We apply a particular form of the inverse scattering theory to turbulent magnetic fluctuations in a plasma. In the present note we develop the theory, formulate the magnetic fluctuation problem in terms of its electrodynamic turbulent response function, and reduce it to the solution of a special form of the famous Gelfand–Levitan–Marchenko equation of quantum mechanical scattering theory. The last of these applies to transmission and reflection in an active medium. The theory of turbulent magnetic fluctuations does not refer to such quantities. It requires a somewhat different formulation. We reduce the theory to the measurement of the low-frequency electromagnetic fluctuation spectrum, which is not the turbulent spectral energy density. The inverse theory in this form enables obtaining information about the turbulent response function of the medium. The dynamic causes of the electromagnetic fluctuations are implicit to it. Thus, it is of vital interest in low-frequency magnetic turbulence. The theory is developed until presentation of the equations in applicable form to observations of turbulent electromagnetic fluctuations as input from measurements. Solution of the final integral equation should be done by standard numerical methods based on iteration. We point to the possibility of treating power law fluctuation spectra as an example. Formulation of the problem to include observations of spectral power densities in turbulence is not attempted. This leads to severe mathematical problems and requires a reformulation of inverse scattering theory. One particular aspect of the present inverse theory of turbulent fluctuations is that its structure naturally leads to spatial information which is obtained from the temporal information that is inherent to the observation of time series. The Taylor assumption is not needed here. This is a consequence of Maxwell's equations, which couple space and time evolution. The inversion procedure takes advantage of a particular mapping from time to space domains. Though the theory is developed for homogeneous stationary non-flowing media, its extension to include flows, anisotropy, non-stationarity, and the presence of spectral lines, i.e. plasma eigenmodes like those present in the foreshock or the magnetosheath, is obvious.

As far as concerns the fluctuations of the magnetic field, magnetic
turbulence

On the shortest scales,

On longer scales

On the other hand, measurement of the magnetic fluctuations is comparably easy in plasma. It would thus be desirable to infer about the turbulent plasma dynamics from the magnetic fluctuations alone, if possible. This is usually done from observation of the magnetic power spectra of turbulence and determination of the spectral index in several ranges of scales, from magnetohydrodynamic scales down into the dissipative range of scales of turbulence. This procedure mainly provides power law indices of the magnetic turbulence and distinguishes between different spectral ranges and between inertial and dissipation scales, while no information about the state of the plasma can be obtained.

In the following we attempt a different approach by formulating a so-called “inverse problem” for the particular case of magnetic turbulence. This is possible when recognising that, as noted above, magnetic turbulence is in fact just a branch of classical electrodynamics. It can thus be formulated in terms of purely electrodynamic quantities with the dynamics implicitly included only.

In a first step of such an approach, we demonstrate, how the problem of magnetic fluctuations can be reduced to the solution of an inverse problem, whose solution is, of course, nontrivial. We develop the theory until the formulation of the final integral equation, whose input is the experimentally obtained field fluctuation spectrum. (This is not the power spectral density usually used in turbulence and inferred by measurements. Instead, it is the full spectrum of electromagnetic fluctuations that is on stake – quite different from the magnetic power spectral densities used in ordinary low-frequency turbulence.) This integral equation will have to be solved for any given observed fluctuation spectrum. This reformulation of magnetic fluctuation theory might provide a new path in the investigation of turbulence as it infers about the dynamics of the plasma which lead to the generation of turbulent fluctuations. In a subsequent step its relation to observed turbulent spectral power densities should be investigated. Here we only develop the inverse magnetic fluctuation theory. A similar approach should, of course, also be possible for genuine kinetic plasma turbulent fluctuations, including electron scales.

We restrict ourselves to purely magnetic turbulence, i.e. turbulent fluctuations

Determination of the conductivity tensor depends on which dynamical reference
model of the plasma is adopted. Herein lies the difficulty of relating the
magnetic fluctuations as well as the magnetic spectral power density to the
turbulent flow. In general,

One may note that this ansatz by no means implies linearity. Through
all the plasma quantities and the electromagnetic fields,

In the following we deal with fluctuations only and for convenience
drop the prefix

With these preliminaries in mind Maxwell's equations for the fluctuating
fields in the absence of charge density fluctuations

The difficulty in solving the above equation for a turbulent plasma lies in
the fact that the generalised conductivity tensor

Before coming to this question, let us rewrite the last above equation in a
different form by introducing the dielectric response tensor of the turbulent
plasma:

One remark on the fluctuation of the dielectric response tensor
(suppressing the

So far we have formulated the fluctuation problem in terms of the
fluctuating induction electric or vector potential fields.
Alternatively, the entire problem can also be given a formulation purely in
terms of the magnetic field. For this we define the inverse transverse
response tensor

Observations are frequently only one-dimensional. Magnetic fluctuations are
strictly transverse with solenoidal fields

In developing the theory, let us for simplicity restrict ourselves in this
work to the case of perpendicular magnetic fluctuations propagating
along the mean field. We assume that the external/mean field is in
direction

The observations refer to the fluctuations of the magnetic field in time from which the temporal spectrum is subsequently obtained. The determination of the spatial spectrum is difficult. It requires the simultaneous measurement of the magnetic field over an extended area in various locations, for various spacings and many frequencies. In near-Earth space, where this is mostly done – a well-known example is the solar wind – frequency spectra have been obtained in multitude. However, so far, multi-spacecraft missions are rare. Only very few direct measurements of spatial spectra have become available in a restricted range of wavenumbers.

Measured time series of the magnetic fluctuation field

No direct spatial dependence of

Techniques for measuring the turbulent electric field by following the electron gyration orbit have been developed and are sometimes available. However, no application to turbulence has been attempted because these techniques are highly complex. It would be of great advantage if they could be transformed into measuring devices for the direct determination of the turbulent vector potential.

To make them efficient they should be applied to ion gyrations. Ion orbits are substantially more stable than electron orbits. They therefore could map the low-frequency turbulent electric field in the solar wind or elsewhere. We will show below that availability of both magnetic and electric time series provides a device for investigation of turbulence.

Temporal spectra are frequently transformed into spatial spectra under the
assumption of Taylor's hypothesis. This hypothesis implies that, for
sufficiently low frequencies and sufficiently fast plasma streams, the
turbulent eddies and fluctuations are swept across the spacecraft,
Galilei-transforming wavenumbers

Spectra propagating perpendicular to the streaming are not affected. Their
wavenumber spectra cannot be obtained in this simple way. These conclusions
are well known. Checking for the solar wind, one immediately finds that all
low-frequency waves obey the Taylor hypothesis if propagating along the solar
wind. Their nominal speed is the Alfvén speed

On the other hand, there is no reason to assume that fluctuations and
turbulent spectra consist of eigenmodes. However, they do not show
indication of line spectra which would correspond to particular plasma wave
modes. Instead, they represent fluctuations transformed into wavenumber and
frequency space which may not have anything to do with eigenmodes even when
resulting originally from the injection of energy at injection wavenumbers by
some plasma eigenmode spectrum. Turbulent cascading is highly nonlinear and
obscures any plasma modes which might be present inside its range of
wavenumbers. For this reason the Taylor mapping may well hold in all cases as
long as the ratio

The Taylor hypothesis becomes problematic in anisotropic turbulence

By the same token it is obvious that the sole measurement of magnetic power spectra provides incomplete information only about magnetic turbulence.

This can be shown more explicitly. The turbulent magnetic power spectrum is the only solid measurement available when no electric field observations are available and no velocity fluctuations can be obtained. The latter could, also in principle, be constructed from measurements of the distribution function if only resolution in velocity/momentum and time would suffice. In that case the turbulence problem would have to be formulated differently by reference to the dynamic equations or, more directly, to the kinetic equations.

In the availability of only magnetic fluctuations, one forms the average time
correlation function of the magnetic field at location

Of course, assuming that

Another promising measurable quantity that can be constructed from observations is the dissipative convolution function

In a magnetically turbulent medium, the convolution function

To demonstrate that

In homogeneous turbulence all quantities depend on spatial differences

In the following we show that the dissipative convolution function

The goal of our present approach is different. We try to use the electromagnetic fluctuation spectra as combined in the dissipative convolution function, a measurable quantity, in order to reconstruct the electromagnetic response of the plasma. This is not impossible because we are in possession of the evolution equation of the fluctuations. The idea is to invert this equation and to infer about the plasma response. This leads us to the inverse fluctuation problem.

We assume that the convolution function is available. It stores the
observational information about the electromagnetic fluctuations. Is there a
way of reconstructing the generalised dielectric response function

The inverse problem theory, known as the inverse scattering problem,
was formulated by

In a first step we transform the induction equation for the fluctuating vector potential
into a Schrödinger equation.
The necessary transformations are

Let

The “equivalent potential” function

One may note that the above stationary Schrödinger equation depends
only on space. Frequency appears as an eigenvalue parameter. Its solution
will depend on full space, while observations exist just in one space point

This requires concluding about spatial variation in the fluctuations from
observation in only one point. As shown below, this becomes indeed
possible due to the transformation of frequency

The standard Schrödinger form has a number of advantages. For one, its solution has been developed to very high standards. This provides a starting point for investigation of turbulence theory. However, ours is only on the inversion of the above equation by making use of its known solutions.

The transformation of Eq. (

The upshot to the above relations is that the turbulent response function

This problem is the inverse fluctuation problem of magnetic
turbulence. The formal problem of inverting the Schrödinger equation
was solved half a century ago

In inverse scattering theory the available observations are the reflection
and transmission coefficients of radiation passing a passive medium.
These coefficients are the ratios of the wave amplitudes of incoming from

One such possibility is reference to the convolution function

The solutions

The left-hand sides of these expressions are considered to be known from the
data. They consist solely of data, while the right-hand sides are obtained
from the linearly independent solutions

We note in passing that a similar procedure can be applied to the
solutions of Eq. (

It may be of some interest to ask for the relation to the observed magnetic
spectrum. We have

The last section summarised all the information needed for formulating
the inverse problem of turbulent magnetic fluctuations. In this
section we reduce the inverse problem to the solution of some version of
Gelfand–Levitan–Marchenko's equation

This has been done
first in the solution of the inverse geomagnetic induction problem

The scattering formulation has mathematically lucidly been presented by

The general solution of the Sturm–Liouville or Schrödinger
Eq. (

The Jost solution obtained in this way (also known as the Schrödinger integral equation) is composed of the two parts

What is interesting about the Jost solution is that the conjugate time
variable, i.e. the frequency

The function

The task is thus to find a way to calculate

For completeness we construct the above equation in a form that is
applicable to our problem of turbulent magnetic fluctuations. This requires
constructing the relation of

In dealing with turbulence we are in a situation quite different from
scattering theory

In turbulence, no such boundaries exist primarily. We are dealing with an
infinitely extended medium filled with turbulence while observations are
performed at a fixed location

In order to derive the Gelfand–Levitan–Marchenko equation we refer to the
general Jost solution and its representation through the turbulent-convolution function

Assume that

Inserting the Schrödinger integral form of the two linearly independent
solutions

The form of this equation is reminiscent of a chain of Fourier transforms with
respect to the dummy variable

With the left-hand side zero the second term on the right is already in the
form of a Fourier transform. Thus, its integral just yields

The integral of the first term on the right can be split into two
integrals by resolving the cosine into exponentials. One then recognises that

Since our reference point

Finally, the last term on the right is the product of Fourier transforms – i.e.
it is the transform of a convolution integral of the data function and
the unknown kernel of the Schrödinger integral equation. By resolving it
into the configuration space convolution integral, we ultimately arrive at
the Gelfand–Levitan–Marchenko Eq. (

In fact, the Fourier transform on

Solution of the Gelfand–Levitan–Marchenko equation finally provides the
wanted information about the unknown turbulent dissipation function

We have, in principle, achieved our goal: finding a method to reconstruct the electromagnetic turbulent response function in magnetic turbulent fluctuations. The theory developed here is rather complex. However, it has a large historical record in quite a different context and has been given a solid mathematical fundament.

This is the maximum that can be achieved for the time being regarding the inverse
problem of turbulent magnetic fluctuations in plasma. As noted earlier, it
requires knowledge of the turbulent-convolution function

Usually, only the turbulent magnetic fluctuations are available; however, in principle, methods could be developed to measure the electric fluctuation field by injecting dilute ion beams into the plasma and monitoring their return fluxes, which provide direct information about the low-frequency electric fluctuations. Such measurements have occasionally been performed using electrons but are polluted by the enormous sensitivity of electrons to the presence of electric and magnetic fluctuation fields. They also suffer from the difficulty of identifying the injected electrons and distinguishing them from the ambient electrons.

Another more promising possibility is the injection of low-energy ion beams in order to measure their distribution function and to calculate from it the fluctuations of the velocity field.

In the absence of either of these, one cannot proceed further. Magnetic field observations alone are insufficient. They cover only half of the information stored in the electromagnetic field.

It is easy to see that, without the independent determination of the
turbulent-convolution (response) function

This simply expresses the above-noted obvious fact that reduction to magnetic measurements alone, lacking the electric field or otherwise the velocity field, implies loss of one-half of the electromagnetic information which is needed in solving the inverse scattering problem. This resembles the inverse scattering case where, without knowledge of the reflection and transmission coefficients which couple the incoming and outgoing waves, no solution exists. Hence, in solving the inverse problem of turbulent magnetic fluctuations, knowledge of either the electric field or velocity fluctuations in addition to the magnetic fluctuations is obligatory.

With the reduction of the inverse problem of turbulent magnetic fluctuations
to the Gelfand–Levitan–Marchenko equation, the formal problem of inversion
of the magnetic fluctuations in a turbulent plasma has been solved. It has
been reduced to the determination of the dissipative convolution function
from observations of the fluctuations of the electromagnetic fields at
observation point

In practice the full solution of the inverse problem which aims at the
determination of the dissipative response function

The form of the convolution function is not known a priori. Its spatial
dependence is not required, however. What is necessary is simply its temporal
spectrum, i.e. its Fourier transform with respect to

We have already noted that it is not expected that the turbulence contains emission or absorption lines corresponding to eigenmodes of the Schrödinger equation. This would imply the presence of distinct plasma waves or turbulent energy losses at some particular frequency as might be present in non-homogeneous plasmas such as near a shock wave in a restricted region of space. Examples are the narrow upstream and downstream regions of shocks in the solar wind, the Earth's magnetospheric foreshock, and those magnetosheath regions where turbulence prevails while distinct plasma modes are excited by some energy source that is related to shock. Moreover, if the turbulence is not fully developed, intermittency might play a role leading to additional structure in the dissipative response function.

These problems are all very interesting and important. However, as noted several times, in order to be included into the inverse problem, they
require precise measurements of both types of electromagnetic fields, as well
as their magnetic and electric components. Formally they introduce discrete
eigenvalues leading to poles in the complex

In the absence of discrete fluctuation modes, we may try an
unstructured power law distribution of the dissipative convolution function
of the kind

Reformulation of the convolution function in terms of turbulent spectral energy densities would require a complete reformulation of the inverse problem, which is beyond the scope of this work. Such a reformulation suppresses the use of Jost functions, which are the solutions of the Schrödinger equation for the mapped fields, not for their spectral energy densities. Presumably, this inhibits reference to the Gelfand–Levitan–Marchenko theory.

The Fourier transforms map the electric and magnetic fluctuation
fields from time into frequency space. The mapped field spectra necessarily
possess some phases

The assumption of a power law of the turbulent-convolution function allows one to write

If we, for simplicity, assume that the power law spectrum extends over
several orders of magnitude in

We write the integral with respect to

Use of a power law spectrum for the temporal fluctuations is justified
by observations which provide the turbulent electromagnetic fluctuations. The
relation to any kind of Kolmogorov spectra

Observations in the solar wind suggest that power laws are realised in the spectral energy densities only over a number of ranges of very limited extension in frequency. Observed spectra contain break points which connect spectral ranges exhibiting different power laws. They also contain more or less well expressed energy injection as well as dissipation ranges of various shapes ranging from exponential decay to exponentials of more complicated arguments or even algebraic decays. Hence, assuming a simple power law in the turbulent-convolution function and extending the range of integration over the entire frequency interval from zero frequency to infinity somehow violates the observational input. The solution for a power law turbulent-convolution function can therefore serve only as an example and has little to do with reality.

In fact, if anyone wants to apply the inverse procedure to power
spectral energy densities, they must refer to Poynting's theorem in
electrodynamics, i.e. use the heat transfer equation for the electromagnetic
field. This equation is nonlinear containing the product

Doing justice to the observations necessarily implies a numerical treatment of the inverse problem of magnetic turbulence. It moreover requires, in addition, the measurement of both the electric and magnetic fluctuation time series and subsequent determination of their spectral equivalents.

The possibility of an application of the inverse problem of scattering to turbulence and fluctuations has not been obvious. It required the transformation of the electromagnetic turbulent fluctuation problem into the Sturm–Liouville–Schrödinger form. This is an interesting turn that might bring a new view on magnetic fluctuation and turbulence theory, possibly opening a path to infer the properties under which magnetic fluctuations in a broad spectral range in plasma develop.

Solving the new integral equation for a given data set is, it seems, not an
easy task. Though the solution of the Gelfand–Levitan–Marchenko equation
should not provide unsurmountable hurdles, it is not particularly simple. The
way to solve it starts from the assumption of any reasonable initial solution
for the unknown function

The general method is not restricted to a simple power law in the
fluctuations just covering the limited inertial range as used in the above
last example. It depends on the precision of the time resolution of the
observations and the related spectral representation of the turbulent
magnetic fluctuations. This spectrum may well extend from the lowest
injection frequencies to deep into the dissipation range

Otherwise, all magnetically active dynamical contributions to the evolution
of turbulence will contribute and are formally included in the theory in the
definition of the response function. Moreover, since the theory is based on
temporal spectra as ingredients of the observational function

The only restrictions on the validity of the Gelfand–Levitan–Marchenko approach in its form of making it available to magnetic turbulent fluctuations are the simplifying assumptions made by us of one-dimensionality, homogeneity of turbulence, restriction to non-expanding turbulence, the uncertainty of observations, and thus the incompleteness of coverage of the frequency domain. Some of these assumptions may prevent application to fast-expanding plasma streams like the solar wind where the observations are performed in one particular spatial point which is about fixed to space and not to the stream, thus violating one of our assumptions.

Though the relation between the observations in our one-dimensional approach and the inverse theory is striking, it is obvious from the last expression that even the solution of the Fourier integral, the input to the kernel of the Gelfand–Levitan–Marchenko equation, cannot be provided in a sufficiently simple form that would allow for an analytical solution. This does not prevent application of the theory; it however suggests that any application to observations requires subsequent numerical treatment of the inverse problem. Whether or not this will be advantageous in investigating turbulence is difficult to estimate. The effort in formulating and solving the inverse problem is large. Its outcome is the maximum available information about the turbulent response function at maximum effect of the fluctuating fields. This function will subsequently have to be interpreted in view of the physical conditions under which magnetic fluctuations have evolved in the turbulence. A reformulation of this theory to the inclusion of the turbulent power spectral densities on which current investigations of magnetic turbulence rely is, however, currently not in site. Its formulation would require reference to Poynting's theorem in turbulence and an attempt to transform it into Schrödinger form, which probably cannot be done.

Here, for completeness and since the algebra is nontrivial, we provide the
transformation of the induction equation

In order to finally, after solving the inverse problem, recover the spatial
coordinate

This work was undertaken as part of a Visiting Scientist Program at the International Space Science Institute Bern, Switzerland. We acknowledge the interest of the ISSI Directorate and thank the ISSI staff for their support, in particular Saliba Saliba for technical help and the librarians Andrea Fischer and Irmela Schweizer for access to the literature. We also thank the anonymous referees for their constructive comments.The topical editor, E. Roussos, thanks R. Schlickeiser and one anonymous referee for help in evaluating this paper.