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By a semi-Lagrangian change of coordinates, the hydrostatic Euler equationsdescribing free-surface sheared flows is rewritten as a system of quasilinearequations, where stability conditions can be determined by the analysis of itshyperbolic structure. This new system can be written as a quasi linear systemin time and horizontal variables and involves no more vertical derivatives.However, the coefficients in front of the horizontal derivatives include anintegral operator acting on the new vertical variable. The spectrum of theseoperators is studied in detail, in particular it includes a continuous part.Riemann invariants are then determined as conserved quantities along thecharacteristic curves. Examples of solutions are provided, in particularstationary solutions and solutions blowing-up in finite time. Eventually, wepropose an exact multi-layer $\mathbb{P}_0$-discretization, which could be usedto solve numerically this semi-Lagrangian system, and analyze the eigenvaluesof the corresponding discretized operator to investigate the hyperbolic natureof the approximated system.

Hyperbolicity of a semi-Lagrangian formulation of the hydrostatic free-surface Euler system