Propagation of a finite-amplitude potential vorticity front along the wall of a stratified fluid

Abstract

A similarity solution to the long-wave shallow-water equations is obtained for a density current (reduced gravity = g′, Coriolis parameter = f) propagating alongshore (y = 0). The potential vorticity q = f/H₁ is uniform in −∞ < x [les ] x_nose(t), 0 < y [les ] L(x, t), and the nose of this advancing potential vorticity front displaces fluid of greater q = f/H₀, which is located at L < y < ∞. If L₀ = L(−∞, t), the nose point with L(x_nose(t), t) = 0 moves with velocity U_nose = √g′H₀ φ, where φ is a function of H₁/H₀, f²L²₀/g′H₀. The assumptions made in the similarity theory are verified by an initial value solution of the complete reduced-gravity shallow-water equations. The latter also reveal the new effect of a Kelvin shock wave colliding with a potential vorticity front, as is confirmed by a laboratory experiment. Also confirmed is the expansion wave structure of the intrusion, but the observed values of U_nose are only in qualitative agreement; the difference is attributed to the presence of small-scale (non-hydrostatic) turbulence in the laboratory experiment but not in the numerical solutions.