  In this article we consider the multi-layer shallow water system for the
propagation of gravity waves in density-stratified flows, with additional terms
introduced by the oceanographers Gent and McWilliams in order to take into
account large-scale isopycnal diffusivity induced by small-scale unresolved
eddies. We establish a bridge between the multi-layer shallow water system and
the corresponding system for continuously stratified flows, that is the
incompressible Euler equations with eddy-induced diffusivity under the
hydrostatic approximation.
  Specifically we prove that, under an assumption of stable stratification,
sufficiently regular solutions to the incompressible Euler equations can be
approximated by solutions to multi-layer shallow water systems as the number of
layers, $N$, 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-layer
system which relies on suitable energy estimates mimicking the ones recently
established 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.
