Computation of a Solenoid as a Ratio of Areas and a Geometric Interpretation of the Arakawa Jacobian

Abstract

Abstract A formula for the computation of a solenoidal term, for example, k · (∇ψ × ∇ϒ) or ∂(ψ,ϒ)/∂(x, y) in Jacobian form, from three or more noncollinear stations or grid points is presented. The formula is based on a geometric interpretation of the solenoid as the ratio of an elementary area (expressed as a line integral) in ψ–ϒ dependent-variable or phase space to the corresponding area in the x–y plane. The Arakawa finite-difference Jacobian on a rectangular grid in physical space is shown to be a linear combination of such ratios. Thus the Arakawa method is a line-integral method. The new interpretation readily provides the forms of the Jacobian at boundary points needed to maintain the integral constraints in a closed domain. Elementary properties of Jacobians ensure that the Arakawa Jacobian can be used on a regular mesh in any general orthogonal curvilinear coordinate system, thus permitting the use of “stretched” grids that have a lesser density of grid points away from boundaries and also away ...