Asymptotic behavior of tidal damping in alluvial estuaries

Tidal wave propagation can be described analytically by a set of four implicit equations, i.e., the phase lag equation, the scaling equation, the damping equation, and the celerity equation. It is demonstrated that this system of equations has an asymptotic solution for an infinite channel, reflecting the balance between friction and channel convergence. Subsequently, explicit expressions for the tidal amplitude and velocity amplitude are derived, which are different from the generally assumed exponential damping equation that follows from linearizing the friction term. Analysis of the asymptotic behavior demonstrates that exponential damping of the tidal amplitude is only correct for a frictionless wave or an ideal estuary (no damping). However, in estuaries with modest damping (near ideal) it provides a reasonable approximation. In natural estuaries, there is generally a need to take account of local variability of, e.g., depth and friction, by subdividing the estuary into multiple reaches. This is illustrated with an example of the Scheldt estuary, which has been gradually deepened for navigation purpose over the last half century. The analytical model is used to study the effect of this deepening on the tidal dynamics in the main navigation channel, demonstrating that the navigation channel will become overamplified when it reaches a depth larger than the critical depth. In the case of overamplification, a further increase of the depth reduces the amplification until critical convergence (condition for a frictionless standing wave) is reached asymptotically. Finally, based on the ratio between the tidal amplitude at the seaward boundary and the asymptotic tidal amplitude, estuaries can be classified into damped, amplified, or ideal estuaries, which is illustrated with 23 real estuaries.