By a semi-Lagrangian change of coordinates, the hydrostatic Euler equations describing free-surface sheared flows is rewritten as a system of quasilinear equations, where stability conditions can be determined by the analysis of its hyperbolic structure. This new system can be written as a quasi linear system in time and horizontal variables and involves no more vertical derivatives. However, the coefficients in front of the horizontal derivatives include an integral operator acting on the new vertical variable. The spectrum of these operators is studied in detail, in particular it includes a continuous part. Riemann invariants are then determined as conserved quantities along the characteristic curves. Examples of solutions are provided, in particular stationary solutions and solutions blowing-up in finite time. Eventually, we propose an exact multi-layer $\mathbb{P}_0$-discretization, which could be used to solve numerically this semi-Lagrangian system, and analyze the eigenvalues of the corresponding discretized operator to investigate the hyperbolic nature of the approximated system.

翻译：通过半拉格朗日坐标变换，描述自由表面剪切流的静水欧拉方程被改写为拟线性方程组，其稳定性条件可通过分析双曲结构确定。该新系统可写为时间与水平变量的拟线性系统，且不再涉及垂直导数。然而，水平导数前的系数包含作用于新垂直变量的积分算子。这些算子的谱被详细研究，特别地，其包含连续谱部分。随后，黎曼不变量被确定为沿特征曲线守恒的量。本文提供了解的示例，特别是定常解和有限时间爆破解。最终，我们提出一种精确的多层$\mathbb{P}_0$-离散化格式，可用于该半拉格朗日系统的数值求解，并通过分析相应离散算子的本征值来研究近似系统的双曲性质。