This article is concerned with the rigorous justification of the hydrostatic limit for continuously stratified incompressible fluids under the influence of gravity. The main difference with respect to earlier works is that we neglect viscosity effects but rather include diffusivity effects induced by an advection component proposed by Gent and McWilliams in the 90's to model effective eddy correlations for non-eddy-resolving systems. We also crucially make use of the assumption that the fluid is stably stratified, work in functional spaces of high (but finite) regularity and keep track of the various parameters at stake. A key ingredient of our analysis is the reformulation of our problem using isopycnal coordinates, on which careful energy estimates are performed.