For the specific evaluation of the integral in Eq. (), spherical coordinates (k,φ,θ) in the k space are introduced: 5k=ksin⁡θcos⁡φsin⁡θsin⁡φcos⁡θ=:kek, where the corresponding unit vectors are given by ek=sin⁡θcos⁡φex+sin⁡θsin⁡φey+cos⁡θez,eφ=-sin⁡φex+cos⁡φey,eθ=cos⁡θcos⁡φex+cos⁡θsin⁡φey-sin⁡θez.

The vectors ex, ey and ez denote the unit vectors of the Cartesian coordinate system.

In this case, the magnetic field in Eq. () can be rewritten as 6Bx,ω0=∫B^0k,ω0eik⋅xd3k=∫B^0k,θ,φ,ω0eikek⋅xd3k=∫0∞∫02π∫0πB^0k,θ,φ,ω0eikek⋅xk2sin⁡θdθdφdk=∫0∞∫02π∫0πB^0k,θ,φ,ω0eikek⋅xsin⁡θdθdφk2dk=∫0∞b̃k,ω0,xk2dk, where 7b̃k,ω0,x=∫02π∫0πB^0k,θ,φ,ω0eikek⋅xsin⁡θdθdφ is the spectral amount of the field corresponding to k.

Due to Maxwell's equations, the magnetic field (as well as the electric field in the absence of free charge carriers) is solenoidal, 8∂x⋅B(x,t)=0, such that 9B^0k,θ,φ,ω0⋅ek=0. To guarantee the solenoidality of the magnetic field, the ansatz 10B^0k,θ,φ,ω0=αk,φ,θ,ω0eφ+βk,φ,θ,ω0eθ is chosen, which results in 11b̃k,ω0,x=∫02π∫0π[αk,φ,θ,ω0eφ+βk,φ,θ,ω0eθ]eikek⋅xsin⁡θdθdφ and 12Bx,ω0=∫0∞∫02π∫0π[αk,φ,θ,ω0eφ+βk,φ,θ,ω0eθ]eikek⋅xsin⁡θdθdφk2dk, where α(k,φ,θ,ω0) and β(k,φ,θ,ω0) are complex functions of (k,φ,θ) and ω0. In the following this ansatz is specified by constraining the geometry of the three-dimensional k space.

Equation () represents the spectral amplitude of the magnetic field for a fixed value of k. Thus, it is useful to separate the angular dependency (φ,θ) of the spectral amplitude from the k dependency by choosing the functions α and β as 13αk,φ,θ,ω0=α′φ,θ,ω0B02πK(k)k2,14βk,φ,θ,ω0=β′φ,θ,ω0B02πK(k)k2, where K(k) is a function of k alone, α′(φ,θ,ω0) and β′(φ,θ,ω0) are complex functions of (φ,θ,ω0), and B0 is a real constant.

In this respect, the spectral amplitude (Eq. ) can be rewritten as 15b̃k,ω0,x=B02π∫02π∫0π[α′(φ,θ)eφ+β′(φ,θ)eθ]eikek⋅xsin⁡θdθdφK(k)k2, where the functions α′(φ,θ) and β′(φ,θ) weight the summation over the k space with respect to the angulars φ and θ. Introducing the abbreviation 16B̃k,ω0,x=B02π∫02π∫0π[α′(φ,θ)eφ+β′(φ,θ)eθ]eikek⋅xsin⁡θdθdφ provides 17b̃k,ω0,x=B̃k,ω0,xK(k)k2 such that 18Bx,ω0=∫0∞B̃k,ω0,xK(k)dk. In the following, the functions α′(φ,θ) and β′(φ,θ) are specified to evaluate the spectral amplitude B̃(k,ω0,x) with regard to electromagnetic knots .

Each spectral amount (corresponding to a fixed k value) of the field may be characterized by a superposition of plane waves with the same amplitude propagating in every direction (independent of φ and θ) such that 19α′(φ,θ)=α0′=const.∈C,β′(φ,θ)=β0′=const.∈C. In this case, the spectral amplitude results in 20B̃(k,ω0,x)=B02π∫02π∫0πα0′eφ+β0′eθeikek⋅xsin⁡θdθdφ, representing a superposition of infinitely many plane waves of the same amplitude with the spectrum 21Sk=k∈R3|kx2+ky2+kz2=k2. Therefore, the distribution in k space is completely characterized by the value k.

Using the definitions of the unit vectors eφ and eθ, the magnetic field can be further expanded into the form 22B̃(k,ω0,x)=B02π{α0′∫02π∫0π[-sin⁡φex+cos⁡φey]eikek⋅xsin⁡θdθdφ+β0′∫02π∫0π[cos⁡θcos⁡φex+cos⁡θsin⁡φey-sin⁡θez]eikek⋅xsin⁡θdθdφ}=B02π{ex∫02π∫0π(-α0′sin⁡φ+β0′cos⁡θcos⁡φ)eikek⋅xsin⁡θdθdφ+ey∫02π∫0π(α0′cos⁡φ+β0′cos⁡θsin⁡φ)eikek⋅xsin⁡θdθdφ-β0′ez∫02π∫0πsin⁡2θeikek⋅xdθdφ}. For the evaluation of the integrals in Eq. () it is useful to introduce a cylindrical coordinate system (ρ,ϕ,z) in the position space: 23x=ρcos⁡ϕρsin⁡ϕz=ρeρ+zez, where ρ=x2+y2. The corresponding unit vectors are given by eρ=cos⁡ϕex+sin⁡ϕey,eϕ=-sin⁡ϕex+cos⁡ϕey,ez=ez. The scalar product of the k vector and the position vector results in 24ek⋅x=ρcos⁡ϕcos⁡φsin⁡θ+ρsin⁡ϕsin⁡φsin⁡θ+zcos⁡θ=ρsin⁡θ(cos⁡ϕcos⁡φ+sin⁡ϕsin⁡φ)+zcos⁡θ. Using x=ρcos⁡ϕ and y=ρsin⁡ϕ provides 25ek⋅x=sin⁡θ(xcos⁡φ+ysin⁡φ)+zcos⁡θ. For the further evaluation of the integrals in each component of Eq. (), the abbreviations η1(θ):=kxsin⁡θandη2(θ):=kysin⁡θ are introduced. By means of these preparations, the φ integration can be solved analytically, delivering the Bessel functions of the first kind: Jn(x)=12π∫-ππei(nτ-xsin⁡τ)dτ. The detailed evaluation of the integrals can be found in the Appendix, resulting in 26B̃k,ω0,x=ReB0iα0′feϕ+β0′geρ+hez and 27B(x,ω0)=∫0∞B̃k,ω0,xK(k)dk=∫0∞Re{B0[iα0′f(x,k)eϕ+β0′g(x,k)eρ+h(x,k)ez]}K(k)dk, where f(x,k):=∫0πsin⁡θcos⁡(kzcos⁡θ)J1(kρsin⁡θ)dθg(x,k):=-∫0πsin⁡θcos⁡θsin⁡(kzcos⁡θ)J1(kρsin⁡θ)dθ and h(x,k):=-∫0πsin⁡2θcos⁡(kzcos⁡θ)J0(kρsin⁡θ)dθ. The complex constants α0′ and β0′ are the free parameters of the magnetic field in Eq. () and can be chosen independently of each other. The first part of the field, 28B̃rk,ω0,x:=ReB0iα0′feϕorBrx,ω0=∫0∞ReB0iα0′feϕK(k)dk, that corresponds to the expansion coefficient α0′ is called the magnetic ring (see Fig. a). The second part, 29B̃g(k,ω0,x):=ReB0β0′geρ+hezorBg(x,ω0)=∫0∞ReB0β0′geρ+hezK(k)dk, corresponding to the expansion coefficient β0′, is the magnetic globule (see Fig. b).

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 α′(φ,θ) and β′(φ,θ) in Eq. () control the angular dependency in the k space. By choosing α′(φ,θ)=const. and β′(φ,θ)=const., infinitely many plane waves propagating in every direction contribute to the field. The resulting field structures are solenoidal and spatially localized. Thus, the magnetic ring and the magnetic globule can be interpreted as a set of basis functions for isotropically localized, divergence-free structures. Choosing different shapes for the functions α′(φ,θ) and β′(φ,θ) enables the modeling of structures beyond electromagnetic knots.

The electric field and the magnetic field are connected via Ampère's law. Under the absence of ohmic currents, Ampère's law reduces to 30∂x×B(x,t)=1cph2∂tE(x,t), where cph is the phase velocity. Fourier transformation provides 31ik×B(x,ω)=-iωcph2E(x,ω). Using k=kek yields 32ek×B(x,ω)=-ωkcph2E(x,ω) such that 33E(x,ω)=-kcph2ωek×B(x,ω). Ampère's law is valid for every k vector that contributes to the spectrum of the field, yielding the ansatz 34Ẽ(k,ω,x)=-kcph2ωek×B̃(k,ω,x). Using ek×eφ=-eθandek×eθ=eφ delivers 35Ẽ(k,ω,x)=-kcph2ωB02π∫02π∫0π[β0′eφ-α0′eθ]eikek⋅xsin⁡θdθdφ such that the real part can be expressed as 36Ẽ(k,ω,x)=-kcph2ωRe{B0[iβ0′f(x,k)eϕ-α0′g(x,k)eρ+h(x,k)ez]}. Thus, the electric field is given by 37E(x,ω0)=-∫0∞kcph2ω0Re{B0[iβ0′f(x,k)eϕ-α0′g(x,k)eρ+h(x,k)ez]}K(k)dk.

When ohmic currents j(x,t)≠0 are present, Ampère's law can be written as 38∂x×B(x,t)=μ0j(x,t) under the assumption of stationarity or if the displacement current is negligible. Again, Fourier transformation provides 39ik×B(x,ω)=μ0j(x,ω) such that 40j̃(k,ω,x)=ikμ0ek×B̃(k,ω,x). In analogy to the electric field, the current density can be calculated via 41j(x,ω0)=∫0∞kμ0Re{iB0[iβ0′f(x,k)eϕ-α0′g(x,k)eρ+h(x,k)ez]}K(k)dk. Thus, the current density of the magnetic ring follows the topology of a globule and vice versa.

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 (ρ,ϕ,z). The resulting structures are also known as (electro)magnetic disturbances of the first kind . However, in general the structures can be defined with respect to different (local) coordinate systems, spanned by the local unit vectors (eρq,eϕq,ezq), where q=1,…,Q, with different origins Oq. The resulting structures, 42Bx,ω0=∫0∞Re{B0∑q=1Q[iα0q′fxq,keϕq+β0q′(gxq,keρq+h(xq,k)ezq)]}K(k)dk, where x=Oq+xq,xq=ρqeρq+zqezq, are a superposition of Q translated and/or rotated (electro)magnetic disturbances of the first kind (see Fig. c) and are also called (electro)magnetic disturbances of the second kind . The field is characterized by 8Q free parameters, i.e., the expansion coefficients α0q′ and β0q′, the origins Oq, and the orientation of the local coordinate system that can be described, for example, via Euler angles .

Within the derivation presented above, the spectral distribution of the field with respect to k is controlled by the function K(k). Electromagnetic knots, as originally described by , are superpositions of infinitely many monochromatic plane waves, i.e., K(k)=δ(k-k0), with the same amplitude, propagating in every direction with the spectrum 43Sk0=k∈R3|kx2+ky2+kz2=k02. In contrast to single plane waves, knots are localized structures, similar to wave packages. The localization of the structures results from the spatial distribution of the wave phases: 44F(θ,φ):=ek⋅x=sin⁡θxcos⁡φ+ysin⁡φ+zcos⁡θ. Thus, the knots are a superposition of plane waves with different phases F(θ,φ) at all points in space despite its central point. At the origin of the structure (x=y=z=0) the phases of the waves are all equal: F(θ,φ)=0, resulting in a constructive interference with a maximum amplitude at the central point. The scale size of the knot is determined by k0, representing a set of infinitely many k vectors with the same length. The superposition of the plane waves is schematically illustrated in Fig. .

Equation () represents the magnetic field with respect to the position vector x an the frequency ω0. However, the spatial structure of the field can also directly be analyzed from the measurement data B(x,t) evaluated at different time steps t, and thus no Fourier transform with respect to time is required.

After performing the temporal Fourier transform, the magnetic field (Eq. ), measured at the position xi, i=1,…,N, can be rewritten as 45Bxi,ω0=∫0∞B̃(k,ω0,xi)K(k)dk=∫0∞ReH(k)B^0(k,ω0)iα0′B^0(k,ω0)β0′K(k)dk, where 46Hi(k)=-f(xi,k)sin⁡ϕg(xi,k)cos⁡ϕf(xi,k)cos⁡ϕg(xi,k)sin⁡ϕ0h(xi,k) is the corresponding shape matrix of the position xi. Summarizing the measurements into a 3N-dimensional vector B(ω0), the magnetic field can be rearranged as 47B(ω0)=∫0∞ReH(k)B^0(k,ω0)iα0′B^0(k,ω0)β0′K(k)dk, where 48H(k):=H1(k)⋮HN(k)∈R3N×2. Again, the determination of the amplitudes α0′B^0(k,ω0) and β0′B^0(k,ω0) results in an overdetermined inversion problem. In analogy to the classical wave telescope technique, the spectrum of the ring can be estimated via 49P(k)=trH†M-1H-1. Since P(k) is a nonlinear function of k, the whole k space has to be scanned to estimate the spectrum of the field .

Solely considering the magnetic ring (Eq. ), the shape matrix transfers onto the shape vector 50hr(k):=f(x,k)eϕ=-f(x,k)sin⁡ϕf(x,k)cos⁡ϕ0. In this case, the spectrum of the ring can be estimated via 51Pr(k)=1hr†(k)M-1hr(k).

For the first application of electromagnetic knots in the context of magnetospheric structures, we consider the modeling of plasmoids via a magnetic ring . Plasmoids are a consequence of magnetic reconnection in the far-tail region of a planetary magnetosphere triggered by the Dungey cycle e.g.,. The structures are characterized by a magnetic ring along the neutral sheet line with a length scale of the order of the solar wind's obstacle e.g.,.

We model the magnetic field in the tail region by superposing a stationary magnetic ring (α0′=-i, Eq. ), 52Br(x,ω0=0)=∫0∞ReB0iα0′f(x,k)eϕδ(k-k0)dk=ReB0iα0′f(x,k0)eϕ=B0hr(k0), composed of monochromatic plane waves, representing the plasmoid, with the field generated by the neutral sheet current (Harris neutral sheet, ) such that 53B(x)=Brx,ω0=0-Bstanh⁡yLex=B0hrx,k0-Bstanh⁡yLex, where the x axis points towards the night side magnetosphere, the y axis points from the southern geographic pole to the northern geographic pole and the z axis completes the right-handed system. Thus, we model the plasmoid as a two-dimensional structure in the x–y plane . The value B0 represents an arbitrarily chosen background amplitude, Bs=0.3B0, and the length scale of the current sheet is chosen to be L=10-3RE, where RE is the planetary radius, e.g., the terrestrial radius. The characteristic length scale of the plasmoid is chosen to be λ0=1.5RE, corresponding to k0=2π/λ0≈4.19RE-1.

The resulting magnetic field data are evaluated at N=7 synthetically generated spacecraft positions, representing a HelioSwarm-like configuration . As plasmoids are highly dynamical, traveling structures, the measurement positions are shifted along the x axis with respect to the origin of the plasmoid (left, mean, right), representing different time steps. The length scale λ0 (or equivalently k0) of the plasmoid is estimated from the virtual spacecraft data via Eq. (). The resulting field geometry (blue arrows) and the measurement positions (red dots) as well as the corresponding spectra are illustrated in Fig. .

When the measurement positions are distributed around the origin of the plasmoid (mean), the implemented value of k0 can be reconstructed with high precision from the data. In the other cases, the spatial length scale is slightly overestimated and the relative error results in about 6% (left) and 4% (right). Thus, the wave telescope technique is capable of (1) separating the plasmoid from the neutral sheet part and (2) estimating the characteristic length scale of the plasmoid from a limited number of measurement positions.

In analogy to the classical wave telescope technique, the accuracy of the reconstruction depends of the relation between the plasmoid's length scale λ0 and the mean distance d between the spacecraft positions e.g.,. For example, if d≪λ0, the measurement positions do not properly cover the spatial extend of the plasmoid, resulting in ambiguities within the reconstruction procedure. In the case of d≫λ0, the magnetic field structure of the plasmoid is not detectable. Thus, the mean distance between the spacecraft positions has to be of the order of the plasmoid's spatial scale d∼λ0, which will be realized by the configuration of the planned HelioSwarm multiscale mission.

Furthermore, the amplitude of the ring B0 has to be of the same order as or larger than the sheath field Bs to guarantee a precise reconstruction result. For example, in the case of Bs=10B0 no peak occurs within the spectrum Pr(k) and the ring cannot be discerned from the background field. On the other hand, the peak within the spectrum becomes sharper in the case of Bs=0.1B0.

The application presented above of electromagnetic knots indicates the potential of the representation. Spatially distributed electromagnetic knots as described by enable the modeling of more complex structures, provide generalized spectral information and open the door for further applications, delivering an alternative interpretation of magnetospheric structures. For example, the magnetic field configuration resulting from a field-aligned current can be modeled as a superposition of magnetic rings stacked on top of each other. Due to Ampère's law, the corresponding current density is given as a superposition of globules. Thus, the inner structure of field-aligned currents can be analyzed directly from the magnetic field measurements . Also, the current system of Alfvén wings can be described as a superposition of rings e.g., so that the corresponding magnetic field topology follows the structure of superposed globules. Furthermore, field line resonances may be described as a special superposition of magnetic rings.