An Existence Result for the Steady Rotating Prandtl Equation
- PDF / 658,265 Bytes
- 27 Pages / 547.087 x 737.008 pts Page_size
- 115 Downloads / 172 Views
Journal of Mathematical Fluid Mechanics
An Existence Result for the Steady Rotating Prandtl Equation Anne-Laure Dalibard
and Matthew Paddick
Communicated by D. Gerard-Varet
Abstract. We consider a steady, geophysical 2D fluid in a domain, and focus on its western boundary layer, which is formally governed by a variant of the Prandtl equation. By using the von Mises change of variables, we show that this equation is well-posed under the assumptions that the trace of the interior stream function is large, and variations in the coastline profile are moderate. Keywords. Steady Prandtl equation, Geophysical fluids, Boundary layer theory.
1. Introduction The goal of this article is to prove the global existence of solutions of the stationary Prandtl-like equation ⎧ λ0 ν(y)(u∂ξ v + v∂y v) + ψ − ν(y)2 ∂ξ2 v ⎪ ⎪ ⎪ ⎪ (u, v) ⎨ v|y=0 ⎪ ⎪ (u, v) ⎪ |ξ=0 ⎪ ⎩ limξ→+∞ ψ(ξ, y)
= ψ 0 (y) = ∇⊥ ψ := (−∂y ψ, ∂ξ ψ) = v0 (ξ) =0 = ψ 0 (y),
(1)
in the domain ΩY = {(ξ, y) ∈ R2 | ξ > 0, 0 < y < Y }. The function ψ 0 is given and smooth, λ0 is a strictly positive parameter, and the function ν is a smooth function with ν ≥ 1. This equation arises in geophysical models to describe the behaviour of western boundary oceanic currents in certain regimes (see [4,12]). In this setting, the function u (resp. v) is the East-West (resp. North-South) component of the velocity of oceanic currents, the variable y is the latitude, and the variable ξ is the scaled longitude within the boundary layer, see Fig. 1. In other words, the rigid wall is here {ξ = 0}, and corresponds to the western coastline, and v (resp. u) is the velocity in the tangential (resp. normal) direction to the wall. We describe the physical assumptions and scaling leading to (1) in paragraph 1.3 after the statement of our main result, but let us merely mention that the role of the function ν is to take into account the geometry of the western coast. We also refer to [14] for a recent work on the well-posedness on the time-dependent version of this equation. Note that Eq. (1) is similar to the stationary Prandtl equation, namely1
1 Note
that here y is the tangential variable and ξ is the rescaled normal variable, in contrast with the usual convention in the study of the Prandtl equation. We have made this choice to stick to the geophysical setting, in which u is the East-West component of the velocity and v its North-South component. 0123456789().: V,-vol
13
Page 2 of 27
Anne-Laure Dalibard and Matthew Paddick
JMFM
Fig. 1. The domain ΩY
⎧ u∂ξ v + v∂y v − ∂ξ2 v ⎪ ⎪ ⎪ ⎪ ⎨ (u, v) v|y=0 ⎪ ⎪ ⎪ (u, v, ψ) |ξ=0 ⎪ ⎩ limξ→+∞ v(ξ, y)
E = − dp dy ⊥ = ∇ ψ := (−∂y ψ, ∂ξ ψ) = v0 (ξ) =0 = vE (y),
(2)
where the functions vE and pE are given (they are the trace of some outer Euler flow) and satisfy E = − dp vE v E dy . Notice that here we have ν(y) = 1, which corresponds to a flat boundary. The main differences between the usual Prandtl equation (2) and Eq. (1) lie in the presence of the additional term ψ in the equation (which is due to rotation, as we will explain in paragraph 1.3) and in
Data Loading...
