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The magnetic field outside the earth is in good approximation a harmonicvector field determined by its values at the earth's surface. The directionproblem seeks to determine harmonic vector fields vanishing at infinity andwith prescribed direction of the field vector at the surface. In general thistype of data does neither guarantee existence nor uniqueness of solutions ofthe corresponding nonlinear boundary value problem. To determine conditions forexistence, to specify the non-uniqueness, and to identify cases of uniquenessis of particular interest when modeling the earth's (or any other celestialbody's) magnetic field from these data.  Here we consider the case of {\em axisymmetric} harmonic fields $\BB$ outsidethe sphere $S^2 \subset \real^3$. We introduce a rotation number $\ro \in\ints$ along a meridian of $S^2$ for any axisymmetric H\"older continuousdirection field $\DD \neq 0$ on $S^2$ and, moreover, the (exact) decay order $3\leq \de \in \ints$ of any axisymmetric harmonic field $\BB$ at infinity.Fixing a meridional plane and in this plane $\ro - \de +1 \geq 0$ points $z_n$(symmetric with respect to the symmetry axis and with $|z_n| > 1$, $n =1,\ldots,\rho-\de +1$), we prove the existence of an (up to a positive constantfactor) unique harmonic field $\BB$ vanishing at $z_n$ and nowhere else, withdecay order $\de$ at infinity, and with direction $\DD$ at $S^2$. The proof isbased on the global solution of a nonlinear elliptic boundary value problem,which arises from a complex analytic ansatz for the axisymmetric harmonic fieldin the meridional plane. The coefficients of the elliptic equation arediscontinuous and singular at the symmetry axis, which requires solutiontechniques that are adapted to this special situation.

Axisymmetric solutions in the geomagnetic direction problem