We rewrite Poynting's theorem, already used in a previous publication

In a recent communication

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

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

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

Since any dissipation of

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.

These are generated progressively by turbulent self-organization in the spectral energy flowWriting all quantities as sums of mean fields plus fluctuations

A difficulty arises in dealing with the electric field. Relativistic
invariance requires its transformation into the rest frame of the flow

In so-called purely Alfvénic turbulence,

Fourier transforming in space and time in the infinitely extended domain,
assuming stationary and homogeneous conditions and constant

There is, of course, no obvious reason for

For non-Alfvénic turbulence

If the turbulence is independent in the parallel direction such that the
parallel turbulent wave vectors

Otherwise, for

Further conclusions can be drawn when considering the propagation of the
turbulent fluctuations. Propagation perpendicular to

Any magnetically compressive turbulence

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. (

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

Observations in the solar wind on comparably large scales indicate that the
velocity and magnetic spectra in the inertial MHD range exhibit different
slopes

The above measurements of the turbulent solar wind velocity spectrum were
restricted to the MHD frequency range

These measurements confirm the

It is also of interest that, in the inertial range, the temperature spectrum
mimics the velocity spectrum

Entering the ion-kinetic range at higher frequencies, the temperature
adjusts to the steeper slope of

It would be desirable to apply the measurements published above to our
theoretical determination of the conductivity spectrum. Unfortunately,
however, the experimental spectral energy densities are available only in
frequency space. Application of the Taylor hypothesis to transfer them into
wavenumber space implies imposing a linear Galilean transformation relation

At this point a general remark on the use of Taylor's hypothesis is in
order. It not only reduces the wavenumber–frequency spectrum to the
inclusion of a delta function, but it also reduces the “turbulent dispersion
relation” to a linear relation. This might indeed hold as long as the flow
velocity is very high,

The observations used above make no difference between the propagation
directions. Thus any distinction is impossible and any application of spatial
scales like gyroradii and inertial scales is questionable because it applies
only to part of the mixture of components which makes up the spectra. In
order to solve this problem, observations should be split into components
perpendicular and parallel to

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

Finally we note that we did not use Elsasser

No data sets were used in this article.

The authors declare that they have no conflict of interest.

This work was part of a Visiting Scientist Programme in 2007 at the International Space Science Institute Bern. We acknowledge the interest of the ISSI directorate and the friendly hospitality of the ISSI staff. We thank the ISSI technical administrator Saliba F. Saliba for help, and the librarians Andrea Fischer and Irmela Schweitzer for access to the library and literature. We thank the anonymous reviewer for the constructive critical comments and for directing our attention to some recent publications on measurements of solar wind turbulent velocity and density spectra. The topical editor, Elias Roussos, thanks one anonymous referee for help in evaluating this paper.