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- W1551159334 abstract "In this paper we study the behaviour of solutions of Navier-Stokes equations with adherence to the bottom and traction by wind at the surface when the aspect ratio = depth=lenght of the domain tends to 0. Precisely, we prove that when wind is moderate, a variational solution converges to the solution of a vertical diffusion model. This model is either quasistationary or nonstationary, according to the characteristic time. 1. Introduction This paper is devoted to the solutions of Navier-Stokes equations with Coriolis force, traction by wind at the surface and adherence to the bottom in a thin domain. Their behaviour when the aspect ratio = depth=lenght of the domain tends to 0 is investigated. At first we prove the existence of a variational solution for a fixed , without regularity assumptions on the surfaceO. We only assume that the depth h is continuous onO and that the force f due to the wind is such that h 1=2 f2 (L 2 ((0,T )O)) 2 . Next we prove that a conveniently normalized variational solution of Navier-Stokes equations con- verges as ! 0 when Reynolds and Rossby numbers satisfy Re ! 0a nd 2 Re=Ro ! e. These assumptions may be considered satisfied in some lakes and seas when wind is moderate, see (3, Sec- tion 11). The limit problem is quasistationary when the characteristic time allows water renewing, that is at the climatic scale; it is nonstationary at a shorter scale. The limit problem is hydrostatic, that is, the vertical component of the momentum equation reduces to @Z p = 0. In the quasistationary case, the limit potential p = p(x) is uniquely determined by a 2D degenerated elliptic equation on the surface, and the velocity is explicitely known in terms of Z and rxp(x), see (1). The equation on p degenerates on the shore because the depth h vanishes on it. In the nonstationary case a similar reduction of dimension can be performed on a semi-discretizated in time approximation (work in preparation)." @default.
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- W1551159334 date "2000-01-01" @default.
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- W1551159334 title "Nonstationary models for shallow lakes" @default.
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