# American Institute of Mathematical Sciences

July  2022, 21(7): 2433-2445. doi: 10.3934/cpaa.2022083

## Stability analysis of the boundary value problem modelling a two-layer ocean

 Delft Institute of Applied Mathematics, Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands

Received  November 2021 Revised  April 2022 Published  July 2022 Early access  April 2022

We study boundedness of solutions to a linear boundary value problem (BVP) modelling a two-layer ocean with a uniform eddy viscosity in the lower layer and variable eddy viscosity in the upper layer. We analyse bounds of solutions to the given problem on the examples of different eddy viscosity profiles in the case of their parameter dependence.

Citation: Kateryna Marynets. Stability analysis of the boundary value problem modelling a two-layer ocean. Communications on Pure and Applied Analysis, 2022, 21 (7) : 2433-2445. doi: 10.3934/cpaa.2022083
##### References:
 [1] J. P. Boyd, Dynamics of the Equatorial Ocean, Springer, Berlin, 2018. [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. [3] A. Constantin, Frictional effects in wind-driven ocean currents, Geophys. Astrophys. Fluid Dyn., 115 (2021), 1-14.  doi: 10.1080/03091929.2020.1748614. [4] A. Constantin, D. G. Dritschel and N. Paldor, The deflection angle between a wind-forced surface current and the overlying wind in an ocean with vertically varying eddy viscosity, Phys. Fluids, 32 (2020), 5 pp. doi: 10.1063/5.0030473. [5] A. Constantin and R. I. Ivanov, Equatorial wave-current interactions, Commun. 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, Bound. Lay. Meteorol., 170 (2019), 395-414. [7] A. Constantin and R. S. Johnson, On the modelling of large-scale atmospheric flow, J. Differ. Equ., 285 (2021), 751-798.  doi: 10.1016/j.jde.2021.03.019. [8] A. Constantin and R. S. Johnson, Ekman-type solutions for shallow-water flows on a rotating sphere: a new perspective on a classical problem, Phys. Fluids, 31 (2019), 021401. [9] A. Constantin and R. S. Johnson, On the nonlinear, three-dimensional structure of equatorial oceanic flows, J. Phys. Oceanogr., 49 (2019), 2029-2042. [10] W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, Heath Mathematical Monographs, Heath, Boston, 1965. [11] D. G. Dritschel, N. Paldor and A. Constantin, The Ekman spiral for piecewise uniform viscosity, Ocean Sci., 16 (2020), 1089-1093. [12] V. W. Ekman, On the influence of the Earth's rotation on ocean-currents, Ark. Mat. Astron. Fys., 2 (1905), 1-52. [13] B. Grisogono, A generalized Ekman layer profile with gradually varying eddy diffusivities, Quart. J. Roy. Meteorol. Soc., 121 (1995), 445-453. [14] O. S. Madsen, A realistic model of the wind-induced Ekman boundary layer, J. Phys. Oceanogr., 7 (1977), 248-255. [15] K. Marynets, A Sturm-Liouville problem arising in the atmospheric boundary-layer dynamics, J. Math. Fluid Mech., 22 (2020), 6 pp doi: 10.1007/s00021-020-00507-5. [16] L. Roberti, The Ekman spiral for piecewise-constant eddy viscosity, Appl. Anal., (2021), 1–9. doi: 10.1080/00036811.2021.1896709. [17] L. Roberti, Perturbation analysis for the surface deflection angle of Ekman-type flows with variable eddy viscosity, J. Math. Fluid Mech., 23 (2021), 1-7.  doi: 10.1007/s00021-021-00586-y. [18] J. Röhrs and K.H. Christensen, Drift in the uppermost part of the ocean, Geophys. Res. Lett., 42 (2015), 10349-10356. [19] T. Ström, On logarithmic norms, SIAM J. Numer. Anal., 12 (1975), 741-753.  doi: 10.1137/0712055. [20] G. K. Vallis, Atmospheric and Oceanic Fluid Dynamics, Cambridge University Press, Cambridge, UK, 2017.

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##### References:
 [1] J. P. Boyd, Dynamics of the Equatorial Ocean, Springer, Berlin, 2018. [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. [3] A. Constantin, Frictional effects in wind-driven ocean currents, Geophys. Astrophys. Fluid Dyn., 115 (2021), 1-14.  doi: 10.1080/03091929.2020.1748614. [4] A. Constantin, D. G. Dritschel and N. Paldor, The deflection angle between a wind-forced surface current and the overlying wind in an ocean with vertically varying eddy viscosity, Phys. Fluids, 32 (2020), 5 pp. doi: 10.1063/5.0030473. [5] A. Constantin and R. I. Ivanov, Equatorial wave-current interactions, Commun. 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, Bound. Lay. Meteorol., 170 (2019), 395-414. [7] A. Constantin and R. S. Johnson, On the modelling of large-scale atmospheric flow, J. Differ. Equ., 285 (2021), 751-798.  doi: 10.1016/j.jde.2021.03.019. [8] A. Constantin and R. S. Johnson, Ekman-type solutions for shallow-water flows on a rotating sphere: a new perspective on a classical problem, Phys. Fluids, 31 (2019), 021401. [9] A. Constantin and R. S. Johnson, On the nonlinear, three-dimensional structure of equatorial oceanic flows, J. Phys. Oceanogr., 49 (2019), 2029-2042. [10] W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, Heath Mathematical Monographs, Heath, Boston, 1965. [11] D. G. Dritschel, N. Paldor and A. Constantin, The Ekman spiral for piecewise uniform viscosity, Ocean Sci., 16 (2020), 1089-1093. [12] V. W. Ekman, On the influence of the Earth's rotation on ocean-currents, Ark. Mat. Astron. Fys., 2 (1905), 1-52. [13] B. Grisogono, A generalized Ekman layer profile with gradually varying eddy diffusivities, Quart. J. Roy. Meteorol. Soc., 121 (1995), 445-453. [14] O. S. Madsen, A realistic model of the wind-induced Ekman boundary layer, J. Phys. Oceanogr., 7 (1977), 248-255. [15] K. Marynets, A Sturm-Liouville problem arising in the atmospheric boundary-layer dynamics, J. Math. Fluid Mech., 22 (2020), 6 pp doi: 10.1007/s00021-020-00507-5. [16] L. Roberti, The Ekman spiral for piecewise-constant eddy viscosity, Appl. Anal., (2021), 1–9. doi: 10.1080/00036811.2021.1896709. [17] L. Roberti, Perturbation analysis for the surface deflection angle of Ekman-type flows with variable eddy viscosity, J. Math. Fluid Mech., 23 (2021), 1-7.  doi: 10.1007/s00021-021-00586-y. [18] J. Röhrs and K.H. Christensen, Drift in the uppermost part of the ocean, Geophys. Res. Lett., 42 (2015), 10349-10356. [19] T. Ström, On logarithmic norms, SIAM J. Numer. Anal., 12 (1975), 741-753.  doi: 10.1137/0712055. [20] G. K. Vallis, Atmospheric and Oceanic Fluid Dynamics, Cambridge University Press, Cambridge, UK, 2017.
Ekman flow and the net water transport in the Northern Hemisphere
Piecewise-constant eddy viscosity profile for a two-layer ocean
Piecewise-constant eddy viscosity profile for a three-layer ocean
Piecewise-constant generalized eddy viscosity profile for a two-layer ocean
Linear eddy viscosity profiles for a two-layer ocean at a fixed ocean depth: red line – the increasing regime and blue points – the decreasing regime
The polynominal eddy viscosity profile at a fixed ocean depth $-h$ for a fixed value of parameter $a$
The exponential eddy viscosity profile at a fixed ocean depth $-h$ for a fixed value of parameter $\nu$
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