Tidal conversion at a very steep ridge

We obtain an analytic solution for the generation of internal gravity waves by tidal flow past a vertical barrier of height b in a uniformly stratified ocean of depth h\,>\,b and buoyancy frequency N . The radiated power (watts per metre of barrier) is 1/4 π ρ₀ b² U² N √1-(f/ω)²M(b/h), where ρ₀ is the mean density of seawater, U cos (ω t) the tidal velocity, and f the Coriolis frequency. The function $M(b/h)$ increases monotonically with M(0)\,=\,1 , M(0.92)\,=\,2 and M(1)\,=\,∞ . As b/h \,→\, 1 , M diverges logarithmically and consequently the radiated power grows as ln[(h-b)/b] . We also calculate the conversion in a realistically stratified ocean with strongly non-uniform buoyancy frequency, $N(z)$ . A rough approximation to the radiated power in this case is 1/4 π ρ₀ b² U² N(b) √1- (f/ω)² M(B/π), where $N(b)$ is the buoyancy frequency at the tip of the ridge and B is the height of the ridge in WKB coordinates. (The WKB coordinate is normalized so that the total depth of the ocean is π .) The approximation above is an over-estimate of the actual radiation by as much as 20% when B/π \,≈\, 0.8 . But the formula correctly indicates the strong dependence of conversion on stratification through the factor $N(b)$ .