Theory

Considering the schematic illustration below, forward- and backward tracking can be formulated respectively as:

\[\begin{split}\begin{align} s_{t+1}=s_t & + F_{we,w}\ast s_{t,w} - F_{we,e}\ast s_t + F_{ew,e}\ast s_{t,e} - F_{ew,w}\ast s_t \\ & + F_{ns,n}\ast s_{t,n} - F_{ns,s}\ast s_t + F_{sn,s}\ast s_{t,s} - F_{sn,n}\ast s_t \\ & + F_{ul,u}\ast s_{t,u} - F_{ul,l}\ast s_t + F_{lu,l}\ast s_{t,l} - F_{lu,u}\ast s_t \\ & + E - P \end{align}\end{split}\]
\[\begin{split}\begin{align} s_{t-1}=s_t & + F_{we,e}\ast s_{t,e} - F_{we,w}\ast s_t + F_{ew,w}\ast s_{t,w} - F_{ew,e}\ast s_t \\ & + F_{ns,s}\ast s_{t,s} - F_{ns,n}\ast s_t + F_{sn,n}\ast s_{t,n} - F_{sn,s}\ast s_t \\ & + F_{ul,l}\ast s_{t,l} - F_{ul,u}\ast s_t + F_{lu,u}\ast s_{t,u} - F_{lu,l}\ast s_t \\ & - E + P \end{align}\end{split}\]

where \(F\) are (total) moisture fluxes and \(s\) represents the amount of tracked relative to total moisture in the grid cells. The vertical fluxes, denoted by underscript \(u\) and \(l\), are not illustrated, but they follow the same systematic as the horizontal fluxes. Note that all fluxes are by definition positive; this is needed because moisture flux is scaled with the relative amount of tracked moisture in the grid cells where it originates from.

Illustration of discretization scheme

In WAM2Layers code, these equations are solved for one upper and one lower layer (such that only two of the 4 vertical transport terms are relevant for each layer). The evaporation term is used for the lower layer only, while the precipitation contribution is distributed across the two layers.