's magnetosheath using
Magnetospheric MultiScale mission data,
Astrophys. J., 859, 127,
10.3847/1538-4357/aabae8, 2018.Chen, C. H. K. and Boldyrev, S.:
Nature of kinetic scale turbulence in the Earth's magnetosheath,
Astrophys. J., 842, 122,
10.3847/1538-4357/aa74e0, 2017.Cohen, R. H. and Kulsrud, R. M.:
Nonlinear evolution of parallel-propagating hydromagnetic waves,
Phys. Fluids, 17, 2215–2225,
10.1063/1.1694695, 1974.Eastwood, J. P., Phan, T. D., Bale, S. D., and Tjulin, A.:
observations of turbulence generated by magnetic reconnection,
Phys. Rev. Lett., 102, 035001,
10.1103/PhysRevLett.102.035001, 2009.Escoubet, C. P., Fehringer, M., and Goldstein, M.: Introduction The Cluster mission, Ann. Geophys., 19, 1197–1200, 10.5194/angeo-19-1197-2001, 2001.Franci, L., Landi, S., Matteini, L., Verdini, A., and Hellinger, P:
High-resolution hybrid simulations of kinetic
plasma turbulence at proton scales,
Astrophys. J., 812, 21,
10.1088/0004-637X/812/1/21, 2015.Hori, D. and Miura, H.:
Spectrum properties of Hall MHD turbulence,
Plasma Fusion Res., 3, S1053,
10.1585/pfr.3.S1053, 2008.Howes, G. G.: Limitations of Hall MHD as a model for turbulence in weakly collisional plasmas, Nonlin. Processes Geophys., 16, 219–232, 10.5194/npg-16-219-2009, 2009.Matsuoka, A., Mukai, T., Hayakawa, H., Kohno, Y.-I.,
Tsuruda, K., Nishida, A., Okada, T., Kaya, N., and Fukunishi, H.:
EXOS-D observations of electric field fluctuations and
charged particle precipitation in the polar cusp,
Geophys. Res. Lett., 18, 305–308,
10.1029/91GL00032, 1991.Matteini, L., Alexandrova, O., Chen, C. H. K., and Lacombe, C.:
Electric and magnetic spectra from MHD to electron scalesin the magnetosheath,
Mon. Not. R. Astron. Soc., 466, 945–951,
10.1093/mnras/stw3163, 2017.Narita, Y.:
Space-time structure and wavevector anisotropy
in space plasma turbulence,
Living Rev. Sol. Phys., 15, 2,
10.1007/s41116-017-0010-0, 2018.Narita, Y. and Hada, T.:
Density response to magnetic field fluctuation in the foreshock plasma,
Earth Planets Space, 70, 171,
10.1186/s40623-018-0943-0, 2018.Narita, Y., Plaschke, F., Nakamura, R., Baumjohann, W., Magnes, W., Fischer, D., Vörös, Z.,
Torbert, R. B., Russell, C. T., Strangeway, R. J., Leinweber, H. K., Bromund, K. R.,
Anderson, B. J., Le, G., Chutter, M., Slavin, J. A., Kepko, E. L., Burch, J. L.,
Motschmann, U., Richter, I., and Glassmeier, K.-H.:
Wave telescope technique for MMS magnetometer,
Geophys. Res. Lett., 43, 4774–4780,
10.1002/2016GL069035, 2016.Perschke, C., Motschmann, U., Narita, Y., and Glassmeier, K.-H.:
Multi-spacecraft observations linear modes and
sideband waves in ion-scale solar wind turbulence.
Astrophys. J., 793, L25,
10.1088/2041-8205/793/2/L25, 2015.Perschke, C., Narita, Y., Motschmann, U., and Glassmeier, K.-H.:
Observational test for a random sweeping model
in solar wind turbulence,
Phys. Rev. Lett., 116, 125101,
10.1103/PhysRevLett.116.125101, 2016.Roberts, O. W., Li, X., and Jeska, L.:
A statistical study of the solar wind turbulence
at ion kinetic scales using the k-filtering technique
and Cluster data,
Astrophys. J., 802, 2,
10.1088/0004-637X/802/1/2, 2015.Roberts, O. W., Toledo-Redondo, S., Perrone, D., Zhao, J., Narita, Y.,
Gershman, D., Nakamura, R., Lavraud, B., Escoubet, C. P., Giles, B.,
Dorelli, J., Pollock, C., and Burch, J.:
Ion-scale kinetic Alfvén turbulence:
MMS measurements of the Alfvén ratio in the magnetosheath,
Geophys. Res. Lett., 45, 7974–7984,
10.1029/2018GL078498, 2018.Šafránková, J., Němeček, Z., Přech, L., and Zastenker, G. N.:
Ion kinetic scale in the solar wind observed,
Phys. Rev. Lett., 110, 25004,
10.1103/PhysRevLett.110.025004, 2013.Šafránková, J., Němeček, Z., Němec, F.,
Přech, L., Chen, C. H. K., and Zastenker, G. N.:
Solar wind density spectra around the ion spectral break,
Astrophys. J., 803, 107,
10.1088/0004-637X/803/2/107, 2015.Sahraoui, F., Goldstein, M. L, Robert, P.,
and Khotyaintsev, Y. V.:
Evidence of a cascade and dissipation of
solar-wind turbulence at the electron gyroscale,
Phys. Rev. Lett., 102, 231102,
10.1103/PhysRevLett.102.231102, 2009.Schekochihin, A. A., Cowley, S. C., Dorland, W., Hammett, G. W., Howes, G. G.,
Quataert, E., and Tatsuno, T.:
Astrophysical gyrokinetics: Kinetic and fluid turbulent cascades in
magnetized weakly collisional plasmas,
Astrophys. J., 182, 310–377,
10.1088/0067-0049/182/1/310, 2009.Stawarz, J. E., Eriksson, S. , Wilder, F. D., Ergun, R. E.,
Schwartz, S. J. , Pouquet, A., Burch, J. L., Giles, B. L.,
Khotyaintsev, Y., Le Contel, O., Lindqvist, P.-A., Magnes, W., Pollock, C. J.,
Russell, C. T., Strangeway, R. J., Torbert, R. B., Avanov, L. A.,
Dorelli, J. C., Eastwood, J. P., Gershman, D. J., Goodrich, K. A.,
Malaspina, D. M., Marklund, G. T., Mirioni, L., and Sturner, A. P.:
Observations of turbulence in a Kelvin-Helmholtz event
on 8 September 2015 by the Magnetospheric Multiscale mission,
J. Geophys. Res.-Space, 121, 11021–11034,
10.1002/2016JA023458, 2016.Treumann, R. A., Baumjohann, W., and Narita, Y.: On the ion-inertial-range density-power spectra in solar wind turbulence, Ann. Geophys., 37, 183–199, 10.5194/angeo-37-183-2019, 2019.