Dispersion and kinematics of multi-layer non-hydrostatic models

- Conversion of multi-layer framework into Boussinesq form.
- Function of dispersion effects in tapering flow profile.
- Analytical solution of dispersion relation for arbitrary number of layers.
- Convergence of wave dispersion and kinematics as well as their inter-relationship from shallow to extreme deep water.
- Implementation criteria for numerical applications oriented to both wave celerity and velocity profile.

Multi-layer non-hydrostatic models are gaining popularity in studies of coastal wave processes owing to the resolution of the flow kinematics, but the linear dispersion relation remains the primary criterion for assessment of model convergence. In this paper, we reformulate the linear governing equations of an N-layer model into Boussinesq form by writing the non-hydrostatic terms as high-order derivatives of the horizontal flow velocity. The equation structure allows implementation of Fourier analysis to provide a [2N−2, 2N] expansion of the velocity at each layer. A variable transformation converts the governing equations into separate flux- and dispersion-dominated systems, which explicitly give an equivalent Pade´ expansion of the wave celerity for examination of the convergence and asymptotic properties. Flow continuity equates the depth-integrated horizontal velocity to the celerity and verifies the analytical solution. The surface-layer velocity, which is driven by the kinematic free surface boundary condition, shows a positive error and converges monotonically to the solution of Airy wave theory. When the depth parameter kd > 2N, flow reversal occurs in the sub-surface layers to offset overestimation of the surface velocity and to better approximate the flux. This model internal mechanism facilitates convergence of the celerity at large kd and benefits applications on wave transformation. Such non-physical flow reversal, however, might complicate studies that require detailed wave kinematics.