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In this article we consider the multi-layer shallow water system for thepropagation of gravity waves in density-stratified flows, with additional termsintroduced by the oceanographers Gent and McWilliams in order to take intoaccount large-scale isopycnal diffusivity induced by small-scale unresolvededdies. We establish a bridge between the multi-layer shallow water system andthe corresponding system for continuously stratified flows, that is theincompressible Euler equations with eddy-induced diffusivity under thehydrostatic approximation.  Specifically we prove that, under an assumption of stable stratification,sufficiently regular solutions to the incompressible Euler equations can beapproximated by solutions to multi-layer shallow water systems as the number oflayers, $N$, increases. Moreover, we provide a convergence rate of order$1/N^2$.  A key ingredient in the proof is a stability estimate for the multi-layersystem which relies on suitable energy estimates mimicking the ones recentlyestablished by Bianchini and Duch\^ene on the continuously stratified system.This requires to compile a dictionary that translates continuous operations(differentiation, integration, etc.) into corresponding discrete operations.

Approximating a continuously stratified hydrostatic system by the multi-layer shallow water system