Tidal Conversion by Subcritical Topography

Analytical estimates of the rate at which energy is extracted from the barotropic tide at topography and converted into internal gravity waves are given. The ocean is idealized as an inviscid, vertically unbounded fluid on the f plane. The gravity waves are treated by linear theory and freely escape to z = ∞. Several topographies are investigated: a sinusoidal ripple, a set of Gaussian bumps, and an ensemble of “random topographies.” In the third case, topographic profiles are generated by randomly selecting the amplitudes of a Fourier superposition so that the power spectral density is similar to that of submarine topography. The authors' focus is the dependence of the conversion rate (watts per square meter of radiated power) on the amplitude of the topography, h0, and on a nondimensional parameter ϵ∗, defined as the ratio of the slope of an internal tidal ray to the maximum slope of the topography. If ϵ∗ ≪ 1, then Bell's theory indicates that the conversion is proportional to h20. The results span the interval 0 ≤ ϵ∗ < 1 and show that the enhancement above Bell's prediction is a smoothly and modestly increasing function of ϵ∗: For ϵ∗ → 1, the conversion of sinusoidal topography is 56% greater than Bell's ϵ∗ ≪ 1 estimate, while the enhancement is only 14% greater for a Gaussian bump. With random topography, the enhancement at ϵ∗ = 0.95 is typically about 6% greater than Bell's formula. The ϵ∗ ≪ 1 approximation is therefore quantitatively accurate over the range 0 < ϵ∗ < 1, implying that the conversion is roughly proportional to h20. As ϵ∗ is increased, the radiated waves develop very small spatial scales that are not present in the underlying topography and, when ϵ∗ approaches unity, the associated spatial gradients become so steep that overturns must occur even if the tidal amplitude is very weak. The solutions formally become singular at ϵ∗ = 1, in a breakdown of linear, inviscid theory.