The Attenuation of Short Gravity Waves Propagating in Turbulent Conditions

Abstract

Summary An application of boundary layer theory is made to determine the attenuation of progressive waves in fluid of infinite depth. The turbulence in the flow is expressed mathematically by means of a system of (variable) coefficients of eddy viscosity. The principal contribution to the damping is found to be independent of surface turbulence and dependent on integrals involving the eddy viscosities over the depth of the fluid. Boundary layer theory has been used by Johns (1968a) and Dore (1968) to develop methods for the determination of viscous damping in a variety of complex oscillatory wave flows involving fluids of small viscosities. In particular, Johns (1968b) has applied his method to calculate the spatial attenuation of small amplitude gravity waves propagating in turbulent conditions in shuZlow water. The effect of the turbulence was represented by coefficients of eddy viscosity which were assumed to depend on the vertical co-ordinate only, whilst the turbulence was supposed to be present only in thin boundary layers adjacent to the rigid horizontal bottom and the free surface. It was then found that the leading term in the damping was independent of surface turbulence and thus resulted solely from energy dissipation in the bottom boundary layer. In the present work, we determine formulae for both the spatial and temporal attenuation of short gravity waves propagating in turbulent conditions. The turbulence will also be represented by coefficients of eddy viscosity which may, however, be such that (i) the turbulence exists throughout the fluid and (ii) the turbulence is weakly dependent on the horizontal co-ordinate and the time. The analytical method we adopt will be based on that of Dore (1968), which was presented to avoid certain unnecessary algebraical complications inherent in the formulation of Johns (1968a). It is again found that the damping is independent of surface turbulence and depends on integrals involving the eddy viscosities over the depth of the fluid. If, therefore, the turbulence is confined to the surface boundary layer, the main contribution to the damping is identical to that of laminar flow.