# American Institute of Mathematical Sciences

July  2022, 21(7): 2463-2477. doi: 10.3934/cpaa.2022064

## The surface current of Ekman flows with time-dependent eddy viscosity

 Faculty of Mathematics, University of Vienna, Oskar–Morgenstern–Platz 1, 1090 Vienna, Austria

Received  September 2021 Revised  February 2022 Published  July 2022 Early access  March 2022

In this paper we investigate transients in the oceanic Ekman layer in the presence of time-varying winds and a constant-in-depth but time dependent eddy viscosity, where the initial state is taken to be the steady state corresponding to the initial wind and the initial eddy viscosity. For this specific situation, a formula for the evolution of the surface current can be derived explicitly. We show that, if the wind and the eddy viscosity converge toward constant values for large times, under mild assumptions on their convergence rate the solution converges toward the corresponding steady state. The time evolution of the surface current and the surface deflection angle is visualized with the aid of simple numerical plots for some specific examples.

Citation: Luigi Roberti. The surface current of Ekman flows with time-dependent eddy viscosity. Communications on Pure and Applied Analysis, 2022, 21 (7) : 2463-2477. doi: 10.3934/cpaa.2022064
##### References:
 [1] 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. [2] A. Constantin, Frictional effects in wind-driven ocean currents, Geophys. Astro. Fluid Dyn., 115 (2021), 1-14.  doi: 10.1080/03091929.2020.1748614. [3] A. Constantin and R. I. Ivanov, Equatorial wave-current interactions, Comm. Math. Phys., 370 (2019), 1-48.  doi: 10.1007/s00220-019-03483-8. [4] A. Constantin and R. S. Johnson, The dynamics of waves interacting with the Equatorial Undercurrent, Geophys. Astro. Fluid Dyn., 109 (2015), 311-358.  doi: 10.1080/03091929.2015.1066785. [5] A. Constantin and R. S. Johnson, Atmospheric Ekman flows with variable eddy viscosity, Bound. Layer Meteorol., 170 (2019), 395-414.  doi: 10.1007/s10546-018-0404-0. [6] 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, 15 pp. doi: 10.1063/1.5083088. [7] A. Constantin and R. S. Johnson, On the modelling of large-scale atmospheric flows, J. Differ. Equ., 285 (2021), 751-798.  doi: 10.1016/j.jde.2021.03.019. [8] M. F. Cronin and W. S. Kessler, Near-surface shear flow in the tropical Pacific cold tongue front, J. Phys. Oceanogr., 39 (2009), 1200-1215.  doi: 10.1175/2008JPO4064.1. [9] D. G. Dritschel, N. Paldor and A. Constantin, The Ekman spiral for piecewise-uniform viscosity, Ocean Sci., 16 (2020), 1089-1093.  doi: 10.5194/os-16-1089-2020. [10] V. W. Ekman, On the influence of the Earth's rotation on ocean-currents, Ark. Mat. Astron. Fys., 2 (1905), 1-52. [11] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th Edition, Academic Press, New York, NY, 2007. [12] B. Grisogono, A generalized Ekman layer profile with gradually varying eddy diffusivities, Quart. J. Roy. Meteorol. Soc., 121 (1995), 445-453.  doi: 10.1002/qj.49712152211. [13] D. Ionescu-Kruse, Analytical atmospheric Ekman-type solutions with height-dependent eddy viscosities, J. Math. Fluid Mech., 23 (2021), 1-11.  doi: 10.1007/s00021-020-00543-1. [14] D. M. Lewis and S. E. Belcher, Time-dependent, coupled, Ekman boundary layer solutions incorporating Stokes drift, Dyn. Atmospheres Oceans, 37 (2004), 313-351.  doi: 10.1016/j.dynatmoce.2003.11.001. [15] T. Lyons, Variable eddy viscosities in the atmospheric boundary layer from ageostrophic wind-speed profiles, J. Math. Fluid Mech., 23 (2021), 16 pp. doi: 10.1007/s00021-021-00575-1. [16] 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. [17] A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, 2$^{\text{nd}}$ Edition, CRC Press LLC, Boca Raton, FL, 2003. [18] J. F. Price and M. A. Sundermeyer, Stratified Ekman layers, J. Geophys. Res.: Oceans, 104, No. C9 (1999), 20467–20494. doi: 10.1029/1999JC900164. [19] J. F. Price, R. A. Weller and R. R. Schudlich, Wind-driven ocean currents and Ekman transport, Science, 238 (1987), 1534-1538.  doi: 10.1126/science.238.4833.1534. [20] L. Roberti, Perturbation analysis for the surface deflection angle of Ekman-type flows with variable eddy viscosity, J. Math. Fluid Mech., 23, No. 57 (2021), 1–7. doi: 10.1007/s00021-021-00586-y. [21] J. Röhrs and K. H. Christensen, Drift in the uppermost part of the ocean, Geophys. Res. Lett., 42 (2015), 10349-10356.  doi: 10.1002/2015GL066733. [22] V. I. Shrira and R. B. Almelah, Upper-ocean Ekman current dynamics: a new perspective, J. Fluid Mech., 887 (2020), 33 pp. doi: 10.1017/jfm.2019.1059. [23] V. I. Shrira and R. B. Almelah, What do we need to probe upper ocean stratification remotely?, Radio. Quant. Electron., 63 (2020), 1-20.  doi: 10.1007/s11141-020-10030-2. [24] G. K. Vallis, Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation, 2nd Edition, Cambridge University Press, Cambridge, 2017. [25] W. Wang and R. X. Huang, Wind energy input to the Ekman layer, J. Phys. Oceanogr., 34 (2004), 1267-1275.  doi: 10.1175/1520-0485(2004)034<1267:WEITTE>2.0.CO;2. [26] 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.

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##### References:
 [1] 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. [2] A. Constantin, Frictional effects in wind-driven ocean currents, Geophys. Astro. Fluid Dyn., 115 (2021), 1-14.  doi: 10.1080/03091929.2020.1748614. [3] A. Constantin and R. I. Ivanov, Equatorial wave-current interactions, Comm. Math. Phys., 370 (2019), 1-48.  doi: 10.1007/s00220-019-03483-8. [4] A. Constantin and R. S. Johnson, The dynamics of waves interacting with the Equatorial Undercurrent, Geophys. Astro. Fluid Dyn., 109 (2015), 311-358.  doi: 10.1080/03091929.2015.1066785. [5] A. Constantin and R. S. Johnson, Atmospheric Ekman flows with variable eddy viscosity, Bound. Layer Meteorol., 170 (2019), 395-414.  doi: 10.1007/s10546-018-0404-0. [6] 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, 15 pp. doi: 10.1063/1.5083088. [7] A. Constantin and R. S. Johnson, On the modelling of large-scale atmospheric flows, J. Differ. Equ., 285 (2021), 751-798.  doi: 10.1016/j.jde.2021.03.019. [8] M. F. Cronin and W. S. Kessler, Near-surface shear flow in the tropical Pacific cold tongue front, J. Phys. Oceanogr., 39 (2009), 1200-1215.  doi: 10.1175/2008JPO4064.1. [9] D. G. Dritschel, N. Paldor and A. Constantin, The Ekman spiral for piecewise-uniform viscosity, Ocean Sci., 16 (2020), 1089-1093.  doi: 10.5194/os-16-1089-2020. [10] V. W. Ekman, On the influence of the Earth's rotation on ocean-currents, Ark. Mat. Astron. Fys., 2 (1905), 1-52. [11] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th Edition, Academic Press, New York, NY, 2007. [12] B. Grisogono, A generalized Ekman layer profile with gradually varying eddy diffusivities, Quart. J. Roy. Meteorol. Soc., 121 (1995), 445-453.  doi: 10.1002/qj.49712152211. [13] D. Ionescu-Kruse, Analytical atmospheric Ekman-type solutions with height-dependent eddy viscosities, J. Math. Fluid Mech., 23 (2021), 1-11.  doi: 10.1007/s00021-020-00543-1. [14] D. M. Lewis and S. E. Belcher, Time-dependent, coupled, Ekman boundary layer solutions incorporating Stokes drift, Dyn. Atmospheres Oceans, 37 (2004), 313-351.  doi: 10.1016/j.dynatmoce.2003.11.001. [15] T. Lyons, Variable eddy viscosities in the atmospheric boundary layer from ageostrophic wind-speed profiles, J. Math. Fluid Mech., 23 (2021), 16 pp. doi: 10.1007/s00021-021-00575-1. [16] 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. [17] A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, 2$^{\text{nd}}$ Edition, CRC Press LLC, Boca Raton, FL, 2003. [18] J. F. Price and M. A. Sundermeyer, Stratified Ekman layers, J. Geophys. Res.: Oceans, 104, No. C9 (1999), 20467–20494. doi: 10.1029/1999JC900164. [19] J. F. Price, R. A. Weller and R. R. Schudlich, Wind-driven ocean currents and Ekman transport, Science, 238 (1987), 1534-1538.  doi: 10.1126/science.238.4833.1534. [20] L. Roberti, Perturbation analysis for the surface deflection angle of Ekman-type flows with variable eddy viscosity, J. Math. Fluid Mech., 23, No. 57 (2021), 1–7. doi: 10.1007/s00021-021-00586-y. [21] J. Röhrs and K. H. Christensen, Drift in the uppermost part of the ocean, Geophys. Res. Lett., 42 (2015), 10349-10356.  doi: 10.1002/2015GL066733. [22] V. I. Shrira and R. B. Almelah, Upper-ocean Ekman current dynamics: a new perspective, J. Fluid Mech., 887 (2020), 33 pp. doi: 10.1017/jfm.2019.1059. [23] V. I. Shrira and R. B. Almelah, What do we need to probe upper ocean stratification remotely?, Radio. Quant. Electron., 63 (2020), 1-20.  doi: 10.1007/s11141-020-10030-2. [24] G. K. Vallis, Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation, 2nd Edition, Cambridge University Press, Cambridge, 2017. [25] W. Wang and R. X. Huang, Wind energy input to the Ekman layer, J. Phys. Oceanogr., 34 (2004), 1267-1275.  doi: 10.1175/1520-0485(2004)034<1267:WEITTE>2.0.CO;2. [26] 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.
Plots of $\tan(\theta_0(t))$ (left) and the (normalized) velocity components (right) for $K(t) = c\vert\tau(t)\vert$ and $\tau(t) = 2- \mathrm{e}^{-t^2}$
Plots of $\tan(\theta_0(t))$ (left) and the (normalized) velocity components (right) for $\tau(t) = 1+ \mathrm{e}^{-t^2}$
Plots of $\tan(\theta_0(t))$ (left) and the (normalized) velocity components (right) for $\tau(t) = 1+2t^2\left(1-\frac{3}{4}t^2\right) \mathrm{e}^{-t^2}$
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