doi: 10.3934/dcdsb.2022012
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Constant vorticity atmospheric Ekman flows in the $ f- $plane approximation

1. 

Department of Mathematics, Guizhou University, Guiyang 550025, China

2. 

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

3. 

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

4. 

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

*Corresponding author: Michal Fečkan

Received  July 2021 Revised  December 2021 Early access January 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), Major Research Project of Innovative Group in Guizhou Education Department ([2018]012), Guizhou Data Driven Modeling Learning and Optimization Innovation Team ([2020]5016), Natural Science Foundation of Guizhou Province ([2020]090), 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)

We study the geophysical fluid dynamical problem of the wind in the steady atmospheric Ekman layer with constant eddy viscosity. Three dimensional Ekman flows with constant vorticity is considered in the $ f- $plane approximation. For non-equatorial $ f- $plane approximation, we show that any bounded solution of the Ekman flow with a flat surface and constant vorticity vector is the stationary flow with vanishing velocity field, while for the equatorial $ f- $plane approximation, we obtain that the pressure presents no variation in the northward direction and the meridional component is constant throughout the fluid domain.

Citation: Jinrong Wang, Michal Fečkan, Yi Guan. Constant vorticity atmospheric Ekman flows in the $ f- $plane approximation. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022012
References:
[1] C. R. Benoit and J. M. Beckers, Introduction to Geophysical Gluid Dynamics: Physical and Numerical Aspects, Academic Press, New York, 2011. 
[2]

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.

[3]

J. F. Chu, I. K. Delia and Y. J. Yang, Exact solution and instability for geophysical waves with centripetal forces and at arbitrary latitude, J. Math. Fluid Mech., 21 (2019), Art.19, 16pp. doi: 10.1007/s00021-019-0423-8.

[4]

J. F. ChuD. K. Ionescu and Y. J. Yang, Exact solution and instability for geophysical waves at arbitrary latitude, Discrete Contin. Dyn. Syst., 39 (2019), 4399-4414.  doi: 10.3934/dcds.2019178.

[5]

J. F. Chu and Y. J. Yang, Constant vorticity water flows in the euqatorial $\beta-$plane approximation with centripetal forces, J. Differential Equations, 269 (2020), 9336-9347.  doi: 10.1016/j.jde.2020.06.044.

[6]

J. F. Chu and Y. J. Yang, A cylindrical coordinates approach to constant vorticity geophysical waves with centripetal forces at arbitrary latitude, J. Differential Equations, 279 (2021), 46-62.  doi: 10.1016/j.jde.2021.01.014.

[7]

A. Constantin, Two-dimensionality of gravity water flows of constant nonzero vorticity beneath a surface wave train, Eur. J. Mech. B Fluids, 30 (2011), 12-16.  doi: 10.1016/j.euromechflu.2010.09.008.

[8]

A. Constantin, On the modelling of equatorial waves, Geophysical Research Letters, 39 (2012), L05602. doi: 10.1029/2012GL051169.

[9]

A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res. Oceans, 117 (2012), C05029. doi: 10.1029/2012JC007879.

[10]

A. Constantin, Some three-dimensional nonlinear equatorial flows, J. Phys. Oceanogr, 43 (2013), 165-175.  doi: 10.1175/JPO-D-12-062.1.

[11]

A. Constantin, Some nonlinear, equatorially trapped, nonhydrostatic internal geophysical waves, J. Phys. Oceanogr, 44 (2014), 781-789.  doi: 10.1175/JPO-D-13-0174.1.

[12]

A. Constantin and P. Germain, Instability of some equatorially trapped waves, J. Geophys. Res. Oceans, 118 (2013), 2802-2810.  doi: 10.1002/jgrc.20219.

[13]

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.

[14]

A. Constantin and R. S. Johnson, An exact, steady, purely azimuthal equatorial flow with a free surface, J. Geophys. Res. Oceans, 46 (2016), 1935-1945.  doi: 10.1175/JPO-D-15-0205.1.

[15]

A. Constantin and R. S. Johnson, A nonlinear, three-dimensional model for ocean flows, motivated by some observations of the pacific equatorial undercurrent and thermocline, Phys. Fluids, 29 (2017), 056604.  doi: 10.1063/1.4984001.

[16]

A. Constantin and E. Kartashova, Effect of non-zero constant vorticity on the nonlinear resonances of capillary water waves, Europhys. Lett., 86 (2009), 29001.  doi: 10.1209/0295-5075/86/29001.

[17]

A. Constantin and S. G. Monismith, Gerstner waves in the presence of mean currents and rotation, J. Fluid Mech., 820 (2017), 511-528.  doi: 10.1017/jfm.2017.223.

[18]

D. G. DritschelN. Paldor and A. Constantin, The Ekman spiral for piecewise-uniform diffusivity, Ocean Science, 16 (2020), 1089-1093.  doi: 10.5194/os-2020-31.

[19]

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

[20]

L. L. Fan and H. J. Gao, On three-dimensional geophysical capillary-gravity water flows with constant vorticity, Ann. Mat. Pura Appl. (4), 200 (2021), 711-720.  doi: 10.1007/s10231-020-01010-4.

[21]

M. FečkanY. GuanD. O'Regan and J. Wang, Existence and uniqueness and first order approximation of solutions to atmospheric Ekman flows, Monatsh. Math., 193 (2020), 623-636.  doi: 10.1007/s00605-020-01414-7.

[22]

Y. Guan, M. Fečkan and J. Wang, Explicit solution of atmospheric Ekman flows with some types of Eddy viscosity, Monatsh. Math., (2021). doi: 10.1007/s00605-021-01551-7.

[23]

Y. GuanM. Fečkan and J. Wang, Explicit solution and dynamical properties of atmospheric Ekman flows with boundary conditions, Electron. J. Qual. Theory Differ. Equ., 2021 (2021), 1-19.  doi: 10.14232/ejqtde.2021.1.3.

[24] G. J. Haltinar and R. T. Williams, Numercial Prediction and Dynamic Meteorology, Wiley Press, New York, 1980. 
[25]

D. Henry, A modified equatorial $\beta-$plane approximation modelling nonlinear wave-current interactions, J. Differential Equations, 263 (2017), 2554-2566.  doi: 10.1016/j.jde.2017.04.007.

[26]

D. Henry, Exact equatorial water waves in the $f-$plane, Nonlinear Anal. Real World Appl., 28 (2016), 284-289.  doi: 10.1016/j.nonrwa.2015.10.003.

[27]

D. Henry, An exact solution for equatorial geophysical water waves with an underlying current, Eur. J. Mech. B Fluids, 38 (2013), 18-21.  doi: 10.1016/j.euromechflu.2012.10.001.

[28]

D. Henry, Equatorially trapped nonlinear water waves in a $\beta-$plane approximation with centripetal forces, J. Fluid Mech., 804 (2016), R1, 11pp. doi: 10.1017/jfm.2016.544.

[29]

D. Henry and C. I. Martin, Exact, free-surface equatorial flows with general stratification in spherical coordinates, Arch. Ration. Mech. Anal., 233 (2019), 497-512.  doi: 10.1007/s00205-019-01362-z.

[30]

D. Henry and C. I. Martin, Stratified equatorial flows in cylindrical coordinates, Nonlinearity, 33 (2020), 3889-3904.  doi: 10.1088/1361-6544/ab801f.

[31] J. R. Holton, An Introduction to Dynamic Meteorology, Academic Press, New York, 2004. 
[32]

D. Ionescu-Kruse, Analytical atmospheric Ekman-type solutions with heght-dependent eddy viscosities, J. Math. Fluid Mech., 23 (2021), Art.18, 11pp. doi: 10.1007/s00021-020-00543-1.

[33]

D. Ionescu-Kruse, Instability of edge waves along a sloping beach, J. Differential Equations, 256 (2014), 3999-4012.  doi: 10.1016/j.jde.2014.03.009.

[34]

D. Ionescu-Kruse, Short-wavelength instability of edge waves in stratified water, Discrete Contin. Dyn. Syst., 35 (2015), 2053-2066.  doi: 10.3934/dcds.2015.35.2053.

[35]

S. Leblanc, Local stability of Gerstner's waves, J. Fluid Mech., 506 (2004), 245-254.  doi: 10.1017/S0022112004008444.

[36]

C. I. Martin, Two-dimensionality of gravity water flows governed by the equatorial $f-$plane approximation, Ann. Mat. Pura Appl.(4), 196 (2017), 2253-2260.  doi: 10.1007/s10231-017-0663-2.

[37]

C. I. Martin, Constant vorticity water flows with full Coriolis term, Nonlinearity, 32 (2019), 2327-2336.  doi: 10.1088/1361-6544/ab1c76.

[38]

C. I. Martin, On constant vorticity water flows in the $\beta-$plane approximation, J. Fluid Mech., 865 (2019), 762-774.  doi: 10.1017/jfm.2019.95.

[39]

C. I. Martin, Geophysical water flows with constant vorticity and centripetal terms, Ann. Mat. Pura Appl.(4), 200 (2021), 101-116.  doi: 10.1007/s10231-020-00985-4.

[40] J. Pedlosky, Geophysical Fluid Dynamic, Springer-Verlag Press, New York, 1987. 
[41]

J. Wang, M. Fečkan and Y. Guan, Constant vorticity Ekman flows in the $\beta-$plane approximation, J. Math. Fluid Mech., 23 (2021), Art.85, 11pp. doi: 10.1007/s00021-021-00612-z.

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

show all references

References:
[1] C. R. Benoit and J. M. Beckers, Introduction to Geophysical Gluid Dynamics: Physical and Numerical Aspects, Academic Press, New York, 2011. 
[2]

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.

[3]

J. F. Chu, I. K. Delia and Y. J. Yang, Exact solution and instability for geophysical waves with centripetal forces and at arbitrary latitude, J. Math. Fluid Mech., 21 (2019), Art.19, 16pp. doi: 10.1007/s00021-019-0423-8.

[4]

J. F. ChuD. K. Ionescu and Y. J. Yang, Exact solution and instability for geophysical waves at arbitrary latitude, Discrete Contin. Dyn. Syst., 39 (2019), 4399-4414.  doi: 10.3934/dcds.2019178.

[5]

J. F. Chu and Y. J. Yang, Constant vorticity water flows in the euqatorial $\beta-$plane approximation with centripetal forces, J. Differential Equations, 269 (2020), 9336-9347.  doi: 10.1016/j.jde.2020.06.044.

[6]

J. F. Chu and Y. J. Yang, A cylindrical coordinates approach to constant vorticity geophysical waves with centripetal forces at arbitrary latitude, J. Differential Equations, 279 (2021), 46-62.  doi: 10.1016/j.jde.2021.01.014.

[7]

A. Constantin, Two-dimensionality of gravity water flows of constant nonzero vorticity beneath a surface wave train, Eur. J. Mech. B Fluids, 30 (2011), 12-16.  doi: 10.1016/j.euromechflu.2010.09.008.

[8]

A. Constantin, On the modelling of equatorial waves, Geophysical Research Letters, 39 (2012), L05602. doi: 10.1029/2012GL051169.

[9]

A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res. Oceans, 117 (2012), C05029. doi: 10.1029/2012JC007879.

[10]

A. Constantin, Some three-dimensional nonlinear equatorial flows, J. Phys. Oceanogr, 43 (2013), 165-175.  doi: 10.1175/JPO-D-12-062.1.

[11]

A. Constantin, Some nonlinear, equatorially trapped, nonhydrostatic internal geophysical waves, J. Phys. Oceanogr, 44 (2014), 781-789.  doi: 10.1175/JPO-D-13-0174.1.

[12]

A. Constantin and P. Germain, Instability of some equatorially trapped waves, J. Geophys. Res. Oceans, 118 (2013), 2802-2810.  doi: 10.1002/jgrc.20219.

[13]

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.

[14]

A. Constantin and R. S. Johnson, An exact, steady, purely azimuthal equatorial flow with a free surface, J. Geophys. Res. Oceans, 46 (2016), 1935-1945.  doi: 10.1175/JPO-D-15-0205.1.

[15]

A. Constantin and R. S. Johnson, A nonlinear, three-dimensional model for ocean flows, motivated by some observations of the pacific equatorial undercurrent and thermocline, Phys. Fluids, 29 (2017), 056604.  doi: 10.1063/1.4984001.

[16]

A. Constantin and E. Kartashova, Effect of non-zero constant vorticity on the nonlinear resonances of capillary water waves, Europhys. Lett., 86 (2009), 29001.  doi: 10.1209/0295-5075/86/29001.

[17]

A. Constantin and S. G. Monismith, Gerstner waves in the presence of mean currents and rotation, J. Fluid Mech., 820 (2017), 511-528.  doi: 10.1017/jfm.2017.223.

[18]

D. G. DritschelN. Paldor and A. Constantin, The Ekman spiral for piecewise-uniform diffusivity, Ocean Science, 16 (2020), 1089-1093.  doi: 10.5194/os-2020-31.

[19]

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

[20]

L. L. Fan and H. J. Gao, On three-dimensional geophysical capillary-gravity water flows with constant vorticity, Ann. Mat. Pura Appl. (4), 200 (2021), 711-720.  doi: 10.1007/s10231-020-01010-4.

[21]

M. FečkanY. GuanD. O'Regan and J. Wang, Existence and uniqueness and first order approximation of solutions to atmospheric Ekman flows, Monatsh. Math., 193 (2020), 623-636.  doi: 10.1007/s00605-020-01414-7.

[22]

Y. Guan, M. Fečkan and J. Wang, Explicit solution of atmospheric Ekman flows with some types of Eddy viscosity, Monatsh. Math., (2021). doi: 10.1007/s00605-021-01551-7.

[23]

Y. GuanM. Fečkan and J. Wang, Explicit solution and dynamical properties of atmospheric Ekman flows with boundary conditions, Electron. J. Qual. Theory Differ. Equ., 2021 (2021), 1-19.  doi: 10.14232/ejqtde.2021.1.3.

[24] G. J. Haltinar and R. T. Williams, Numercial Prediction and Dynamic Meteorology, Wiley Press, New York, 1980. 
[25]

D. Henry, A modified equatorial $\beta-$plane approximation modelling nonlinear wave-current interactions, J. Differential Equations, 263 (2017), 2554-2566.  doi: 10.1016/j.jde.2017.04.007.

[26]

D. Henry, Exact equatorial water waves in the $f-$plane, Nonlinear Anal. Real World Appl., 28 (2016), 284-289.  doi: 10.1016/j.nonrwa.2015.10.003.

[27]

D. Henry, An exact solution for equatorial geophysical water waves with an underlying current, Eur. J. Mech. B Fluids, 38 (2013), 18-21.  doi: 10.1016/j.euromechflu.2012.10.001.

[28]

D. Henry, Equatorially trapped nonlinear water waves in a $\beta-$plane approximation with centripetal forces, J. Fluid Mech., 804 (2016), R1, 11pp. doi: 10.1017/jfm.2016.544.

[29]

D. Henry and C. I. Martin, Exact, free-surface equatorial flows with general stratification in spherical coordinates, Arch. Ration. Mech. Anal., 233 (2019), 497-512.  doi: 10.1007/s00205-019-01362-z.

[30]

D. Henry and C. I. Martin, Stratified equatorial flows in cylindrical coordinates, Nonlinearity, 33 (2020), 3889-3904.  doi: 10.1088/1361-6544/ab801f.

[31] J. R. Holton, An Introduction to Dynamic Meteorology, Academic Press, New York, 2004. 
[32]

D. Ionescu-Kruse, Analytical atmospheric Ekman-type solutions with heght-dependent eddy viscosities, J. Math. Fluid Mech., 23 (2021), Art.18, 11pp. doi: 10.1007/s00021-020-00543-1.

[33]

D. Ionescu-Kruse, Instability of edge waves along a sloping beach, J. Differential Equations, 256 (2014), 3999-4012.  doi: 10.1016/j.jde.2014.03.009.

[34]

D. Ionescu-Kruse, Short-wavelength instability of edge waves in stratified water, Discrete Contin. Dyn. Syst., 35 (2015), 2053-2066.  doi: 10.3934/dcds.2015.35.2053.

[35]

S. Leblanc, Local stability of Gerstner's waves, J. Fluid Mech., 506 (2004), 245-254.  doi: 10.1017/S0022112004008444.

[36]

C. I. Martin, Two-dimensionality of gravity water flows governed by the equatorial $f-$plane approximation, Ann. Mat. Pura Appl.(4), 196 (2017), 2253-2260.  doi: 10.1007/s10231-017-0663-2.

[37]

C. I. Martin, Constant vorticity water flows with full Coriolis term, Nonlinearity, 32 (2019), 2327-2336.  doi: 10.1088/1361-6544/ab1c76.

[38]

C. I. Martin, On constant vorticity water flows in the $\beta-$plane approximation, J. Fluid Mech., 865 (2019), 762-774.  doi: 10.1017/jfm.2019.95.

[39]

C. I. Martin, Geophysical water flows with constant vorticity and centripetal terms, Ann. Mat. Pura Appl.(4), 200 (2021), 101-116.  doi: 10.1007/s10231-020-00985-4.

[40] J. Pedlosky, Geophysical Fluid Dynamic, Springer-Verlag Press, New York, 1987. 
[41]

J. Wang, M. Fečkan and Y. Guan, Constant vorticity Ekman flows in the $\beta-$plane approximation, J. Math. Fluid Mech., 23 (2021), Art.85, 11pp. doi: 10.1007/s00021-021-00612-z.

[42] W. Zdunkowski and A. Bott, Dynamics of the Atmosphere, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511805462.
Figure 1.  The flow domain
Table 1.  A comparison between the non-equatorial and the equatorial $ f- $plane approximation
Situation Coriolis force The velocity The vorticity
non-equational consider $ (0,0,0) $ $ (0,0,0) $
equational don't consider $ (u,v,w) $ $ (0,\Omega_{2},0) $
Situation Coriolis force The velocity The vorticity
non-equational consider $ (0,0,0) $ $ (0,0,0) $
equational don't consider $ (u,v,w) $ $ (0,\Omega_{2},0) $
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