A Finite Element Algorithm of a Nonlinear Diffusive Climate Energy Balance Model

Abstract

We present a finite element algorithm of a climate diagnostic model that takes as a climate indicator the atmospheric sea-level temperature. This model belongs to the category of energy balance models introduced independently by the climatologists M.I. Budyko and W.D. Sellers in 1969 to study the influence of certain geophysical mechanisms on the Earth climate. The energy balance model we are dealing with consists of a two-dimensional nonlinear parabolic problem on the 2-sphere with the albedo terms formulated according to Budyko as a bounded maximal monotone graph in \({\mathbb{R}}^{2}.\) The numerical model combines the first-order Euler implicit time discretization scheme with linear finite elements for space discretization, the latter is carried out for the special case of a spherical Earth and uses quasi-uniform spherical triangles as finite elements. The numerical formulation yields a nonlinear problem that is solved by an iterative procedure. We performed different numerical simulations starting with an initial datum consisting of a monthly average temperature field, calculated from the temperature field obtained from 50 years of simulations, corresponding to the period 1950–2000, carried out by the Atmosphere General Circulation Model HIRLAM.