Energetics of lateral eddy diffusion/advection：Part Ⅲ. Energetics of horizontal and isopycnal diffusion/advection

Gravitational Potential Energy (GPE) change due to horizontal/isopycnal eddy diffusion and advection is examined. Horizontal/isopycnal eddy diffusion is conceptually separated into two steps: stirring and subscale diffusion. GPE changes associated with these two steps are analyzed. In addition, GPE changes due to stirring and subscale diffusion associated with horizontal/isopycnal advection in the Eulerian coordinates are analyzed. These formulae are applied to the SODA data for the world oceans. Our analysis indicates that horizontal/isopycnal advection in Eulerian coordinates can introduce large artificial diffusion in the model. It is shown that GPE source/sink in isopycnal coordinates is closely linked to physical property distribution, such as temperature, salinity and velocity. In comparison with z-coordinates, GPE source/sink due to stirring/cabbeling associated with isopycnal diffusion/advection is much smaller. Although isopycnal coordinates may be a better choice in terms of handling lateral diffusion, advection terms in the traditional Eulerian coordinates can produce artificial source of GPE due to cabbeling associated with advection. Reducing such numerical errors remains a grand challenge.