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March  2021, 41(3): 1157-1176. doi: 10.3934/dcds.2020313

Periodic solutions and Hyers-Ulam stability of atmospheric Ekman flows

Yi Guan 1,2, , Michal Fečkan 3,4, and  Jinrong Wang 1,5,,

1. 

Department of Mathematics, Guizhou University, Guiyang 550025, China
2. 

College of Mathematics and Information Science, Guiyang University, Guiyang 550005, China
3. 

Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava, Mlynská dolina, 842 48 Bratislava, Slovakia
4. 

Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49,814 73 Bratislava, Slovakia
5. 

School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China

* Corresponding author: Jinrong Wang

Received  May 2020 Revised  July 2020 Published  March 2021 Early access  August 2020

Fund Project: 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 (grant Nos. 1/0358/20 and 2/0127/20)

In this paper, we study the classical problem of the wind in the steady atmospheric Ekman layer with constant eddy viscosity. Different from the well-known homogeneous system in [14,20], we retain the turbulent fluxes and establish a new nonhomogeneous system of first order differential equations involving a term with the horizontal dependent. We present the existence and uniqueness of periodic solutions and show the Hyers-Ulam stability results for the nonhomogeneous systems under the mild conditions via the matrix theory. Further, we consider the nonhomogeneous systems with varying eddy viscosity coefficient and study systems with piecewise constants, systems with small oscillations, systems with rapidly varying coefficients and systems with slowly varying coefficients and give more continued results.

Keywords: Ekman layer, periodic solution, Hyers-Ulam stability, monodromy matrix, Cauchy matrix.
Mathematics Subject Classification: Primary: 34B05, 34C25; Secondary: 34D20.
Citation: Yi Guan, Michal Fečkan, Jinrong Wang. Periodic solutions and Hyers-Ulam stability of atmospheric Ekman flows. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1157-1176. doi: 10.3934/dcds.2020313
References:
[1]

C. Alsina and R. Ger, On some inequalities and stability results related to the exponential function, J. Inequal. Appl., 2 (1998), 373-380.  doi: 10.1155/S102558349800023X.

[2]

S. Blanes and F. Casas, On the convergence and optimization of the Baker-Campbell-Hausdorff formula, Linear Algebra Appl., 378 (2004), 135-158.  doi: 10.1016/j.laa.2003.09.010.

[3]

A. Bressan and A. Constantin, The deflection angle of surface ocean currents from the wind direction, J. Geophys. Res. Oceans, 124 (2019), 7412-7420.  doi: 10.1029/2019JC015454.

[4]

C. Chicone, Ordinary Differential Equations with Applications, Texts in Applied Mathematics, 34, Springer-Verlag, New York, 1999.

[5]

A. Constantin and R. I. Ivanov, Equatorial wave-current interactions, Comm. Math. Phys, 370 (2019), 1-48.  doi: 10.1007/s00220-019-03483-8.

[6]

A. Constantin and R. S. Johnson, Atmospheric Ekman flows with variable eddy viscosity, Boundary-Layer Meteorology, 170 (2019), 395-414.  doi: 10.1007/s10546-018-0404-0.

[7]

A. Constantin and R. S. Johnson, On the nonlinear, three-dimensional structure of equatorial oceanic flows, J. Phys. Oceanogr, 49 (2019), 2029-2042.  doi: 10.1175/JPO-D-19-0079.1.

[8]

W. A. Coppel, Dichotomies in Stability Theory, Lecture Notes in Mathematics, 629, Springer-Verlag, Berlin/New York, 1978. doi: 10.1007/BFb0067780.

[9]

V. W. Ekman, On the influence of the Earth's rotation on ocean-currents, Arkiv Matematik Astronmi Och Fysik, 2 (1905), 1-52. 

[10]

M. Fečkan, Y. Guan, D. O'Regan and J. Wang, Existence and uniqueness and first order approximation of solutions to atmospheric Ekman flows, Monatshefte für Mathematik, 115 (2020). doi: 10.1007/s00605-020-01414-7.

[11]

B. Grisogono, A generalized Ekman layer profile with gradually varying eddy diffusivities, Quart. J. Roy. Meteor. Soc., 121 (1995), 445-453.  doi: 10.1002/qj.49712152211.

[12] G. J. Haltinar and R. T. Williams, Numerical Prediction and Dynamic Meteorology, Wiley Press, New York, 1980. 
[13]

D. Henry, Nonlinear features of equatorial ocean flows, Oceanography, 31 (2018), 22-27.  doi: 10.5670/oceanog.2018.305.

[14] J. R. Holton, An Introduction to Dynamic Meteorology, Academic Press, New York, 2004. 
[15]

D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A., 27 (1941), 222-224.  doi: 10.1073/pnas.27.4.222.

[16]

S. M. Jung, Hyers-Ulam stability of a system of first order linear differential equations with constant coefficients, J. Math. Anal. Appl., 320 (2006), 549-561.  doi: 10.1016/j.jmaa.2005.07.032.

[17]

S. M. Jung, Hyers-Ulam stability of linear differential equations of first order, Appl. Math. Lett., 17 (2004), 1135-1140.  doi: 10.1016/j.aml.2003.11.004.

[18]

S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Inc., Palm Harbor, FL, 2001.

[19]

O. S. Madsen, A realistic model of the wind-induced Ekman boundary layer, J. Phys. Oceanogr., 7 (1977), 248-255.  doi: 10.1175/1520-0485(1977)007<0248:ARMOTW>2.0.CO;2.

[20] J. Marshall and R. A. Plumb, Atmosphere, Ocean and Climate Dynamic, Academic Press, New York, 2008. 
[21]

F. T. M. Nieuwstadt, On the solution of the stationary, baroclinic Ekman-layer equations with a finite boundary-layer height, Boundary-Layer Meteorology, 26 (1983), 377-390.  doi: 10.1007/BF00119534.

[22]

O. Parmhed, I. Kos and B. Grisogono, An improved Ekman layer approximation for smooth eddy diffusivity profiles, Boundary-Layer Meteorology, 115 (2005), 399-407.  doi: 10.1007/s10546-004-5940-0.

[23] J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag Press, New York, 1987.  doi: 10.1007/978-1-4612-4650-3.
[24]

J. A. Sanders, F. Verhulst and J. Murdock, Averaging Methods in Nonlinear Dynamical Systems, Applied Mathematical Sciences, 59, Springer, New York, 2007. doi: 10.1007/978-0-387-48918-6.

[25]

A. E. Taylor and D. C. Lay, Introduction to Functional Analysis, John Wiley & Sons, New York-Chichester-Brisbane, 1980.

[26]

J. Wang and M. Fečkan, Ulam-Hyers-Rassias stability for semilinear equations, Discontinuity Nonlinear Complexity, 3 (2014), 379-388.  doi: 10.5890/DNC.2014.12.002.

[27]

J. O. Wenegrat and M. J. McPhaden, Wind, waves, and fronts: Frictional effects in a generalized Ekman model, J. Phys. Oceanogr., 46 (2016), 371-394.  doi: 10.1175/JPO-D-15-0162.1.

[28] W. Zdunkowski and A. Bott, Dynamic of the Atmosphere, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511805462.

show all references

References:
[1]

C. Alsina and R. Ger, On some inequalities and stability results related to the exponential function, J. Inequal. Appl., 2 (1998), 373-380.  doi: 10.1155/S102558349800023X.

[2]

S. Blanes and F. Casas, On the convergence and optimization of the Baker-Campbell-Hausdorff formula, Linear Algebra Appl., 378 (2004), 135-158.  doi: 10.1016/j.laa.2003.09.010.

[3]

A. Bressan and A. Constantin, The deflection angle of surface ocean currents from the wind direction, J. Geophys. Res. Oceans, 124 (2019), 7412-7420.  doi: 10.1029/2019JC015454.

[4]

C. Chicone, Ordinary Differential Equations with Applications, Texts in Applied Mathematics, 34, Springer-Verlag, New York, 1999.

[5]

A. Constantin and R. I. Ivanov, Equatorial wave-current interactions, Comm. Math. Phys, 370 (2019), 1-48.  doi: 10.1007/s00220-019-03483-8.

[6]

A. Constantin and R. S. Johnson, Atmospheric Ekman flows with variable eddy viscosity, Boundary-Layer Meteorology, 170 (2019), 395-414.  doi: 10.1007/s10546-018-0404-0.

[7]

A. Constantin and R. S. Johnson, On the nonlinear, three-dimensional structure of equatorial oceanic flows, J. Phys. Oceanogr, 49 (2019), 2029-2042.  doi: 10.1175/JPO-D-19-0079.1.

[8]

W. A. Coppel, Dichotomies in Stability Theory, Lecture Notes in Mathematics, 629, Springer-Verlag, Berlin/New York, 1978. doi: 10.1007/BFb0067780.

[9]

V. W. Ekman, On the influence of the Earth's rotation on ocean-currents, Arkiv Matematik Astronmi Och Fysik, 2 (1905), 1-52. 

[10]

M. Fečkan, Y. Guan, D. O'Regan and J. Wang, Existence and uniqueness and first order approximation of solutions to atmospheric Ekman flows, Monatshefte für Mathematik, 115 (2020). doi: 10.1007/s00605-020-01414-7.

[11]

B. Grisogono, A generalized Ekman layer profile with gradually varying eddy diffusivities, Quart. J. Roy. Meteor. Soc., 121 (1995), 445-453.  doi: 10.1002/qj.49712152211.

[12] G. J. Haltinar and R. T. Williams, Numerical Prediction and Dynamic Meteorology, Wiley Press, New York, 1980. 
[13]

D. Henry, Nonlinear features of equatorial ocean flows, Oceanography, 31 (2018), 22-27.  doi: 10.5670/oceanog.2018.305.

[14] J. R. Holton, An Introduction to Dynamic Meteorology, Academic Press, New York, 2004. 
[15]

D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A., 27 (1941), 222-224.  doi: 10.1073/pnas.27.4.222.

[16]

S. M. Jung, Hyers-Ulam stability of a system of first order linear differential equations with constant coefficients, J. Math. Anal. Appl., 320 (2006), 549-561.  doi: 10.1016/j.jmaa.2005.07.032.

[17]

S. M. Jung, Hyers-Ulam stability of linear differential equations of first order, Appl. Math. Lett., 17 (2004), 1135-1140.  doi: 10.1016/j.aml.2003.11.004.

[18]

S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Inc., Palm Harbor, FL, 2001.

[19]

O. S. Madsen, A realistic model of the wind-induced Ekman boundary layer, J. Phys. Oceanogr., 7 (1977), 248-255.  doi: 10.1175/1520-0485(1977)007<0248:ARMOTW>2.0.CO;2.

[20] J. Marshall and R. A. Plumb, Atmosphere, Ocean and Climate Dynamic, Academic Press, New York, 2008. 
[21]

F. T. M. Nieuwstadt, On the solution of the stationary, baroclinic Ekman-layer equations with a finite boundary-layer height, Boundary-Layer Meteorology, 26 (1983), 377-390.  doi: 10.1007/BF00119534.

[22]

O. Parmhed, I. Kos and B. Grisogono, An improved Ekman layer approximation for smooth eddy diffusivity profiles, Boundary-Layer Meteorology, 115 (2005), 399-407.  doi: 10.1007/s10546-004-5940-0.

[23] J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag Press, New York, 1987.  doi: 10.1007/978-1-4612-4650-3.
[24]

J. A. Sanders, F. Verhulst and J. Murdock, Averaging Methods in Nonlinear Dynamical Systems, Applied Mathematical Sciences, 59, Springer, New York, 2007. doi: 10.1007/978-0-387-48918-6.

[25]

A. E. Taylor and D. C. Lay, Introduction to Functional Analysis, John Wiley & Sons, New York-Chichester-Brisbane, 1980.

[26]

J. Wang and M. Fečkan, Ulam-Hyers-Rassias stability for semilinear equations, Discontinuity Nonlinear Complexity, 3 (2014), 379-388.  doi: 10.5890/DNC.2014.12.002.

[27]

J. O. Wenegrat and M. J. McPhaden, Wind, waves, and fronts: Frictional effects in a generalized Ekman model, J. Phys. Oceanogr., 46 (2016), 371-394.  doi: 10.1175/JPO-D-15-0162.1.

[28] W. Zdunkowski and A. Bott, Dynamic of the Atmosphere, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511805462.
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