Author(s): Victor J. Llorente; Enrique M. Padilla; Manuel Diez-Minguito

Linked Author(s):

Keywords: Wind-driven currents; Ekman’s model; High-order finite difference methods; Observations

Abstract: Ekman (1905)’s theory of wind-driven currents in a surface boundary layer is a building-block of modern oceanography and helps to explain (e.g.) coastal upwelling, which have high impact on biological productivity. Ekman deduced that the momentum balance for steady horizontal currents must be between Coriolis force and divergence of turbulent shear induced by the wind. Assuming an infinite homogeneous water column of a constant eddy viscosity, the vertical profile of the horizontal current features it maximum current at the surface, which is deflected 45⁰ to the right (left) from the wind direction in the Northern (Southern) Hemisphere, and a net transport at 90⁰ to the right (left) of the wind direction. However, Ekman’s model is highly idealized and can not be expected to closely match actual observations of wind-driven circulation. In this contribution, his theory is extended considering (1) a nonhomogeneous ocean of finite depth, (2) realistic vertical eddy viscosity profiles, and (3) a vertically variable horizontal baroclinic pressure gradient. Extensive sensitivity numerical tests of surface deflection angles, maximum currents, and transport direction to changes in (1), (2), and (3) are considered. Numerical tests are performed both by using a classical finite differentiation of the momentum balance equation and by a high-order scheme based on compact differentiation. Compact differentiation was introduced by Lele in the 1990s as a linear combination of derivatives, df(z)/dz, at grid points and its adjacent ones, written as a stencil of grid-point values of f(z). In such a way, compact differentiation is more accurate than that of the classical differentiation due to its wider spectral resolution. Compact differentiation of the extended Ekman’s model leads to an implicit method in which a tridiagonal matrix system is solved to compute df(z)/dz. The results show that numerical performance improves in two orders of magnitude or above and reduce CPU time in almost a ratio of 2.5 seconds at the beginning of the convergence region. Regarding the physical solution, using this numerical approach, the greatest deviations from the Ekman classical solutions are obtained when shallower depth, higher stratification, and higher turbulence stress rate are considered. While acknowledging additional deviations due to other processes to the horizontal momentum balance, the results show that the improved numerical solutions of the extended wind-driven Ekman boundary layer model successfully reproduce observations in several study areas of the Northern Hemisphere.

DOI: https://doi.org/10.3850/IAHR-39WC2521711920221208

Year: 2022