A kinetic theory for internal waves in a randomly stratified fluid

We discuss the transport of energy of internal waves propagating in a stratified unbounded fluid with randomly varying buoyancy frequency N of the form $\text{N}{}^2\text{= N}{}_0{}^2\text{[1 + ?Ξ(}\textcolor{red}{}\text{)]}$. Here N₀ = constant, 0 < ? ? 1 and Ξ is a zero-mean stationary random function of $\textcolor{red}{}\text{= (x,z)}$ where x and z are respectively horizontal and vertical coordinates. In the limit of small ?, a linear kinetic equation (transport equation) is derived which describes the space—time evolution of the mean wave energy density in such a medium. When it is integrated over all wave number space, the kinetic equation implies the total conservation of wave energy. The approach used is reminiscent of the nonlinear wave interaction theories of Phillips (1960), Benny (1962) and Hasselmann (1966) and others, in which the random microstructure Ξ would be regarded as a nonpropagating (zero-frequency) random wave field. Our analysis is based on the Eulerian equations of motion in which no a priori assumptions are made regarding the scattered wave field — on the contrary, it is a deduction of the theory presented here that only the propagating internal wave modes participate in the energy exchange processes. In particular we do not assume that the internal wave field constitutes a homogenous assembly of random wave packets evolving in time alone — unlike much of the earlier work — and this enables us to treat the scattering of individual internal waves by the microstructure.The kinetic equation is used to determine the energy transmitted through and reflected by a horizontally oriented random slab which models a layer of microstructure of finite thickness in the ocean. Specifically, we show that significant reflection can occur when Ψ(2k_0z) is sufficiently large, where Ψ is the vertical wavenumber spectrum of the microstructure fluctuations Ξ(z) and k_0z is the vertical wavenumber of the incident wave. We also show that the reflection coefficient increases monotonically with increasing frequency, which is in qualitative agreement with recent measurements at site D(39°N, 70°W) which indicate that in regions where density inhomogeneities are present, the vertical coherence decreases with increasing frequency. Actual numerical estimates for the reflection coefficient are obtained for vertical microstructure data from station P(50°N, 145°W). It is found that for intermediate wavelengths — 0(10²m) — and a broad band of frequencies (0.6 ? ω/N₀ < 1), the reflection coefficient is greater than 0.5. Finally, the qualitative behaviour of the kinetic equation for two-dimensional microstructure is examined in the geometric optics limit: wavelength much less than (the integral) correlation scale. In this case the integro-differential kinetic equation reduces to a Fokker—Planck diffusion equation. From the latter we infer that at high frequencies, a wave packet becomes incoherent after propagating a distance that is less than a typical correlation scale associated with the fluctuations Ξ.