Axisymmetric Solutions in the Geomagnetic Direction Problem

Abstract

The magnetic field outside the earth is in good approximation a harmonic vector field determined by its values at the earth’s surface. The direction problem seeks to determine harmonic vector fields vanishing at infinity and with the prescribed direction of the field vector at the surface. In general this type of data neither guarantees the existence nor the uniqueness of solutions of the corresponding nonlinear boundary value problem. To determine conditions for existence, to specify the non-uniqueness and to identify cases of uniqueness is of particular interest when modeling the earth’s (or any other celestial body’s) magnetic field from these data. Here we consider the case of axisymmetric harmonic fields mathbf{B} outside the sphere S^2 subset {{mathbb {R}}}^3. We introduce a rotation number {r!o}in {{mathbb {Z}}} along a meridian of S^2 for any axisymmetric Hölder continuous direction field mathbf{D}ne 0 on S^2 and, moreover, the (exact) decay order 3 le delta in {{mathbb {Z}}} of any axisymmetric harmonic field mathbf{B} at infinity. Fixing a meridional plane and in this plane {r!o}- delta +1 geqq 0 points z_n (symmetric with respect to the symmetry axis and with |z_n| > 1, n = 1,ldots ,{r!o}-delta +1), we prove the existence of an (up to a positive constant factor) unique harmonic field mathbf{B} vanishing at z_n and nowhere else, with decay order delta at infinity, and with direction mathbf{D} at S^2. The proof is based on the global solution of a nonlinear elliptic boundary value problem, which arises from a complex analytic ansatz for the axisymmetric harmonic field in the meridional plane. The coefficients of the elliptic equation are discontinuous and singular at the symmetry axis, and this requires solution techniques that are adapted to this special situation.