Tidal propagation in strongly convergent channels

TLDR

Analytic solutions for tidal propagation in strongly convergent channels diverge from classical cooscillating tide theory. Scaling the governing equations shows that first‑order dynamics are dominated by cross‑sectional area gradients and friction, yielding a first‑order wave equation, while second‑order terms introduce finite‑amplitude effects, wave‑propagation velocity gradients, and local acceleration. The solutions predict constant‑amplitude, 90° phase‑shifted waves with speeds near the frictionless wave speed, and second‑order harmonics whose amplitudes grow or decay with channel convergence, matching observations and linking phase speed, amplitude growth, and tidal harmonics through morphodynamic feedback.

Abstract

Simple first‐ and second‐order analytic solutions, which diverge markedly from classical views of cooscillating tides, are derived for tidal propagation in strongly convergent channels. Theoretical predictions compare well with observations from typical examples of shallow, “funnel‐shaped” tidal estuaries. A scaling of the governing equations appropriate to these channels indicates that at first order, gradients in cross‐sectional area dominate velocity gradients in the continuity equation and the friction term dominates acceleration in the momentum equation. Finite amplitude effects, velocity gradients due to wave propagation, and local acceleration enter the equations at second order. Applying this scaling, the first‐order governing equation becomes a first‐order wave equation, which is inconsistent with the presence of a reflected wave. The solution is of constant amplitude and has a phase speed near the frictionless wave speed, like a classical progressive wave, yet velocity leads elevation by 90°, like a classical standing wave. The second‐order solution at the dominant frequency is also a unidirectional wave; however, its amplitude is exponentially modulated. If inertia is finite and convergence is strong, amplitude increases along channel, whereas if inertia is weak and convergence is limited, amplitude decays. Compact solutions for second‐order tidal harmonics quantify the partially canceling effects of (1) time variations in channel depth, which slow the propagation of low water, and (2) time variations in channel width, which slow the propagation of high water. Finally, it is suggested that phase speed, along‐channel amplitude growth, and tidal harmonics in strongly convergent channels are all linked by morphodynamic feedback.