Surface gravity waves in the presence of vertically shearing current and eddy viscosity

Abstract

Conventional models of surface gravity waves usually use potential theory in their calculations where the governing equations are the inviscid Euler equations. Nevertheless, turbulence may play a significant role for short waves or highly energetic sea or river conditions and, therefore, must be taken into account in the flow's formulation. This work extends the model of linear monochromatic waves in the presence of vertically shearing currents to account for turbulence in the form of eddy viscosity varying in depth. The boundary value problem is found to be governed by an augmented Orr–Sommerfeld(–Squire) equation(s) in two-(three-)dimensional with additional terms of viscosity derivatives. The free-surface conditions are extended to account for shearing currents with turbulent viscosity and external stress for the air layer. This provides a fundamental model for investigating the influence of the turbulent viscosity on the oscillatory wave flow. Examples of semi-analytical and numerical solutions show a fundamentally different dynamical behavior with respect to the known non-viscous solutions. These differences include two regimes in the dispersion relation with non-dispersive short waves and dispersive longer waves with a reduced celerity. They also include tilted stream function contours (i.e., a phase which is a function of depth), much deeper penetration depth, and more. These preliminary results show that turbulence potentially has great importance in wave generation mechanisms as well as in highly turbulent regions such as under strong storms or cyclones.