  This article is concerned with the well-posedness of the incompressible Euler
equations describing a stably stratified ocean, reformulated in isopycnal
coordinates. Our motivation for using this reformulation is twofold: first, its
quasi-2D structure renders some parts of the analysis easier. Second, it closes
a gap between the analysis performed in the paper by Bianchini and Duch{\^e}ne
in 2022 in isopycnal coordinates, with shear velocity but with a regularizing
term, and the analysis performed in the paper by Desjardins, Lannes, Saut in
2020 in Eulerian coordinates, without any regularizing term but without shear
velocity. Our main result is an energy estimate in Sobolev spaces on the system
in isopycnal coordinates, with shear velocity, without any regularizing term.
With additional assumptions, it is uniform in the shallow-water parameter. The
main difficulty consists in transposing to the isopycnal reformulation the
symmetric structure of the system which is more straightforward in Eulerian
coordinates.
