Transformed Flux-form Semi-Lagrangian Scheme for Coastal Ocean Modeling

Numerical approaches in atmospheric and oceanic modeling inevitably introduce diffusion (or dissipation) and dispersion into the approximate solution. From a physical point of view, advection of a passive tracer is the simple transition of a quantity. Therefore, dispersion, the propagation of different spatial scales at different phase speed, and diffusion are processes that are aliens to the process that is being modeled (Chu and Fan 1998, 1999). As applied to constituent advection problem, these numerical artifacts manifest themselves as nonphysical mixing by numerical diffusion, nonphysical highs and lows in the constituent field caused by dispersion, and nonphysical tracer spectra caused by the trapping of tracer in nonpropagating small spatial scales (Rood 1987). The less the numerical diffusion and dispersion errors, the better the model performance is. Propagation of a Rossby soliton on an equatorial beta-plane is treated as an asymptotic solution, which exists to the inviscid, nonlinear shallow water equations. In principle, the soliton propagates to the west at fixed phase speed, without change of shape. Since the uniform propagation and shape preservation of the soliton are achieved through a delicate balance between linear wave dynamics and nonlinearity. In other words, the Rossby soliton is non-diffusive and non-dispersive (Boyd 1980), which makes it a perfect test case for verification of numerical schemes in ocean models since any diffusion and dispersion in the numerical solution of the Rossby soliton are computational errors. Interested readers are referred to the website: http://marine.rutgers.edu/po/index.php?model=te st-problems.