A Nonlinear Model of Monochromatic Waves on Steady Currents Over Gradually Varying Bottoms

Abstract

A new set of differential equations with the free surface elevation and the depth-integrated volume flux being the dependent variables is derived for description of monochromatic waves on steady currents over gradually varying bottoms. The derivation is based on the vertical integration of the continuity equation and the equation of motion for general free-surface flows. A similarity law on the dynamic pressure and the horizontal velocity, modified from a relevant relation for small amplitude waves on uniform currents, has been introduced. A finite-difference scheme with second-order accuracy is proposed for solutions of the equations. The computational algorithm is proved to be conditionally stable and easy to implement. Key points on specification of boundary and initial conditions are discussed in details. Computations are carried out to investigate the nonlinear effects on wave shoaling over a uniform slope with following or opposing current in presence. A case test of the proposed model is also performed and satisfactory agreement obtained between the computational results and reported laboratory data.

Keywords:

• Nonlinear wave
• current
• vertically-integrated model
• finite-difference method