In a recent communication we used Poynting's theorem in electrodynamics in order to construct an experimentally accessible expression for the spectral energy density of the electromagnetic field in collisionless magnetic turbulence. That attempt turned out much simpler and therefore also more effective than our previous fairly involved inverse scattering theory of electromagnetic fluctuations in magnetic turbulence. Since we used only electromagnetic theory, not referring to any mechanical fluid turbulence, it remained unclear to what extent an approach in turbulence like that one was justified. Magnetic turbulence at low frequencies – scales longer than the electron gyro-radius – involves both the electromagnetic and mechanical flow fields. Restriction to one of these components only apparently neglects an important part of the turbulence. This argument also applies to any experiments which use just measurements of magnetic fluctuations, calculate spectral energy densities, and possibly do not refer to electric field or velocity fluctuations. Determination of the power law shape of those spectra contains information about the turbulence, but its physical content remains inaccessible. Spectral slopes are sensitive to varying physical conditions . Small changes in the slope, which within experimental errors are difficult to detect, may indicate completely different physics.

Observations of magnetic turbulence in the solar wind take advantage of their easy accessibility in order to determine spectral slopes of the turbulent magnetic energy densities cf. e.g.for early reviews in the frequency domain. They enable us to distinguish between Kolmogorov's spectral ranges of energy injection, constant energy flux, and dissipation in frequency space cf. e.g.and references therein. Sometimes they enable distinction between Kolmogorov and Kraichnan regimes. They also provide absolute values of the turbulent magnetic energy density. Applying the Taylor hypothesis, limited information about the corresponding spatial scales has been obtained and, in a few cases, spectra of the electric field and streaming velocity fluctuations have been added. Measurements of turbulent density fluctuations in the solar wind have also been published.

In the present note, following our previous attempt, Poynting's theorem is briefly re-examined in order to relate it to the inclusion of the mechanical part of turbulence and to clarify the effect of the electric and velocity fluctuations.

Measurement of the Poynting flux in order to infer the plasma wave energy flow in near-Earth space has a long history. One of the first attempts was to determine its direction and absolute value in plasmaspheric electromagnetic ion-cyclotron waves. More recently it was used to detect dispersive whistlers in Earth's bow shock which are expected to contribute to shock reformation in quasi-perpendicular shocks cf. e.g.for a rather complete account and to the investigation of the energy flow in kinetic Alfvén waves near the plasma sheet boundary as a source of the auroral energy flow which often is attributed to the inflow of kinetic Alfvén waves cf. e.g. causing particle acceleration and radio emission . These works deal with the Poynting flux in particular waves only.

In magnetic/magnetohydrodynamic turbulence (at non-relativistic speeds) the equation of energy conservation, which is the generalization of Poynting's theorem in electrodynamics to the inclusion of mechanical energy transport, is quite generally written in the form ∂∂t(12ρv2+ρε+B22μ0)=-∇⋅q. The vector q is the energy flux density, ρ is the plasma mass density, v is the velocity, and ε=w-P/ρ is the internal energy, with w the internal enthalpy and P=trP≡13Pii the (scalar) pressure, and the relativistically small electric field density has been suppressed in the time-derivative term on the left. In this form the energy law accounts for all the energy in the turbulence. The energy flux vector q contains all the dissipative processes, mechanical and electromagnetic, in particular all anomalous processes which contribute to dissipation. The former (mechanical) terms contain a mechanical dissipation tensor, with bulk and shear viscosity coefficients. The latter (electromagnetic) terms are inherent to a conductivity tensor σij, which enters Ohm's law and which can always be written in its simplest form, such that the current is given by Ji=σijEj, where E=-v×B is the (relativistically correct) electric field. For finite electrical resistance, the relation between the electric field and current J becomes E+v×B=σ-1⋅J, an expression which is general in the sense that the various dissipative processes contributing to this generalized Ohm's law cf. e.g. are included in the definition of the conductivity tensor σ which in all realistic cases, if made explicit, becomes an involved expression. Any dissipation of electromagnetic energy is given by the product -E⋅J. Neglecting collisional dissipation, as is usually done in ordinary MHD, one has q=ρv(12v2+w)+1μ0B×(v×B) Written in terms of the electromagnetic field, energy conservation takes the form ∂∂tB22μ0=-1μ0∇⋅(E×B)-∇⋅qm-∂∂tEm with qm=ρv(12v2+w),Em=12ρv2+ρε. Any possibly occurring dissipation is solely due to turbulent mixing and in this sense is “anomalous”. This is Poynting's theorem completed with the two mechanical terms on the right. On the left is the time variation of the magnetic energy density. The first term on the right is the divergence of the electromagnetic energy flux vector, a familiar quantity. The other two terms, depending on their signs, either pump energy into the magnetic field by mechanical motion, as in the case of a dynamo, or dissipate magnetic energy.

Since any dissipation of magnetic energy, either positive or negative, can always be written as the above product -E⋅J, Poynting's theorem for the electromagnetic field under ideal dissipationless conditions in magnetic/magnetohydrodynamic turbulence can be written as ∂∂tB22μ0=-E⋅J-1μ0∇⋅(E×B), which is its familiar version in electrodynamics, and E⋅J=∇⋅qm+∂∂tEm+E⋅σan-1⋅E. A possibly present anomalous conductivity σan caused by kinetic processes on scales shorter than the ion or electron inertial lengths or gyroradii <λi,e,rci,ce would appear as the last term in Eq. (), but is not explicitly considered in the following. In collisionless and non-viscous turbulent plasmas the latter form applies at scales exceeding the Debye length and is far away from any molecular scale to which dissipation of the turbulent mechanical energy is attributed. In contrast, the turbulent electromagnetic energy is ultimately dissipated at least already at electron scales <λe,rce by spontaneous reconnection in small-scale current filaments.

At scales shorter than the electron gyroradius, electrons demagnetize and no longer contribute to magnetic fluctuations, electron thermal pressure does not balance the Lorentz force which contracts the current, and collisionless reconnection is spontaneous and explosive, causing electron exhausts, strongly deformed electron distributions, and electron beams. Dissipation here is kinetically and electrostatically provided by plasma waves (Langmuir, ion sound, Bernstein, electron holes). Except for a possible filamentary Weibel mode which causes further filamentation of the current and turbulence, no non-radiative magnetic fields are generated here. Hence, the magnetic turbulence spectrum should decay at those scales. High-frequency and thus weak-radiative fields can be produced in addition by the electron cyclotron maser instability inside the exhaust.

Writing all quantities as sums of mean fields plus fluctuations F=F‾+δF with average F‾=F‾ and δF‾=0 in Eq. (), averaging, subtracting the scale-averaged equation, and dropping the averaged products of the fluctuations as these depend only on the mean-field scale, we find ∂∂t(2B‾⋅δB2μ0+(δB)22μ0)=-δE⋅J‾-δE⋅δJ-1μ0∇⋅[δE×B‾+δE×δB] with mean electric field E‾=0 (see the next section below), and defining δJ=σT⋅δE, where σT is an equivalent turbulent conductivity tensor chosen such that when it relates the turbulent current to the turbulent electric field, the mean current vanishes. The result is ∂∂t(2B‾⋅δB2μ0+(δB)22μ0)=-δE⋅σT⋅δE-1μ0∇⋅[δE×B‾+δE×δB], which is the basic equation used in . Restricting it to magnetically non-compressive turbulence δB⋅B‾=0 makes the first term on the left vanish. The first term in the brackets on the right vanishes for δE‖B‾, the case k⟂B‾ of propagation of the turbulent fluctuations perpendicular to the mean field. For parallel propagation this term contains the excluded compressive magnetic component. We are thus left with the simplified Poynting equation ∂∂t(δB⟂)22μ0=-δE⋅σT⋅δE-1μ0∇⋅[δE×δB⟂]. All dynamics of the turbulent mechanical flow is implicit in σT, which (keeping an anomalous conductivity σan) is formally defined as σT=(δE)-1⋅[σan+∇⋅δqm+∂∂tδEm]⋅(δE)-1 where δqm, δEm are the fluctuations of qm, Em. Once, by the means of measuring the electromagnetic turbulent fluctuation spectrum, the turbulent conductivity spectrum σωkT has been determined as a function of fluctuation frequency ω and wavenumber k, its transformation back into real space provides a relation to the turbulent mechanical quantities.

A difficulty arises in dealing with the electric field. Relativistic invariance requires its transformation into the rest frame of the flow E′=E+v×B. In an ideal turbulent medium the moving frame speed depends on the fluctuation scale, which in general makes it difficult (if not impossible) to define a common moving frame valid on all scales. Splitting into mean and fluctuating quantities yields the averaged field E′‾=E‾+v‾×B‾+δv×δB‾, which in the moving frame must vanish. This gives the mean electric field E‾=-v‾×B‾-δv×δB‾. Measurement of the velocity fluctuations δv in the scale range of interest is required in the averaged second term. The fluctuating primed electric field becomes δE′=δE+δv×B‾+δv×δB+v‾×δB-δv×δB‾. The mean magnetic field B‾ and the last averaged term are constant on the fluctuation scale. In an infinitely extended medium without boundaries the last term can be dropped, yielding δE=-δv×B‾-δv×δB-v‾×δB, which is to be used in Eq. (). It requires knowledge of the velocity fluctuations on the same scales (and with same resolution) as the magnetic fluctuations. The second term on the right measures the “alignment” of the magnetic and velocity fluctuations.

In so-called purely Alfvénic turbulence, δv‖δB and the cross-helicity (normalized to the total energy) is close to unity, resulting in a linear relation for the fluctuating electric field δE=-δv×B‾-v‾×δB. The electric fluctuations are perpendicular to both δB,δv in this case. The Poynting flux vector term in Eq. () assumes the form -B‾⋅∇(δv⋅δB⟂)/μ0, which eliminates the electric fluctuations in favour of the velocity field and reduces Eq. () to ∂∂t(δB⟂)22μ0=-σ⟂TB‾2(δv)2-1μ0B‾⋅∇(δv⋅δB⟂)-σ⟂T[v‾2(δB⟂)2-(v‾⋅δB⟂)2] where σ⟂T is the turbulent conductivity parallel to δE, i.e. perpendicular to both δB⟂ and B‾. The last terms contain only the mean flow components v‾⟂ perpendicular to δB⟂. The complications they introduce disappear when transforming to the easily determined mean flow v‾=0. The Poynting term vanishes when considering spatial dependencies perpendicular to the mean field. More generally, since in Alfvénic turbulence δv=αδB⟂ with α some angular-dependent scalar factor (which can, in principle, be determined from the fluctuations), the argument of the Poynting vector can be expressed by (δB⟂)2. Except for any spatial dependence of α, the magnetic and velocity fluctuation spectra should thus be comparable in Alfvénic turbulence for either parallel or perpendicular propagation. (One may note that for cross-helicity δv⋅δB⟂/|δv⋅δB⟂|≈±1 the second term on the right in Eq. () disappears.)

Fourier transforming in space and time in the infinitely extended domain, assuming stationary and homogeneous conditions and constant α yields σ⟂ωkT=iω2μ0B‾2(1-2αk⋅B‾ω)(δB⟂)ωk2(δv⟂)ωk2. This holds in Alfvénic turbulence. (The contribution of a finite mean speed may be retained any time when wanted.) For cross-helicity one, the expression in parentheses in the first term on the right reduces to unity.

There is, of course, no obvious reason for α to be constant. In general it will depend on space and time, which is suggested by the radial variation of the solar wind spectra with increasing solar distance . Locally, the assumption of constancy is well justified, however, as is also confirmed by solar wind observations at 1 AU of the constancy of the cross-helicity .

Poynting's theorem provides additional information about turbulence which so far had not been exploited. It allows us to account for the relativistic effect in the electric field and reduces it to a measurement of the turbulent velocity and magnetic fields as suggested by Eq. (). This cannot be circumvented by no means. It is interesting to briefly discuss more recent measurements of electric field, velocity, and also density fluctuations in this light.

The specifications of Sect. 4 show that, as expected from electrodynamics, replacing the electric fluctuations in electromagnetic turbulence, the magnetic and velocity fields become related. This follows from relativity. The electric fluctuation field plays an intermediate role of an mediator only. The versions of Poynting's theorem given above explicate the interrelation. They can be applied to stationary homogeneous turbulence providing expressions for the spectrum of the turbulent conductivity as a functional of the magnetic and velocity power spectral densities similar to those given previously but expressed here in terms of the velocity fields. There we insisted on the independent determination of the electric and magnetic power spectral densities. It turns out that determination of the spectrum of turbulent velocities on all scales is more important.

Observations in the solar wind on comparably large scales indicate that the velocity and magnetic spectra in the inertial MHD range exhibit different slopes . Velocity power spectra are typically flatter, of slope -32 (2-D or Kraichnan), than magnetic spectra at 1 AU, which are close to the 3-D-Kolmogorov -53 with apparently less power in the kinetic than magnetic energy fluctuations. In fact, there is no obvious reason why they should be similar. Any magnetic fluctuations δB are, through Ampère's law, related to fluctuations of the electric current δJ=eN‾(δvi-δve)+eδN(v‾i-v‾e) assuming quasi-neutrality in turbulence. Examples are diamagnetic currents in pressure gradients. Under stationary conditions this reduces to pressure balance. It is the difference in the fluctuations of the ion and electron velocities and the density fluctuations which both contribute. At long MHD scales the average velocities cancel and the last term in the current disappears, but in the first term the ion and electron velocity fluctuations are not aligned and contribute differently to the spectra. Measured fluctuations in the flow δv have little in common with the fluctuations of the current. At short scales the second term on the right in the current contributes through the density fluctuations which are caused mainly by fluctuations of the plasma pressure and thus are related to the transverse magnetic pressure. With increasing solar distance in the solar wind, the velocity spectra though in the inertial scale range, still being of lower spectral density than the magnetic spectra, seem to approach the Kolmogorov slope while at the same time intensifying. If confirmed, a simple explanation is that in solar wind turbulence the effect of decreasing magnetic field on the flow weakens with increasing solar distance, thus gradually losing dominance.

In the previous section we applied Poynting's theorem to derive expressions between the turbulent conductivity and measurable spectral energy densities. These expressions are formulated in terms of the magnetic and velocity spectra. The electric field appears just on an intermediate step, becoming eliminated by the relativistic transformation. These expressions may be useful in application to observations, but require precise measurements of the velocity field fluctuations. This is the main experimental difficulty. Their knowledge is of general interest in turbulence theory as they allow construction of a turbulent dispersion relation which is not a solution of an eigenmode equation but determines the relation between observed frequencies and wavenumbers. This should provide a useful experimental input into the conventional approach to both fully developed strong and weak stationary and homogeneous magnetohydrodynamic turbulence.

Finally we note that we did not use Elsasser variables here, the mixed magnetic and flow fields which are usually used in magnetohydrodynamic turbulence theory . Reformulation of the results in these variables is a simple matter. This will be left for a separate investigation.