Minimal model for tidal bore revisited

Abstract

This develops a recent analysis of gentle undular tidal bores (2018 New J. Phys. 20 053066) and corrects an error. The simplest linear-wave superposition, of monochromatic waves propagating according to the shallow-water dispersion relation, leads to a family of profiles satisfying natural tidal bore boundary conditions, involving initial smoothed steps with different shapes. These profiles can be uniformly approximated to high accuracy in terms of the integral of an Airy function with deformed argument. For the long times corresponding to realistic bores, the profiles condense asymptotically onto the previously obtained integral-Airy function with linear argument: as the bore propagates, it forgets the shape of the initial step. The integral-Airy profile expands slowly, as the cube root of time, rather than advancing rigidly. This 'minimal model' leads to simple formulas for the main properties of the profile: amplitude, maximum slope, 'wavelength', and steepness; and an assumption about energy loss suggests how the bore weakens as it propagates.

Undular bores generated by fracture

“…From this we find the duration as the difference between the values of the first and third stationary points to be gently increasing as t (x) ≈ 3.643 c 0 δ 2 x 1/3 . This is an extension of the result given in [77] for the integral Airy solution. Importantly, this result is not dependent on the height or gradient of the initial slope as there is no dependence on κ, κ t , or L. Thus the expansion of the duration is the same for the integral Airy solution (22) and the solution (21).…”

“…This function gradually decreases with propagation distance, thus the front slope of the bore gets less steep. This formula is an extension of the counterpart of the result for the integral Airy solution in [77], which can be recovered by taking the limit L → 0 as g s (x) ≈ 0.3101c 0 (κ − κ t ) 1 δ 2 x 1/3 .…”

“…The amplitude of the oscillatory part of the solution (21), characterized by the vertical distance between the first minimum and first maximum, will grow and approach the limiting value as x → ∞. Using the result for this limiting value in [77], we find that a approaches the limiting value of a ≈ 0.466(κ − κ t ).…”

“…maps the problem (19) to that given in [77] (up to the change in notations). We used the analytical solution given in [77] to obtain the solution to (19) as…”

“…regardless of the gradient of the initial condition, i.e., the linear solution forgets its initial slope as it propagates [77].…”