ANGEOAnnales GeophysicaeANGEOAnn. Geophys.1432-0576Copernicus PublicationsGöttingen, Germany10.5194/angeo-35-1353-2017The usefulness of Poynting's theorem in magnetic turbulenceTreumannRudolf A.https://orcid.org/0000-0002-9783-994XBaumjohannWolfgangwolfgang.baumjohann@oeaw.ac.athttps://orcid.org/0000-0001-6271-0110Department of Geophysics and Environmental Sciences, Munich University, Munich, GermanySpace Research Institute, Austrian Academy of Sciences, Graz, Austriavisiting scientist at: the International Space Science Institute, Bern,
SwitzerlandWolfgang Baumjohann (wolfgang.baumjohann@oeaw.ac.at)15December2017356135313607September201721November201729November2017This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/This article is available from https://angeo.copernicus.org/articles/35/1353/2017/angeo-35-1353-2017.htmlThe full text article is available as a PDF file from https://angeo.copernicus.org/articles/35/1353/2017/angeo-35-1353-2017.pdf
We rewrite Poynting's theorem, already used in a previous publication
to derive relations between the turbulent magnetic and
electric power spectral densities, to make explicit where the mechanical
contributions enter. We then make explicit use of the relativistic
transformation of the turbulent electric fluctuations to obtain expressions
which depend only on the magnetic and velocity fluctuations. Any electric
fluctuations play just an intermediate role. Equations are constructed for
the turbulent conductivity spectrum in Alfvénic and non-Alfvénic
turbulence in extension of the results in the above citation. An
observation-based discussion of their use in application to solar wind
turbulence is given. The inertial range solar wind turbulence exhibits signs
of chaos and self-organization.
Space plasma physics (kinetic and MHD theory; turbulence)Introduction
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.
Poynting's theorem in magnetic turbulence
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.
These are generated progressively by
turbulent self-organization in the spectral energy flow
towards the short scales. There dissipation is anomalous, mediated by
plasma-kinetic processes.
Application to turbulent fluctuations
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.
Turbulent electric and velocity fields
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 .
The turbulent response
of the plasma contained in the conductivity spectrum
σ⟂ωkT is, under stationary and homogeneous
conditions, given by the ratio of the spectral energy densities of the
turbulent magnetic and velocity fields. (We note in passing that this
expression can also be exploited when constructing a
low-frequency “turbulent dispersion relation”
N2≡k2c2/ω2=iσ⟂ωkT/ωϵ0,
which is not the solution of a linear eigenmode problem, but determines the
nonlinear relation between the turbulent frequencies ω and wavenumbers
k.)
For non-Alfvénic turbulence δv⟂δB, i.e.
δv⋅δB=0, which means that the cross-helicity
vanishes. It is convenient to distinguish velocity fluctuations parallel and
perpendicular to the mean field. If δv‖B‾, the
turbulent electric, magnetic, and velocity fluctuations form a mutually
orthogonal system δE=-δv×δB⟂.
Hence Poynting's vector becomes δE×δB⟂=δv‖(δB⟂)2, giving from ()
∂∂t(δB⟂)22μ0=-σ⟂T(δB⟂)2(δv‖)2-1μ0∇‖[δv‖(δB⟂)2]
for non-compressive non-Alfvénic magnetic turbulence. It is obvious that in
this case the cross-helicity contributes through the (parallel) divergence of
the Poynting flux. Unlike the Alfvénic case, the last term in the above
expression generally cannot be reduced further. Moreover, the first term on
the right is a triple product, which makes any further treatment difficult.
If the turbulence is independent in the parallel direction such that the
parallel turbulent wave vectors k‖=0 vanish, then the last equation
simplifies and can be solved for the perpendicular non-Alfvénic
conductivity spectrum:
σ⟂ωk⟂T=iω2μ0[log(δB⟂)2]ωk⟂[(δv‖)ωk⟂2]-1.
The logarithmic dependence on the spectral energy density of the magnetic
turbulence implies that the conductivity spectrum is mainly determined by the
spectral energy density in the turbulence of the mechanical flow. This is
also the case when k‖≠0, because then the above equation can be
brought into the form
(∂∂t+2δv‖∇‖)log(δB⟂)22μ0=-1μ0∇‖δv‖-σ⟂T(δv‖)2
where the dependence on the magnetic fluctuation spectrum remains logarithmic
as well. Again, in homogeneous stationary turbulence this can be reduced to
an equation for the spectral density of σ⟂T.
Otherwise, for δv⟂B‾, one has
δB‖B‾ as a consequence of
δv⟂δB. We called this case compressive magnetic
turbulence and, for our purposes, excluded it from
consideration.
Further conclusions can be drawn when considering the propagation of the
turbulent fluctuations. Propagation perpendicular to B‾ of
magnetically non-compressive fluctuations (δB‖=0) implies
δE‖B‾. Hence the first term on the right in
Eq. () is zero, and since the magnetic and electric fluctuation
fields are orthogonal, lying both in the plane perpendicular to the mean
field, one has δv‖B‾, i.e. all velocity
fluctuations which contribute are parallel to the mean field. Moreover, in
Eq. () the last term on the right thus disappears and, after
Fourier transformation, one obtains a simple expression for the turbulent
conductivity spectrum in homogeneous stationary turbulence in this case
.
Any magnetically compressive turbulence δB‖B‾,
which so far has been excluded here, requires a separate investigation. In
this case, still considering only electromagnetic fluctuations with
δE⋅δB=0, the electric fluctuations corresponding
to δB‖ are perpendicular to B‾, in agreement
with Eq. (). One obtains after some simple algebra that
δE×δB‖=δb‖(1+δb‖)B‾2δv⟂⟂
where δv⟂⟂ is the velocity fluctuation perpendicular
to the mean magnetic and turbulent electric fields, and δb‖=δB‖/B‾ is the ratio of the compressive amplitude of the magnetic
fluctuations to the mean field. The divergence of this expression is the
contribution of the compressive part of the magnetic turbulence. It vanishes
for parallel propagation, contributing only for propagation
k=k⟂ perpendicular to the mean field. Combining all the
terms produces the equation
∂∂t(δb‖)22μ0=-σ‖T(δv⟂⟂)2[1+(δb‖)2]-1μ0∇⟂⋅[δv⟂⟂δb‖(1+δb‖)]
for the magnetically compressive component. Experimentally it is a simple
matter to separate out δB‖. We do not invest further in any
discussion of this case.
Discussion and conclusions
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.
Data-based thermodynamic considerations
The above measurements of the turbulent solar wind velocity spectrum were
restricted to the MHD frequency range ≲10-2 Hz. More recent
observations based on a sophisticated
technique aboard the Spektr-R spacecraft, extended to higher frequencies into
the range ≲2 Hz, presumably scale below the ion gyroradius, where
ion kinetic effects become important, for instance in supporting kinetic
Alfvén waves, and the ions demagnetize.
These measurements confirm the ∼-32 slope of the turbulent
velocity spectrum in the MHD range at frequencies below the ion-cyclotron
frequency (scales, presumably longer than the ion gyro and/or inertial
scales), thus being more 2-D and flatter than those observed about
Kolmogorov-turbulent magnetic spectra. At their higher frequencies they
partially cover the kinetic non-magnetized ion range spectra and exhibit
power laws of a steeper slope close to -3, indicating that the turbulent
(ion) velocity fluctuations enter a different, presumably still inertial
fluid regime when decoupling from the magnetic field. Currents which
contribute to the magnetic fluctuations here are carried by magnetized
electrons either perpendicularly, as drift currents in the density and
temperature gradients of the turbulent eddies, thereby forming narrow scale
current filaments, or along the magnetic field as kinetic Alfvén waves
. Signatures of the proximity to this regime are
visible as undulations in the velocity spectrum above say
3×10-2 Hz already where they form a weak bump on the spectrum
which is even more expressed in the density spectrum first observed
already bytheir Fig. 1 which, in general, does not follow
either Kraichnan's or Kolmogorov's prescription.
It is also of interest that, in the inertial range, the temperature spectrum
mimics the velocity spectrum . Thus inertial range
kinetic energy dϵ and thermal energy dT follow
each other. Assuming ideal gas conditions implies that
dϵ=cvdT.
Therefore, the specific heat cv≈ const (within the uncertainty of
the measurements) does not change across the inertial range. Such processes
are isentropic with
T∼Nγ-1
where γ=cp/cv is the ratio of specific heats. Using the average
inertial range slopes seeFig. 1, we then find from
the general adiabatic (isentropic) equation cf. e.g.dlogT-(γ-1)dlogN=0
that in the solar wind inertial range the ratio of specific heats as
determined from the fluctuations in density and thermal speed
is γ≈1.82, which implies that under the
ideal gas assumption one finds from the relation
γ=2+DD
between γ and the number of dimensions Dcf.
e.g. that the inertial range has fractal dimensionD≈2.46, which again implies deterministic chaos,
self-organization, and structure formation cf.
e.g. in this range.
Since, at least in part of the inertial range, the density and magnetic
spectra behave similarly, this reasoning also applies to the turbulent
magnetic field.
Entering the ion-kinetic range at higher frequencies, the temperature
adjusts to the steeper slope of ∼-52, suggesting
non-adiabaticity and heating over the velocity spectrum, as is of course
expected when ion-kinetic processes like heating by kinetic Alfvén wave
turbulence take over in this range.
Application to Alfvénic solar wind turbulence
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
ω=v‾⋅k which may hold for very high
nonrelativistic average speeds (see also the brief discussion below) and thus
in frequency–wavenumber space restricts one to multiplication of the
conductivity spectrum with a Dirac function
δ(ω-v‾⋅k). In the Alfvénic turbulence
case one may formally obtain from the measurements of, say,
, and using Eq. () that
σ⟂ωk∼(1-2B‾αk‖ω)ω-(sδB-sδv-1)δ(ω-v‾⋅k)
where sδv,sδB are the respective experimental slopes of
the velocity and magnetic field spectra. Since these are about
∼32 and ∼53 respectively, the conductivity
spectrum in the inertial range is also the power law of the index (up to the
factor in brackets and the Dirac function) sσ=sδB-sδv-1≈-56, indicating an increase in conductivity
σ⟂ωk∼ω5/6δ(ω-v‾⋅k) with frequency
(shrinking temporal scale). Applying the Dirac function which the Taylor
hypothesis in addition imposes yields the wavenumber dependence
σ⟂k∼(1-αB‾k‖v‾⋅k)(v‾⋅k)56.
(Note that k‖ refers to the mean magnetic field, while in the denominator
the wavenumber is parallel to the average flow through Taylor's hypothesis
which artificially reintroduces v‾ at this late place after
developing the theory!) If this finding is confirmed and applies, the
inertial range turbulent resistance drops in frequency and wavenumber,
meaning that the inertial range in Alfvénic turbulence behaves increasingly
less dissipative towards shorter scales. The system is
collisionless, so this contradicts the expected self-organization
and structure formation (formation of progressively shorter scale current
filaments, eddies, etc.) which we have inferred above from fundamental
thermodynamic arguments without making any reference to any additional
hypothesis. This should not be the case. So this result may provide a strong
argument against the application of the Taylor hypothesis, at least at short
scales, i.e. large wavenumbers and frequencies. For the above-mentioned
reasons concerning observations, such a conclusion must, however, be drawn
with care.
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, |v‾|≫sup|δv|, a
trivial condition. Instead, the “correct” turbulent dispersion relation for
magnetic turbulence is given through the frequency–wavenumber spectrum of
the turbulent conductivity see. In addition, the
Taylor hypothesis applies only to turbulent structures which propagate
along the mean flow such that k‖v‾. Any
turbulence propagating at an angle, for instance the rotational velocity
component of a turbulent eddy, is thus affected only up to an angle where the
projection of the mean speed of the flow onto the wavenumber vector still by
far exceeds the turbulent speed. Any strictly perpendicular wave is not
affected by Taylor's hypothesis and thus principally cannot become
transformed into wavenumber space.
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 v‾ and the Taylor hypothesis
should be applied to the parallel component only.
Conclusions
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.
No data sets were used in this article.
The authors declare that they have no conflict of interest.
Acknowledgement
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.
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