Numerical Solution of the Boundary Value Problem for Internal Inertia-Gravity Waves

Abstract

The initial and boundary value problem for the equations of free internal inertia-gravity waves in an unconfined basin of constant depth is numerically calculated in the Boussinesq approximation in the presence of a two-dimensional, vertically-inhomogeneous flow. The boundary value problem for the vertical velocity amplitude includes complex coefficients and is solved both numerically and within the framework of perturbation theory. With reference to the example of the calculations of the decay rate of internal waves and wave-induced momentum fluxes it is shown that the exact numerical calculations provide considerably better estimates than those obtained using the perturbation method. In particular, at minimum disagreement of the dispersion curves obtained using the two calculation methods the imaginary parts of the wave frequency interpreted as the decay rates can differ by two-three orders. The vertical wave-induced momentum fluxes are comparable with turbulent fluxes and can be even greater than those. In this case, the results obtained using numerical methods are almost an order smaller than those calculated by the method of perturbation theory.