Magnetic Curvatures of a Uniformly Magnetized Tesseroid Using the Cartesian Kernels

In recent years, the gravitational curvatures, the third-order derivatives of the gravitational potential (GP), of a tesseroid have been introduced in the context of gravity field modeling. Analogous to the gravity field, magnetic field modeling can be expanded by magnetic curvatures (MC), the third-order derivatives of the magnetic potential (MP), which are the change rates of the magnetic gradient tensor (MGT). Exploiting Poisson's relations between $(n+1)$ th-order derivatives of the GP and nth-order derivatives of the MP, this paper derives expressions for the MC of a uniformly magnetized tesseroid using the fourth-order derivatives of the GP of a uniform tesseroid expressed in terms of the Cartesian kernel functions. Based on the magnetic effects of a uniform spherical shell, all expressions for the MP, magnetic vector (MV), MGT and MC of tesseroids have been examined for numerical problems due to singularity of the respective integral kernels (i.e., near zone and polar singularity problems). For this, the closed analytical expressions for the MP, MV, MGT and MC of the uniform spherical shell have been provided and used to generate singularity-free reference values. Varying both height and latitude of the computation point, it is found numerically that the near zone problem also exists for all magnetic quantities (i.e., MP, MV, MGT and MC). The numerical tests also reveal that the polar singularity problems do not occur for the magnetic quantity as a result of the use of Cartesian as opposed to spherical integral kernels. This demonstrates that the magnetic quantity including the newly derived MC 'inherit' the same numerical properties as the corresponding gravitational functional. Possible future applications (e.g., geophysical information) of the MC formulas of a uniformly magnetized tesseroid could be improved modeling of the Earth's magnetic field by dedicated satellite missions.