The Semi-Lagrangian Discretization

The height equation may be rewritten as

We discretize () along a Lagrangian trajectory T, which connects the departure point (, t₀), (where in general is not a grid point) to the arrival point (, t), where is always a grid point, deriving

This discretization is centered on the midpoint of T and we have used harmonic averaging in the midpoint evaluation of the first term on the right hand side of () to linearize the equation in the as yet unknown values of the fluid variables at the arrival point. is a short-hand notation for , and is derived from a second-order interpolation procedure , where is the nondimensional separation of the departure point from its nearest grid point , and represents the Courant number for the advection problem. We use a second-order Lax-Wendroff scheme for .

A similar discretization of () centered on the midpoint of T leads to

where , etc., and where is the predicted value of at the arrival point. Note that in the above discretization, the last term which is nonlinear in has been evaluated wholly at the departure point. Also the horizontal rotation terms (the last three terms on the second line of ()) have been represented by for brevity in (). Equations () and () form the basis of our semi-Lagrangian model. We use a centered spatial differencing of the operator in () and () and a second-order accurate interpolation scheme to obtain field values at the departure point at the time level t₀ in those same equations, resulting in a semi-Lagrangian discretization that is second-order accurate in both space and time.