Green's law revisited: Tidal long‐wave propagation in channels with strong topography

Abstract

Green's Law states that tidal long‐wave elevation ζ and tidal transport Q vary with width b and depth h according to ζ ≌ b−1/2h−1/4 and Q ≌ b+1/2h+/4. This solution is of limited utility because it is restricted to inviscid, infinitesimal waves in channels with no mean flow and weak topography (those with topographic scale L ≫ wavelength λ). An analytical perturbation model including finite‐amplitude effects, river flow, and tidal flats has been used to show that (1) wave behavior to lowest order is a function of only two nondimensional parameters representing, respectively, the strength of friction at the bed and the rate of topographic convergence/divergence; (2) two different wave equations with nearly constant coefficients can be derived that together cover most physically relevant values of these parameters, even very strong topography; (3) a single, incident wave in a strongly convergent or divergent geometry may mimic a standing wave by having a ≡ 90° phase difference between Q and ζ and a very large phase speed, without the presence of a reflected wave; (4) channels with strong friction and/or strong topography (L ≪ λ) show very large deviations from Green/s Law; and (5) these deviations arise because both frictional damping and the direct dependence of |Q| and |ζ| on topography (topographic funnelling) must be considered.