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Large-Eddy Simulations and Damped-Oscillator Models of the Unsteady Ekman Boundary Layer

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  • 1 Department of Civil and Environmental Engineering, Princeton University, Princeton, New Jersey
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Abstract

The Ekman boundary layer (EBL) is a central problem in geophysical fluid dynamics that emerges when the pressure gradient force, the Coriolis force, and the frictional force interact in a flow. The unsteady version of the problem, which occurs when these forces are not in equilibrium, is solvable analytically only for a limited set of forcing variability regimes, and the resulting solutions are intricate and not always easy to interpret. In this paper, large-eddy simulations (LESs) of neutral atmospheric EBLs are conducted under various unsteady forcings to reveal the range of physical characteristics of the flow. Subsequently, it is demonstrated that the dynamics of the unsteady EBL can be reduced to a second-order ordinary differential equation that is very similar to the dynamical equation of a damped oscillator, such as a mass–spring–damper system. The validation of the proposed reduced model is performed by comparing its analytical solutions to LES results, revealing very good agreement. The reduced model can be solved for a wide range of variable forcing conditions, and this feature is exploited in the paper to elucidate the physical origin of the inertia (mass), energy storage (spring), and energy dissipation (damper) attributes of Ekman flows.

Supplemental information related to this paper is available at the Journals Online website: http://dx.doi.org/10.1175/JAS-D-15-0038.s1.

Corresponding author address: Elie Bou-Zeid, E-414 E-Quad, Dept. of Civil and Environmental Engineering, Princeton University, 59 Olden Street, Princeton, NJ 08544. E-mail: ebouzeid@princeton.edu

Abstract

The Ekman boundary layer (EBL) is a central problem in geophysical fluid dynamics that emerges when the pressure gradient force, the Coriolis force, and the frictional force interact in a flow. The unsteady version of the problem, which occurs when these forces are not in equilibrium, is solvable analytically only for a limited set of forcing variability regimes, and the resulting solutions are intricate and not always easy to interpret. In this paper, large-eddy simulations (LESs) of neutral atmospheric EBLs are conducted under various unsteady forcings to reveal the range of physical characteristics of the flow. Subsequently, it is demonstrated that the dynamics of the unsteady EBL can be reduced to a second-order ordinary differential equation that is very similar to the dynamical equation of a damped oscillator, such as a mass–spring–damper system. The validation of the proposed reduced model is performed by comparing its analytical solutions to LES results, revealing very good agreement. The reduced model can be solved for a wide range of variable forcing conditions, and this feature is exploited in the paper to elucidate the physical origin of the inertia (mass), energy storage (spring), and energy dissipation (damper) attributes of Ekman flows.

Supplemental information related to this paper is available at the Journals Online website: http://dx.doi.org/10.1175/JAS-D-15-0038.s1.

Corresponding author address: Elie Bou-Zeid, E-414 E-Quad, Dept. of Civil and Environmental Engineering, Princeton University, 59 Olden Street, Princeton, NJ 08544. E-mail: ebouzeid@princeton.edu

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