Explicit solution and dynamical properties of atmospheric Ekman flows with boundary conditions

Yi Guan,Michal Fečkan,Jinrong Wang

doi:10.14232/ejqtde.2021.1.30

Abstract

In this paper, we study the classical problem of the wind in the steady atmospheric Ekman layer with the constant eddy viscosity. Different from the previous work, we modify the boundary conditions and derive the explicit solution by using the notation of matrix cosine and matrix sine. For the arbitrary height-dependent eddy viscosity, we get the solution of the classical problem with zero velocity and acceleration at the bottom of the layer. In addition, uniqueness is shown and dynamical properties of solution are characterized.

Highlights

The Earth’s atmosphere can be divided into several layers based on the behaviour of its temperature [11], these layers are, starting from ground level upwards, the troposphere, the stratosphere, the mesosphere and the thermosphere, A further region, beginning about 500 km above the ground level, is the exosphere, which fades away into the realm of interplanetary space
Within the surface layer, confined to 20–100 meters of the atmosphere, the velocity profile is adjusted so that the horizontal frictional stress is nearly independent of height
In the Ekman layer, located on top of the surface layer and extending to a height of about 1 km, on average, the flow is governed by a three-way balance among frictional effects, pressure gradient and the influence of the coriolis force [5, 8, 21]

Summary

Introduction

The Earth’s atmosphere can be divided into several layers based on the behaviour of its temperature [11], these layers are, starting from ground level upwards, the troposphere, the stratosphere, the mesosphere and the thermosphere, A further region, beginning about 500 km above the ground level, is the exosphere, which fades away into the realm of interplanetary space. The dynamics of the atmospheric boundary-layer is very important in applications, for example, other than meteorology (weather prediction and climate studies), in the control and management of air pollution (since the dispersal of smog in urban environments depends strongly on meteorological conditions) and in agriculture (e.g. dewfall and frost formation). For this reason, it is important, both from the theoretical as well as from the practical point of view, to understand the flow dynamics of the atmospheric boundary-layer in the context of height-dependent eddy viscosities. Some dynamical properties of solution like asymptotic property, Lyapunov exponents, and stable manifold are characterized

Model description

Existence of explicit solution

Uniqueness

Findings

Dynamical properties