The wave telescope technique is broadly established in the analysis of spacecraft data and serves as a bridge between local measurements and the global picture of spatial structures. The technique is originally based on plane waves and has been extended to spherical waves, phase-shifted waves and planetary magnetic field representation. The goal of the present study is the extension of the wave telescope technique using electromagnetic knot structures as a basis. As the knots are an exact solution of Maxwell's equations they open the door for a new modeling and interpretation of magnetospheric structures, such as plasmoids.

The classification and mathematical modeling of spatial structures are among the major missions of theoretical physics. Our extraterrestrial space environment in particular provides a diversity of spatial structures with different characteristics. For example, oscillating structures can be classified into plane waves (e.g., MHD waves), spherical waves generated at the bow shock, surface waves triggered by instabilities at the magnetopause and phase-shifted waves caused by field line resonances

In general, any spatial structure can be expanded into a set of mathematical basis functions, such as plane waves or spherical harmonics. Plane waves are the simplest spatial structures forming a basis for the representation of spatial fields. The contribution of any plane wave with its characteristic spatial scale to the total field is described by the spectrum of the field. However, in the worst case, infinitely many elements forming the basis have to be incorporated to describe the structure, resulting in an infinite set of expansion coefficients that have to be determined from the measurements. In this case, it is desirable to choose a new representation based on a new set of basis functions that are well-adjusted to the symmetry of the structure with fewer unknown parameters.

Electromagnetic knots, proposed by

Vector representation of the electromagnetic ring

The electromagnetic ring and the electromagnetic globule are an exact solution of Maxwell's equations and provide a new tool in the context of plasma physical and electrodynamical modeling. Based on the elaboration of

Maxwell's equations represent a set of coupled partial differential equations for the magnetic field

The measurement position

Suppose that the magnetic field vector

The classical wave telescope technique does not assume any symmetry or relation between different

For the specific evaluation of the integral in Eq. (

The vectors

In this case, the magnetic field in Eq. (

Due to Maxwell's equations, the magnetic field (as well as the electric field in the absence of free charge carriers) is solenoidal,

Equation (

In this respect, the spectral amplitude (Eq.

Each spectral amount (corresponding to a fixed

Using the definitions of the unit vectors

It should be noted that the electromagnetic knot structures do not form an entire set of mathematical basis functions. Regarding the derivation presented here, electromagnetic knots can be written as a superposition of infinitely many plane waves, as plane waves represent an entire set of basis functions. However, the inverse is not true. The functions

The electric field and the magnetic field are connected via Ampère's law. Under the absence of ohmic currents, Ampère's law reduces to

When ohmic currents

Within the derivation of the knot structures, the magnetic ring and the magnetic globule are defined with respect to the same origin of the cylindrical coordinate system

Within the derivation presented above, the spectral distribution of the field with respect to

Illustration of superposed, monochromatic plane wave fronts (gray lines) with the wave length

Equation (

Following this short derivation and discussion of the electromagnetic knots, the knot model needs to be reformulated in terms of the wave telescope technique to estimate the spectrum of the knots.

After performing the temporal Fourier transform, the magnetic field (Eq.

Solely considering the magnetic ring (Eq.

Reconstructed spectrum

For the first application of electromagnetic knots in the context of magnetospheric structures, we consider the modeling of plasmoids via a magnetic ring

We model the magnetic field in the tail region by superposing a stationary magnetic ring (

The resulting magnetic field data are evaluated at

When the measurement positions are distributed around the origin of the plasmoid (mean), the implemented value of

In analogy to the classical wave telescope technique, the accuracy of the reconstruction depends of the relation between the plasmoid's length scale

Furthermore, the amplitude of the ring

The application presented above of electromagnetic knots indicates the potential of the representation. Spatially distributed electromagnetic knots as described by

Electromagnetic knots are a superposition of infinitely many monochromatic plane waves with a spherical symmetric spectrum and represent an exact solution of Maxwell's equation. The resulting basis elements, i.e., the electromagnetic ring and the globule, form a basis set for localized, divergence-free spatial structures. For this reason, the concept of electromagnetic knots opens the door for a completely new description and interpretation of spatial structures in planetary magnetospheres.

The classification of spatial structures evaluated at a limited number of measurement points describes an overdetermined inversion problem. The wave telescope technique serves as a robust data analysis tool for the global interpretation of spacecraft measurements in terms of expected physical structures. By reformulating the formalism of electromagnetic knots in terms of the wave telescope technique, we extended the zoo of spatial structures that can be analyzed by the method. In this sense, the present study can be interpreted as a generalization of the wave telescope technique to a structure telescope technique.

For a first validation, the concept of electromagnetic knots has been applied to the modeling of a plasmoid. Using a HelioSwarm-like satellite configuration, the wave telescope technique is capable of separating the plasmoid, modeled as a magnetic ring, from the field generated by the neutral sheet current and enables the estimation of the length scale of the ring. Thus, the presented extension of the wave telescope technique serves as a new data analysis tool for multi-spacecraft missions, such as the planned HelioSwarm mission. However, the application of electromagnetic knots for characterizing further structures, such as field-aligned currents or Alfvén wings, should be analyzed in future studies. In general, we conclude that the modified wave telescope technique outlined here bears the potential for a new representation and physical description of complex spatial structures existing in space plasmas.

The

Here, the magnetic field is given by

For the evaluation of the integral

The remaining integrals can be expanded into the form

Real (blue) and imaginary part (orange) of the integrand of

Data can be provided upon request.

ST performed the mathematical derivations. KHG came up with the idea of extending the wave telescope technique. ST and UM wrote the first draft of the manuscript. All authors contributed equally to the conception and design of the study; all authors read and approved the submitted version.

The contact author has declared that none of the authors has any competing interests.

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We are grateful to Norbert Fürstenau (DLR Braunschweig) for pointing our interest to electromagnetic knots. We acknowledge support by the German Research Foundation and the Open Access Publication Fund of the Technische Universität Braunschweig.

This work has been supported by the German Research Foundation and the Open Access Publication Fund of the Technische Universität Braunschweig. The contribution by Karl-Heinz Glassmeier was financially supported by the German Bundesministerium für Wirtschaft und Klimaschutz and the Deutsches Zentrum für Luft- und Raumfahrt under 50OC1803.This open-access publication was funded by Technische Universität Braunschweig.

This paper was edited by Elena Kronberg and reviewed by two anonymous referees.