Ocean Science www.ocean-sci.net 1812-0784 1812-0792 2 2 2006 10.5194/os-2-161-2006 http://www.ocean-sci.net/2/161/2006/ http://www.ocean-sci.net/2/161/2006/os-2-161-2006.html http://www.ocean-sci.net/2/161/2006/os-2-161-2006.pdf 161 171 2006-10-12 Energetics of the layer-thickness form drag based on an integral identity H. Aiki aiki@jamstec.go.jp T. Yamagata Frontier Research Center for Global Change, Japan Agency for Marine-Earth Science and Technology, Yokohama-city 236-0001, Japan Department of Earth and Planetary Science, Graduate School of Science, University of Tokyo, Tokyo 113-0033, Japan The vertical redistribution of the geostrophic momentum by the residual effects of pressure perturbations (called the layer-thickness form drag) is investigated using thickness-weighted temporal-averaged mean primitive equations for a continuously stratified fluid in an adiabatic formulation. A four-box energy diagram, in which the mean and eddy kinetic energies are defined by the thickness-weighted mean velocity and the deviation from it, respectively, shows that the layer-thickness form drag reduces the mean kinetic energy and endows the eddy field with an energy cascade. The energy equations are derived using an identity (called the "pile-up rule") between cumulative sums of the Eulerian mean quantity and the thickness-weighted mean quantity in each vertical column. The pile-up rule shows that the thickness-weighted mean velocity satisfies a no-normal-flow boundary condition at the top and bottom of the ocean, which enables the volume budget of pressure flux divergence in the energy diagram to be determined. With the pile-up rule, the total kinetic energy based on the Eulerian mean can be rewritten in a thickness-weighted form. 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