A note on the dipole coordinates

by, for example, a finite difference method in the computational (μ, χ, φ) space. The above standard dipole coordinates is convenient and certainly appropriate for analytical studies in which the Earth’s dipolar field plays central roles. It also works as a base coordinates for the node and cell generation of the finite element method in the dipole geometry [2, 3]. However, when one tries to use other numerical methods in which analytical expression of the metric terms are important for preserving numerical simplicity and accuracy, as in the case of the finite difference method, the standard dipole coordinate (μ, χ, φ) cannot be used in its original form since the metric hμ changes intensely along the field lines. It should be noted that hμ ∝ |Bd| from the above definitions, which means that hμ is roughly proportional to r. Therefore, the metric hμ at r = 1 is O(10 ) smaller than that at r = 10. Fig. 1(a) shows the hμ profile along a field line starting from 70N as a function of μ. (We suppose that the north pole is located in θ = 0 in this note.) This field line goes through the equator (μ = 0) at r = 8.55. Note the sharp peak in Fig. 1(a) at the equator. When one uses the finite central difference method, the grid spacing along the field line is given by ∆sμ = hμ∆μ. Fig. 2 shows grid point distribution in the standard dipole coordinates. The grid size in the figure isNμ×Nχ = 101×10. (101 grids along each field line and 10 grids in the perpendicular direction.) The starting points of the field lines are