Advanced Ocean Modelling: Using Open-Source Software

154 5 3D Level ModellingFig. 5.24 Definition of the distance δx ∗ .Thecircle indicates the float location. The subscript “i”refers to the start location of the floatThe displacement equation is now given by:dx ∗dt= u(x ∗ ) (5.19)With the abbreviation u ′ x = (u e − u w )(Δx), Eqs. (5.18) and (5.19) can be combinedand integrated to yield:∫ t+Δttdt =∫ eidx ∗u(x ∗ )where the integral boundaries “i” and “e” refer to the start and end locations of afloat. This integral gives:(u ′ x Δt = ln uw + δxe ∗ )u′ xu w + δxi ∗u′ xFinally, the distance traveled by the float over a time span of Δt can then becalculated from:Δx ∗ = x ∗ e − x ∗ i =(uwu ′ x+ δx ∗ i) [exp(u′x Δt) − 1 ] (5.20)Notice that this scheme collapses when u ′ x approaches zero. In this case, thesimple averaging method must be used instead. Errors arise when a float crosses acell boundary during a time step. This can be accommodated in the code by splittingthe method into parts, whereby the first sub-time step is based on the time is takesfor a float to approach the eastern or western cell face, and the remaining time stepcontinues with the adjacent scalar grid point as a new cell centre. To this end, the