Internal Undular Bores in the Coastal Ocean

Abstract

In the coastal ocean, large amplitude, horizontally propagating internal wave trains are commonly observed. These are long nonlinear waves and are often modelled by equations of the Korteweg-de Vries type, such as the variable-coefficient Korteweg-de Vries equation when the background topography varies as the waves propagate shoreward. Most emphasis has been placed on the solitary wave solutions of these model equations, whereas in reality, wave trains are more usually observed. In this review article we examine the undular bore asymptotic representation of wave trains in the framework of the variable-coefficient Korteweg-de Vries equation, placing a special emphasis on the front of the undular bore which can be represented by a simplified model as a solitary wave train. We consider applications for both propagation shorewards whenw nonlinearity increases, and for cases when the wave train passes through a critical point of polarity change, when the nonlinear coefficient in the Korteweg-de Vries equation changes sign.

Appendix

Here we summarise the derivation of the equations (18, 19, 20) presented in [16]. When the coefficient $\alpha$ in (12) is a constant the KdV equation supports a periodic travelling wave, $U(X-V\tau )$, the well-known cnoidal wave,

$$\begin{aligned} U = a\, \{b(m)+ \hbox {cn}^2 (\gamma \theta ; m)\} + d \,, \quad \theta = k(X-V\tau ) \,, \end{aligned}$$

(28)

$$\begin{aligned} V - \alpha d = \frac{\alpha a}{3}\left\{ \frac{2-m}{m}-\frac{3E(m)}{mK(m)}\right\} = 4\gamma ^2 k^2\left\{ 2-m-\frac{3E(m)}{K(m)}\right\} \,. \end{aligned}$$

(29)

Here cn(x; m) is the Jacobian elliptic function of modulus $m, 0< m < 1$, and K(m) and E(m) are the elliptic integrals of the first and second kind, The expression (28) has period $2\pi$ in $\theta$ so that $\gamma = K(m)/\pi$, while the spatial period is $2\pi / k$. The (trough-to-crest) amplitude is a and the mean value over one period is d. It is a three-parameter family with parameters k, m, d say. As the modulus $m \rightarrow 1$, this becomes a solitary wave, since then $b \rightarrow 0$ and $\hbox {cn} (x) \rightarrow \hbox {sech} (x)$, while $\gamma \rightarrow \infty$, $k \rightarrow 0$ with $\gamma k = \Gamma$ fixed. As $m \rightarrow 0$, $b \rightarrow - 1/2$, $\gamma \rightarrow 1/2$, $\hbox {cn} (x) \rightarrow \cos {(x)}$, and it reduces to a sinusoidal wave $(a/2) \cos {(\theta )}$ of small amplitude $a \sim m$ and wavenumber k.

The Whitham modulation theory allows this cnoidal wave to vary slowly with $\tau , X$, that is the wavenumber k, modulus m and mean level d vary slowly with $\tau , X$. The Whitham modulation equations describing this variation can be obtained by averaging conservation laws, the original Whitham method, see [28, 29] or by exploiting the integrability of the constant-coefficient KdV equation, see [21] for instance. Because here we are concerned with the case when $\alpha = \alpha (\tau )$ varies slowly with $\tau$, and so the variable-coefficient KdV equation (13) is not integrable, we will use the original Whitham method, readily adapted to this present case. A similar strategy was used by [24] for a frictionally perturbed KdV equation. An alternative method developed by [22] for a perturbed KdV equation is not available here because to use it one must make a change of variable in (13) ${\tilde{\!U}} = \alpha U$ to generate a KdV equation for $\tilde{\!U}$ with a perturbation term of the form $\alpha _{\tau }\tilde{\!U}/\alpha$. But as one of our concerns is with the situation when $\alpha$ passes through zero, this approach cannot be used here.

As three modulation equations are needed, we supplement (15, 16) with the equation for conservation of waves,

$$\begin{aligned} k_{\tau } + (kV)_{X} =0 \,. \end{aligned}$$

(30)

The remaining two modulation equations are obtained by inserting the cnoidal wave solution into the conservation laws (15, 16) and averaging over the phase $\theta$. The outcomes are

$$\begin{aligned} d_{\tau } + \alpha M_{X} = 0 \,, \quad M = {<}\frac{U^2 }{2}{>} \,, \end{aligned}$$

(31)

$$\begin{aligned} M_{\tau } + P_{X} = 0 \,, \quad P = {<}\frac{\alpha U^3}{3} - \frac{3U_{X}^2 }{2}{>} \,, \end{aligned}$$

(32)

where the ${<}\cdots {>}$ denotes a $2\pi$-average over $\theta$. The expression M is given by

$$ M = \frac{d^2 }{2} + \frac{a^2 }{2} \{C_4 -b^2 \}\,, $$

$$\begin{aligned} C_4 = \frac{1}{3m^2 K(m) }\{3m^2 K(m) - 5mK(m) + 4mE(m) +2K(m)-2E(m) \} \,, \end{aligned}$$

(33)

while that for P is given by

$$ P = \alpha \{ -\frac{2d^3 }{3} + 2dM +a^3 \{-\frac{2b^3}{3} +\frac{(1-m)b}{2m} + (b + \frac{1-2m}{2m})\,C_4 +\frac{5}{6}C_6 \}\,, $$

$$ C_6 = \frac{1}{15m^{3}K(m)} \{15m^3 K(m) - 34m^2 K(m) + 23m^2 E(m) $$

$$\begin{aligned} \quad \quad +27mK(m) - 23mE(m) - 8K(m) + 8E(m)\}\,. \end{aligned}$$

(34)

Here the notation $C_{4}, C_{6}$ denote ${<}cn^{4}{>}, {<}cn^6{>}$ respectively, and like $b = -C_2 = -{<}cn^2{>}$ depend on the modulus m only.

To obtain the modulation equations for a solitary wave train, we take the limit $m \rightarrow 1$ and then $b \sim -1/K(m)$, $C_4 \sim 2/3K(m)$, and $C_6 \sim 8/15K(m)$. To leading order $M \sim d^2/2$ and $P \sim \alpha d^3 /3$ and then both equations (31, 32) reduce to the same equation for d alone,

$$\begin{aligned} d_\tau + \alpha dd_X =0 \,, \end{aligned}$$

(35)

and so d can be regarded as a known quantity. At the same time, the cnoidal wave expression (28) reduces to

$$\begin{aligned} U = a\,\hbox {sech}^2 (\gamma \theta ; m)\} + d \,, \quad \theta = k(X-V\tau ) \,, \quad V -\alpha d = \frac{\alpha a }{3} = 12 \gamma ^2 k^2 \,, \end{aligned}$$

(36)

with two parameters still to be determined. The equation for conservation of waves (30) provides one equation for k and the second equation is

$$\begin{aligned} \{\frac{a^2}{k\gamma }\}_{\tau } +V \{\frac{a^2}{k\gamma }\}_X + \frac{a^2}{k\gamma }\alpha d_X = 0 \,, \end{aligned}$$

(37)

This can be obtained by a more careful consideration of the limit $m \rightarrow 1$ in the modulation equations (31, 32) by retaining the terms in 1 / K(m), or more directly by averaging the wave action conservation law (16) directly for a solitary wave, see [7] and the discussion in [5].