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Minimal model for tidal bore revisited

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.


Introduction
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. My purpose here is to present the theory ab initio, and extend it, avoiding the error. The simplest model now 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.
There is a substantial literature on tidal bores [3][4][5][6][7][8][9], in which the emphasis is on effects associated with nonlinearity, often modelled by the Korteweg-de Vries (KdV) equation [10][11][12][13][14][15][16][17][18]. The distinctive feature of the present version of the minimal model is the systematic exploration of the description based solely on linear waves. The outcome will be a family of bore shapes (section 2), uniformly approximated to high accuracy (section 3), with the expanding integral-Airy profile (section 4) emerging as a realistic approximation for natural bores, stable in the sense that the details of the initial step get forgotten as the bores propagate. The theory makes a number of quantitative predictions, listed in the concluding section 5.
Referring to figure 1, schematically depicting the bore in a frame moving upstream with it, the aim is to calculate the water depth d(x, t) between its value d 0 in the downflowing river and its value d 1 (>d 0 ) in the advancing tide. For the gentle bores described by linear-wave theory, the important parameter is only slightly greater than unity. The starting-point is the dispersion relation relating the frequency ω and wavenumber k for monochromatic waves on water of depth d: Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence.
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This describes waves on water that is not flowing, but in the frame moving with the bore, water flows downstream, i.e. towards positive x. The bore profile will be built as a superposition of monochromatic waves. For downflowing speed v, transformation to the bore frame gives, for waves that advance upstream in the land frame, the monochromatic wave and the dispersion relation (Hamiltonian) What is v? 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) v 0 and v 1 to the corresponding depths d 0 and d 1 : For r only slightly greater than unity, v 0 (supercritical) is only slightly greater than v 1 (subcritical), and both values are close to 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).

Family of bore profiles
It is convenient to transform to the following dimensionless variables: The last equality shows that T is the distance travelled by the bore since its nominal birth at T=0, divided by the depth. For all realistic bores, T is very large, a fact we will exploit later (for a bore that has travelled 10 km on a river of depth 2 m, T=5000). We will create wave superpositions that describe the bore profile above the downflowing depth d 0 , thus: d e x pi t a n h , 2 . 3 C Figure 1. Definition sketch of the bore in the frame in which its upstream-moving front is stationary; upstream (i.e. before the bore arrives), is x<0, and downstream (after the bore front has passed) is x>0.
with amplitude g(K ) and contour C to be determined. This type of superposition has been systematically explored for tsunami propagation [19]. The difference here is the need to accommodate the transition from η=0 to η=1 between T=-∞ and T=+∞; it means that g(K ) must have a simple pole at K=0, with residue unity, that is and C must pass below the real axis. This leads to the following family of bore profiles: In this theory, T=0 represents the birth of the bore, when the profile is a simple jump, with no waves. The shape of this inital jump is represented by the function f (K ). For the simplest choice f (K )=1, the jump is a sharp step; if f (K ) is an increasing function, the jump is smooth. Later (section 4) we will find that the evolution after long times is independent of f (K ). Figure 2(a) shows the profile for f (K )=1, for T=100. The oscillations get faster as X increases, stopping abruptly near X=T. These very fast oscillations are unphysical because X=T corresponds to following the bore all the way down-river to the place where it was born, where numerous factors are likely to spoil the idealisation in this minimal model. These far-downstream oscillations can be suppressed by choosing f (K ) as an increasing function of K, as figure 2 The oscillations can be understood by approximating the integral in (2.5) asymptotically for X?0. There is a pole, contributing 1/2, and a stationary-phase point (saddle) that describes the oscillations. The stationaryphase point is at K c , involving the group velocity V g and determined by t a n h 1 .

. 6
K g c 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 (in this case, after transformation to the bore frame). Figure 3 illustrates this construction. The standard stationary-phase method [20] now leads to the approximate profile h h As figure 4 illustrates, this accurately describes the oscillations, even for rather small X, but of course it fails near the front, i.e. X=0, and upstream, i.e. X<0.

Uniform approximation for profile
We now obtain an approximation to the exact linear-wave profile (2.5) that incorporates both the large X approximation (2.7) and the behaviour near the front and upstream, that is, a large T approximation uniformly valid for all X. Reverting to the contour integral  where To simplify this integral, we apply the now-standard procedure [21][22][23] of transforming the integration variable K to a new variable Q, replacing the exponent by a simpler function with the same qualitative behaviour. In this case the behaviour is linear+cubic, so we choose the transformation The new function ξ(u, t) must be chosen to make the transformation smooth and reversible, so the stationary point K=K c (u) of the left side, defined by (2.6), must correspond to the stationary point Q=√ξ of the right side. This leads to The same transformation has been used to simplify tsunami propagation [19].) For u<0, i.e. X<0 (upstream of the bore), the correct branch is the one for which ξ is negative real. Thus 3 This is still exact. The technique of uniform approximation, whose lowest order (in T) will suffice here, consists in replacing K(Q) in the prefactor by the stationary point K c (u). The derivative is determined by differentiating the transformation (3.3) twice: 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. There are small discrepancies; some are barely visible in figure 5(b).

Asymptotic emergence of expanding integral-Airy profile
For the physical situation T?1, with X fixed, a simplified version of the uniform approximation is valid. This could be derived by approximating (3.7), but it is simpler to exploit the fact that the oscillations are determined by the small K region in the integrand of (2.5). Then it is natural to approximate the dispersion relation in (1.3) (in the scaled variables) by This is our main result: the same integral-Airy profile shape obtained in [1], but now slowly expanding as the bore travels upstream.
In water-wave theory, the nonlinear dynamics with the dispersion approximated as cubic can be described [12] by the KdV equation, which in the dimensionless variables (2.1), using (2.2), and with the sign chosen to represent waves travelling upstream (negative x), is The expanding Airy-integral profile emerges when the nonlinear term can be neglected, i.e. r-1=1; this solution is known [13,25], as is its (nonunform) asymptotics [26], but its application as a minimal model for tidal bores does not seem to have been systematically explored. The T 1/3 expansion of an Airy profile (not integrated) is familiar in the theory of tsunami propagation [27].
In the language of asymptotics, the expanding integral-Airy profile (4.2) is a transitional approximation; the range of X for which it is applicable-near the front-gets larger as T increases. Figure 6 illustrates this; for the realistic value T=5000, the agreement over the range depicted is almost perfect. This indicates that the expanding Airy-integral profile is the definitive linear-wave representation of the bore profile. For later reference, figure 7 shows detail of this profile near the bore front.
The emergence of the integral-Airy profile can be illustrated explicitly for the initial condition  It is clear that when T?L 3 this reduces to (4.2), whatever the value of the initial smoothing L: the wave forgets the shape of its initial step.

Consequences
The expanding integral-Airy profile (4.2) is surprisingly rich in predicting features of gentle tidal bores. Here is a list, extending and correcting section 4 of [1].
(a) A consequence of dispersion is that the bore profile (2.2) expands as it propagates. The expansion is slow: as T 1/3 . For example, the Severn bore takes about T 1 =1.75 h to travel from Awre, where it starts, to Stonebench, a popular viewing location (a distance of 24.1 km), and T 2 =2.17 h to travel from Awre to Maisemore, where a weir destroys it (a distance of 31.8 km); therefore the expansion ratio is only about (T 2 /T 1 ) 1/3 =1.07, which would be hard to see amid the many complicating factors associated with real bores;  I know of no relevant observations. A factor that would tend to inhibit the expansion is nonlinearity, for example as embodied in the term omitted in the KdV equation (4.3). It is hard to see how this would be relevant for gentle bores, but nonlinearity probably explains why ripples do not expand in the analogous stationary hydraulic jump in a sink [28], generated by water flowing outwards while replenished from a tap above; the depth contrast seems substantial, so this analogue bore is probably not gentle.
(b) The depth at the front of the bore, defined as X=0, follows from the integral of Ai(u) over 0<u<∞: Thus at X=0 the water surface has risen by 1/3 of its total rise between d 0 and d 1 .
(c) The slope at X=0 is (The maximum slope θ max , at the inflection of the profile ( figure 7), is greater by a factor 1.509 (see e.g. equation (4.4) of [1]).) Thus the slope decreases as the bore propagates, assuming the depth ratio r remains constant (for a conjecture about this, see (g) below).
(d) The 'wavelength' λ near the front, defined as the distance between the first two wave crests, is (see figure 7) Ai 0  The increasing wavelength, as well as the decreasing slope (5.2), predicts that the bore gets gentler as it advances, reinforcing the linear-wave approximation.
(e) The 'amplitude' a, defined as the vertical distance between the first maximum and the first minimum, is Intriguingly, a different argument, for the amplitude of waves in a monochromatic train [10], gives the same functional dependence and the slightly different constant 1/√3=0.577; and our earlier argument (involving the erroneous assumption), gave the same result (equation 4.9) of [1], after a factor 1/2 between the two definitions is included.