Two-layer and multi-layer depth-averaged models have become popular for simulating exchange flows, seawater currents and geophysical flows. The partial differential equation systems associated with these models are similar to the single-layer shallow-water model. Yet, their eigenstructures are more complex owing to the pressure coupling between the layers. Such models occasionally lose their hyperbolic character, which may lead to numerical issues. A physical explanation is that Kelvin-Helmholtz type instabilities arise at the layers' interface, if the velocity difference between the layers becomes sufficiently large. A way to avoid the hyperbolicity loss is to locally introduce an extra momentum exchange between the layers, assessable from the system eigenstructure and aimed at roughly mimicking the dynamical effects of such instabilities. To better understand the hyperbolicity conditions, the eigenstructure of the two-layer model is methodically studied by an asymptotic analysis. The analysis for the limiting cases, where the layers' thicknesses are either comparable or very different from each other, reveals new stability criteria. These analytical criteria are, then, exploited to design a new family of approximate criteria, valid for any flow condition. Numerical investigations demonstrate the reliability of this approach, which can be easily implemented in numerical schemes for preserving the hyperbolicity.

两层和多层深度平均模型已成为模拟交换流、海水流和地球物理流的流行方法。与这些模型相关的偏微分方程系统类似于单层浅水模型。然而，由于层之间的压力耦合，它们的本征结构更加复杂。此类模型有时会失去其双曲线特征，这可能会导致数值问题。物理解释是，如果层之间的速度差变得足够大，则在层的界面处会出现开尔文-亥姆霍兹型不稳定性。避免双曲线损失的一种方法是在层之间局部引入额外的动量交换，可从系统本征结构进行评估，旨在粗略地模拟这种不稳定性的动力学效应。为了更好地理解双曲线条件，通过渐近分析有条不紊地研究了两层模型的特征结构。对极限情况的分析，其中层的厚度彼此相当或非常不同，揭示了新的稳定性标准。然后，利用这些分析标准来设计一系列新的近似标准，适用于任何流动条件。数值研究证明了这种方法的可靠性，它可以很容易地在数值方案中实现，以保持双曲线性。揭示了新的稳定性标准。然后，利用这些分析标准来设计一系列新的近似标准，适用于任何流动条件。数值研究证明了这种方法的可靠性，它可以很容易地在数值方案中实现，以保持双曲线性。揭示了新的稳定性标准。然后，利用这些分析标准来设计一系列新的近似标准，适用于任何流动条件。数值研究证明了这种方法的可靠性，可以很容易地在数值方案中实现以保持双曲线性。