A model-independent first-principle first-order investigation of the shape of turbulent density-power spectra in the ion-inertial range of the solar wind at 1 AU is presented. Demagnetised ions in the ion-inertial range of quasi-neutral plasmas respond to Kolmogorov (K) or Iroshnikov–Kraichnan (IK) inertial-range velocity–turbulence power spectra via the spectrum of the velocity–turbulence-related random-mean-square induction–electric field. Maintenance of electrical quasi-neutrality by the ions causes deformations in the power spectral density of the turbulent density fluctuations. Assuming inertial-range K (IK) spectra in solar wind velocity turbulence and referring to observations of density-power spectra suggest that the occasionally observed scale-limited bumps in the density-power spectrum may be traced back to the electric ion response. Magnetic power spectra react passively to the density spectrum by warranting pressure balance. This approach still neglects contribution of Hall currents and is restricted to the ion-inertial-range scale. While both density and magnetic turbulence spectra in the affected range of ion-inertial scales deviate from K or IK power law shapes, the velocity turbulence preserves its inertial-range shape in the process to which spectral advection turns out to be secondary but may become observable under special external conditions. One such case observed by WIND is analysed. We discuss various aspects of this effect, including the affected wave-number scale range, dependence on the angle between mean flow velocity and wave numbers, and, for a radially expanding solar wind flow, assuming adiabatic expansion at fast solar wind speeds and a Parker dependence of the solar wind magnetic field on radius, also the presumable limitations on the radial location of the turbulent source region.

The solar wind is a turbulent flow with an origin in the solar corona. It is
believed to become accelerated within a few solar radii in the coronal
low-beta region. Though this awaits approval, it is also believed that its
turbulence originates there. Turbulent power spectral densities in the solar
wind have been measured in situ at around 1 AU for several decades already.
They include spectra of the magnetic field

Complementary to the measurements in situ, the solar wind, ground-based
observations of radio scintillations from distant stars, originally applied

We do not touch on the subtle
question of whether any frozen turbulence on MHD-scales above the
ion-cyclotron radius in a low-beta or strong-field plasma can evolve.
According to inferred spatial anisotropies, it seems that close to the Sun,
turbulence in the density is almost field-aligned. On the other hand,
ion-inertial-range turbulence at shorter scales will be much less affected.
It can be considered to be isotropic. Near 1 AU, where most in situ
observations take place, one has

Density fluctuations

Inertial-range velocity turbulence is subject to Kolmogorov

An example is shown in Fig.

Normalised solar wind
power spectra of turbulent temperature and density fluctuations. The curves
are based on data from

The data in Fig.

Inertial-range power spectra of turbulent density fluctuations are power
laws. Occasionally they exhibit pronounced spectral excursions from their
monotonic course prior to dropping into the dissipative range. Whenever this
happens, the spectrum flattens or, in a narrow range of scales, even turns to
positive slopes, sometimes dubbed spectral “bumps”. The reason for such
spectral excesses still remains unclear. Similar bumps have also been seen in
electric field spectra

In the present note we take a completely different

In the next section we discuss the response of demagnetised ions to the
presence of turbulence on scales between the ion and electron inertial
lengths. We interpret this response as the consequence of electric field
fluctuations in relation to the turbulent velocity field. The requirement of
charge neutrality maps them to the density field via Poisson's equation. The
additional contribution of the Hall effect can be separated. We then refer to
turbulence theory, assuming that the mechanical inertial-range
velocity–turbulence spectrum is either Kolmogorov (K) or
Iroshnikov–Kraichnan (IK) and, in a fast-streaming solar wind under
relatively weak conditions

In order to be more general, we split the mean flow velocity into bulk

Our main question concerns the cause of the occasionally observed scale-limited bumps in the turbulent density-power spectra, in particular their deviation from the expected monotonic inertial-range power law decay towards high wave numbers prior to entering the presumable dissipation range.

The philosophy of our approach is the following. Turbulence is always mechanical, i.e. in the velocity. It obeys a turbulent spectrum which extends over all scales of the turbulence. In a plasma, containing charged particles of different mass, these scales for the particles divide into magnetised, inertial, unmagnetised, and dissipative groups. On each of these intervals, the particles behave differently, reacting to the turbulence in the velocity. In the inertial range, the particles lose their magnetic property. They do not react to the magnetic field. They, however, are sensitive to the presence of electric fields, independent of their origin. Turbulence in velocity in a conducting medium in the presence of external magnetic fields is always accompanied by turbulence in the electric field due to gauge invariance, namely the Lorentz force. This electric field affects the unmagnetised component of the plasma, the ions in our case, which to maintain quasi-neutrality tend to compensate it. Below we deal with this effect and its consequences for the density-power spectrum.

The steep decay of the normalised fluctuations in ion temperature above
frequencies

This
equation is easily obtained by standard methods when splitting the fields in
the ideal (collisionless) Hall-MHD Ohm's law:

The last three averaged nonlinear terms within the angular brackets

This leaves us with the fluctuating perpendicular induction field in the
first Eq. (

The complete Hall contribution to the electric field, viz. the last term in
the brackets in Eq. (

Let us assume that advection by large-scale energy-carrying eddies is
perpendicular

On scales in the ion-inertial range shorter than either the ion thermal
gyroradius

We are interested in the power spectrum of the turbulent density fluctuations in the proper frame of the turbulence.

Multiplication of the only remaining first term in the electric induction
field Eq. (

The procedure of obtaining the power spectrum is standard, so we skip the formal steps which lead to this expression.

in wave-number spaceOne may
object that, at smaller wave numbers outside the ion-inertial range, this
would also be the case, which is true. There, reference to the continuity
equation, for advection speeds

The last equation is the main formal result. It is the wanted relation between the power spectra of density and velocity fluctuations. It contains the response of the unmagnetised ions to the mechanical turbulence.

The density response demands that the ions are unmagnetised. This implies that

The range of permitted values of

The power spectrum of the Poisson-modified ion-inertial-range density turbulence can be inferred once the power spectral density of the velocity is given. This spectrum must either be known a priori or requires reference to some model of turbulence.

We do not develop any model of turbulence here. In application to
the solar wind we just make us, in the following, of the Kolmogorov (K)
spectrum

We shall make use of those spectra in two forms: the original ones which just
assume stationarity and absence of any bulk flows and their modified
advected extensions. The latter account for a distinction between a small
number of large energy-carrying eddies with mean eddy vortex speed

Solar wind power spectra of turbulent density fluctuations

The stationary velocity spectrum of turbulent eddies at energy injection rate

This changes drastically when referring to an advected K spectrum of
velocity turbulence

The notion of a turbulent dispersion relation is
alien to turbulence theory, which refers to stationary turbulence,
conveniently collecting any temporal changes under the loosely defined term
intermittency. However, observation of stationary turbulence shows that
eddies come and go on an internal timescale, which stationary theory
integrates out. In Fourier representation this corresponds to an integration
of the spectral density

It is of particular interest to note that solar wind turbulent power spectra
at high frequency repeatedly obey spectral indices very close to this number.
Boldly referring to Taylor's hypothesis where

If this is true, then the corresponding observed spectral transition (or
break point) from the spectral K index

Inspecting the behaviour in the long-wavelength range, one finds that the
exponential dependence

In the stationary turbulence frame the frequency spectrum is obtained when
integrated with respect to

It is most interesting that spectral broadening, when transformed into the spacecraft frame in streaming turbulence, causes that strong of a difference between the original Kolmogorov and the advected Kolmogorov spectrum. This spectral behaviour is still independent of the Poisson modification, which we are going to investigate in the next section.

Here we apply the Poisson-modified expressions to the theoretical inertial-range K and IK turbulence models. We concentrate on the inertial-range K spectrum and rewrite the result subsequently to the IK spectrum.

For the simple inertial-range K spectrum, we know from Eqs. (

Following exactly the same reasoning when dealing with the IK spectrum, which
has power index

We now proceed to the investigation of the effect of advection.

Use of the advected power spectral density Eq. (

In contrast to the Kolmogorov law, the

The case of an IK spectrum leads to an advected velocity spectrum

Solar wind power spectra of the turbulent magnetic field for the
same time interval as in Fig.

Turning to the fast-streaming solar wind, we find that with

The same reasoning produces, for the Poisson-modified advected IK spectrum, the
Taylor's Galilei-transformed spacecraft frequency spectrum

In the following two subsections we apply the above theory to real observations made in situ in the solar wind. We first consider density-power spectra exhibiting well-expressed spectral bumps of positive slope. We then show two examples where no bump is present but where the power spectra exhibit a scale-limited excess and consequently a scale-limited spectral flattening.

Figure

In order to check pressure balance between the density and magnetic field
fluctuations, we refer to turbulent magnetic power spectra obtained at the
WIND spacecraft

Figure

K and IK ion-inertial-range spectral indices

The majority of observed density-power spectra in the solar wind do not
exhibit positive slopes. Such spectra are of monotonic negative slope. In
this sense they are normal. They frequently possess break points in an
intermediate range where the slopes flatten. Two typical examples are shown
in Fig.

Two (redrawn on same scale) cases of normal solar wind density-power
spectra measured by Spektr-R-BMSW

Their flattened spectral intervals each extend roughly over 1 decade in
frequency. The BMSW spectrum is shifted by 1 order of magnitude in
frequency to higher frequencies than the WIND spectrum. Its low-frequency
part below the ion-cyclotron frequency

The slope of the WIND spectrum above its break point at frequency

When crossing the cyclotron frequency

The obvious difference between the two plasma states is not in the Mach
numbers but rather in

In this communication we dealt with the power spectra of density in low-frequency plasma turbulence. We did not develop any new theory of turbulence. We showed that, in the ion-inertial scale range of non-magnetised ions, the electric response of the ion population to a given theoretical turbulent K or IK spectrum of velocity may contribute to a scale-limited excess in the density fluctuation spectrum with a positive or flattened slope. We demonstrated that the obtained inertial-range spectral slopes within experimental uncertainty are not in disagreement with observations in the solar wind, but we could not decide between the models of turbulence. This may be considered a minor contribution only; it shows, however, that correct inclusion of the electrodynamic transformation property is important and suffices for reproducing an observational fact without any need to invoke higher-order interactions, any instability, or nonlinear theory. We also inferred the limitations and scale ranges for the response to cause an effect. However, a substantial number of unsolved problems remain. Below we discuss some of them.

The main problem concerns the agreement with observations. Determination and confirmation of spectral slopes is a necessary condition. However, how should the observed frequency range be adjusted?

Inspecting Fig.

With angles of this kind the positive slope spectral range can be explained.
The lower frequencies then correspond to eddies which propagate nearly
perpendicular. Since our theory is generally restricted to wave numbers
perpendicular to the ambient magnetic field, the eddies which contribute to
the bumps are perpendicular to

The assumption of Taylor's Galilei transformation in the way we used it (and is
generally applied to turbulent solar wind power spectra) is valid only in
stationary homogeneous turbulent flows of spatially constant plasma and field
parameters

For general restrictions on its applicability already in
homogeneous MHD, see

This is a strong assumption. In the absence of dissipation, individual frequencies are conserved. They correspond to energy. Wave numbers correspond to momenta which do not obey a separate conservation law.

Under the fast flow conditions of Fig.

So far we have referred to the inertial length as limiting the frequency
range. We now ask, for the more stringent condition

The wave number

A similar reasoning can be applied to the upper frequency bound

We then have the following relation for the maximum wave number:

Reconciling the observed range of the bump poses a tantalising problem. Our
theoretical approach would suggest that the bump develops between the two
cyclotron frequencies of ions and electrons in the spacecraft frame. This
would correspond to a range of the order of the mass ratio

The latter estimate is, however, quite speculative. Thus the narrowness of
the observed bump in frequency poses a serious problem. Its solution is not
obvious. The most honest conclusion is that little can be said about the
observed upper frequency termination of the bump in Fig.

One may, however, argue that in a high-

In order to get an idea of the distance between source and spacecraft, we
assume that in the interval between the minimum and maximum frequencies, the
ion-cyclotron frequency is crossed. Hence the corresponding wave number is
contained in the spectrum though it is invisible. This fact, however, enables
us to refer to the difference in the ion-inertial length scale and the ion
gyroradius. The total difference in frequency amounts to roughly 1 order of
magnitude. The ratio of both lengths is

In this paper, we considered the cases

Table

The obtained advected slopes in the stationary turbulence frame are also too
far away from the flattest notorious and badly understood negative slope

Generally the form of a distorted power spectrum in density depends on the external solar wind conditions. The reconciliation of these with the theoretical predictions and the observation of the spectral range of the distortion is a difficult, mostly observational task. We have attempted it in the discussion section. In particular the proposed bending of the power spectral density in the direction of lower frequencies requires identification of the maximum point of the advected spectrum in frequency and the transition to the undisturbed K or IK inertial ranges.

We tentatively tried taking thermodynamic effects in an expanding solar wind
into account. This led to preliminary information about the angle between
flow and the turbulent wave numbers which contribute to deformation of the
spectrum. Some tentative information could also be retrieved in this case
about the radial solar distance of the turbulent source region. When
thermodynamics come in, one may raise the important question for the
collisionless turbulent ion heating

So far we have not taken into account the contribution of Hall spectra. These affect the shape of the density spectrum via the Hall-magnetic field, a second-order effect indeed, though it might contribute to additional spectral deformation. Inclusion of the Hall effect requires a separate investigation with reference to magnetic fluctuations. On those scales the Hall currents should provide a free energy source internal to the turbulence, which is not included in K and IK theory.

Hall fields are closely related to kinetic effects in the ion-inertial range.
Among them are kinetic Alfvén waves whose perpendicular scales

Similarly,

Inclusion of all these effects is a difficult task. It still opens up a wide
field for investigation of turbulence on the ion-inertial scale not yet
entering the

No data sets were used in this article.

All authors contributed equally to this paper.

The authors declare that they have no conflict of interest.

This work was part of a brief Visiting Scientist Programme at the International Space Science Institute Bern. We acknowledge the interest of the ISSI directorate as well as the generous hospitality of the ISSI staff, in particular the assistance of the librarians Andrea Fischer and Irmela Schweitzer, and the system administrator Saliba F. Saliba. We also thank the anonymous reviewer for intriguing comments and criticism.

This paper was edited by Elias Roussos and reviewed by one anonymous referee.