Effect of viscosity on gravity waves and the upper boundary condition

Abstract

The linearized problem of two-dimensional gravity waves in a viscous incompressible stratified fluid occupying the upper half-space z > 0 is investigated. It is assumed that the dynamic viscosity coefficient μ is constant and that the density distribution ρ(z) is exponential. This leads to a fourth-order differential equation in the z co-ordinate, the coefficients of which depend on ρ(z) and on a dimensionless parameter ε which is proportional to μ/σ, σ being the frequency of the oscillation. The problem is solved for small ε. It is found that there is a region in which the solutions behave like certain solutions of the inviscid problem (with ε = 0). However, when the solutions of the inviscid problem are wave-like in z, they do not satisfy the radiation condition. This is because the viscosity, in addition to damping the motion for large z, reflects waves. The appropriate solution of the inviscid problem consists, therefore, of an incident and a reflected wave. As μ → 0, the ratio of the amplitudes of the reflected and the incident wave approaches exp (− 2π2H/Λ), where Λ is the vertical wavelength, and H the density scale height. The solution, however, does not have a limit since the reflecting layer shifts, altering the phase of the reflected wave. The results of the analysis are supplemented by a number of numerically computed solutions, which are then used to discuss the validity of the linearization.