دانلود رایگان مقاله بررسی خواص موج چگالی انرژی، شار انرژی، تنش تابش و رانش استوکس

Abstract

This work analyses basic wave properties originating from the interaction between waves and submerged rigid vegetation. First of all, an analytical framework is presented that describes the propagation and dissipation of waves over a rigid and submerged canopy, where the flow resistance is linearised. A nonlinear closure term is introduced to ensure that the work done by the linearised flow resistance equals that of the nonlinear flow resistance. The anisotropic flow resistance is found to have an impact on both the distribution of velocities and pressure inside the canopy and it partly explains the small decrease in flow velocities inside the canopy, which was previously observed experimentally. The following second order wave properties are derived: the wave energy density, the wave energy flux, the vegetated group velocity of the wave energy density, the radiation stress components parallel and perpendicular to the direction of wave propagation, the Eulerian and Lagrangian Stokes velocities and fluxes. The additional Stokes drift due to the discontinuity in the velocity field at the top of the vegetation is derived; the inclusion of this mass flux in the Lagrangian formulation of the Stokes drift is important for the ratio between the Lagrangian and Eulerian Stokes drifts. The relation between the wave energy density and the wave energy flux, i.e. the vegetated group velocity of the wave energy density, is of practical importance for large scale wave modelling. The modification to the vegetated group velocity relative to that derived from linear wave theory on non-dissipative waves is described. It is seen that the corrections to linear wave theory are of Hγ, where γ is in the interval 1.5–2.0.

9. Conclusion

The present work presents a description of the propagation of waves in the interior of a submerged canopy with anisotropic flow resistance; diffusion was neglected. The theory is based on the assumptions that boundary layer effects on top of the canopy can be neglected, that the waves are (almost) linear and there is no wave breaking. Furthermore, it is assumed that the outer flow is irrotational, while the inner flow is subject to friction. The second order wave properties such as wave energy density, wave energy flux, the corresponding vegetated group velocity, the radiation stress tensor and the Eulerian and Lagrangian Stokes drifts were derived. It was shown that the vegetated group velocity does not equal the classical group velocity from non-dissipative linear wave theory and there is an non-equal splitting between the potential, Ep, and kinetic, Ek, wave energy densities. Both of these properties have a direct implication for the large scale modelling of waves in vegetation, since they can cause shoaling, refraction and conservative decay in the wave height due to changes in the ratio Ek/Ep. The derivations of the Stokes drift have shown that the Eulerian and Lagrangian formulations are identical even in the presence of vegetation, but it is important to include the Lagrangian flux at the top of the canopy due to the discontinuity in the horizontal velocity. The expressions are not easily comparable due to their complexity, but through an evaluation of the finite Stokes velocity at the free surface, it was seen that the horizontal Eulerian and Lagrangian volume fluxes are identical. The Stokes drift increases in the presence of vegetation. The finite vertical Stokes drift must be balanced by a gradient in the Eulerian, mean velocity field (return flow) in the limit of a steady solution. It was hypothesised that the Stokes velocity is established faster than the return flow, thus the vertical Stokes velocity is (partly) responsible for the mass flux needed to establish the mean waveinduced setup. This mean wave-induced setup is required to balance the gradient in the radiation stress tensor. Further validation of the theory is needed. The validation material should preferably include data on (i) spatial variation in the surface elevation and (ii) orbital velocities over the height of the canopy for rigid stems. This will allow for a direct evaluation of the drag coefficients in a reduced velocity field.