Minimal analytical model for undular tidal bore profile; quantum and Hawking effect analogies

Abstract

Waves travelling up-river, driven by high tides, often consist of a smooth front followed by a series of undulations. A simple approximate theory gives the rigidly travelling profile of such 'undular hydraulic jumps', up to scaling, as the integral of the Airy function; applying self-consistency fixes the scaling. The theory combines the standard hydraulic jump with ideas borrowed from quantum physics: Hamiltonian operators and zero-energy eigenfunctions. There is an analogy between undular bores and the Hawking effect in relativity: both concern waves associated with horizons.

Minimal model for tidal bore revisited

“…This is our main result: the same integral-Airy profile shape obtained in [1], but now slowly expanding as the bore travels upstream.…”

“…which is the value that will be used in the dispersion relation (1.3). The supercritical flow speed upstream and the subcritical speed downstream formed the basis of the analogy with horizons in relativity physics, described in [1] (the new feature described here, that the bore expands as it propagates, seems to have no counterpart for relativistic horizons).…”

“…We will create wave superpositions that describe the bore profile above the downflowing depth d 0 , thus: (The error in [1] was to employ linear waves to represent the entire depth d(x), rather than the difference d(x)-d 0 .) Using (1.3) and (2.1), the superposition can be written as…”

“…For the correct branch, this is positive real for all u. With the prefactor thus extracted, the integral is an Airy function, leading to the uniform approximation for the profile (3.1): Since the bore front can be regarded as a caustic [1], where two 'water rays' (K c (u) in (2.6) and its negative) coalesce, the appearance of the Airy function, common to two-ray caustics [24] is unsurprising. The extraordinary accuracy of this large T approximation over the whole range of X is illustrated in figure 5, even for T=20 which is far smaller than any value relevant to the modelling of natural bores.…”

“…In some of the world's rivers open to an ocean, the advancing tide develops into a smooth front followed by a train of waves with a characteristic shape, travelling far upstream: an undular bore. In a recent paper [1], I presented a 'minimal model' [2] in which the bore profile is the integral of an Airy function. That theory predicted, erroneously, that the profile would propagate rigidly, that is, without change of scale.…”