Low-frequency electrostatic drift wave turbulence has been studied in both laboratory plasmas and in space. The present review describes a number of such laboratory experiments together with results obtained by instrumented spacecraft in the Earth's near and distant ionospheres. The summary emphasizes readily measurable quantities, such as the turbulent power spectra for the fluctuations in plasma density, potential and electric fields. The agreement between power spectra measured in the laboratory and in space seems to be acceptable, but there are sufficiently frequent counterexamples to justify a future dedicated analysis, for instance by numerical tools, to explain deviations. When interpreting spectra at low ionospheric altitudes, it is necessary to give attention to the DC ionospheric electric fields and the differences in the physics of electron–ion collisions and collisions of charged particles with neutrals for cases with significant Hall drifts. These effects modify the drift wave spectra. A dedicated laboratory experiment accounted for some of these differences.

Turbulence in neutral flows has been studied extensively, in part because of
its significance for industry, the environment, etc., but also because of the
theoretical intricacies which the phenomenon represents. An understanding of
turbulent flows is important for weather forecasting, environmental
pollution, windmill design, the transport of material by water flows in
industrial plants, cooling and many other applications

Electrostatic drift waves become unstable when a certain mechanism inhibits
the free flow of electrons along magnetic field lines from wave crest to wave
trough. Electron–ion collisions have this effect, but in kinetic models
electrons in resonance with the drift waves will also induce linear
instability, although with modest growth rates. Plasma currents can enhance
the instability

Spectra are often easy to measure, and many studies both in the laboratory and in space have reported such results. It is a possibility that turbulent power spectra can be used to identify some of the underlying mechanisms and instabilities that give rise to the enhanced fluctuation levels. For the time being, this remains speculative, but there seems to be some basis for the argument. Drift wave turbulence, when it is fully developed, seems to posses certain universal features, best found by considering a fit in terms of a power law in a wave number representation. The present review will discuss this possibility, illustrated by results from laboratory and from space observations, with reference also to some of the analytical results. Comparatively smaller attention is given to numerical results in order to limit the exposition.

Higher-order spectral methods, such as bispectra

Following a brief summary of some basic results concerning neutral flow
turbulence, the present review will emphasize drift wave turbulence
supplemented by a shorter discussion related to current-driven ion acoustic
instabilities. This approach may appear unduly restrictive, but for
magnetized low-

The ideas of turbulence (strong turbulence, in particular) were first
formulated for incompressible neutral flows

It is generally believed that the Navier–Stokes equation is adequate for
describing all the scales that are important and relevant for modelling
incompressible turbulent neutral flows. This is a relatively simple
differential equation, after all, being of first order in time and second order
in spatial differentials. Some of the solutions of this equation are simple
as well. The solution represented by a turbulent flow has, however, so far
escaped a complete understanding. Indeed, there was a time when it was expected that
turbulence phenomena were to remain incomprehensible, and it probably came as
a surprise that while understanding an individual turbulent flow is beyond
our reach, some simple laws could be predicted for statistical averages

For incompressible neutral flows the mass density does not explicitly play a
part, and the kinematic viscosity

We want first to obtain an expression for the wave number power spectrum

We have the relation

The foregoing arguments emphasized spectra in terms of wave vectors. It is
easily demonstrated that similar arguments can be used to find an inertial
subrange for frequency spectra, giving

When it was realized that plasmas can also develop a turbulent state, it was
argued that the dimensional arguments found for fluid turbulence might be
applied also to the plasma case and electrostatic drift wave turbulence in
particular

If we are to have a universal long-wavelength range independent of

However, this result cannot possibly be correct, since it predicts a
turbulent spectrum independent of any energy input, i.e. also in the absence
of a density gradient (i.e. no drift waves), for instance. The shortcoming is
due to an oversimplification of the parameters needed to account for the
basic properties of drift wave turbulence. Although the result is in error,
the paper by

Other analytical works

The result Eq. (

A transition wave number separating the production and coupling subranges
(i.e. the

It is by no means obvious that results from strong-turbulence studies apply
to plasmas as well

Recalling the dimension

Within fluid models a class of nonlinear equations for plasma dynamics can be
written

In many studies of seemingly turbulent plasmas it is possible, by suitable
chosen multiprobe diagnostic methods

Summary figure from experimental studies of plasma density
fluctuations in radio frequency (RF) discharges, obtaining dispersion
relations against a noisy background

It turns out that drift waves offer one example of plasma waves where a
strongly turbulent state may develop. A close analogy can be found, for
instance, with waveforms in the Earth's neutral atmosphere, the Rossby
waves. A simple model equation was derived by

The Hasegawa and Mima equation accounts for the nonlinear coupling of modes,
but describes linearly stable plasma, with no energy input or dissipation.
An extended and generalized model of the Hasegawa and Mima equation

Summary of the normalized dispersion relation Eq. (

These models, and those derived from them,
formed the starting points of many studies (although not all)
of strong drift wave turbulence. Investigations of turbulent drift waves
have been stimulated by their importance for anomalous transport
in magnetized plasmas and fusion plasma experiments in particular

None of the early analytical studies of drift wave turbulence

Schematic illustration of spectra observed by

Universal spectral laws can be derived also within a weak-turbulence theory.
One of these results seems particularly relevant here. The current generated
ion acoustic instability (i.e. an instability generated by an electron flow
with velocity

The fluctuation level obtained from Eq. (

Experiments in the laboratory and in space obtain frequency spectra which are
subsequently interpreted as wave number spectra with reference to the Taylor
hypothesis (or the frozen turbulence approximation) often used in studies
of fluid and plasma turbulence

For one-dimensional turbulence (should it exist), the relations between the
observed power spectrum

It might be instructive to show the result for a case where the turbulence is
confined to a part of wave number space: assume we have

For three-dimensional fluid turbulence, we have an isotropic velocity power spectrum

Taylor's hypothesis is not restricted to power law spectra, although
this form is often implied. Since these forms for spectra are found so often,
they are also emphasized in this review. It should be noted, however, that
arguments have been offered also for other spectral laws

A number of laboratory experiments have been carried out for studying drift
wave turbulence

For cold plasmas, as in Q machines

Studies of fully developed turbulent limits with power law wave number
spectra

Spectral results from laboratory experiments.
The data from Tore Supra identify a production and a
coupling subrange as indicated

Systematic studies were carried out in a rotating caesium plasma column in a
linear Q device

Schematic
illustration of a Q-machine set-up with a nearly “solid body”

Summary of basic Q-machine caesium plasma conditions

In some cases it has been possible to determine both wave number and frequency
spectra simultaneously and thus test the accuracy of the Taylor hypothesis

For scales comparable to the density gradient scale length, the assumption of
local isotropy can be difficult to justify. For short wavelengths it was
found by microwave scattering

For neutral turbulence it has been argued by

Experimentally obtained power spectra for fluctuations in density
and potential

The variation in the spectral index in the coupling subrange in the
terminology of

In laboratory experiments we have the additional constraint that the average
collisional mean free path

In comparison to the spectral index

Variation in
the spectral index

A finite length of an experimental device imposes certain restrictions on the
wavelengths parallel to

Results from laboratory studies of plasmas with varying neutral background
densities have also been carried out

Two features are conspicuous in Fig.

Power spectra for the electrostatic potential and plasma density
at different neutral background pressures are shown in

For low neutral densities we find that the spectral indices for density and
potential are nearly equal, consistent with a model where

The RMS–density fluctuation level

When the linear Farley–Buneman instability

By a narrow-band filtering of the data at enhanced pressures, a dispersion
relation between frequency and propagation velocity could be obtained

The discrepancy between the two data sets for RF discharges noted in
Table

Wave studies in the ionosphere have the attractive feature of allowing
boundary conditions to be ignored in many cases. The analysis of collisional
or resistive drift waves applies to the Earth's ionosphere. Collisions between
charged particles and neutrals dominate collisions between electrons and ions
at low altitudes, while results for resistive drift waves will apply at
higher altitudes. In the equatorial regions, the geometry will be the
standard, with density gradients perpendicular to the magnetic field lines.
In polar regions there will be a density gradient component along
magnetic field lines as well, but simplified models have been suggested by

The ionospheric plasma in the E and lower parts of the F region has a
significant neutral component. For a wide altitude range up to 110–120 km,
the collisions between neutrals and charged particles will completely
dominate electron–ion collisions so that the ions are in effect unmagnetized. When we have

For the present problem we find a linear dispersion relation for low-frequency electrostatic waves in the form

To the same approximation, the imaginary part of

A model consistent with the observations is one where we have a

Turbulent spectra have been observed in situ in the Earth's ionosphere by use
of instrumented rockets and satellites. Following the set-up used elsewhere

Since rockets or satellites do not have an absolute ground, potential variations are usually detected by using probe potential differences. For scales or wavelengths much larger than the probe separation, the measurements can be interpreted in terms of an electric field component taken along the direction of the boom connecting the probes. Very often the situation is the opposite: the scales are shorter than the probe separations. This has consequences for the interpretation of turbulent potential spectra detected by probes. The Appendix offers arguments for interpreting high-frequency, short-wavelength parts of experimentally obtained spectra as power spectra for the electric potential.

A vast number of numerical simulations contain partial results for spectral
power laws: there are too many to be summarized here, in part also because it
is not always evident which ionospheric parameter range (if relevant at all)
the analysis refers to. There is an early summary of studies of the
type II irregularities

Observations made by instrumented rockets. If two values for a spectral index are listed, then the lowest frequency range is listed first. Note that the uncertainty on the spectral estimates is not always given in the literature. The data are sorted according to average altitudes, with the lowest altitudes first.

An overview and synthesis of plasma irregularities and turbulence
was given by

Some reports seem to distinguish drift and transverse velocity shear-driven
turbulence

Observations made by instrumented satellites. The results are
obtained by determining the exponent

The results of

Colour-coded representation of parts of the information in
Table

Black- and
white-coded representation of parts of the information in
Table

The lists in Tables

The altitude range with parameters

Using dimensional analysis

If we also include data from satellites (see Fig,

Can it be that a universal power law may simply not exist? This conclusion
seems somewhat pessimistic, and in the opinion of the present author it is in
part a question of sorting the data according to an effective Reynolds number
as argued in Sect.

Most laboratory results refer to large intensity fluctuations, and there indications of universal power law spectra were found. It could also be that the instrumentations of the rockets and satellites is insufficient to distinguish between differences in the nature of the turbulence. By applying Taylor's hypothesis (or the frozen turbulence hypothesis) to a “one-point signal” alone, it is not logically possible to distinguish between different dimensions of the turbulence field. It is particularly unfortunate that with a one-point measurements, it is not possible to estimate the density gradient either; at best only one component of it can be found: for studies of gradient-driven drift wave turbulence this information would be essential, in particular for quantifying a spectral “production subrange”.

It is possible that some space observations deal with plasma turbulence that is almost two-dimensional and perpendicular to

The present summary discussed observations of turbulent fluctuations in
magnetized plasmas, studied in the laboratory and in space. It was demonstrated that
turbulent spectra can be observed and that these spectra can be characterized
by well-defined power laws that can span several orders of magnitude, as in
Fig.

Turbulence in plasmas dominated by collisions between charged particles and
neutrals seem to offer a problem that is not readily accounted for by strong-turbulence models for drift waves. Analytical studies

It is interesting to note that the rocket data in
Table

Laboratory experiments seem also to be in reasonable agreement with rocket data concerning
spectral indices for low neutral pressures. Detailed studies with enhanced
neutral pressures have been carried out but seemingly only for conditions
where the Farley–Buneman instability is excited. It would be valuable to
investigate cases where only the gradient instability is present, in
particular in order to verify that the short-wavelength damping effect of the
first term in Eq. (

Turbulent signals can also be found at altitudes below the E region

Except for a few cases

Spectra are in general presented with the implied assumption that the
turbulence can be considered to be locally homogeneous to a good
approximation, possibly also isotropic in two or three spatial dimensions. In this
context, it is interesting that non-ideal laboratory studies with widely
different plasma conditions give results that show general agreement,
indicating that the homogeneity requirement is relatively mild. See for
instance Fig.

An interesting possibility would be if a spectral index could be used as a
means of identifying the instability causing the observed turbulence, but so
far this remains speculative. It is interesting to note that turbulence in
the high-

For ionospheric and magnetospheric studies in particular, the question of
waves excited by moving rockets and satellites also needs to be addressed.
Such perturbations can influence the observed wave spectra

Low-frequency electrostatic waves in space plasmas, pure drift waves or their
modified versions due to the Hall current instability, are important for
turbulent transport. For the resistive drift waves, classical models based on
transport due to turbulent

Zonal flows

The author acknowledges valuable discussions with Lars Dyrud, Bård Krane,
Søren Larsen, Torben Mikkelsen, Chan-Mou Tchen, Jan Trulsen and
Andrzej Wernik. Thanks also to Edward Powers for permission to reproduce
Fig.