Existence and uniqueness and first order approximation of solutions to atmospheric Ekman flows
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Existence and uniqueness and first order approximation of solutions to atmospheric Ekman flows Michal Feˇckan1,2 · Yi Guan3 · Donal O’Regan4 · JinRong Wang3,5 Received: 12 March 2020 / Accepted: 9 April 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract In this paper, we study the classical problem of the wind in the steady atmospheric Ekman layer with classical boundary conditions and the eddy viscosity is an arbitrary height-dependent function with a finite limit value. We present existence and uniqueness and smooth results to justify computing first order approximation of solutions. Using a different argument that in previous works, we construct the Green’s function to derive the solution by a perturbation approach. Keywords Ekman layer · Variable eddy viscosity · Explicit solutions · Green function Mathematics Subject Classification 34B05
1 Introduction The atmospheric boundary layer has three parts [1,2], i.e., the lamina sublayer, surface (Prandtl) layer and the Ekman layer. The Ekman layer covers 90% of the atmospheric boundary layer and it is driven by a three-way balance among frictional effects, pressure gradient and the influence of the coriolis force [1,3,4]. In general, textbooks on geophysical fluid dynamics and dynamic meteorology contain the derivation for the Ekman layer with a constant eddy viscosity k [5,6]. However k usually varies with
Communicated by Adrian Constantin. This work is partially supported by the National Natural Science Foundation of China (11661016), Training Object of High Level and Innovative Talents of Guizhou Province ((2016)4006), Major Research Project of Innovative Group in Guizhou Education Department ([2018]012), Guizhou Data Driven Modeling Learning and Optimization Innovation Team ([2020]5016), the Slovak Research and Development Agency under the Contract No. APVV-18-0308, and the Slovak Grant Agency VEGA No. 1/0358/20 and No. 2/0127/20.
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JinRong Wang [email protected]
Extended author information available on the last page of the article
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the height. As a result it is necessary to find the explicit solution of Ekman flows with a non-constant eddy viscosity, but unfortunately explicit solutions are scare in the literature and are restricted to linear [7,8] or quadratic and cubic poly-nominal [9]. For arbitrary k(z) or k(z, t), we often have to rely on approximation and numerical simulation; the authors in [10–13] apply the Wentzel, Kramers and Brillouin’s method to get the approximation solution. Constantin and Johnson [14] studied Ekman flows with variable eddy viscosity k(z) and they derived the explicit solution through an unclosed form and verified the existence of the solution by transformation and the iterative technique. The authors in [15] studied the horizontal wind drift currents which spiral and decay with depth and they obtained the solution by a perturbation approach. In this paper, we adopt the linearization approach to establish existence and uniqueness results and we consider the smoothness of soluti
