It is increasingly common for models of shallow-layer overland flows to include equations for the evolution of the underlying bed (morphodynamics) and the motion of an associated sedimentary phase. We investigate the linear stability properties of these systems in considerable generality. Naive formulations of the morphodynamics, featuring exchange of sediment between a well-mixed suspended load and the bed, lead to mathematically ill-posed governing equations. This is traced to a singularity in the linearised system at Froude number $Fr = 1$ that causes unbounded unstable growth of short-wavelength disturbances. The inclusion of neglected physical processes can restore well posedness. Turbulent momentum diffusion (eddy viscosity) and a suitably parametrised bed load sediment transport are shown separately to be sufficient in this regard. However, we demonstrate that such models typically inherit an associated instability that is absent from non-morphodynamic settings. Implications of our analyses are considered for simple generic closures, including a drag law that switches between fluid and granular behaviour, depending on the sediment concentration. Steady morphodynamic flows bifurcate into two states: dilute flows, which are stable at low $Fr$, and concentrated flows which are always unstable to disturbances in concentration. By computing the growth rates of linear modes across a wide region of parameter space, we examine in detail the effects of specific model parameters including the choices of sediment erodibility, eddy viscosity and bed load flux. These analyses may be used to inform the ongoing development of operational models in engineering and geosciences.
A two-layer shallow water model for bedload sediment transport: convergence to Saint-Venant-Exner model
本文提出了一个两层浅水型推移质输沙模型。上层为水充填，下层为泥沙充填。关键点在于两层之间摩擦定律的定义，这是Fernán-Nieto等人（ESAIM：M2 AN，51：115-145，2017）中介绍的摩擦定律的推广。这一定义允许适当地应用两层浅水模式的情况下，强烈和缓慢的推移质泥沙输移。此外，我们证明了两层模型收敛到一个圣维南Exner系统（SVE），包括重力的影响时，流体动力学和形态动力学的时间尺度之间的比例是小的。具有引力效应的SVE是一个退化的非线性抛物型方程组。这意味着从计算的角度来看，它的数值近似是非常昂贵的，例如参见T。Morales de Luna等人（J. Sci.比较器，48（1）：258-273，2011）。在这项工作中，引力效应被引入到两层系统中，而没有这样的额外计算成本。最后，我们还考虑了一个泛化的模型，其中包括一个非静水压力校正的流体层和沉积物表面的边界条件。数值试验表明，该模型提供了有前途的结果，表现良好，在低输运率制度以及在许多其他情况下。
Strongly stable dual-pairing summation by parts finite difference schemes for the vector invariant nonlinear shallow water equations -- I: Numerical scheme and validation on the plane
