Scale-Dependent Eddy Diffusivities at the Kuroshio Extension: A Particle-Based Estimate and Comparison to Theory

For eddy-permitting climate models, only eddies smaller than the smallest resolvable scale need to be parameterized. Therefore, it is important to study the diffusivities induced by eddies smaller than a specific separation scale L`, that is, the scale-dependent eddy diffusivities. Using a submesoscale-permitting model solution (MITgcm llc4320), we estimate the scale-dependent eddy diffusivity in the Kuroshio Extension. We find that, as the separation scale L` increases, the diffusivity increases, and the spatial structure approaches that of the total eddy diffusivity. We quantify this scale dependence through fitting the diffusivity to L`n. Our derivation shows that n is approximately (a 1 1)/2, where a is the eddy kinetic energy spectral slope. For domain-averaged diffusivity, n is 1.33. We then extend four existing mixing theories by including scale dependence. Our results show that both of the theories designed for intense-jet regions, the suppressed mixing length theory and the multiwavenumber theory, closely match the magnitude of the scale-dependent diffusivity but fail to capture well the diffusivity's spatial structure. However, the other two theories based on eddy size and Rhines scale can reasonably represent the spatial structure. Based on this finding, we propose an empirical formula for scale-dependent eddy diffusivity that well represents both the magnitude and the spatial structure of the eddy diffusivity. Our work demonstrates that climate models should use scale-dependent diffusivity, and designing appropriate empirical formulas may be a reasonable approach to represent these scale-dependent diffusivities. Also, our diagnostic framework and theories for scale-dependent eddy diffusivity may be applicable to the global ocean.