Abstract
Abstract. We report results gained with a three-dimensional, semi-implicit, semi-spectral model of the shallow water equations on the rotating Earth that allowed one to compute the wind-induced motion in lakes. The barotropic response to unidirectional, uniform winds, Heaviside in time, is determined in a rectangular basin with constant depth, and in Lake Constance, for different values and vertical distributions of the vertical eddy viscosities. It is computationally demonstrated that both the transitory oscillating, as well as the steady state current distribution, depends strongly upon the absolute value and vertical shape of the vertical eddy viscosity. In particular, the excitation and attenuation in time of the inertial waves, the structure of the Ekman spiral, the thickness of the Ekman layer, and the exact distribution and magnitude of the upwelling and downwelling zones are all significantly affected by the eddy viscosities. Observations indicate that the eddy viscosities must be sufficiently small so that the oscillatory behaviour can be adequately modelled. Comparison of the measured current-time series at depth in one position of Lake Constance with those computed on the basis of the measured wind demonstrates fair agreement, including the rotation-induced inertial oscillation.Key words. Oceanography: general (limnology) – Oceanography: physical (Coriolis effects; general circulation)
Highlights
Knowledge of water movements is a prerequisite for the study of a multitude of water quality problems of natural and artificial lakes
The computation of the current distribution in homogeneous lakes is commonly performed with the shallow water equations on the rotating Earth
It is common knowledge that numerical codes integrating these spatially three-dimensional shallow water equations, explicitly in time, are conditionally stable; due to the explicit treatment of the friction terms, the time step is limited by the smallest mesh size
Summary
With regard to non-linear advection terms, if the second-order centered finite difference scheme is used (or other traditional high-order schemes, e.g. the spectral method used here), numerical oscillations always occur. For most of these works, if the threedimensional field equations including the nonlinear advection terms are numerically solved, the inertial waves seem to be hardly observable in the numerical results, or they remain only for a fairly short time and are rapidly damped out. This fact is due to the much larger eddy viscosities used in the models in order to restrain the numerical (not physical) oscillations and assure numerical stability.
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