PROGRAM | Ocean Engineering

By: Zhifei Dong Chair: James Kirby

ABSTRACT

Existing theories of wave-current interaction mostly assume that waves propagate on currents which vary weakly in the vertical direction. In most cases, the assumption is satisfied. However, at the mouth of rivers with very energetic discharge such as the Columbia River, the current becomes strongly sheared due to stratification and tidal effects. The wave-current interaction for waves on strongly sheared current needs to be discussed. In this study, a new theory is first developed to formulate the interaction of small-amplitude surface gravity waves with strongly sheared current in finite-depth water. In contrast to existing formulations, where waves at the leading order respond to a depth-uniform current field, the present formulation allows for an arbitrary degree of vertical current shear, leading to a description of wave vertical structure in terms of solutions to the Rayleigh stability equation. The Rayleigh equation is then solved using both numerical and perturbation methods. The perturbation solutions are recommended in numerical modeling to avoid directly solving Rayleigh equation for each coupling time step (Dong and Kirby, 2012). As a special case, the constantly sheared current profile is used to provide analytical wave solutions and to evaluate the performance of numerical solver and different orders of perturbation solutions. The leading order wave vorticity for constantly sheared current is discussed. The magnitude of wave vorticity is determined by the current vertical shear and the oblique wave angle to current direction. Wave orbital velocity and vorticity are calculated using the current velocity profile measured at the mouth of Columbia River (MCR). The comparison of numerical solutions with perturbation solutions suggests that the second order perturbation solution successfully captures the features of current shear effect on the wave vertical structures. Furthermore, the solvability condition for the second order inhomogeneous Rayleigh equation leads to the wave action conservation equation. The wave action equation is evaluated using wave solutions for constantly sheared currents. Results show that both the wave action and action flux are modified by the current vertical shear effect. For the mean flow part, the wave-averaged forcing terms for the description of the mean flow dynamics are presented using the Craik-Leibovich vortex force formalism. The present vortex force formulation is compared with the Uchiyama et al. (2010) formulation using wave solutions for constantly sheared current. The wave-current interaction theory is then applied to a coupled system of NHWAVE and SWAN, which extends the existing formalism to include the strong shear effect. Three cases have been tested in the numerical application part. Test case (1) is the obliquely incident waves on a planar beach. As waves propagate onshore with an oblique angle, wave breaks near the beach face and generates offshore undertow in the cross-shore direction and strong current in the alongshore direction. This case is used to compare the present vortex force formulation with Uchiyama et al. (2010) without activating SWAN. The waves are provided in an input file. Test case (2) is wave propagation on highly stratified, vertically sheared current at the mouth of Columbia River (MCR). During the ebb tide, the strong fresh water discharge and ebb tidal current creates a strongly sheared offshore current at MCR. The strongly sheared current meets and interacts with the incident waves. This is case designed for the analysis of wave effects on currents and current effects on waves. Both NHWAVE and SWAN are activated and coupled in the simulation. Test case (3) is the formation of Langmuir circulation in the presence of wind-driven current and waves. The water surface wind creates a wind-driven flow, which interacts with surface gravity waves also propagating in downwind direction. The wave-current interaction generates Langmuir cells in cross-wind direction. This case is another application of the present formulation.

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