Higher-order Boussinesq-type equations for surface gravity waves: derivation and analysis

Boussinesq-type equations of higher order in dispersion as well as in nonli nearity are derived for waves land wave-current interaction) over an uneven bottom. Formulations are given in terms of various velocity variables such as the depth-averaged velocity and the particle velocity at the still wate r level, and at an arbitrary vertical location. The equations are enhanced and analysed with emphasis on linear dispersion, shoaling and nonlinear pro perties for large wavenumbers. As a starting point the velocity potential is expanded as a power series in the vertical coordinate measured from the still water level (SWL). Substit uting this expansion into the Laplace equation leads to a velocity field ex pressed in terms of spatial derivatives of the vertical velocity (w) over c ap and the horizontal velocity vector (u) over cap at the SWL. The series e xpressions are given to infinite order in the dispersion parameter, mu. Sat isfying the kinematic bottom boundary condition defines an implicit relatio n between (w) over cap and (u) over cap, which is recast as an explicit rec ursive expression for (w) over cap in terms of (u) over cap under the assum ption that mu << 1. Boussinesq equations are then derived from the dynamic and kinematic boundary conditions at the free surface. In this process the infinite series solutions are truncated at O(mu(6)), while all orders of th e nonlinearity parameter, epsilon are included to that order in dispersion. This leads to a set of higher-order Boussinesq equations in terms of the s urface elevation eta and the horizontal velocity vector (u) over cap at the SWL. The equations are recast in terms of the depth-averaged velocity, U leaving out O(epsilon(2)mu(4)), which corresponds to assuming epsilon = O(mu). Thi s formulation turns out to include singularities in linear dispersion as we ll as in nonlinearity. Next, the technique introduced by Madsen et al. in 1 991 and Schaffer & Madsen in 1995 is invoked, and this results in It set of enhanced equations formulated in U and including O(mu(4), epsilon mu(4)) t erms. These equations contain no singularities and the embedded linear and nonlinear properties are shown to be significantly improved. To quantify th e accuracy, Stokes's third-order theory is used as reference and Fourier an alyses of the new equations are carried out to third order tin nonlinearity ) for regular waves on a constant depth and to first order for shoaling cha racteristics. Furthermore, analyses are carried out to second order for bic hromatic waves and to first order for waves in ambient currents. These anal yses are not restricted to small values of the linear dispersion parameter, mu. In conclusion, the new equations are shown to have linear dispersion c haracteristics corresponding to a Pade [4,4] expansion in k'h' (wavenumber times depth) of the squared celerity according to Stokes's linear theory. T his corresponds to a quite high accuracy in linear dispersion up to approxi mately k'h' = 6. The high quality of dispersion is also achieved for the Do ppler shift in connection with wave-current interaction and it allows for a study of wave blocking due to opposing currents. Also, the linear shoaling characteristics are shown to be excellent, and the accuracy of nonlinear t ransfer of energy to sub- and superharmonics is found to be superior to pre vious formulations. The equations are then recast in terms of the particle velocity, (u) over t ilde, at an arbitrary vertical location including O(mu(4), epsilon(5)mu(4)) terms. This formulation includes, as special subsets, Boussinesq equations in terms of the bottom velocity or the surface velocity Furthermore, the a rbitrary location of the velocity variable can be used to optimize the embe dded linear and nonlinear characteristics. A Fourier analysis is again carr ied out to third order (in epsilon) for regular waves. It turns out that Pa de [4,4] linear dispersion characteristics can not be achieved for any choi ce of the location of the velocity variable. However, for an optimized loca tion we achieve fairly good linear characteristics and very good nonlinear characteristics. Finally the formulation in terms of ii is modified by introducing the techn ique of dispersion enhancement while retaining only O(mu(2), epsilon(3)mu(2 )) terms. Now the resulting set of equations do show Pade [4,4] dispersion characteristics in the case of pure waves as well as in connection with amb ient currents, and again the nonlinear properties (such as second- and thir d-order transfer functions and amplitude dispersion) are shown to be superi or to those of existing formulations of Boussinesq-type equations.