Nonlinear diffusion of the tidal signal in frictionally dominated embayments

Abstract

The dynamics of many shallow tidal embayments may be usefully represented by a single “zero‐inertia” equation for tidal elevation which has the form of a nonlinear diffusion equation. The zero‐inertia equation clarifies the lowest order dynamics, namely, a balance between pressure gradient and friction. It also provides insight into the properties of higher‐order harmonic components via the identification of compact approximate solutions and governing nondimensional parameters. Approximate analytic solutions which assume a constant diffusion coefficient are governed by the nondimensional parameters x/L and ‖k0‖L, where L is the length of the embayment, and ‖k0‖−1 scales both the length of factional dissipation and the physical length of the diffusive waveform. As ‖k0‖L increases, the speed at which the tidal signal diffuses decreases, and the rate of decay of tidal amplitude with distance increases. The parameter ‖k0‖L increases as depth is reduced, friction is increased, forcing amplitude or frequency is increased, or total embayment width is increased relative to the width of the channel. Approximate analytic solutions which assume a time‐varying diffusion coefficient result in additional components at the zeroth, second, and third harmonic frequencies. The zeroth and second harmonics are governed by the parameter γ, as well as x/L and ‖k0‖L. Parameter γ measures the relative importance of time variations of channel depth (γ> 0) versus time variations in embayment width (γ< 0). If γ > 0, the diffusion coefficient is larger near the crest of the tidal waveform, causing the rising tide to be of shorter duration and mean elevation to be set up. If γ< 0, the diffusion coefficient is larger near the trough, causing the falling tide to be shorter and elevation to be set down. The third harmonic is produced by fluctuations in the diffusion coefficient associated with times of greatest surface gradient. The third harmonic is governed only by the parameters x/L and ‖k0‖L, which indicates the third harmonic is insensitive to time variations in cross‐sectional geometry. Comparisons to field observations and to numerical solutions of the full equations including inertia terms indicate that the zero‐inertia equation (1) reproduces the results of the more general one‐dimensional equations to within the accuracy predicted by scaling arguments and (2) reproduces the main features of the nonlinear tidal signal observed in many shallow tidal embayments.