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.
Highlights
The simplest linear-wave superposition, of monochromatic waves propagating according to work may be used under the terms of the Creative the shallow-water dispersion relation, leads to a family of profiles satisfying natural tidal bore
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
K tanh K o 1 - Vg (K ) K = Kc (u). This exemplifies a familiar wave phenomenon: the wavenumber of the local oscillations near X, T is that of the monochromatic wave that travels to X in time T with the group velocity
Summary
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. The simplest model predicts the same integral-Airy profile, but with a nonlinearly stretched argument. For long times, this reduces to the earlier integral-airy profile, i.e. without the nonlinear stretching, but instead of propagating rigidly the profile slowly expands as the bore advances. Transformation to the bore frame gives, for waves that advance upstream in the land frame, the monochromatic wave and the dispersion relation (Hamiltonian) exp (i(kx - w (k)t)), w (k) = vk - gk tanh dk. Standard hydraulic jump theory [7], in which incompressibility of water is combined with Newtonian dynamics (‘momentum equation’) relates the downflowing upstream and downstream speeds (in the bore frame) v0 and v1 to the corresponding depths d0 and d1: v0 =. 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)
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