# American Institute of Mathematical Sciences

September  2022, 14(3): 473-490. doi: 10.3934/jgm.2022015

## Atmospheric Ekman flows with uniform density in ellipsoidal coordinates: Explicit solution and dynamical properties

 1 Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, China 2 Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University, Mlynská dolina, 842 48 Bratislava, Slovakia 3 Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73 Bratislava, Slovakia

*Corresponding author: JinRong Wang

Received  February 2022 Revised  May 2022 Published  September 2022 Early access  June 2022

Fund Project: This work is partially supported by the National Natural Science Foundation of China (12161015), Training Object of High Level and Innovative Talents of Guizhou Province ((2016)4006), Guizhou Data Driven Modeling Learning and Optimization Innovation Team ([2020]5016), Major Project of Guizhou Postgraduate Education and Teaching Reform (YJSJGKT[2021]041), 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

In this paper, we present a new general system of equations describing the steady motion of atmosphere with uniform density in ellipsoidal coordinates, which is derived from the general governing equations for viscous fluids. We first show that this new system can be reduced to the classic Ekman equations. Secondly, we obtain the explicit solution of the Ekman equations in ellipsoidal coordinates. Thirdly, for the viscosity related to the height, we obtain the solution of the classical problem with zero acceleration at the bottom of Ekman layer. Finally, the uniqueness and dynamical properties of solution are demonstrated.

Citation: Taoyu Yang, Michal Fečkan, JinRong Wang. Atmospheric Ekman flows with uniform density in ellipsoidal coordinates: Explicit solution and dynamical properties. Journal of Geometric Mechanics, 2022, 14 (3) : 473-490. doi: 10.3934/jgm.2022015
##### References:
 [1] C. Chicone, Ordinary Differential Equations with Applications, Springer-Verlag, New York, 2006. [2] A. Constantin and R. S. Johnson, Atmospheric Ekman flows with variable eddy viscosity, Boundary-Layer Meteorol., 170 (2019), 395-414. [3] A. Constantin and R. S. Johnson, On the modelling of large-scale atmospheric flows, J. Differential Equations, 285 (2021), 751-798.  doi: 10.1016/j.jde.2021.03.019. [4] A. Constantin and R. S. Johnson, On the propagation of waves in the atmosphere, Proc. R. Soc. A, 477 (2021), 20200424. [5] A. Constantin and R. S. Johnson, On the propagation of nonlinear waves in the atmosphere, Proc. R. Soc. A, 478 (2022), 20210895.  doi: 10.1098/rspa.2021.0895. [6] W. A. Coppel, Dichotomies in Stability Theory, Springer-Verlag, Berlin, 1978. [7] Y. Du and R. Rotumno, A simple analytical model of the nocturnal low-level jet over the great plains of the United States, J. Atmos. Sci., 71 (2014), 3674-3683. [8] V. W. Ekman, On the influence of the earth's rotation on ocean-currents, Ark. Mat. Astr. Fys., 2 (1905), 1-52. [9] B. Gayen, S. Sarkar and J. R. Taylor, Large eddy simulation of oscillating boundary layer under an oscillatory current, J. Fluid Mech., 643 (2010), 233-266. [10] Y. Guan, J. Wang and M. Fečkan, Explicit solution and dynamical properties of Atmospheric Ekman flows with boundary conditions, Electron. J. Qual. Theory Differ. Equ., 30 (2021), 1-19.  doi: 10.14232/ejqtde.2021.1.3. [11] Y. Guan, M. Fečkan and J. Wang, Periodic solutions and Hyers-Ulam stability of atmospheric Ekman flows, Discrete Contin. Dyn. Syst., 41 (2021), 1157-1176.  doi: 10.3934/dcds.2020313. [12] J. R. Holton, An Introduction to Dynamic Meteorology, Academic Press, New York, 2004. [13] C. T. Hsu, X. Y. Lu and M. K. Kwan, LES and RANS studies of oscillating flows over flat plate, J. Eng. Mech., 126 (2000), 186-193. [14] D. Ionescu-Kruse, Analytical atmospheric Ekman-type solutions with height-dependent eddy viscosities, J. Math. Fluid Mech., 23 (2021), 18.  doi: 10.1007/s00021-020-00543-1. [15] O. S. Madsen, A realistic model of the wind-induced boundary layer, J. Atmos. Sci., 7 (1977), 248-255. [16] J. Marshall and R. A. Plumb, Atmosphere Ocean and Climate Dynamic: An Introduction Text, Academic Press, New York, 2016. [17] J. Miles, Analytical solutions for the Ekman layer, Boundary-Layer Meteorol., 67 (1994), 1-10. [18] M. Mostafa and E. Bou-Zeid, Anaylitical reduced models for the non-stationary diabatic atmospheric boundary layer, Boundary-Layer Meteorol., 164 (2017), 383-399. [19] M. Mostafa and E. Bou-Zeid, Large-eddy simulations and damped-oscillator models of the unsteady Ekman boundary layer, J. Atmos. Sci., 73 (2016), 25-40. [20] F. T. M. Nieuwstadt, On the solution of the stationary, baroclinic Ekman-layer equations with a finite boundary-layer height, Boundary-Layer Meteorol., 26 (1983), 377-390. [21] S. Radhakrishnan and U. Piomelli, Large-eddy simulation of oscillating boundary layers: Model comparison and validation, J. Geophys. Res., 113 (2008), 1-14. [22] R. C. Robinson, An Introduction to Dynamical Systems: Continuous and Discrete, Prentice Hall, Upper Saddle River, 2004. [23] W. J. Rugh, Linear System Theory, Prentice Hall, Upper Saddle River, 1996. [24] A. E. Taylor and D. C. Lay, Introduction to Functional Analysis, Wiley, New Jersey, 1980. [25] A. Viúdez and D. G. Dritschel, Vertical velocity in mesoscale geophysical flows, J. Fluid Mech., 483 (2003), 199-223.  doi: 10.1017/S0022112003004191. [26] J. Wang, M. Fečkan and Y. Guan, Local and global analysis for discontinuous atmospheric Ekman equations, to appear, J. Dynam. Differential Equations. doi: 10.1007/s10884-021-10037-x. [27] J. Wang, M. Fečkan and W. Zhang, On the nonlocal boundary value problem of geophysical fluid flows, Z. Angew. Math. Phys., 72 (2021), 27.  doi: 10.1007/s00033-020-01452-z. [28] W. Zdunkowski and A. Bott, Dynamics of the Atmosphere, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511805462.

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##### References:
 [1] C. Chicone, Ordinary Differential Equations with Applications, Springer-Verlag, New York, 2006. [2] A. Constantin and R. S. Johnson, Atmospheric Ekman flows with variable eddy viscosity, Boundary-Layer Meteorol., 170 (2019), 395-414. [3] A. Constantin and R. S. Johnson, On the modelling of large-scale atmospheric flows, J. Differential Equations, 285 (2021), 751-798.  doi: 10.1016/j.jde.2021.03.019. [4] A. Constantin and R. S. Johnson, On the propagation of waves in the atmosphere, Proc. R. Soc. A, 477 (2021), 20200424. [5] A. Constantin and R. S. Johnson, On the propagation of nonlinear waves in the atmosphere, Proc. R. Soc. A, 478 (2022), 20210895.  doi: 10.1098/rspa.2021.0895. [6] W. A. Coppel, Dichotomies in Stability Theory, Springer-Verlag, Berlin, 1978. [7] Y. Du and R. Rotumno, A simple analytical model of the nocturnal low-level jet over the great plains of the United States, J. Atmos. Sci., 71 (2014), 3674-3683. [8] V. W. Ekman, On the influence of the earth's rotation on ocean-currents, Ark. Mat. Astr. Fys., 2 (1905), 1-52. [9] B. Gayen, S. Sarkar and J. R. Taylor, Large eddy simulation of oscillating boundary layer under an oscillatory current, J. Fluid Mech., 643 (2010), 233-266. [10] Y. Guan, J. Wang and M. Fečkan, Explicit solution and dynamical properties of Atmospheric Ekman flows with boundary conditions, Electron. J. Qual. Theory Differ. Equ., 30 (2021), 1-19.  doi: 10.14232/ejqtde.2021.1.3. [11] Y. Guan, M. Fečkan and J. Wang, Periodic solutions and Hyers-Ulam stability of atmospheric Ekman flows, Discrete Contin. Dyn. Syst., 41 (2021), 1157-1176.  doi: 10.3934/dcds.2020313. [12] J. R. Holton, An Introduction to Dynamic Meteorology, Academic Press, New York, 2004. [13] C. T. Hsu, X. Y. Lu and M. K. Kwan, LES and RANS studies of oscillating flows over flat plate, J. Eng. Mech., 126 (2000), 186-193. [14] D. Ionescu-Kruse, Analytical atmospheric Ekman-type solutions with height-dependent eddy viscosities, J. Math. Fluid Mech., 23 (2021), 18.  doi: 10.1007/s00021-020-00543-1. [15] O. S. Madsen, A realistic model of the wind-induced boundary layer, J. Atmos. Sci., 7 (1977), 248-255. [16] J. Marshall and R. A. Plumb, Atmosphere Ocean and Climate Dynamic: An Introduction Text, Academic Press, New York, 2016. [17] J. Miles, Analytical solutions for the Ekman layer, Boundary-Layer Meteorol., 67 (1994), 1-10. [18] M. Mostafa and E. Bou-Zeid, Anaylitical reduced models for the non-stationary diabatic atmospheric boundary layer, Boundary-Layer Meteorol., 164 (2017), 383-399. [19] M. Mostafa and E. Bou-Zeid, Large-eddy simulations and damped-oscillator models of the unsteady Ekman boundary layer, J. Atmos. Sci., 73 (2016), 25-40. [20] F. T. M. Nieuwstadt, On the solution of the stationary, baroclinic Ekman-layer equations with a finite boundary-layer height, Boundary-Layer Meteorol., 26 (1983), 377-390. [21] S. Radhakrishnan and U. Piomelli, Large-eddy simulation of oscillating boundary layers: Model comparison and validation, J. Geophys. Res., 113 (2008), 1-14. [22] R. C. Robinson, An Introduction to Dynamical Systems: Continuous and Discrete, Prentice Hall, Upper Saddle River, 2004. [23] W. J. Rugh, Linear System Theory, Prentice Hall, Upper Saddle River, 1996. [24] A. E. Taylor and D. C. Lay, Introduction to Functional Analysis, Wiley, New Jersey, 1980. [25] A. Viúdez and D. G. Dritschel, Vertical velocity in mesoscale geophysical flows, J. Fluid Mech., 483 (2003), 199-223.  doi: 10.1017/S0022112003004191. [26] J. Wang, M. Fečkan and Y. Guan, Local and global analysis for discontinuous atmospheric Ekman equations, to appear, J. Dynam. Differential Equations. doi: 10.1007/s10884-021-10037-x. [27] J. Wang, M. Fečkan and W. Zhang, On the nonlocal boundary value problem of geophysical fluid flows, Z. Angew. Math. Phys., 72 (2021), 27.  doi: 10.1007/s00033-020-01452-z. [28] W. Zdunkowski and A. Bott, Dynamics of the Atmosphere, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511805462.
Away from the polar axis, we represent a point $P$ in the atmosphere using the hybrid spherical-geopotential rotating coordinate system $(\varphi,\theta,\tilde{z})$, obtained from the spherical system $(\bf{e}_\varphi,\bf{e}_\theta,\bf{e}_r)$ and the geopotential system $(\bf{e}_\varphi,\bf{e}_\beta,\bf{e}_z)$. Here, $\varphi$ and $\theta$ are the longitude and geocentric latitude of $P$, respectively, $\beta$ and $\alpha$ are the geodetic and geocentric latitude of the projection $P^*$ of $P$ on the ellipsoidal geoid, respectively, and $\bf{e}_z$ points upwards along the normal $P^*P$ to the geoid (which intersects the equatorial plane in the point $P_e$). The unit vectors $(\bf{e}_\theta,\bf{e}_r)$ are obtained by rotating the unit vectors $(\bf{e}_\beta,\bf{e}_z)$ by the angle $(\beta-\theta)$, in the plane of fixed longitude $\varphi$
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