Comments on the “shear effect” and diffusion in the Lagrangian framework |
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| Authors: | Brian G Sanderson Akira Okubo |
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| Institution: | (1) Department of Physics and Newfoundland Institute of Cold Ocean Science, Memorial University of Newfoundland, A1B 3X7 St John's, Newfoundland, Canada;(2) Marine Sciences Research Center, State University of New York, 11794-5000 Stony Brook, New York, USA |
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| Abstract: | We indentify three different types of Lagrangian coordinate systems that are used in oceanography. These are: true Lagrangian coordinates (TLC), Lagrangian coordinates (LC), and averaged Lagrangian coordinates (ALC). The diffusion process is studied in each of these coordinate systems. At  large scales the eddydiffusivity is proven to be independent of molecular diffusivity, providing the spectrum of turbulent kinetic energy varies as scale raised to a power less than 4 1/3. The shear effect is examined using solutions to the averaged Lagrangian diffusion equation obtained by Okuboet al. (1983). In Eulerian coordinates both advection and diffusion are necessary for the occurrence of the shear effect, while in ALC timedependent dispersion coefficients are necessary for the process. In TLC we use the method of Taylor (1921) to study the dispersion of material by a velocity field, that from the Eulerian perspective, consists of turbulent motion across a uniform shear. The transformation of the above Eulerian velocity field into TLC results in a uniform deformation field and turbulent motion both along and across the shear. This work shows how dispersion of material is related to the turbulent Eulerian velocity and uniform velocity gradients. The instantaneous rate of change of variance of a spreading patch of material is completely specified by the instantaneous divergence obtained over the area occupied by the patch (Kawai, 1976). This relationship is shown to depend upon the fact that at any particular instant it is possible to define TLC that are equivalent to the Eulerian coordinates. In order to describe patch spreading from divergence measured over longer periods it is also necessary to consider other dispersive processes.Contribution number of the Newfoundland Institute of Cold Ocean Science. |
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