The extraction of the magnetic signal induced by the oceanic M2 tide is typically based solely on the temporal periodicity of the signal. Here, we propose a system of tailored trial functions that additionally takes the spatial constraint into account that the sources of the signal are localized within the oceans. This construction requires knowledge of the underlying conductivity model but not of the inducing tidal current velocity. Approximations of existing tidal magnetic field models with these trial functions and comparisons with approximations based on other localized and nonlocalized trial functions are illustrated.

Conductive seawater moving through the ambient Earth's main magnetic field

In this short paper, we want to illustrate the effects that different (spatially localized) sets of trial functions can have on the approximation of the magnetic field induced by the M2 tide in the first place.

In particular, we describe a possible setup for the inclusion of spatial
localization constraints (in addition to the constraint of temporal
periodicity) for the approximation of ocean-tide-induced magnetic fields.
Clearly, the velocity field

In Sect.

The kernel

Given a so-called dictionary

In our case, we use the norm

In the following, we briefly introduce some function systems that can be used
for the constitution of

Absolute value of the vectorial Slepian function

We briefly recapitulate the notion of classical vector spherical harmonics in
a form that we need at a few occasions later on. By

The involved associated Legendre functions are, for

Going over to the vectorial setting, it is well known that every
square-integrable vector field

The vector spherical harmonics from above are defined solely on the unit
sphere and can, therefore, only be used for the expansion of vector-valued
functions on

While the set of functions

Absolute value of the trial function

The corresponding vectorial Abel–Poisson kernel is simply defined by

While the localization of Abel–Poisson kernels is of radially symmetric
nature, one is often interested in regions of more complex geometry, e.g.,
continents or oceans. Spherical Slepian functions, for instance, provide an
orthonormal system of functions that can reflect localization in such general
predefined regions

Specifically, the function

Typically, the order is

Accumulated energy for

In typical scenarios, it turns out that the eigenvalues are clustered close
to 1 and close to zero. Those eigenvalues

Absolute value of the radial part of the tidal model

We start with the time-harmonic Maxwell equations as already indicated in
Eq. (

Absolute value of the radial part of approximations of

For the generation of the tailored trial functions

Absolute value of the radial part of approximations of

Figure

Absolute value of the radial part of

For our experiments we rely on the CM5 geomagnetic field model (cf.

Absolute value of the radial part of approximations of superposition

The actual signals that we want to approximate are indicated in
Fig.

The difference between the approximations

Root mean square errors (RMSEs) corresponding to the approximations
of the undisturbed forward model

In the case of

The residuals of the approximations of the forward model

In Fig.

The main goal of this paper is to study the errors that are made by the
approximation of tidal magnetic fields by use of different sets of trial
functions. While, e.g.,

The authors would like to thank the following providers of software and models: Alexey Kuvshinov for the code of the X3DG solver and a processed version of the depth-integrated tidal ocean velocities from TPXO8-ATLAS, Nils Olsen for the coefficients of the M2 tidal contribution in the CM5 model, Alain Plattner for the code for the generation of vectorial Slepian functions.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Dynamics and interaction of processes in the Earth and its space environment: the perspective from low Earth orbiting satellites and beyond”. It is not associated with a conference.

This work was supported by DFG grant GE 2781/1-1 within the Research Priority Program SPP 1788 “DynamicEarth”. Edited by: Juergen Kusche Reviewed by: two anonymous referees