Quantification of Numerical and Physical Mixing in Coastal Ocean Model Applications

In this study, the method of numerical mixing analysis is presented for three-dimensional ocean models with general vertical coordinates. Numerical mixing of a scalar is defined as the decay of the square of the scalar due to the three-dimensional advection discretisation. It is shown that for any advection scheme the numerical mixing can be calculated as the difference between the advected square of the scalar and the square of the advected tracer, divided by the time step. Special emphasis on directional-split advection schemes is made. It is shown that for those directional-split schemes the numerical analysis method is exact only when the involved advection of the square of the scalar is carried out individually for each split step. As applications, an idealised meso-scale eddy test scenario without any explicit mixing is calculated. It is shown that only for high-order advection schemes for the scalar (salinity in that case) and the momentum, a physically reasonable solution is obtained. Finally, the method is demonstrated for a fully realistic application to the dynamics of the Western Baltic Sea. Here it becomes clear that physical and numerical mixing depend on each others (increased physical mixing leads to decreased numerical mixing) with the dynamically most relevant mixing being the effective mixing, i.e., the sum of the physical and the numerical mixing.