The slow downstream (x) variation of a dense and inviscid bottom current (u) in a parabolic strait with a sill at y = 0 is investigated. Vanishing potential vorticity is assumed and the density interface in the 1 1/2-layer model intersects the bottom at y = y1 and y = y2 < y1, where the vanishing layer thickness (h) provides the free dynamical boundary condition. For time-dependent finite-amplitude waves, the nonlinear hyperbolic equations obtained here give the wave velocity and indicate the sense in which lateral wave steepening occurs. The long-wave perturbations of y1(x,t), y2(x,t) are stationary if
where g′ is the reduced gravity, μ = ∂2M/∂y2 is the parabolic curvature of the bottom elevation (M), and f is the Coriolis parameter. This controls the upstream–downstream flow, and the downstream nonlinearity generates ‘short’ waves which may initiate lateral mixing with the adjacent (less dense) water mass.
It is also shown that short waves are exponentially amplified with a maximum growth rate (about 1/day) depending only on g′μ/f2. When g′μ/f2 = 1 (a narrow strait) the instability is suppressed, but for small g′μ/f2≪1 the growth rate is comparable to the flat bottom case μ = 0, studied by Griffiths, Killworth & Stern (J. Fluid Mech. Vol. 117, 1982, p. 343.).