Steady flows drawn from a stably stratified reservoir

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Abstract
Perfect-fluid theory is applied to the description of steady motions that can be generated as the outflow into a horizontal channel from a large reservoir of incompressible heavy fluid whose density is an arbitrary decreasing function of height. A particular aim is to pinpoint the significance of an already known class of flows, called self similar, which satisfy the approximate (shallow-water) equations applicable when the horizontal scale of the motion greatly exceeds its vertical scale, but which have not until now been shown to match the downstream conditions that primarily determine the motion in practice.New variational principles are introduced characterizing the class of self-similar flows: in §2 there is a characterization in terms of flow force among parallel flows realized asymptotically in a uniform channel, in §3 among a wider range of possibilities including periodic flows, and in §6 among supercritical flows realized in a convergent-divergent channel. Aspects of general flows in channels of gradually varying breadth are treated in §§4 and 5, including the remarkable fact, proven in §5, that every steady flow outside but close to the self-similar class must somewhere undergo a local crisis unaccountable by the shallow-water approximation. Practical interpretations afforded by the theoretical results are noted in §7.

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