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**Annales Geophysicae**
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**ANGEO Communicates**
03 Apr 2019

**ANGEO Communicates** | 03 Apr 2019

On the ion-inertial-range density-power spectra in solar wind turbulence

^{1}International Space Science Institute, Bern, Switzerland^{2}Space Research Institute, Austrian Academy of Sciences, Graz, Austria^{3}Geophysics Department, Ludwig Maximilian University of Munich, Munich, Germany

^{1}International Space Science Institute, Bern, Switzerland^{2}Space Research Institute, Austrian Academy of Sciences, Graz, Austria^{3}Geophysics Department, Ludwig Maximilian University of Munich, Munich, Germany

**Correspondence**: Wolfgang Baumjohann (wolfgang.baumjohann@oeaw.ac.at)

**Correspondence**: Wolfgang Baumjohann (wolfgang.baumjohann@oeaw.ac.at)

Abstract

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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.

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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, https://doi.org/10.5194/angeo-37-183-2019, 2019.

1 Introduction

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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 (e.g. Goldstein et al., 1995; Tu and Marsch, 1995; Zhou et al., 2004; Podesta, 2011, for reviews, among others), but with improved instrumentation also of the fluid velocity (Podesta et al., 2007; Podesta, 2009; Šafránková et al., 2013), electric field (Chen et al., 2011, 2012, 2014a, b), temperature (Šafránková et al., 2016), and (starting with Celnikier et al., 1983, who already reported its main properties) also of the (quasi-neutral) solar wind density (Chen et al., 2012, 2013; Šafránková et al., 2013, 2015, 2016).

Complementary to the measurements in situ, the solar wind, ground-based
observations of radio scintillations from distant stars, originally applied
(Lee and Jokipii, 1975, 1976; Cordes et al., 1991; Armstrong et al., 1995) to the interstellar medium
(ISM; for early reviews, e.g. Coles, 1978; Armstrong et al., 1981) and used for
extra-heliospheric plasma diagnosis (cf., Haverkorn and Spangler, 2013), also
provided information about the solar wind density turbulence
(Coles and Harmon, 1989; Armstrong et al., 1990; Spangler and Sakurai, 1995;
Harmon and Coles, 2005) mostly at solar radial distances $<\mathrm{60}{R}_{\odot}\approx \mathrm{0.25}$ AU in the innermost very low solar wind
($\mathrm{0.1}<{\mathit{\beta}}_{\mathrm{i}}<\mathrm{1}$; e.g. the model of McKenzie et al., 1995) region,
which is of particular interest because it is the presumable source region of
the solar wind, being accessible only remotely. Solar wind turbulence
generated here seems to freeze^{1} and is transported radially
outward afterwards by the flow.
Radio-phase scintillation of spacecraft signals from Viking, Helios, and
Pioneer have been used
early on (Woo and Armstrong, 1979) to determine solar wind density-power spectra in the
radial interval ≤1 AU, reporting mean spectral Kolmogorov slopes
$\sim -\mathrm{5}/\mathrm{3}$ with a strong flattening of the spectrum near the Sun at distances
$<\mathrm{30}{R}_{\odot}$ where the slope flattens down to $\sim -\mathrm{7}/\mathrm{6}=-\mathrm{1.1}$, a finding
which suggests evolution of the density turbulence with solar distance. In
the ISM radio scintillation, observations covered a huge range of decades,
from wavelength scales *λ*≈15 AU down to close to the Debye
length *λ*_{D}≈50 m, suggesting an approximate
Kolmogorov spectrum over 7 decades. From recent in situ Voyager 1
observations of ISM electron densities (Gurnett et al., 2013) a Kolmogorov
spectrum has been inferred down to wavelengths of *λ*∼10^{6} m that
is followed by an adjacent spectral intensity excess on the assumed kinetic
scales for wavelengths *λ**≳**λ*_{D} (Lee and Lee, 2019).

Density fluctuations *δ**N* are generally inherent to pressure
fluctuations *δ**P*. From fundamental physical principles, it follows
that density turbulence does not evolve by itself. Through the continuity
equation, it is related to velocity turbulence, which in its course requires
the presence of free energy, being driven by external forces. It is primary,
while turbulence in density, temperature, and the magnetic field is
secondary (for a different claim, see Howes and Nielson, 2013; Nielson et al., 2013). Density turbulence may
signal the presence of a population of compressive (magneto-acoustic-like)
fluctuations in addition to the usually assumed
(e.g. Biskamp, 2003; Howes, 2015) alfvénic turbulence, the dominant
fluid–magnetic fluctuation family dealing with the mutually related
alfvénic velocity and magnetic fields made use of in
magnetohydrodynamic (MHD) theory based on
Elsasser variables (Elsasser, 1950).

Inertial-range velocity turbulence is subject to Kolmogorov (Kolmogorov, 1941a, b, 1962) or Iroshnikov–Kraichnan (Iroshnikov, 1964; Kraichnan, 1965, 1966, 1967) turbulence spectra. (Regarding their generalisation to anisotropy with respect to any mean magnetic field, see Goldreich and Sridhar, 1995.) In the solar wind, Kolmogorov inertial-range spectra reaching down into the presumable dissipation range have been confirmed by a wealth of in situ observations (e.g. Goldstein et al., 1995; Tu and Marsch, 1995; Zhou et al., 2004; Alexandrova et al., 2009; Boldyrev et al., 2011; Matthaeus et al., 2016; Lugones et al., 2016; Podesta, 2011; Podesta et al., 2006, 2007; Sahraoui et al., 2009, and others). Since the mean fields ${B}_{\mathrm{0}},{T}_{\mathrm{0}},{N}_{\mathrm{0}},{U}_{\mathrm{0}}$ themselves obey pressure balance, one has the following for pressure balance among the turbulent fluctuations:

$$\begin{array}{}\text{(1)}& {\displaystyle \frac{\langle |\mathit{\delta}\mathit{B}{|}^{\mathrm{2}}\rangle}{{B}_{\mathrm{0}}^{\mathrm{2}}}}={\displaystyle \frac{\sqrt{\langle |\mathit{\delta}N{|}^{\mathrm{2}}\rangle}}{{N}_{\mathrm{0}}}}+{\displaystyle \frac{\sqrt{\langle |\mathit{\delta}T{|}^{\mathrm{2}}\rangle}}{{T}_{\mathrm{0}}}}.\end{array}$$

The angular brackets 〈…〉 indicate averaging over the
spatial scales of the turbulence with respect to turbulent fluctuations.
Alfvénic fluctuations (e.g. Howes, 2015, for a recent theoretical account
of their importance in MHD turbulence) compensate separately due
to their magnetic and velocity fluctuations being related; they do not
contribute to extra compression. In order to infer the contribution of
density fluctuations, one compares their spectral densities with those of the
temperature *δ**T* or magnetic field *δ*** B**. This requires
normalisation to the means. Solar wind densities at 1 AU are of the order of

An example is shown in Fig. 1 based on solar wind
measurements on 6 July 2012 (Šafránková et al., 2015, 2016). There is
not much freedom left in choosing the mean densities and temperatures in
Fig. 1. Densities at 1 AU barely exceed 10 cm^{−3}.
Electron temperatures are insensitive to those low-frequency density
fluctuations. High mobility makes electron reaction isothermal.

The data in Fig. 1 show the relative dominance of density fluctuations over ion temperature fluctuations under moderately low-speed solar wind conditions at all frequencies larger than the lowest accessible MHD frequencies. This is not surprising because one would not expect large temperature effects. Ion heating is a slow process which does not react to any fast pressure fluctuations caused by density or magnetic turbulence. It just shows that the turbulent thermal pressure is mainly due to density fluctuations over most of the frequency range. In the low-frequency MHD range the kinetic pressure of large-scale turbulent eddies dominates.

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 (e.g. Chen et al., 2012), where they have tentatively
been suggested to indicate the presence of kinetic Alfvén waves which may
be excited in the Hall-MHD (e.g. Huba, 2003) range as eigenmodes of
the plasma. Models including Alfvén ion-cyclotron waves (Harmon and Coles, 2005)
or kinetic Alfvén waves (Chandran et al., 2009) have been proposed to cause
spectral flattening. Kinetic Alfvén waves may also lead to bumps if only
*β*_{i}≪1. In fact, kinetic Alfvén waves possess a large
perpendicular wave number ${k}_{\u27c2}{\mathit{\lambda}}_{\mathrm{i}}\sim \mathrm{1}$ of the order of
the inverse ion-inertial length (e.g. Baumjohann and Treumann, 1996), the scale on
which ions demagnetise. If sufficient free energy is available, they can thus
be excited and propagate in this regime
(e.g. Gary, 1993; Treumann and Baumjohann, 1997). Recently Wu et al. (2019) provided
kinetic-theoretical arguments for kinetic Alfvén waves contributing to
turbulent dissipation in the ion-inertial scale region. Causing bumps, the
waves should develop large amplitudes on the background of general
turbulence, i.e. causing intermittency. This requires the presence of a
substantial amount of unidentified free energy, for instance in the form of
intense plasma beams, which are very well known in relation to collisionless
shocks both upstream and downstream (e.g. Balogh and Treumann, 2013). If kinetic
Alfvén waves are unambiguously confirmed (see, e.g. Salem et al., 2012), the inner solar wind at *≲*0.6 AU could be subject to the
continuous presence of small-scale collisionless shocks, a assumption that is
not unreasonable and which would be supported by observation of sporadic
nonthermal coronal radio emissions (type I through type IV solar radio
bursts).

In the present note we take a completely different *model-independent*
point of view, avoiding reference to any superimposed plasma instabilities or
intermittency (e.g. Chen et al., 2014b). We do not develop any “new theory”
of turbulence. Instead, we remain in the realm of turbulent fluctuations,
asking for the effect of ion inertia, with respect to ion demagnetisation in the ion-inertial Hall-MHD range,
on the shape of the inertial-range power spectral density which will be
illustrated referring to a few selected observations. To demonstrate pressure
balance we refer to related magnetic power spectra, both measured in situ
aboard spacecraft, which require a rather sophisticated instrumentation.
Those measurements were anticipated by indirectly inferred density spectra in
the solar wind (Woo, 1981; Coles and Filice, 1985; Bourgeois et al., 1985) and the interstellar
plasma (Coles, 1978; Armstrong et al., 1981, 1990) from detection of
ground-based radio scintillations.

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 (Treumann et al., 2019), maps from wave number
** k** into a stationary observer's frequency

In order to be more general, we split the mean flow velocity into bulk
*V*_{0} and large-eddy *U*_{0} velocities, the latter being known
(Tennekes, 1975) to cause Doppler broadening of the local velocity
spectrum at a fixed wave number (reviewed and backed by numerical
simulations by Fung et al., 1992; Kaneda, 1993). Imposing the theoretical K or IK
inertial-range spectra, we then find the deformed power density spectra of
density turbulence versus spacecraft frequency. We apply these to some
observed spectral density bumps which we check on a measured magnetic power
spectrum for pressure balance. The results are tabulated. Since bumpy spectra
are rather rare, we also consider two more “normal” bumpless spectra. Such
deformed density-power spectra which exhibit some typical spectral flattening
were obtained under different solar wind conditions. The paper concludes with
a brief discussion of the results.

2 Inertial-range ion response

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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 $>{\mathrm{10}}^{-\mathrm{1}}$ Hz is certainly due to the drop in ion dynamics at
frequencies close to and exceeding the ion-cyclotron frequency, which at
1 AU distance from the Sun is of the order of
${f}_{\mathrm{ci}}={\mathit{\omega}}_{\mathrm{ci}}/\mathrm{2}\mathit{\pi}\sim \mathrm{1}$ Hz for a nominal magnetic
field of ∼10 nT. In this range we enter the (dissipationless)
ion-inertial or Hall (electron-MHD) domain where ions demagnetise, currents
are carried by magnetised electrons, both species decouple magnetically, and
Hall currents arise. At those frequencies, far below the electron
${f}_{\mathrm{e}}={\mathit{\omega}}_{\mathrm{e}}/\mathrm{2}\mathit{\pi}\sim \mathrm{35}$ kHz and (assuming protons) ion
${f}_{\mathrm{i}}={\mathit{\omega}}_{\mathrm{i}}/\mathrm{2}\mathit{\pi}\sim \mathrm{0.8}$ kHz plasma frequencies, ions
and electrons couple mainly through the condition of quasi-neutrality, i.e.
via the turbulent induction–electric field which becomes^{2}

$$\begin{array}{ll}{\displaystyle}\mathit{\delta}\mathit{E}& {\displaystyle}=-\mathit{\delta}{\mathit{V}}_{\u27c2}\times {\mathit{B}}_{\mathrm{0}}-({\mathit{V}}_{\mathrm{0}}+{\mathit{U}}_{\mathrm{0}})\times \mathit{\delta}\mathit{B}-\\ {\displaystyle}& {\displaystyle}-{\displaystyle \frac{\mathrm{1}}{e{N}_{\mathrm{0}}}}{\mathit{B}}_{\mathrm{0}}\times \mathit{\delta}\mathit{J}+\\ \text{(2)}& {\displaystyle}& {\displaystyle}+\langle \mathit{\delta}\mathit{V}\times \mathit{\delta}\mathit{B}+{\displaystyle \frac{\mathrm{1}}{e{N}_{\mathrm{0}}}}\mathit{\delta}\mathit{J}\times ({\displaystyle \frac{\mathit{\delta}N}{{N}_{\mathrm{0}}}}{\mathit{B}}_{\mathrm{0}}-\mathit{\delta}\mathit{B})\rangle .\end{array}$$

For later use, we split the main velocity field
$\langle \mathit{V}\rangle ={\mathit{V}}_{\mathrm{0}}+{\mathit{U}}_{\mathrm{0}}$ into the bulk flow (convection)
*V*_{0} and an advection velocity *U*_{0}. The latter is the mean
velocity of a small number of large eddies which carry the main energy of the
turbulence. Even for stationary turbulence, they advect the bulk of
small-scale eddies around at speed *U*_{0} (Tennekes, 1975; Fung et al., 1992).

The last three averaged nonlinear terms within the angular brackets
〈…〉 on the right are the nonlinear contributions of the
fluctuations to the mean fields yielding an electromotive force which
contributes to mean-field processes like convection, dynamo action, and
turbulent diffusion. They vary only on the large mean-field scale. On the
fluctuation scale they are constant and can be dropped, unless the turbulence
is bounded, in which case boundary effects must be taken into account at the
large scales of the system. Generally, in the solar wind this is not the
case. The remaining three linear terms distinguish between directions
parallel and perpendicular to the main magnetic field *B*_{0}. The third
linear term is the genuine perpendicular Hall contribution. From Ampere's
law for the current fluctuation
${\mathit{\mu}}_{\mathrm{0}}\mathit{\delta}\mathit{J}=\mathrm{\nabla}\times \mathit{\delta}\mathit{B}$ we have the following for the perpendicular
and parallel components of the turbulent electric field:

$$\begin{array}{ll}{\displaystyle}\mathit{\delta}{\mathit{E}}_{\u27c2}& {\displaystyle}={\mathit{B}}_{\mathrm{0}}\times [\mathit{\delta}{\mathit{V}}_{\u27c2}-{\displaystyle \frac{{U}_{\mathrm{0}\Vert}}{{B}_{\mathrm{0}}}}\mathit{\delta}{\mathit{B}}_{\u27c2}-\\ {\displaystyle}& {\displaystyle}-{\displaystyle \frac{\mathrm{1}}{e{\mathit{\mu}}_{\mathrm{0}}{N}_{\mathrm{0}}}}(\mathrm{\nabla}\times \mathit{\delta}\mathit{B}{)}_{\u27c2}]-{\mathit{U}}_{\mathrm{0}\u27c2}\times \mathit{\delta}{\mathit{B}}_{\Vert}\\ \text{(3)}& {\displaystyle}\mathit{\delta}{\mathit{E}}_{\Vert}& {\displaystyle}=-\phantom{\rule{0.33em}{0ex}}{\mathit{U}}_{\mathrm{0}\u27c2}\times \mathit{\delta}{\mathit{B}}_{\u27c2}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}=\phantom{\rule{-0.125em}{0ex}}=\phantom{\rule{-0.125em}{0ex}}\u27f9\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\mathrm{0}.\end{array}$$

The second of these equations is of no interest, because the low-frequency
parallel electric field its right-hand side produces is readily compensated
by electron displacements along *B*_{0}.

This leaves us with the fluctuating perpendicular induction field in the
first Eq. (2). Here, any parallel advection *U*_{0‖} attributes
to the perpendicular velocity fluctuations from perpendicular magnetic
fluctuations *δ**B*_{⟂}. On the other hand, any present parallel
compressive magnetic fluctuations $\mathit{\delta}{\mathit{B}}_{\Vert}={\mathit{B}}_{\mathrm{0}}(\mathit{\delta}{B}_{\Vert}/{B}_{\mathrm{0}})$ contribute through perpendicular advection *U*_{0⟂}. In their
absence, when the magnetic field is non-compressive, the last term
disappears.

The complete Hall contribution to the electric field, viz. the last term in the brackets in Eq. (2), can be written as

$$\begin{array}{}\text{(4)}& \mathit{\delta}{\mathit{E}}_{\u27c2}^{\mathrm{H}}=-{\displaystyle \frac{{B}_{\mathrm{0}}}{e{\mathit{\mu}}_{\mathrm{0}}{N}_{\mathrm{0}}}}({\mathbf{\nabla}}_{\u27c2}\mathit{\delta}{B}_{\Vert}-{\mathrm{\nabla}}_{\Vert}\mathit{\delta}{\mathit{B}}_{\u27c2}).\end{array}$$

Even for ${U}_{\mathrm{0}\Vert}=\mathrm{0}$, it contributes through the turbulent fluctuations in the
magnetic field. As both these contributions depend only on *δ*** B**, we
can isolate them for separate consideration. One observes that, in the
absence of any compressive magnetic components

Let us assume that advection by large-scale energy-carrying eddies is perpendicular ${\mathit{U}}_{\mathrm{0}}={\mathit{U}}_{\mathrm{0}\u27c2}$, and there are no compressive magnetic fluctuations $\mathit{\delta}{\mathit{B}}_{\Vert}=\mathrm{0}$. In Eq. (2) this reduces to considering only the first term containing the velocity fluctuations. We ask for its effect on the density fluctuations in the ion-inertial domain on scales where the ions demagnetise.

On scales in the ion-inertial range shorter than either the ion thermal
gyroradius ${\mathit{\rho}}_{\mathrm{i}}={v}_{\mathrm{i}}/{\mathit{\omega}}_{\mathrm{ci}}$ or – depending
on the direction to the mean magnetic field *B*_{0} and the value of
plasma beta $\mathit{\beta}=\mathrm{2}{\mathit{\mu}}_{\mathrm{0}}{N}_{\mathrm{0}}{T}_{\mathrm{0}}/{B}_{\mathrm{0}}^{\mathrm{2}}$, with
${\mathit{\omega}}_{\mathrm{ci}}=e{B}_{\mathrm{0}}/{m}_{\mathrm{i}}$ ion-cyclotron and
${\mathit{\omega}}_{\mathrm{i}}=e\sqrt{{N}_{\mathrm{0}}/{\mathit{\u03f5}}_{\mathrm{0}}{m}_{\mathrm{i}}}$ ion plasma frequency,
respectively – inertial length ${\mathit{\lambda}}_{\mathrm{i}}=c/{\mathit{\omega}}_{\mathrm{i}}$, the
ions demagnetise. Being non-magnetic, they do not distinguish between
potential and induction–electric fields. They experience the induction field
caused by the spectrum of velocity fluctuations as an external electric field
which, in an electron-proton plasma, causes a charge density fluctuation
*e**δ**N*_{i}=*e**δ**N*_{e} and thus a density fluctuation
*δ**N*. Poisson's equation implies that

$$\begin{array}{}\text{(5)}& \mathrm{\nabla}\cdot \mathit{\delta}\mathit{E}={\displaystyle \frac{e}{{\mathit{\u03f5}}_{\mathrm{0}}}}\mathit{\delta}N\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}=\phantom{\rule{-0.125em}{0ex}}=\phantom{\rule{-0.125em}{0ex}}\u27f9\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}i\mathit{k}\cdot \mathit{\delta}{\mathit{E}}_{k}={\displaystyle \frac{e}{{\mathit{\u03f5}}_{\mathrm{0}}}}\mathit{\delta}{N}_{k}.\end{array}$$

The right expression is its Fourier transform. For completeness we note that the Hall contribution to the Poisson equation in Fourier space reads

$$\begin{array}{}\text{(6)}& i{\mathit{k}}_{\u27c2}\cdot \mathit{\delta}{\mathit{E}}_{\u27c2k}^{\mathrm{H}}={\displaystyle \frac{{B}_{\mathrm{0}}}{e{\mathit{\mu}}_{\mathrm{0}}{N}_{\mathrm{0}}}}({k}_{\u27c2}^{\mathrm{2}}\mathit{\delta}{B}_{\Vert k}-{k}_{\Vert}{\mathit{k}}_{\u27c2}\cdot \mathit{\delta}{\mathit{B}}_{\u27c2k})={\displaystyle \frac{e}{{\mathit{\u03f5}}_{\mathrm{0}}}}\mathit{\delta}{N}_{k}^{\mathrm{H}}.\end{array}$$

Again it becomes obvious that absence of parallel (compressive) magnetic turbulence eliminates the first term in this expression, while purely perpendicular propagation eliminates the second term. Alfvénic turbulence, for instance, with $\mathit{\delta}{B}_{\Vert}=\mathrm{0}$ and ${k}_{\u27c2}=\mathrm{0}$, has no Hall effect on the modulation of the density spectrum, a fact which is well known. On the other hand, for perpendicular wave numbers $k={k}_{\u27c2}$ only compressive Hall-magnetic fluctuations $\mathit{\delta}{B}_{\Vert {k}_{\u27c2}}$ contribute to the Hall fluctuations in the density $\mathit{\delta}{N}_{{k}_{\u27c2}}^{\mathrm{H}}$.

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. (2) with wave number ** k** selects wave numbers

$$\begin{array}{}\text{(7)}& \langle |\mathit{\delta}N{|}^{\mathrm{2}}{\rangle}_{{k}_{\u27c2}}=({\displaystyle \frac{{\mathit{\u03f5}}_{\mathrm{0}}{B}_{\mathrm{0}}}{e}}{)}^{\mathrm{2}}{k}_{\u27c2}^{\mathrm{2}}\langle |\mathit{\delta}\mathit{V}{|}^{\mathrm{2}}{\rangle}_{{k}_{\u27c2}},\end{array}$$

where we from now on drop the index ⟂ on the velocity
*δ**V*_{⟂}. Angular brackets again symbolise spatial averaging
over the fluctuation scale. The functional dependence on the wave number is
indicated by the index *k*_{⟂}. It is obvious that the power spectrum of
density fluctuations in the ion-inertial Hall-MHD domain is completely
determined by the power spectrum of the turbulent velocity^{4}. This can
be written as

$$\begin{array}{ll}{\displaystyle \frac{\langle |\mathit{\delta}N{|}^{\mathrm{2}}{\rangle}_{{k}_{\u27c2}}}{{N}_{\mathrm{0}}^{\mathrm{2}}}}& {\displaystyle}=\left({\displaystyle \frac{{V}_{\mathrm{A}}}{c}}{)}^{\phantom{\rule{-0.125em}{0ex}}\phantom{\rule{-0.125em}{0ex}}\mathrm{2}}\right({\displaystyle \frac{{k}_{\u27c2}}{{\mathit{\omega}}_{\mathrm{i}}}}{)}^{\phantom{\rule{-0.125em}{0ex}}\phantom{\rule{-0.125em}{0ex}}\mathrm{2}}\langle |\mathit{\delta}\mathit{V}{|}^{\mathrm{2}}{\rangle}_{{k}_{\u27c2}}\\ \text{(8)}& {\displaystyle}& {\displaystyle}={\displaystyle \frac{\langle |\mathit{\delta}\mathit{V}{|}^{\mathrm{2}}{\rangle}_{{k}_{\u27c2}}}{{c}^{\mathrm{2}}}}\left({\displaystyle \frac{{V}_{\mathrm{A}}}{c}}{)}^{\phantom{\rule{-0.125em}{0ex}}\phantom{\rule{-0.125em}{0ex}}\mathrm{2}}\right({k}_{\u27c2}{\mathit{\lambda}}_{\mathrm{i}}{)}^{\phantom{\rule{-0.125em}{0ex}}\mathrm{2}},\end{array}$$

where ${V}_{\mathrm{A}}^{\mathrm{2}}={B}_{\mathrm{0}}^{\mathrm{2}}/{\mathit{\mu}}_{\mathrm{0}}{m}_{\mathrm{i}}{N}_{\mathrm{0}}$ is the squared Alfvén speed, and ${\mathit{\omega}}_{\mathrm{i}}^{\mathrm{2}}={e}^{\mathrm{2}}{N}_{\mathrm{0}}/{\mathit{\u03f5}}_{\mathrm{0}}{m}_{\mathrm{i}}$ is the squared proton plasma frequency. As expected, in order to contribute to density fluctuations, perpendicular scales ${\mathit{\lambda}}_{\u27c2}<{\mathit{\lambda}}_{\mathrm{i}}$ smaller than the ion-inertial length ${\mathit{\lambda}}_{\mathrm{i}}=c/{\mathit{\omega}}_{\mathrm{i}}$ are required, while in the long-wavelength range ${k}_{\u27c2}{\mathit{\lambda}}_{\mathrm{i}}<\mathrm{1}$, there is no effect on the spectrum. This is in agreement with the assumption that any spectral modification is expected only in the ion-inertial range.

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
${k}_{\u27c2}{\mathit{\rho}}_{\mathrm{i}}<\mathrm{1}$, where
${\mathit{\rho}}_{\mathrm{i}}={v}_{\mathrm{i}}/{\mathit{\omega}}_{\mathrm{ci}}={\mathit{\lambda}}_{\mathrm{i}}{v}_{\mathrm{i}}/{V}_{\mathrm{A}}$
is the ion gyroradius, with *v*_{i} as the thermal speed. Thus we have
two conditions which must simultaneously be satisfied:

$$\begin{array}{}\text{(9)}& {k}_{\u27c2}{\mathit{\lambda}}_{\mathrm{i}}>\mathrm{1}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\mathrm{and}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}{k}_{\u27c2}{\mathit{\lambda}}_{\mathrm{i}}>{\displaystyle \frac{{V}_{\mathrm{A}}}{{v}_{\mathrm{i}}}}\equiv {\mathit{\beta}}_{\mathrm{i}}^{-{\scriptscriptstyle \frac{\mathrm{1}}{\mathrm{2}}}}.\end{array}$$

For *V*_{A}<*v*_{i} the second condition is trivial. This is,
however, a rare case, so the more realistic restriction is the opposite small
ion-beta case when *V*_{A}>*v*_{i} and hence
*β*_{i}<1. It must, however, be combined with another condition
which requires that the wave numbers be smaller than the inverse
electron gyroradius ${\mathit{\rho}}_{\mathrm{e}}={v}_{\mathrm{e}}/{\mathit{\omega}}_{\mathrm{ce}}$. The
relation between *ρ*_{e} and *ρ*_{i} is
${\mathit{\rho}}_{\mathrm{e}}^{\mathrm{2}}/{\mathit{\rho}}_{\mathrm{i}}^{\mathrm{2}}={m}_{\mathrm{e}}{T}_{\mathrm{e}}/{m}_{\mathrm{i}}{T}_{\mathrm{i}}$.
Moreover we have
${\mathit{\rho}}_{\mathrm{i}}/{\mathit{\lambda}}_{\mathrm{i}}={v}_{\mathrm{i}}/{V}_{\mathrm{A}}$, and in
addition, ${\mathit{\beta}}_{\mathrm{i}}={v}_{\mathrm{i}}^{\mathrm{2}}/{V}_{\mathrm{A}}^{\mathrm{2}}=({T}_{\mathrm{i}}/{T}_{\mathrm{e}}){\mathit{\beta}}_{\mathrm{e}}$.
Using all these relations, we obtain finally that

$$\begin{array}{}\text{(10)}& \mathrm{1}<{k}_{\u27c2}^{\mathrm{2}}{\mathit{\lambda}}_{\mathrm{i}}^{\mathrm{2}}{\mathit{\beta}}_{\mathrm{i}}<{\displaystyle \frac{{m}_{\mathrm{i}}}{{m}_{\mathrm{e}}}}{\displaystyle \frac{{T}_{\mathrm{i}}}{{T}_{\mathrm{e}}}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\mathrm{for}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}{\mathit{\beta}}_{\mathrm{i}}<\mathrm{1}.\end{array}$$

This expression defines the marginal condition for the existence of a range
in wave numbers where the ions respond to the spectrum of the turbulent
electric field *δ*** E**:

$$\begin{array}{}\text{(11)}& {\displaystyle \frac{{T}_{\mathrm{i}}}{{T}_{\mathrm{e}}}}\mathit{\gtrsim}{\displaystyle \frac{{m}_{\mathrm{e}}}{{m}_{\mathrm{i}}}}\sim \mathrm{0.001}.\end{array}$$

Because of the smallness of the right-hand side, this is a weak restriction.
As expected, any effect on the density-power spectrum will disappear at
wave numbers *k*_{⟂}*ρ*_{e} where the electrons demagnetise. On the
other hand, the lower wave-number limit is a sensitive function of the
external conditions. This becomes clear when writing it in the form

$$\begin{array}{}\text{(12)}& {T}_{\mathrm{e}}/{T}_{\mathrm{i}}{\mathit{\beta}}_{\mathrm{e}}<{k}_{\u27c2}^{\mathrm{2}}{\mathit{\lambda}}_{\mathrm{i}}^{\mathrm{2}}.\end{array}$$

The electron plasma beta in the solar wind is of the order of
*β*_{e}*≳**O*(1). However, the temperature ratio
*T*_{e}∕*T*_{i} is variable and usually large, varying between a
few and a few tens. Thus usually *β*_{i}<1. Figure 2
shows a graph of this dependence.

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 (or its anisotropic extension by Goldreich and Sridhar, 1995, abbreviated KGS) but will also refer to the IK spectrum, which both have previously been found to be of relevance in solar wind turbulence.

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
*U*_{0} and bulk turbulence consisting of large numbers of small
energy-poor eddies which are frozen to the large eddies. The large eddies
stir the small-scale turbulence, forcing it into *advective* motion
(Tennekes, 1975). This causes a Doppler broadening of the wave-number
spectrum at fixed *k* and has been confirmed by numerical simulations
(Fung et al., 1992; Kaneda, 1993). Below, it will be found that this advection
cannot be resolved in bulk convective flow which buries the subtle effect of
Doppler broadening. A probable counterexample is shown in
Fig. 5.

The stationary velocity spectrum of turbulent eddies at energy injection rate
*ϵ* exhibits a broad inertial power law range in ** k**
(Kolmogorov, 1941a, b, 1962; Obukhov, 1941) which,
between injection

$$\begin{array}{}\text{(13)}& \langle |\mathit{\delta}\mathit{V}{|}^{\mathrm{2}}{\rangle}_{k}\equiv {\mathcal{E}}_{\mathrm{K}}(k)={C}_{\mathrm{K}}{\mathit{\u03f5}}^{{\scriptscriptstyle \frac{\mathrm{2}}{\mathrm{3}}}}{k}^{-{\scriptscriptstyle \frac{\mathrm{5}}{\mathrm{3}}}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\mathrm{for}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}{k}_{\mathrm{in}}<k<{k}_{\mathrm{d}},\end{array}$$

with *C*_{K}≈1.65 as Kolmogorov's constant of proportionality
(as determined by Gotoh and Fukayama, 2001, using numerical simulations). Clearly,
in a fast-streaming solar wind, when straightforwardly mapping this K spectrum
by the Taylor hypothesis (Taylor, 1938) into the stationary spacecraft
frame, the spectral index is unchanged, and one trivially recovers the
${\mathit{\omega}}_{\mathrm{s}}^{-\frac{\mathrm{5}}{\mathrm{3}}}$ Kolmogorov slope in frequency space.

This changes drastically when referring to an advected K spectrum of
velocity turbulence (Fung et al., 1992; Kaneda, 1993) which yields the above-mentioned spectral Doppler broadening at fixed
*k*,

$$\begin{array}{ll}{\displaystyle}{\mathcal{E}}_{k{\mathit{\omega}}_{k}}^{\mathrm{ad}}& {\displaystyle}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\displaystyle \frac{{\mathcal{E}}_{\mathrm{K}}\left(k\right)}{\sqrt{\mathrm{2}\mathit{\pi}}k{U}_{\mathrm{0}}}}\sum _{\pm}\mathrm{exp}[-{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\displaystyle \frac{{\mathit{\omega}}_{\pm}^{\mathrm{2}}}{(k{U}_{\mathrm{0}}{)}^{\mathrm{2}}}}],\\ \text{(14)}& {\displaystyle}{\mathit{\omega}}_{\pm}& {\displaystyle}={\mathit{\omega}}_{k}\pm {\mathrm{\ell}}_{\mathrm{K}}{k}^{\frac{\mathrm{2}}{\mathrm{3}}},\end{array}$$

which is due to decorrelation of the small eddies in advective transport,
with ℓ_{K}∼*O*(1) being some constant. The *k*^{⅔} dependence
in the argument of the exponential results from advection ** k**⋅

$$\begin{array}{}\text{(15)}& {\displaystyle}{\mathcal{E}}_{k{\mathit{\omega}}_{k}}^{\mathrm{ad}}& {\displaystyle}\propto {\displaystyle \frac{{\mathcal{E}}_{\mathrm{K}}\left(k\right)}{k}}\mathrm{exp}(-{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\mathrm{cos}}^{\mathrm{2}}{\mathit{\gamma}}_{k})\sim {k}^{-\frac{\mathrm{8}}{\mathrm{3}}},\text{(16)}& {\displaystyle}{\mathit{\omega}}_{\pm}& {\displaystyle}\approx \mathit{k}\cdot {\mathit{U}}_{\mathrm{0}}=k{U}_{\mathrm{0}}\mathrm{cos}\phantom{\rule{0.125em}{0ex}}{\mathit{\gamma}}_{k}.\end{array}$$

In the stationary turbulence frame the power spectrum of turbulence in the velocity decays to $\propto {k}^{-\frac{\mathrm{8}}{\mathrm{3}}}$ with non-Kolmogorov spectral index $\mathrm{8}/\mathrm{3}\approx \mathrm{2.7}$.

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 *ω*_{s}∝*k*,
one might conclude then that a convective flow maps this spectral range of
the advected turbulent K spectrum into the spacecraft frame where it
appears as an ${\mathit{\omega}}_{\mathrm{s}}^{-\frac{\mathrm{8}}{\mathrm{3}}}$ spectrum.

If this is true, then the corresponding observed spectral transition (or
break point) from the spectral K index $\sim \mathrm{5}/\mathrm{3}$ to the steeper index $\sim \mathrm{8}/\mathrm{3}$ observed in the large-wave-number power spectra indicates the division
between large-scale energy-carrying, energy-rich turbulent eddies and the
bulk of energy-poor small-scale eddies in the mechanical turbulence. It thus
provides a simple explanation of the change in spectral index from $\sim \mathrm{5}/\mathrm{3}$
(K spectrum) to *≲*3 (advected K turbulence spectrum) without
invoking any sophisticated turbulence theory as well as having no effects of dissipation.

Inspecting the behaviour in the long-wavelength range, one finds that the
exponential dependence $\mathrm{exp}(-{\mathrm{\ell}}_{\mathrm{K}}^{\mathrm{2}}/{U}_{\mathrm{0}}^{\mathrm{2}}{k}^{\frac{\mathrm{2}}{\mathrm{3}}})$
suppresses the spectrum here. This flattens the inertial-range spectrum
towards small wave numbers *k*_{in} into the large-eddy range where it
causes bending of the spectrum. The wave number at spectral maximum is

$$\begin{array}{}\text{(17)}& {k}_{\mathrm{min}}\mathit{\lesssim}{\mathrm{\ell}}_{\mathrm{K}}^{\mathrm{3}}/\mathrm{16}{U}_{\mathrm{0}}^{\mathrm{3}}\sqrt{\mathrm{2}}.\end{array}$$

Approaching from the Kolmogorov inertial range towards a smaller *k*, one observes
flattening until *k*_{min}<*k*_{in}. In most cases this point
will lie outside the observation range.

In the stationary turbulence frame the frequency spectrum is obtained when
integrated with respect to *k* (Biskamp, 2003). It then maps the Doppler
broadened advected velocity power spectrum (Fung et al., 1992; Kaneda, 1993) to the
Kolmogorov law in the source-region frequency space:

$$\begin{array}{}\text{(18)}& \underset{{k}_{\mathrm{in}}}{\overset{{k}_{\mathrm{d}}}{\int}}\mathrm{d}k{\mathcal{E}}_{k{\mathit{\omega}}_{k}}^{\mathrm{ad}}\sim {\mathcal{E}}_{\mathrm{K}}^{\mathrm{ad}}\left(\mathit{\omega}\right)\phantom{\rule{0.25em}{0ex}}\propto \phantom{\rule{0.25em}{0ex}}{\mathit{\omega}}^{-{\scriptscriptstyle \frac{\mathrm{5}}{\mathrm{3}}}}.\end{array}$$

This mapping is independent of Taylor's hypothesis. It strictly applies only
to the turbulent reference frame. When attempting to map it into the
spacecraft frame via Taylor's Galilei transformation, referring to solar wind
flow at finite *V*_{0}≠0, one must return to its wave-number representation
in Eq. (11). This transformation, though straightforward, is obscured
by the appearance of *k* in the exponential through *ω*_{±}. According
to Taylor the turbulence frame frequency transforms as

$$\begin{array}{}\text{(19)}& {\mathit{\omega}}_{k}={\mathit{\omega}}_{\mathrm{s}}-k{V}_{\mathrm{0}}\mathrm{cos}\mathit{\alpha}.\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\mathit{\alpha}=\mathrm{\angle}(\mathit{k}\mathbf{,}{\mathit{V}}_{\mathrm{0}}).\end{array}$$

This is Taylor's Galilei transformation. Neglecting *ω*_{k} implies
that *ω*_{s}=*k**V*_{0}cos *α*. The exponential reduces to

$$\begin{array}{}\text{(20)}& {\displaystyle}& {\displaystyle}\mathrm{exp}[-{\displaystyle \frac{\mathrm{1}}{\mathrm{4}}}({\displaystyle \frac{{\mathit{\omega}}_{k}+k{V}_{\mathrm{0}}\mathrm{cos}\phantom{\rule{0.125em}{0ex}}\mathit{\alpha}\pm {\mathrm{\ell}}_{\mathrm{K}}{k}^{\frac{\mathrm{2}}{\mathrm{3}}}}{k{U}_{\mathrm{0}}}}{)}^{\mathrm{2}}]=\text{(21)}& {\displaystyle}& {\displaystyle}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}=\mathrm{exp}[-{\displaystyle \frac{\mathrm{1}}{\mathrm{4}(k{\mathit{\lambda}}_{\mathrm{i}}{)}^{\frac{\mathrm{2}}{\mathrm{3}}}}}({\displaystyle \frac{{\mathit{\lambda}}_{\mathrm{i}}^{\frac{\mathrm{1}}{\mathrm{3}}}{\mathrm{\ell}}_{\mathrm{K}}}{{U}_{\mathrm{0}}}}{)}^{\mathrm{2}}]\text{(22)}& {\displaystyle}& {\displaystyle}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\u27f6\mathrm{1}-{\displaystyle \frac{\mathrm{1}}{\mathrm{4}(k{\mathit{\lambda}}_{\mathrm{i}}{)}^{\frac{\mathrm{2}}{\mathrm{3}}}}}({\displaystyle \frac{{\mathit{\lambda}}_{\mathrm{i}}^{\frac{\mathrm{1}}{\mathrm{3}}}{\mathrm{\ell}}_{\mathrm{K}}}{{U}_{\mathrm{0}}}}{)}^{\mathrm{2}},\end{array}$$

with ${\mathit{\lambda}}_{\mathrm{i}}^{\frac{\mathrm{1}}{\mathrm{3}}}\mathrm{\ell}/{U}_{\mathrm{0}}\equiv {U}_{\mathrm{\ell}}^{\mathrm{K}}/{U}_{\mathrm{0}}$ being a
velocity ratio. The arrow holds for the ion-inertial range
*k**λ*_{i}>1 and ${U}_{\mathrm{\ell}}^{\mathrm{K}}/{U}_{\mathrm{0}}<\mathrm{1}$. The exponential
expression leads to an advected K spectrum as observed by the spacecraft in
frequency space:

$$\begin{array}{}\text{(23)}& {\mathcal{E}}_{{\mathit{\omega}}_{\mathrm{s}}}^{\mathrm{ad}}\propto {\mathit{\omega}}_{\mathrm{s}}^{-{\scriptscriptstyle \frac{\mathrm{8}}{\mathrm{3}}}}\mathrm{exp}[-{\displaystyle \frac{\mathrm{1}}{\mathrm{4}}}({\displaystyle \frac{{V}_{\mathrm{0}}\mathrm{cos}\phantom{\rule{0.125em}{0ex}}\mathit{\alpha}}{{\mathit{\omega}}_{\mathrm{s}}{\mathit{\lambda}}_{\mathrm{i}}}}{)}^{{\scriptscriptstyle \frac{\mathrm{2}}{\mathrm{3}}}}\left({\displaystyle \frac{{U}_{\mathrm{\ell}}^{\mathrm{K}}}{{U}_{\mathrm{0}}}}{)}^{\mathrm{2}}\right],\end{array}$$

which, as before for large *ω*_{s}, is of the spectral index
8∕3. With decreasing spacecraft frequency *ω*_{s}, the
exponential correction factor acts to suppress the
spectrum. This corresponds to a
spectral flattening towards smaller *ω*_{s}. It might even cause
a spectral dip, depending on the parameters and velocities involved. The
effect is strongest for aligned streaming and the eddy wave number. For
*α*∼90^{∘} one recovers the index 8∕3.

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.

3 Ion-inertial-range density-power spectrum

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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. (5) and (10) that

$$\begin{array}{}\text{(24)}& \langle |\mathit{\delta}N{|}^{\mathrm{2}}{\rangle}_{{k}_{\u27c2}}={C}_{\mathrm{K}}({\displaystyle \frac{{\mathit{\u03f5}}_{\mathrm{0}}{B}_{\mathrm{0}}}{e}}{)}^{\mathrm{2}}{\mathit{\u03f5}}^{{\scriptscriptstyle \frac{\mathrm{2}}{\mathrm{3}}}}{k}_{\u27c2}^{{\scriptscriptstyle \frac{\mathrm{1}}{\mathrm{3}}}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\mathrm{for}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}{k}_{\u27c2\mathrm{in}}<{k}_{\u27c2}<{k}_{\u27c2\mathrm{d}}.\end{array}$$

This is a very simple wave-number dependence of the power spectrum of density turbulence, permitting (Treumann et al., 2019) Taylor's Galilei transformation into the spacecraft frame. Setting ${k}_{\u27c2}={\mathit{\omega}}_{\mathrm{s}}/{V}_{\mathrm{0}}\mathrm{cos}\phantom{\rule{0.125em}{0ex}}\mathit{\alpha}$ we immediately obtain

$$\begin{array}{}\text{(25)}& \langle |\mathit{\delta}N{|}^{\mathrm{2}}{\rangle}_{{k}_{\u27c2}}\propto \phantom{\rule{0.25em}{0ex}}{\mathit{\omega}}_{\mathrm{s}}^{{\scriptscriptstyle \frac{\mathrm{1}}{\mathrm{3}}}},\end{array}$$

with factor of proportionality ${C}_{\mathrm{K}}({\mathit{\u03f5}}_{\mathrm{0}}{B}_{\mathrm{0}}/e{)}^{\mathrm{2}}({\mathit{\u03f5}}^{\mathrm{2}}/{V}_{\mathrm{0}}\mathrm{cos}\phantom{\rule{0.125em}{0ex}}\mathit{\alpha}{)}^{\frac{\mathrm{1}}{\mathrm{3}}}$.

Following exactly the same reasoning when dealing with the IK spectrum, which has power index 3∕2, we obtain

$$\begin{array}{}\text{(26)}& \langle |\mathit{\delta}N{|}^{\mathrm{2}}{\rangle}_{{k}_{\u27c2}}\propto \phantom{\rule{0.25em}{0ex}}{\mathit{\omega}}_{\mathrm{s}}^{{\scriptscriptstyle \frac{\mathrm{1}}{\mathrm{2}}}}.\end{array}$$

Hence, the effect of the Poisson response of the plasma to the inertial-range power spectra of K and IK turbulence in the velocity is to generate a positive slope in the density-power spectrum when transformed by Taylor's Galilei transformation into the spacecraft frame.

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

Use of the advected power spectral density Eq. (11) of the
velocity field for *V*_{0}=0 in the transformed Poisson equation, with $k\to {k}_{\u27c2}$ being perpendicular to the mean magnetic field *B*_{0}, yields the following for the
non-convected advected turbulent ion-inertial-range Poisson-modified
density-power spectrum in the stationary large-eddy turbulence frame:

$$\begin{array}{ll}{\displaystyle}\langle |\mathit{\delta}N{|}^{\mathrm{2}}{\rangle}_{{\mathit{\omega}}_{k}{k}_{\u27c2}}^{\mathrm{ad}}& {\displaystyle}={\displaystyle \frac{{\mathit{\u03f5}}_{\mathrm{0}}^{\mathrm{2}}{B}_{\mathrm{0}}^{\mathrm{2}}}{{e}^{\mathrm{2}}}}{k}_{\u27c2}^{\mathrm{2}}\langle |\mathit{\delta}\mathit{V}{|}^{\mathrm{2}}{\rangle}_{{\mathit{\omega}}_{k}{k}_{\u27c2}}\\ {\displaystyle}& {\displaystyle}={\displaystyle \frac{{\mathit{\u03f5}}_{\mathrm{0}}^{\mathrm{2}}{B}_{\mathrm{0}}^{\mathrm{2}}}{{e}^{\mathrm{2}}}}{k}_{\u27c2}^{\mathrm{2}}{\mathcal{E}}_{{k}_{\u27c2}{\mathit{\omega}}_{k}}^{\mathrm{ad}}\\ \text{(27)}& {\displaystyle}& {\displaystyle}\propto {k}_{\u27c2}^{-\frac{\mathrm{2}}{\mathrm{3}}}\sum _{\pm}\mathrm{exp}[-{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\displaystyle \frac{{\mathit{\omega}}_{\pm}^{\mathrm{2}}}{(k{U}_{\mathrm{0}}{)}^{\mathrm{2}}}}].\end{array}$$

Integration with respect to *k*_{⟂} under the above assumption on
${\mathit{\omega}}_{\pm}\approx {k}_{\u27c2}{U}_{\mathrm{0}}$ yields the following for the Eulerian (Fung et al., 1992)
density-power spectrum in frequency space ${\mathit{\omega}}_{\mathrm{\ell}}<\mathit{\omega}<{\mathit{\omega}}_{u}$ in
the ion-inertial domain of the turbulent inertial range:

$$\begin{array}{}\text{(28)}& \langle |\mathit{\delta}N{|}^{\mathrm{2}}{\rangle}_{\mathit{\omega}}^{\mathrm{ad}}\phantom{\rule{0.25em}{0ex}}\sim \phantom{\rule{0.25em}{0ex}}{\mathit{\omega}}^{{\scriptscriptstyle \frac{\mathrm{1}}{\mathrm{3}}}},\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}{k}_{\mathrm{ir}}^{{\scriptscriptstyle \frac{\mathrm{2}}{\mathrm{3}}}}{\mathit{\u03f5}}^{{\scriptscriptstyle \frac{\mathrm{1}}{\mathrm{3}}}}={\mathit{\omega}}_{\mathrm{\ell}}<\mathit{\omega}<{\mathit{\omega}}_{u}.\end{array}$$

This is the proper frequency dependence of the advected turbulent density
spectrum *in the turbulence frame*. Here ${k}_{\mathrm{ir}}\approx \mathrm{2}\mathit{\pi}{\mathit{\omega}}_{\mathrm{i}}/c$ (or 2*π**v*_{i}∕*ω*_{ci}) is the
wave number presumably corresponding to the lower end of the ion-inertial
range. The upper bound on the frequency *ω*_{u} remains undetermined. One
assumption would be that *ω*_{u} is the lower-hybrid frequency which is
intermediate to the ion and electron cyclotron frequencies. At this frequency
electrons become capable of discharging the electric induction field, thus
breaking the spectrum to return to its Kolmogorov slope at increasing
frequency.

In contrast to the Kolmogorov law, the *Poisson-mediated proper advected* density-power spectrum Eq. (18) increases with
frequency in the proper stationary frame of the turbulence. This increase is
restricted to that part of the inertial K range which corresponds to the
ion-inertial scale and frequency range.

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

$$\begin{array}{}\text{(29)}& \langle |\mathit{\delta}\mathit{V}{|}^{\mathrm{2}}{\rangle}_{{\mathit{\omega}}_{k}{k}_{\u27c2}}\propto {k}_{\u27c2}^{-{\scriptscriptstyle \frac{\mathrm{3}}{\mathrm{2}}}},\end{array}$$

which yields

$$\begin{array}{ll}{\displaystyle}\langle |\mathit{\delta}N{|}^{\mathrm{2}}{\rangle}_{{\mathit{\omega}}_{k}{k}_{\u27c2}}^{\mathrm{ad}}& {\displaystyle}\propto {k}_{\u27c2}^{-\frac{\mathrm{1}}{\mathrm{2}}}\sum _{\pm}\mathrm{exp}[-{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\displaystyle \frac{{\mathit{\omega}}_{\pm}^{\mathrm{2}}}{(k{U}_{\mathrm{0}}{)}^{\mathrm{2}}}}],\\ \text{(30)}& {\displaystyle}{\mathit{\omega}}_{\pm}& {\displaystyle}={\mathit{\omega}}_{k}\pm {\mathrm{\ell}}_{\mathrm{IK}}{k}_{\u27c2}^{\frac{\mathrm{3}}{\mathrm{4}}}.\end{array}$$

Integration with respect to *k*_{⟂} then gives the proper advected
frequency spectrum in the stationary frame of IK turbulence:

$$\begin{array}{}\text{(31)}& \langle |\mathit{\delta}N{|}^{\mathrm{2}}{\rangle}_{\mathit{\omega}}^{\mathrm{ad}}\phantom{\rule{0.25em}{0ex}}\sim \phantom{\rule{0.25em}{0ex}}{\mathit{\omega}}^{{\scriptscriptstyle \frac{\mathrm{1}}{\mathrm{2}}}}.\end{array}$$

This proper IK density spectrum increases with frequency like the root of the proper frequency.

Turning to the fast-streaming solar wind, we find that with ${k}_{\u27c2}={\mathit{\omega}}_{\mathrm{s}}/{V}_{\mathrm{0}}\mathrm{cos}\mathit{\alpha}$ for the Poisson-modified advected and convected K density spectrum,

$$\begin{array}{}\text{(32)}& \langle |\mathit{\delta}N{|}^{\mathrm{2}}{\rangle}_{{\mathit{\omega}}_{\mathrm{s}}}^{\mathrm{K},\mathrm{ad}}\propto {\mathit{\omega}}_{\mathrm{s}}^{-{\scriptscriptstyle \frac{\mathrm{2}}{\mathrm{3}}}}\mathrm{exp}[-{\displaystyle \frac{\mathrm{1}}{\mathrm{4}}}\left({\displaystyle \frac{{V}_{\mathrm{0}}\mathrm{cos}\phantom{\rule{0.125em}{0ex}}\mathit{\alpha}}{{\mathit{\omega}}_{\mathrm{s}}{\mathit{\lambda}}_{\mathrm{i}}}}{)}^{{\scriptscriptstyle \frac{\mathrm{2}}{\mathrm{3}}}}\right({\displaystyle \frac{{U}_{\mathrm{\ell}}^{\mathrm{K}}}{{U}_{\mathrm{0}}}}{)}^{\mathrm{2}}],\end{array}$$

where we again neglected the proper frequency dependence. This Taylor's Galilei-transformed density spectrum decays with increasing frequency, albeit at a
weak power $\sim \mathrm{2}/\mathrm{3}$. At large frequency *ω*_{s} the exponent
is 1, and the spectrum becomes $\propto {\mathit{\omega}}_{\mathrm{s}}^{-\frac{\mathrm{2}}{\mathrm{3}}}$.
Towards smaller *ω*_{s} the spectrum flattens and assumes its
maximum at

$$\begin{array}{}\text{(33)}& {\mathit{\omega}}_{\mathrm{sm}}^{\mathrm{K}}={\displaystyle \frac{\mathrm{3}}{\mathrm{8}}}\left({\displaystyle \frac{{V}_{\mathrm{0}}\mathrm{cos}\mathit{\alpha}}{{\mathit{\lambda}}_{\mathrm{i}}}}\right)({\displaystyle \frac{{U}_{\mathrm{\ell}}^{\mathrm{K}}}{{U}_{\mathrm{0}}}}{)}^{\mathrm{3}}.\end{array}$$

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

$$\begin{array}{}\text{(34)}& \langle |\mathit{\delta}N{|}^{\mathrm{2}}{\rangle}_{{\mathit{\omega}}_{\mathrm{s}}}^{\mathrm{IK},\mathrm{ad}}\propto {\mathit{\omega}}_{\mathrm{s}}^{-{\scriptscriptstyle \frac{\mathrm{1}}{\mathrm{2}}}}\mathrm{exp}[-{\displaystyle \frac{\mathrm{1}}{\mathrm{4}}}\left({\displaystyle \frac{{V}_{\mathrm{0}}\mathrm{cos}\phantom{\rule{0.125em}{0ex}}\mathit{\alpha}}{{\mathit{\omega}}_{\mathrm{s}}{\mathit{\lambda}}_{\mathrm{i}}}}{)}^{{\scriptscriptstyle \frac{\mathrm{3}}{\mathrm{4}}}}\right({\displaystyle \frac{{U}_{\mathrm{\ell}}^{\mathrm{IK}}}{{U}_{\mathrm{0}}}}{)}^{\mathrm{2}}].\end{array}$$

Both advected K and IK spectra have negative slopes in spacecraft frequency
*ω*_{s}. Like in the case of a K spectrum, this spectrum
approaches its steepest slope of 1∕2 at large spacecraft frequencies
*ω*_{s}, while in the direction of small frequencies, it flattens
out to assume its maximum value at

$$\begin{array}{}\text{(35)}& {\mathit{\omega}}_{\mathrm{m}}^{\mathrm{IK}}=\left({\displaystyle \frac{\mathrm{7}}{\mathrm{8}}}{\displaystyle \frac{{V}_{\mathrm{0}}\mathrm{cos}\phantom{\rule{0.125em}{0ex}}\mathit{\alpha}}{{\mathit{\lambda}}_{\mathrm{i}}}}{)}^{{\scriptscriptstyle \frac{\mathrm{3}}{\mathrm{2}}}}\right({\displaystyle \frac{{U}_{\mathrm{\ell}}^{\mathrm{IK}}}{{U}_{\mathrm{0}}}}{)}^{\mathrm{3}}.\end{array}$$

In both cases of advected K and IK spectra the Taylor's Galilei transformation
from the proper frame of turbulence into the spacecraft frame is permitted
because it applies to the velocity and density spectra (Treumann et al., 2019).
It maps the wave-number spectrum into the spacecraft frame frequency spectrum.
However, in both cases we recover frequency spectra which decrease with
frequency though weakly approaching the steepest slope at large frequencies. They
flatten out towards low frequencies and may assume maxima only if these
maxima are still in the inertial range of the advected K or IK spectrum. Only
in this case does the spacecraft frequency spectrum exhibit a bump at their
nominal maximum frequencies *ω*_{sm}. When the maximum
frequency falls outside the ion-inertial range the bump will be absent, while
the spectrum will be flatter than at large frequencies. Such flattened
bumpless spectra have been observed. The next subsections provide examples
of observed bumpy and bumpless spectra in the spacecraft frequency frame.

4 Application to selected observations in the solar wind

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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 3 is an example of a density spectrum with respect to
spacecraft frequency which exhibits a positive slope (or bump) on the
otherwise negative slope of the main spectrum. The data in this figure were
taken from published spectra (Šafránková et al., 2013) in the solar wind at an
average bulk velocity of *V*_{0}≈534 km s^{−1}, density ${N}_{\mathrm{0}}\approx \mathrm{3}\times {\mathrm{10}}^{\mathrm{6}}$ m^{−3}, and magnetic field *B*_{0}≈8 nT, yielding a
super-alfvénic Alfvén Mach number *M*_{A}≈6, ion
temperature *T*_{i}*≲*3 eV, and total plasma *β*≈0.3, i.e. low-beta conditions. The straight solid and broken lines drawn
across this slope correspond to the predicted ∼*ω*^{⅓} K
and ∼*ω*^{½} IK slopes under convection-dominated conditions.
Both these lines fit the shape very well though it cannot be decided which of
the inertial-range turbulence models provides a better fit, as the large
scatter of the data mimicked by the line width inhibits any distinction. It
is however obvious from Table 1 that advection plays no role in
this case.

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 Šafránková et al. (2013). WIND was located in the L1 Lagrange point. Magnetic field fluctuations were related in time to the Bright Monitor of the Solar Wind (BMSW) observations by the solar wind flow. In spite of their scatter, the data were sufficiently stationary for comparison to the density measurements.

Figure 4 shows the WIND magnetic power spectral densities. For transformation of the point cloud into a continuous line, we applied the same technique (Šafránková et al., 2016) as that for the density spectrum. The spectrum exhibits the expected positive slope in the BMSW frequency interval. The straight solid and broken lines along the positive slope correspond (within the uncertainty of the observations) to the root slopes of K and IK density inertial-range spectra $\langle |\mathit{\delta}B{|}^{\mathrm{2}}{\rangle}_{{\mathit{\omega}}_{\mathrm{s}}}\sim {\mathit{\omega}}_{\mathrm{s}}^{\frac{\mathrm{1}}{\mathrm{6}}}$ and $\sim {\mathit{\omega}}_{\mathrm{s}}^{\frac{\mathrm{1}}{\mathrm{4}}}$, respectively. The magnetic spectrum is the consequence of the K or IK density spectrum $\sim {\mathit{\omega}}_{\mathrm{s}}^{\frac{\mathrm{1}}{\mathrm{3}}}$ and $\sim {\mathit{\omega}}_{\mathrm{s}}^{\frac{\mathrm{1}}{\mathrm{2}}}$, respectively. Fluctuations in temperature do not, within experimental uncertainty, play any susceptible role. Comparing absolute powers is inhibited by the ungauged differences in instrumentation. (One may note that power spectral densities are positive definite quantities. Measuring their slopes is sufficient indication of pressure balance. Detailed pressure balance can only be seen when checking the phases of the fluctuations. Density and magnetic field would then be found in the antiphase.)

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. 5, combined from unrelated BMSW and WIND data (Šafránková et al., 2013; Podesta and Borovsky, 2010).

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 *f*<*f*_{ci} has slope
$\sim {\mathit{\omega}}^{-\frac{\mathrm{7}}{\mathrm{4}}}$, close to a K spectrum
$\sim {\mathit{\omega}}^{-\frac{\mathrm{5}}{\mathrm{3}}}$. The slope of the flat section is $\sim {\mathit{\omega}}^{-\mathrm{1}}$ which is about the same as the slope of the entire low-frequency
WIND spectrum before its spectral break. None of the Poisson-modified K or IK
spectral slopes fit these flattened regions. At higher frequencies the BMSW
spectrum steepens and presumably enters the dissipative range.

The slope of the WIND spectrum above its break point at frequency
$\sim {\mathrm{10}}^{-\mathrm{2}}$ Hz decreases to $\sim {\mathit{\omega}}^{-\frac{\mathrm{1}}{\mathrm{2}}}$. This corresponds
perfectly to an advected Taylor's Galilei-transformed IK spectrum, suggesting
that WIND detected such a spectrum in the ion-inertial range which maps to
those spacecraft frequencies. The pronounced *ω*^{−1} spectrum at lower
frequencies remains, however, unexplained for both spacecraft.

When crossing the cyclotron frequency *f*_{ci}, the WIND spectrum
steepens. We also note that the normalised power spectral densities of WIND
at $\langle |\mathit{\delta}N{|}^{\mathrm{2}}\rangle /{N}_{\mathrm{0}}^{\mathrm{2}}>\mathrm{0.3}$ and BMSW at $\mathrm{0.005}<\langle |\mathit{\delta}N{|}^{\mathrm{2}}\rangle /{N}_{\mathrm{0}}^{\mathrm{2}}<\mathrm{0.05}$ in the common slope $\sim {\mathit{\omega}}^{-\mathrm{1}}$ interval are
roughly 2 orders of magnitude apart. This can hardly be traced back to the
radial difference of 0.01 AU between L1 and 1 AU.

The obvious difference between the two plasma states is not in the Mach
numbers but rather in *β* and *V*_{0}. The BMSW observed, under moderately
high-*β* low-*V*_{0} conditions, WIND under moderately low-*β*
high-*V*_{0} conditions at similar densities and Mach
numbers. Because of the Galilean
relation $k={\mathit{\omega}}_{\mathrm{s}}/{V}_{\mathrm{0}}\mathrm{cos}\mathit{\alpha}$, the high speed in the case of
WIND seems responsible for the spectral shift in the ${\mathit{\omega}}_{\mathrm{s}}^{-\mathrm{1}}$
spectral range to lower than BMSW frequencies. This, however, comes up merely
for a factor 2 which does not cover the frequency shift of more than 1 order
of magnitude. Rather it is the angle between mean speed and the wave-number
spectrum which displaces the spectra in frequency. If this is the case, then
the WIND spectrum was about parallel to the solar wind velocity with WIND
angle *α*≈0^{∘}, while the BMSW spectrum was close to being
perpendicular with angle *α*≈90^{∘}, and it is the BMSW
spectrum which has been shifted by Taylor's Galilei transformation into the
high-frequency domain, while the WIND spectrum is about original. This may
also be the reason why BMSW does not see the narrow, flattened spectral part
while compressing the ${\mathit{\omega}}_{\mathrm{s}}^{-\mathrm{1}}$ part into just 1 order of
magnitude in frequency. The near-perpendicular angle *α* will also be
confirmed below in the bumpy BMSW spectral case.

5 Discussion

Back to toptop
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. 3, where we included the local ion-cyclotron frequency ${f}_{\mathrm{ci}}={\mathit{\omega}}_{\mathrm{ci}}/\mathrm{2}\mathit{\pi}$, we find the scale-limited positive slope (bump) of the density-power spectrum at spacecraft frequencies ${f}_{\mathrm{m}}\sim {\mathit{\omega}}_{\mathrm{m}}<{\mathit{\omega}}_{\mathrm{ci}}\sim \mathrm{0.22}$ Hz. According to Taylor, we have

$$\begin{array}{}\text{(36)}& {\mathit{\omega}}_{\mathrm{m}}={k}_{\mathrm{m}}{V}_{\mathrm{0}}\mathrm{cos}{\mathit{\alpha}}_{\mathrm{m}},\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\mathrm{and}\phantom{\rule{0.25em}{0ex}}\mathrm{also}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}{k}_{\mathrm{m}}{\mathit{\lambda}}_{\mathrm{i}}>\mathrm{1},\end{array}$$

where *α*_{m} is the angle between *k*_{m} and
velocity *V*_{0}, and ${\mathit{\lambda}}_{\mathrm{i}}=c/{\mathit{\omega}}_{\mathrm{i}}$. The first
expression yields Taylor's Galilei-transformed wave number
${k}_{\mathrm{m}}\sim \mathrm{2.3}\times {\mathrm{10}}^{-\mathrm{6}}/\mathrm{cos}\phantom{\rule{0.125em}{0ex}}{\mathit{\alpha}}_{\mathrm{m}}$ m^{−1}. From the
second, we have, with the observed ion plasma frequency, ${k}_{\mathrm{m}}\sim \mathrm{2}\mathit{\pi}\times {\mathrm{10}}^{-\mathrm{6}}$ m^{−1}. Hence we find that
cos *α*_{m}<0.37 or *α*_{m}>69^{∘}. The
turbulent eddies are at highly oblique angles with respect to the flow
velocity.

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 *B*_{0} and highly oblique with respect to
the flow. Similar arguments apply to the high-frequency excess in the WIND
observations of Fig. 5. Referring to Table 1
this excess is explained as survival of the advected spectrum when
Taylor's Galilei transformed into the spacecraft frame.

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^{6}, which in the solar wind is not
the case. It also assumes that wave numbers *k* are conserved by the
flow^{7}. Thus
the above conclusion is correct only if the turbulence is generated locally
and is transported over a distance where the radial variation of the solar
wind is negligible. If it is assumed that the turbulence is generated in the
innermost heliosphere at a fraction of 1 AU (e.g. McKenzie et al., 1995), any simple application of Taylor's Galilei
transformation and thus the above interpretation break down.

Under the fast flow conditions of Fig. 3 it is reasonable to
assume that the solar wind expands isentropically, denoting the turbulent
source and spacecraft locations by indices *q* and *s*, respectively. The
turbulent inertial range is assumed to be collisionless, dissipationless, and in
ideal gas conditions. For simplicity assume that the expansion is stationary
and purely radial. Under Taylor's assumption each eddy maintains its
identity, which implies that the number of eddies is constant, and the eddy
flux ${F}_{s}\left({r}_{s}\right)/{F}_{q}\left({r}_{q}\right)={r}_{q}^{\mathrm{2}}/{r}_{s}^{\mathrm{2}}$, i.e. the turbulent power, decreases as
the square of the radius. For the plasma we have the isentropic condition
(e.g. Kittel and Kroemer, 1980, p. 174)

$$\begin{array}{}\text{(37)}& {\displaystyle \frac{{T}_{s}\left({r}_{s}\right)}{{T}_{q}\left({r}_{q}\right)}}=[{\displaystyle \frac{{N}_{s}\left({r}_{s}\right)}{{N}_{q}\left({r}_{q}\right)}}{]}^{\mathit{\gamma}-\mathrm{1}},\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\mathit{\gamma}={\displaystyle \frac{\mathrm{5}}{\mathrm{3}}},\end{array}$$

which gives ${N}_{s}\left({r}_{s}\right)/{N}_{q}\left({r}_{q}\right)=({r}_{q}/{r}_{s}{)}^{\mathrm{3}}$, and thus ${T}_{s}\left({r}_{s}\right)/{T}_{q}\left({r}_{q}\right)=({r}_{q}/{r}_{s}{)}^{\mathrm{2}}$. One requires that ${k}_{q}>{\mathit{\lambda}}_{\mathrm{i}q}^{-\mathrm{1}}={\mathit{\omega}}_{\mathrm{i}q}\left({r}_{q}\right)/c$. By the same reasoning as that in the homogeneous case, one finds that

$$\begin{array}{ll}{\displaystyle \frac{{f}_{\mathrm{m}}}{{f}_{\mathrm{is}}}}& {\displaystyle}={\displaystyle \frac{{k}_{\mathrm{m}}{V}_{\mathrm{0}}}{{\mathit{\omega}}_{\mathrm{is}}}}\mathrm{cos}\phantom{\rule{0.125em}{0ex}}{\mathit{\alpha}}_{\mathrm{m}}\\ {\displaystyle}& {\displaystyle}={k}_{q}{\mathit{\lambda}}_{\mathrm{i}q}{\displaystyle \frac{{V}_{\mathrm{0}}}{c}}({\displaystyle \frac{{r}_{q}}{{r}_{s}}}{)}^{\frac{\mathrm{3}}{\mathrm{2}}}\mathrm{cos}\phantom{\rule{0.125em}{0ex}}{\mathit{\alpha}}_{\mathrm{m}}\\ \text{(38)}& {\displaystyle}& {\displaystyle}\mathit{\gtrsim}{\displaystyle \frac{{V}_{\mathrm{0}}}{c}}({\displaystyle \frac{{r}_{q}}{{r}_{s}}}{)}^{\frac{\mathrm{3}}{\mathrm{2}}}\mathrm{cos}\phantom{\rule{0.125em}{0ex}}{\mathit{\alpha}}_{\mathrm{m}},\end{array}$$

inserting for the left-hand side and *V*_{0}, we find, with *r*_{s}=1 AU, that

$$\begin{array}{}\text{(39)}& {r}_{q}<{\displaystyle \frac{\mathrm{0.1}}{(\mathrm{cos}\phantom{\rule{0.125em}{0ex}}{\mathit{\alpha}}_{\mathrm{m}}{)}^{\frac{\mathrm{2}}{\mathrm{3}}}}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\mathrm{AU}.\end{array}$$

We conclude that under the assumption of isentropic expansion of the solar
wind and Taylor's Galilei transport of turbulent eddies from the source region
to the observation site at 1 AU, the generation region of the turbulent
eddies which contribute to the bump in the K or IK density-power spectrum
must be located close to the Sun. The marginally permitted angle
*α*_{m} between wave number and mean flow is obtained by using
*r*_{q}=1 AU, yielding *α*_{m}>47^{∘}, meaning that the flow
must be oblique for the effect to develop, a conclusion already found above
for homogeneous flow. These numbers are obtained under the unproven
assumption that Taylor's Galilei transport conserves turbulent wave numbers in
the inhomogeneous solar wind.

So far we have referred to the inertial length as limiting the frequency
range. We now ask, for the more stringent condition *k**ρ*_{ic}>1,
that the responsible length be the ion gyroradius
${\mathit{\rho}}_{\mathrm{ic}}={v}_{\mathrm{i}}/{\mathit{\omega}}_{\mathrm{ci}}$. In this case reference to
the adiabatic conditions becomes necessary. We also need a model of the
radial variation of the solar wind magnetic field. The field inside *r*_{s}=1 AU is about radial. Magnetic flux conservation yields the Parker model
${B}_{s}\left({r}_{s}\right)={B}_{q}\left({r}_{q}\right)({r}_{q}/{r}_{s}{)}^{\mathrm{2}}$. A more modern empirical model instead
proposes a weaker radial decay of the power 5∕3 (for a review,
e.g. Khabarova, 2013). With these dependences, we have

$$\begin{array}{}\text{(40)}& {\displaystyle}k{\mathit{\rho}}_{s}\left({r}_{s}\right)& {\displaystyle}=k{\mathit{\rho}}_{q}\left({r}_{q}\right){\displaystyle \frac{{B}_{q}\left({r}_{q}\right)}{{B}_{s}\left({r}_{s}\right)}}\sqrt{{\displaystyle \frac{{T}_{\mathrm{i}s}\left({r}_{s}\right)}{{T}_{\mathrm{i}q}\left({r}_{q}\right)}}}\text{(41)}& {\displaystyle}& {\displaystyle}=k{\mathit{\rho}}_{q}\left({r}_{q}\right)({\displaystyle \frac{{r}_{s}}{{r}_{q}}}{)}^{\frac{\mathrm{2}}{\mathrm{3}}}\phantom{\rule{0.33em}{0ex}}>\phantom{\rule{0.33em}{0ex}}({\displaystyle \frac{{r}_{s}}{{r}_{q}}}{)}^{\frac{\mathrm{2}}{\mathrm{3}}},\end{array}$$

where the necessary condition *k**ρ*_{q}>1 has been used. Referring again to
the observed minimum frequency *f*_{m} yields

$$\begin{array}{}\text{(42)}& {\displaystyle \frac{{f}_{\mathrm{m}}}{{f}_{\mathrm{ic},s}}}& {\displaystyle}={\displaystyle \frac{{k}_{\mathrm{m}}{V}_{\mathrm{0}}}{{\mathit{\omega}}_{\mathrm{ic},s}}}\mathrm{cos}\phantom{\rule{0.125em}{0ex}}{\mathit{\alpha}}_{\mathrm{m}}\text{(43)}& {\displaystyle}& {\displaystyle}=k{\mathit{\rho}}_{q}{\displaystyle \frac{{V}_{\mathrm{0}}}{{v}_{\mathrm{i}}}}\left({\displaystyle \frac{{r}_{s}}{{r}_{q}}}{)}^{\frac{\mathrm{2}}{\mathrm{3}}}\mathrm{cos}\phantom{\rule{0.125em}{0ex}}{\mathit{\alpha}}_{\mathrm{m}}\mathit{\gtrsim}{\displaystyle \frac{{V}_{\mathrm{0}}}{{v}_{\mathrm{i}}}}\right({\displaystyle \frac{{r}_{s}}{{r}_{q}}}{)}^{\frac{\mathrm{2}}{\mathrm{3}}}\mathrm{cos}\phantom{\rule{0.125em}{0ex}}{\mathit{\alpha}}_{\mathrm{m}}.\end{array}$$

Inserting for the frequency ratio ${f}_{\mathrm{m}}/{f}_{\mathrm{ic},s}\sim \mathrm{0.1}$ and
the ratio of mean to thermal velocities ${V}_{\mathrm{0}}/{v}_{\mathrm{i}}\approx \mathrm{18}$, and
setting *r*_{s}=1 AU, we obtain, for the source radius lying inside 1 AU,

$$\begin{array}{}\text{(44)}& \mathrm{1}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\mathrm{AU}>{r}_{q\mathrm{m}}>\mathrm{300}(\mathrm{cos}\phantom{\rule{0.125em}{0ex}}{\mathit{\alpha}}_{\mathrm{m}}{)}^{{\scriptscriptstyle \frac{\mathrm{3}}{\mathrm{2}}}},\end{array}$$

which gives the result *α*_{m}*≳*89^{∘} for the
propagation angle obtained above. According to both these estimates eddy
propagation is required to be quasi-perpendicular to the flow. This holds
under the strong condition that the wave number is conserved during outward
propagation.

The wave number $k\sim {\mathit{\lambda}}^{-\mathrm{1}}$ is an inverse wavelength. Let us assume
that *λ*∼*r* stretches linearly when the volume expands, thereby
reducing *k* hyperbolically. The eddies, which are frozen to the volume,
also stretch linearly. In this case the ratio *r*_{s}∕*r*_{q} in Eq. (43) is raised
to the power 1∕3, and we find instead that

$$\begin{array}{}\text{(45)}& {r}_{q\mathrm{m}}\mathit{\lesssim}({\displaystyle \frac{\mathrm{cos}\phantom{\rule{0.125em}{0ex}}{\mathit{\alpha}}_{\mathrm{m}}}{\mathrm{18}}}{)}^{\mathrm{3}}<\mathrm{1}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\mathrm{AU}.\end{array}$$

This gives *α*_{m}*≳*87^{∘} which is not too different
from the above case. Thus the angle between mean speed and the turbulent
wave number is close to perpendicular in order to reconcile the lower observed
limit in spacecraft frequency with the wave number in the source region.

A similar reasoning can be applied to the upper frequency bound
*ω*_{M}. Following the discussion in the Introduction, this bound
is caused by the truncation of the ion-inertial range at large wave numbers
when the scale approaches the electron scale, electron inertia takes over,
and electrons demagnetise. The condition in this case is that
*k**ρ*_{e}<1, which defines the maximum frequency
*ω*_{M}.

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

$$\begin{array}{}\text{(46)}& k{\mathit{\rho}}_{\mathrm{Me}}\left({r}_{s}\right)=k{\mathit{\rho}}_{\mathrm{Me}}\left({r}_{q}\right){\displaystyle \frac{{B}_{q}\left({r}_{q}\right)}{{B}_{s}\left({r}_{s}\right)}}\sqrt{{\displaystyle \frac{{T}_{\mathrm{e}s}\left({r}_{s}\right)}{{T}_{\mathrm{e}q}\left({r}_{s}\right)}}}<({\displaystyle \frac{{r}_{s}}{{r}_{q}}}{)}^{{\scriptscriptstyle \frac{\mathrm{2}}{\mathrm{3}}}}.\end{array}$$

From the maximum observed frequency, we find that, with *f*_{M}*≲**f*_{ce,s},

$$\begin{array}{ll}{\displaystyle \frac{{f}_{\mathrm{M}}}{{f}_{\mathrm{ce},s}}}& {\displaystyle}={\displaystyle \frac{{k}_{\mathrm{M}}{V}_{\mathrm{0}}}{{\mathit{\omega}}_{\mathrm{ce},s}}}\mathrm{cos}\phantom{\rule{0.125em}{0ex}}{\mathit{\alpha}}_{\mathrm{M}}\\ \text{(47)}& {\displaystyle}& {\displaystyle}=k{\mathit{\rho}}_{q}{\displaystyle \frac{{V}_{\mathrm{0}}}{{v}_{\mathrm{e}}}}\left({\displaystyle \frac{{r}_{s}}{{r}_{q}}}{)}^{\frac{\mathrm{2}}{\mathrm{3}}}\mathrm{cos}\phantom{\rule{0.125em}{0ex}}{\mathit{\alpha}}_{\mathrm{M}}\mathit{\lesssim}{\displaystyle \frac{{V}_{\mathrm{0}}}{{v}_{\mathrm{e}}}}\right({\displaystyle \frac{{r}_{s}}{{r}_{q}}}{)}^{\frac{\mathrm{2}}{\mathrm{3}}}\mathit{\lesssim}\mathrm{1},\end{array}$$

which, when inserting *k**ρ*_{q}*≲*1, adopting the main plasma
parameters, and with maximum frequency ${f}_{\mathrm{M}}/{f}_{\mathrm{ce},s}\sim \mathrm{1}$
and *r*_{s}∼1 AU, yields

$$\begin{array}{}\text{(48)}& {r}_{q\mathrm{M}}>({\displaystyle \frac{{V}_{\mathrm{0}}}{{v}_{\mathrm{e}}}}{)}^{{\scriptscriptstyle \frac{\mathrm{3}}{\mathrm{2}}}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\mathrm{AU}\sim \mathrm{0.05}\phantom{\rule{0.25em}{0ex}}\mathrm{AU}.\end{array}$$

Taking the two results for this case together, the observations map to an
angle of propagation *α*_{m}>49^{∘} and places the turbulent
source close to the Sun but outside 11 R_{⊙}*≲**r*_{q}*≲*1 AU. It occurs only if the turbulence contains a dominant population of
eddies obeying wave-number vectors ** k** which are oblique to the mean
flow velocity

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
*m*_{i}∕*m*_{e} which would be 3 orders of magnitude. The
actually observed range *f*_{m}*≲**f*_{s}*≲**f*_{M} is much narrower, being just 1 order of magnitude. Given the
uncertainties of measurement and instrumentation this can be extended at most
to the root of the mass ratio, which in a proton-electron plasma amounts to a
factor of ${f}_{\mathrm{M}}/{f}_{\mathrm{s}}\sim \mathrm{43}$ only. In addition,
unfortunately, the observed local maximum frequency in Fig. 3
is far less than the local electron cyclotron frequency ${f}_{\mathrm{M}}\ll {f}_{\mathrm{ce},s}$. The affected wave number and frequency ranges are very
narrow and at the wrong place. Thus in the given version, the reasoning above
does not apply. Already in the source region, the effect must be bound to a
narrow domain in wave number. The mass ratio might suggest coincidence with
the lower-hybrid frequency of a low-*β* proton-electron plasma which,
when raised to the power 3∕2, yields

$$\begin{array}{}\text{(49)}& {r}_{q\mathrm{M}}\mathit{\gtrsim}\mathrm{0.6}\phantom{\rule{0.125em}{0ex}}\mathrm{AU},\end{array}$$

putting the source region substantially farther out to *≳*45*R*_{⊙}.

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. 3 unless an additional assumption is made.

One may, however, argue that in a high-*β*_{i} plasma the
gyroradius of the ions is large. The ions are non-magnetic, but the effect
can arise only when the wavelength becomes less than the inertial length
${\mathit{\lambda}}_{\mathrm{m}}<{\mathit{\lambda}}_{\mathrm{i}}=c/{\mathit{\omega}}_{\mathrm{i}}$. Similarly the
effect will disappear when the wavelength crosses the electron inertial
length ${\mathit{\lambda}}_{\mathrm{M}}<{\mathit{\lambda}}_{\mathrm{e}}=c/{\mathit{\omega}}_{\mathrm{e}}$. The ratio
of these two limits is
${\mathit{\lambda}}_{\mathrm{M}}/{\mathit{\lambda}}_{\mathrm{m}}={f}_{\mathrm{M}}/{f}_{\mathrm{m}}=\sqrt{{m}_{\mathrm{i}}/{m}_{\mathrm{e}}}\approx \mathrm{43}$. This agrees approximately with the observation. This interpretation then
identifies the range of the effect in spacecraft frequency and source
wave number with the range between electron and ion-inertial lengths. Since
both evolve radially with the ratio of the root of densities, the relative
spectral width should not change from source to spacecraft.

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
${\mathit{\rho}}_{\mathrm{i}}/{\mathit{\lambda}}_{\mathrm{i}}=\sqrt{{\mathit{\beta}}_{\mathrm{i}}}$, with
*β*_{i} being the ion *β*. In isentropic expansion, the evolution of
*β*_{i}, assuming a Parker model, is

$$\begin{array}{}\text{(50)}& ({\displaystyle \frac{{\mathit{\rho}}_{\mathrm{i}s}}{{\mathit{\rho}}_{\mathrm{i}q}}}{\displaystyle \frac{{\mathit{\lambda}}_{\mathrm{i}q}}{{\mathit{\lambda}}_{\mathrm{i}s}}}{)}^{\mathrm{2}}={\displaystyle \frac{{\mathit{\beta}}_{\mathrm{i}s}}{{\mathit{\beta}}_{\mathrm{i}q}}}\propto ({\displaystyle \frac{{r}_{q}}{{r}_{s}}}{)}^{{\scriptscriptstyle \frac{\mathrm{5}}{\mathrm{3}}}}.\end{array}$$

From observations we have a total *β*>1. We expect
*ρ*_{i}*≳**λ*_{i} and assume that
*β*_{i}*≳*1. Figure 3 suggests a frequency
ratio ${f}_{\mathrm{m}}/{f}_{\mathrm{M}}\sim {\mathit{\beta}}_{\mathrm{i}s}\sim \mathrm{0.7}$ larger than
$\sqrt{{m}_{\mathrm{e}}/{m}_{\mathrm{i}}}\approx \mathrm{0.025}$. The affected frequency and
wave-number ranges are limited from above when the scale approaches the
electron gyroradius. In that case, the upper bound is not determined by the
mass ratio alone. With the measured frequency ratio, the location of the
source should then be outside a shortest distance of

$$\begin{array}{}\text{(51)}& {r}_{q}\mathit{\gtrsim}\mathrm{0.24}\phantom{\rule{0.33em}{0ex}}{\mathit{\beta}}_{\mathrm{i}q}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\mathrm{AU}.\end{array}$$

This value corresponds to $>\mathrm{50}{\mathit{\beta}}_{\mathrm{i}q}{R}_{\odot}$ from the Sun.
Since the source must lie inside *r*_{q}<1 AU, we conclude that
*β*_{iq}*≲*4.15. This number is just an upper limit. It is
consistent with model calculations (McKenzie et al., 1995) which predict
*β*_{iq}<1 shifting the inner boundary of the turbulent source
region further in.

In this paper, we considered the cases ${V}_{\mathrm{0}}={U}_{\mathrm{0}}=\mathrm{0}$, ${V}_{\mathrm{0}}=\mathrm{0},\phantom{\rule{0.125em}{0ex}}{U}_{\mathrm{0}}\ne \mathrm{0}$, and
*V*_{0}≠0 for K and IK velocity spectra, where *V*_{0} is the velocity of the
mean solar wind stream, and *U*_{0} is the mean speed of the energy-carrying
largest turbulent MHD vortices which advect the bulk of small-scale
turbulence around (Tennekes, 1975). In the K and IK models of turbulence,
they, in addition, play the role of the energy injectors. The resulting
spectral slopes are given as *b* in the fourth column of Table 1. The
input spectral power densities are ℰ_{IK} and
${\mathcal{E}}_{\mathrm{IK}}^{\mathrm{ad}}$. Each of them yields a different ion-inertial scale range power spectrum in *k* space and, consequently, also a
different power law spectrum in *ω*_{s} space.

Table 1 shows that the ordinary spectra acquire positive slopes
in wave number *k* in the frame of stationary and homogeneous turbulence in
the turbulence frame. However, observations of this slope in frequency
undermine this conclusion, suggesting that it is the ordinary K (IK) velocity
turbulence (or if anisotropy is taken into account, the
KGS)
spectrum in the ion-inertial range which, when convected by the solar wind
flow across the spacecraft, deforms the density spectrum. All advected
spectra have, in contrast, a negative slope in frequency which in this form
disagrees with observation of the spectral bumps.

The obtained advected slopes in the stationary turbulence frame are also too far away from the flattest notorious and badly understood negative slope ${\mathit{\omega}}_{\mathrm{s}}^{-\mathrm{1}}$ for being related. Their nominal K and IK slopes are $-\mathrm{2}/\mathrm{3}$ and $-\mathrm{1}/\mathrm{2}$, respectively. This implies that spacecraft observations interpreted as observing the local stationary turbulence do not, in the majority of cases, detect an advected convected spectrum in the ion-inertial K (IK) inertial range. They are, however, well capable of explaining the high-frequency flattened spectral excursion in the WIND spectrum which is shown in Fig. 5. It has the correct advective IK spectral index $-\mathrm{1}/\mathrm{2}$ when convected across the WIND spacecraft before the onset of spectral decay.

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
$\langle \mathit{\delta}{\dot{Q}}_{\mathrm{i}}\rangle =-\langle \mathit{\delta}{\dot{Q}}_{\mathrm{em}}\rangle =\langle \mathit{\delta}\mathit{J}\cdot \mathit{\delta}\mathit{E}\rangle $
in the ion-inertial range, the negative of the mean loss in electromagnetic
energy density per time $\langle \mathit{\delta}{\dot{Q}}_{\mathrm{em}}\rangle $,
proportional to the product of current vortices *δ*** J** and the
turbulent electric field

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 ${k}_{\u27c2}\sim {\mathit{\lambda}}_{\mathrm{i}}^{-\mathrm{1}}$ agree with the scale of the ion-inertial range. Possibly they can grow on the expense of the Hall field which in this case plays the role of free energy for them. If they can grow to sufficiently large amplitudes, they contribute to further deforming K and IK ion-inertial-range density spectra.

Similarly, *small-scale shock waves* might evolve at the inferred high
Mach numbers when turbulent eddies grow and steepen in the small-scale range.
These necessarily become sources of electron beams, reflect ions, and
transfer their energy in a kinetic-turbulent way to the particle population.
Such beams act as sources of particular wave populations which contribute to
turbulence, preferably at the kinetic scales of interest.

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 (Treumann and Baumjohann, 2015) collisionless dissipation scale where electrons
demagnetise as well and the current filaments dissipate their energy in the
process of *spontaneous collisionless reconnection* as the most
probable ultimate energy sink of otherwise collisionless turbulence. The
scales of this dissipation process are still far away from any molecular
scales. The resulting dissipation is justifiably anomalous.

Author contributions

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Author contributions.

All authors contributed equally to this paper.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgement

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Acknowledgement.

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.

Review statement

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Review statement.

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

References

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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 *β**≳*1. One may expect that
turbulence here also contains contributions which are generated locally, if
only some free energy would become available.

This
equation is easily obtained by standard methods when splitting the fields in
the ideal (collisionless) Hall-MHD Ohm's law:
$\mathit{E}=-\mathit{V}\times \mathit{B}+(\mathrm{1}/eN)\mathit{J}\times \mathit{B}$, with
$\mathit{E}\mathbf{,}\mathit{B}\mathbf{,}\mathit{V}\mathbf{,}\mathit{J}$ electric, magnetic, and current fields, into mean (index 0) and
fluctuating fields according to $\mathit{E}={\mathit{E}}_{\mathrm{0}}+\mathit{\delta}\mathit{E}$ etc.;
averaging over the fluctuation scales, with 〈…〉 indicating
the averaging procedure, yields the mean-field electric field equation.
Subtracting it from the original equation produces the wanted expression of
the turbulent electric fluctuations *δ*** E** through the mean and
fluctuating velocity and magnetic fields.

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

One 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 *U*_{0}≠0, yields $\langle |\mathit{\delta}N{|}^{\mathrm{2}}{\rangle}_{k}={N}_{\mathrm{0}}^{\mathrm{2}}\langle |\mathit{k}\cdot \mathit{\delta}\mathit{V}{|}^{\mathrm{2}}{\rangle}_{k}/(\mathit{k}\cdot {\mathit{U}}_{\mathrm{0}}{)}^{\mathrm{2}}$,
which is obtained without reference to Poisson's equation. However, its
dependence on the wave number is different, and, in addition, it is undefined for
vanishing advection. In the absence of advection the density spectrum is
determined from the equation of motion by simple pressure balance.

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 *S*(*ω*_{k},** k**) with respect to frequency

For general restrictions on its applicability already in homogeneous MHD, see Treumann et al. (2019).

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.

Short summary

Occasional deviations in density and magnetic power spectral densities in an intermediate frequency range are interpreted as an ion-inertial-range response to either the Kolmogorov or Iroshnikov–Kraichnan inertial-range turbulent velocity spectrum.

Occasional deviations in density and magnetic power spectral densities in an intermediate...

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