Atmospheric Ekman-type solutions with some eddy viscosities in ellipsoidal coordinates

Abstract

In this paper, we investigate the motion of the wind in the steady atmospheric Ekman layer with suitable boundary conditions in ellipsoidal coordinates. Firstly, we consider the system with the piecewise eddy viscosity coefficients. When the eddy viscosity of the upper layer is constant and the eddy viscosity of the bottom layer is related to the height, we transform the system to a first-order Riccati equation with a suitable initial value. Then, we also construct the solution to the system with the piecewise-constant eddy viscosity. Secondly, we derive the explicit solution to the system with a small perturbation of a constant. Finally, by choosing an exponential variation of the eddy viscosity, we get the explicit solution to the system.

Keywords:

• Ellipsoidal coordinates
• Ekman layer
• variable eddy viscosity
• explicit solutions

MSC 2020:

• 34B05
• 35Q35

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is partially supported by the National Natural Science Foundation of China [grant number 12161015], Guizhou Data Driven Modeling Learning and Optimization Innovation Team (grant number [2020]5016), Postgraduate Education Innovation Program in Guizhou Province [grant number YJSKYJJ(2021)061], Super Computing Algorithm and Application Laboratory of Guizhou University and Gui'an Scientifific Innovation Company (grant number K22-0116-003), 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.