We consider the Saint-Venant system for shallow water flows, with non-flat bottom. It is a hyperbolic system of conservation laws that approximately describes various geophysical flows, such as rivers, coastal areas, oceans when completed with a Coriolis term, or granular flows when completed with friction. Numerical approximate solutions to this system may be generated using conservative finite volume methods, which are known to properly handle shocks and contact discontinuities. However, in general these schemes are known to be quite inaccurate for near steady states, as the structure of their numerical truncation errors is generally not compatible with exact physical steady state conditions. This difficulty can be overcome by using the so called well-balanced schemes. We describe a general strategy, based on a local hydrostatic reconstruction, that allows to derive a well-balanced scheme from any given numerical flux for the homogeneous problem. Whenever the initial solver satisfies some classical stability properties, it yields a simple and fast well-balanced scheme that preserves the nonnegativity of the water height and satisfies a semi-discrete entropy inequality.

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Atmospheric flows feature length scales from 10−5 to 105 m and timescales from microseconds to weeks or more. For scales above several kilometers and minutes, there is a natural scale separation induced by the atmosphere's thermal stratification, together with the influences of gravity and Earth's rotation, and the fact that atmospheric-flow Mach numbers are typically small. A central aim of theoretical meteorology is to understand the associated scale-specific flow phenomena, such as internal gravity waves, baroclinic instabilities, Rossby waves, cloud formation and moist convection, (anti-)cyclonic weather patterns, hurricanes, and a variety of interacting waves in the tropics. Single-scale asymptotics yields reduced sets of equations that capture the essence of these scale-specific processes. For studies of interactions across scales, techniques of multiple-scales asymptotics have received increasing recognition in recent years. This article recounts the most prominent scales and associated scale-dependent models and summarizes recent multiple-scales developments.

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