| Abstract: | Numerical models of shelf seas must handle sharp gradients in thermoclines and fronts, and at the edges of patches of passive tracer, often in the presence of strong tidal currents, without introducing excessive numerical diffusion or spurious oscillations. In a sigma coordinate model there is an additional problem, since then purely horizontal motion over a sloping sea bed may introduce strong numerical diffusion in the vertical, which can artificially erode thermoclines.In this paper, two advection schemes applicable to shelf sea models are examined. They are based on TVD (Total Variation Diminishing) and PPM (Piecewise Parabolic Method) techniques. They are demonstrated to give satisfactory performance for tracer advection in one, two and three dimensions. They are both monotonic, and the PPM scheme has particularly low numerical diffusion. In two and three dimensions directional splitting is used, which is handled by following advection with each horizontal velocity component by adjustment of the sigma levels in each water column using the same advection scheme. The artificial vertical diffusion is then small, particularly with PPM.The two schemes are also compared in a three-dimensional model of a cylindrical eddy of relatively fresh water, released from rest in an open sea region. Here, both salinity and momentum are advected. Laboratory experiments show that after an initial period of adjustment the eddy should become unstable, with a growth of cyclonic-anticyclonic eddy pairs. This is reproduced in the model, with the PPM scheme again producing sharper results than the TVD scheme.Integrated second moment calculations are used to compare the schemes. These demonstrate the lower numerical diffusion of the PPM scheme. This advantage is achieved at the cost of greater computing time. |
