The steep decay of the normalised fluctuations in ion temperature above frequencies >10-1 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 fci=ωci/2π∼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 fe=ωe/2π∼35 kHz and (assuming protons) ion fi=ωi/2π∼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

This equation is easily obtained by standard methods when splitting the fields in the ideal (collisionless) Hall-MHD Ohm's law: E=-V×B+(1/eN)J×B, with E,B,V,J electric, magnetic, and current fields, into mean (index 0) and fluctuating fields according to E=E0+δE 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 δE through the mean and fluctuating velocity and magnetic fields.

Let us assume that advection by large-scale energy-carrying eddies is perpendicular U0=U0⟂, and there are no compressive magnetic fluctuations δB‖=0. In Eq. () 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 ρi=vi/ωci or – depending on the direction to the mean magnetic field B0 and the value of plasma beta β=2μ0N0T0/B02, with ωci=eB0/mi ion-cyclotron and ωi=eN0/ϵ0mi ion plasma frequency, respectively – inertial length λi=c/ωi, 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 eδNi=eδNe and thus a density fluctuation δN. Poisson's equation implies that 5∇⋅δE=eϵ0δN==⟹ik⋅δEk=eϵ0δNk. The right expression is its Fourier transform. For completeness we note that the Hall contribution to the Poisson equation in Fourier space reads 6ik⟂⋅δE⟂kH=B0eμ0N0(k⟂2δB‖k-k‖k⟂⋅δB⟂k)=eϵ0δNkH. 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 δB‖=0 and k⟂=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⟂ only compressive Hall-magnetic fluctuations δB‖k⟂ contribute to the Hall fluctuations in the density δNk⟂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. () with wave number k selects wave numbers k⟂ perpendicular to B0. The combination of Eq. () and the Poisson equation then yields an expression for the power spectrum of the turbulent density fluctuations

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

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⟂ρi<1, where ρi=vi/ωci=λivi/VA is the ion gyroradius, with vi as the thermal speed. Thus we have two conditions which must simultaneously be satisfied: 9k⟂λi>1andk⟂λi>VAvi≡βi-12. For VA<vi the second condition is trivial. This is, however, a rare case, so the more realistic restriction is the opposite small ion-beta case when VA>vi and hence βi<1. It must, however, be combined with another condition which requires that the wave numbers be smaller than the inverse electron gyroradius ρe=ve/ωce. The relation between ρe and ρi is ρe2/ρi2=meTe/miTi. Moreover we have ρi/λi=vi/VA, and in addition, βi=vi2/VA2=(Ti/Te)βe. Using all these relations, we obtain finally that 101<k⟂2λi2βi<mimeTiTeforβi<1. 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 δE: 11TiTe≳memi∼0.001. 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⟂ρe 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 12Te/Tiβe<k⟂2λi2. The electron plasma beta in the solar wind is of the order of βe≳O(1). However, the temperature ratio Te/Ti is variable and usually large, varying between a few and a few tens. Thus usually βi<1. Figure  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 byabbreviated 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 U0 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 . This causes a Doppler broadening of the wave-number spectrum at fixed k and has been confirmed by numerical simulations . 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. .

The stationary velocity spectrum of turbulent eddies at energy injection rate ϵ exhibits a broad inertial power law range in k which, between injection kin and dissipation at kd wave numbers, obeys the famous isotropic Kolmogorov power spectral density law in wave-number space: 13〈|δV|2〉k≡EK(k)=CKϵ23k-53forkin<k<kd, with CK≈1.65 as Kolmogorov's constant of proportionality as determined byusing numerical simulations. Clearly, in a fast-streaming solar wind, when straightforwardly mapping this K spectrum by the Taylor hypothesis into the stationary spacecraft frame, the spectral index is unchanged, and one trivially recovers the ωs-53 Kolmogorov slope in frequency space.

This changes drastically when referring to an advected K spectrum of velocity turbulence which yields the above-mentioned spectral Doppler broadening at fixed k, Ekωkad=12EK(k)2πkU0∑±exp⁡[-12ω±2(kU0)2],14ω±=ωk±ℓKk23, which is due to decorrelation of the small eddies in advective transport, with ℓK∼O(1) being some constant. The k23 dependence in the argument of the exponential results from advection k⋅δV of neighbouring eddies at velocity of δV∝k-13 . The frequency ωk stands for the internal dependence of the turbulent frequency on the turbulent wave number k. It can be understood as an internal “turbulent dispersion relation”, which is neglected in turbulence theory.

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(ωk,k) with respect to frequency ωk e.g., which leaves only the wave-number dependence. The spectral density S occupies a volume in (ω,k) space. Resolved for ω=ωk(k), it yields a complex multiply connected surface, the turbulent dispersion relation, which has nothing in common with a linear dispersion relation resulting from the solution of a linear eigenmode wave equation. It contains the dependence of Fourier frequency ωk on Fourier wave number k. Though this should be common sense, we feel obliged to note this here because of the confusion caused when speaking about a “dispersion relation” in turbulence.

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 ωs∝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 ωs-83 spectrum.

If this is true, then the corresponding observed spectral transition (or break point) from the spectral K index ∼5/3 to the steeper index ∼8/3 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 ∼5/3 (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 exp⁡(-ℓK2/U02k23) suppresses the spectrum here. This flattens the inertial-range spectrum towards small wave numbers kin into the large-eddy range where it causes bending of the spectrum. The wave number at spectral maximum is 17kmin≲ℓK3/16U032. Approaching from the Kolmogorov inertial range towards a smaller k, one observes flattening until kmin<kin. 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 . It then maps the Doppler broadened advected velocity power spectrum to the Kolmogorov law in the source-region frequency space: 18∫kinkddkEkωkad∼EKad(ω)∝ω-53. 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 V0≠0, one must return to its wave-number representation in Eq. (). 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 19ωk=ωs-kV0cos⁡α.α=∠(k,V0). This is Taylor's Galilei transformation. Neglecting ωk implies that ωs=kV0cos⁡α. The exponential reduces to 20exp⁡[-14(ωk+kV0cos⁡α±ℓKk23kU0)2]=21=exp⁡[-14(kλi)23(λi13ℓKU0)2]22⟶1-14(kλi)23(λi13ℓKU0)2, with λi13ℓ/U0≡UℓK/U0 being a velocity ratio. The arrow holds for the ion-inertial range kλi>1 and UℓK/U0<1. The exponential expression leads to an advected K spectrum as observed by the spacecraft in frequency space: 23Eωsad∝ωs-83exp⁡[-14(V0cos⁡αωsλi)23(UℓKU0)2], which, as before for large ωs, is of the spectral index 8/3. With decreasing spacecraft frequency ωs, the exponential correction factor acts to suppress the spectrum. This corresponds to a spectral flattening towards smaller ωs. 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.

For the simple inertial-range K spectrum, we know from Eqs. () and () that 24〈|δN|2〉k⟂=CK(ϵ0B0e)2ϵ23k⟂13fork⟂in<k⟂<k⟂d. This is a very simple wave-number dependence of the power spectrum of density turbulence, permitting Taylor's Galilei transformation into the spacecraft frame. Setting k⟂=ωs/V0cos⁡α we immediately obtain 25〈|δN|2〉k⟂∝ωs13, with factor of proportionality CK(ϵ0B0/e)2(ϵ2/V0cos⁡α)13.

Following exactly the same reasoning when dealing with the IK spectrum, which has power index 3/2, we obtain 26〈|δN|2〉k⟂∝ωs12. 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. () of the velocity field for V0=0 in the transformed Poisson equation, with k→k⟂ being perpendicular to the mean magnetic field B0, yields the following for the non-convected advected turbulent ion-inertial-range Poisson-modified density-power spectrum in the stationary large-eddy turbulence frame: 〈|δN|2〉ωkk⟂ad=ϵ02B02e2k⟂2〈|δV|2〉ωkk⟂=ϵ02B02e2k⟂2Ek⟂ωkad27∝k⟂-23∑±exp⁡[-12ω±2(kU0)2]. Integration with respect to k⟂ under the above assumption on ω±≈k⟂U0 yields the following for the Eulerian density-power spectrum in frequency space ωℓ<ω<ωu in the ion-inertial domain of the turbulent inertial range: 28〈|δN|2〉ωad∼ω13,kir23ϵ13=ωℓ<ω<ωu. This is the proper frequency dependence of the advected turbulent density spectrum in the turbulence frame. Here kir≈2πωi/c (or 2πvi/ωci) is the wave number presumably corresponding to the lower end of the ion-inertial range. The upper bound on the frequency ωu remains undetermined. One assumption would be that ωu 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. () 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 29〈|δV|2〉ωkk⟂∝k⟂-32, which yields 〈|δN|2〉ωkk⟂ad∝k⟂-12∑±exp⁡[-12ω±2(kU0)2],30ω±=ωk±ℓIKk⟂34. Integration with respect to k⟂ then gives the proper advected frequency spectrum in the stationary frame of IK turbulence: 31〈|δN|2〉ωad∼ω12. 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⟂=ωs/V0cos⁡α for the Poisson-modified advected and convected K density spectrum, 32〈|δN|2〉ωsK,ad∝ωs-23exp⁡[-14(V0cos⁡αωsλi)23(UℓKU0)2], where we again neglected the proper frequency dependence. This Taylor's Galilei-transformed density spectrum decays with increasing frequency, albeit at a weak power ∼2/3. At large frequency ωs the exponent is 1, and the spectrum becomes ∝ωs-23. Towards smaller ωs the spectrum flattens and assumes its maximum at 33ωsmK=38(V0cos⁡αλi)(UℓKU0)3.

The same reasoning produces, for the Poisson-modified advected IK spectrum, the Taylor's Galilei-transformed spacecraft frequency spectrum 34〈|δN|2〉ωsIK,ad∝ωs-12exp⁡[-14(V0cos⁡αωsλi)34(UℓIKU0)2]. Both advected K and IK spectra have negative slopes in spacecraft frequency ωs. Like in the case of a K spectrum, this spectrum approaches its steepest slope of 1/2 at large spacecraft frequencies ωs, while in the direction of small frequencies, it flattens out to assume its maximum value at 35ωmIK=(78V0cos⁡αλi)32(UℓIKU0)3. 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 . 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 ωsm. 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.

Figure 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 in the solar wind at an average bulk velocity of V0≈534 km s-1, density N0≈3×106 m-3, and magnetic field B0≈8 nT, yielding a super-alfvénic Alfvén Mach number MA≈6, ion temperature Ti≲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 ∼ω13 K and ∼ω12 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  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 . 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 shows the WIND magnetic power spectral densities. For transformation of the point cloud into a continuous line, we applied the same technique 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 〈|δB|2〉ωs∼ωs16 and ∼ωs14, respectively. The magnetic spectrum is the consequence of the K or IK density spectrum ∼ωs13 and ∼ωs12, 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. , combined from unrelated BMSW and WIND data .

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<fci has slope ∼ω-74, close to a K spectrum ∼ω-53. The slope of the flat section is ∼ω-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 ∼10-2 Hz decreases to ∼ω-12. 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 fci, the WIND spectrum steepens. We also note that the normalised power spectral densities of WIND at 〈|δN|2〉/N02>0.3 and BMSW at 0.005<〈|δN|2〉/N02<0.05 in the common slope ∼ω-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 V0. The BMSW observed, under moderately high-β low-V0 conditions, WIND under moderately low-β high-V0 conditions at similar densities and Mach numbers. Because of the Galilean relation k=ωs/V0cos⁡α, the high speed in the case of WIND seems responsible for the spectral shift in the ωs-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 ωs-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.

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. , where we included the local ion-cyclotron frequency fci=ωci/2π, we find the scale-limited positive slope (bump) of the density-power spectrum at spacecraft frequencies fm∼ωm<ωci∼0.22 Hz. According to Taylor, we have 36ωm=kmV0cos⁡αm,andalsokmλi>1, where αm is the angle between km and velocity V0, and λi=c/ωi. The first expression yields Taylor's Galilei-transformed wave number km∼2.3×10-6/cos⁡αm m-1. From the second, we have, with the observed ion plasma frequency, km∼2π×10-6 m-1. Hence we find that cos⁡αm<0.37 or αm>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 B0 and highly oblique with respect to the flow. Similar arguments apply to the high-frequency excess in the WIND observations of Fig. . Referring to Table  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

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

Under the fast flow conditions of Fig.  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 Fs(rs)/Fq(rq)=rq2/rs2, i.e. the turbulent power, decreases as the square of the radius. For the plasma we have the isentropic condition e.g.p. 174 37Ts(rs)Tq(rq)=[Ns(rs)Nq(rq)]γ-1,γ=53, which gives Ns(rs)/Nq(rq)=(rq/rs)3, and thus Ts(rs)/Tq(rq)=(rq/rs)2. One requires that kq>λiq-1=ωiq(rq)/c. By the same reasoning as that in the homogeneous case, one finds that fmfis=kmV0ωiscos⁡αm=kqλiqV0c(rqrs)32cos⁡αm38≳V0c(rqrs)32cos⁡αm, inserting for the left-hand side and V0, we find, with rs=1 AU, that 39rq<0.1(cos⁡αm)23AU. 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 αm between wave number and mean flow is obtained by using rq=1 AU, yielding αm>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 kρic>1, that the responsible length be the ion gyroradius ρic=vi/ωci. 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 rs=1 AU is about radial. Magnetic flux conservation yields the Parker model Bs(rs)=Bq(rq)(rq/rs)2. A more modern empirical model instead proposes a weaker radial decay of the power 5/3 for a review, e.g.. With these dependences, we have 40kρs(rs)=kρq(rq)Bq(rq)Bs(rs)Tis(rs)Tiq(rq)41=kρq(rq)(rsrq)23>(rsrq)23, where the necessary condition kρq>1 has been used. Referring again to the observed minimum frequency fm yields 42fmfic,s=kmV0ωic,scos⁡αm43=kρqV0vi(rsrq)23cos⁡αm≳V0vi(rsrq)23cos⁡αm. Inserting for the frequency ratio fm/fic,s∼0.1 and the ratio of mean to thermal velocities V0/vi≈18, and setting rs=1 AU, we obtain, for the source radius lying inside 1 AU, 441AU>rqm>300(cos⁡αm)32, which gives the result αm≳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∼λ-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 rs/rq in Eq. (43) is raised to the power 1/3, and we find instead that 45rqm≲(cos⁡αm18)3<1AU. This gives αm≳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 ωM. 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 kρe<1, which defines the maximum frequency ωM.

We then have the following relation for the maximum wave number: 46kρMe(rs)=kρMe(rq)Bq(rq)Bs(rs)Tes(rs)Teq(rs)<(rsrq)23. From the maximum observed frequency, we find that, with fM≲fce,s, fMfce,s=kMV0ωce,scos⁡αM47=kρqV0ve(rsrq)23cos⁡αM≲V0ve(rsrq)23≲1, which, when inserting kρq≲1, adopting the main plasma parameters, and with maximum frequency fM/fce,s∼1 and rs∼1 AU, yields 48rqM>(V0ve)32AU∼0.05AU. Taking the two results for this case together, the observations map to an angle of propagation αm>49∘ and places the turbulent source close to the Sun but outside 11R⊙≲rq≲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 V0. This is in agreement with our given estimate above on the theoretical limits and explains the relative rarity of its observation. Unfortunately, based on the observations, the desired location of the turbulent source region in space cannot be localised more precisely.

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 mi/me which would be 3 orders of magnitude. The actually observed range fm≲fs≲fM 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 fM/fs∼43 only. In addition, unfortunately, the observed local maximum frequency in Fig.  is far less than the local electron cyclotron frequency fM≪fce,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 49rqM≳0.6AU, putting the source region substantially farther out to ≳45R⊙.

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.  unless an additional assumption is made.

One may, however, argue that in a high-βi 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 λm<λi=c/ωi. Similarly the effect will disappear when the wavelength crosses the electron inertial length λM<λe=c/ωe. The ratio of these two limits is λM/λm=fM/fm=mi/me≈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 ρi/λi=βi, with βi being the ion β. In isentropic expansion, the evolution of βi, assuming a Parker model, is 50(ρisρiqλiqλis)2=βisβiq∝(rqrs)53. From observations we have a total β>1. We expect ρi≳λi and assume that βi≳1. Figure  suggests a frequency ratio fm/fM∼βis∼0.7 larger than me/mi≈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 51rq≳0.24βiqAU. This value corresponds to >50βiqR⊙ from the Sun. Since the source must lie inside rq<1 AU, we conclude that βiq≲4.15. This number is just an upper limit. It is consistent with model calculations which predict βiq<1 shifting the inner boundary of the turbulent source region further in.

In this paper, we considered the cases V0=U0=0, V0=0,U0≠0, and V0≠0 for K and IK velocity spectra, where V0 is the velocity of the mean solar wind stream, and U0 is the mean speed of the energy-carrying largest turbulent MHD vortices which advect the bulk of small-scale turbulence around . 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 . The input spectral power densities are EIK and EIKad. Each of them yields a different ion-inertial scale range power spectrum in k space and, consequently, also a different power law spectrum in ωs space.

Table 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 ωs-1 for being related. Their nominal K and IK slopes are -2/3 and -1/2, 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. . It has the correct advective IK spectral index -1/2 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 〈δQ˙i〉=-〈δQ˙em〉=〈δJ⋅δE〉 in the ion-inertial range, the negative of the mean loss in electromagnetic energy density per time 〈δQ˙em〉, proportional to the product of current vortices δJ and the turbulent electric field δE. Though of finite magnitude, it is second order. This is left for future investigation. Hall currents do not contribute to any 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 k⟂∼λi-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 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.