Shoaling on Steep Continental Slopes: Relating Transmission and Reflection Coefficients to Green’s Law

Abstract

The propagation of long waves onto a continental shelf is of great interest in tsunami modeling and other applications where understanding the amplification of waves during shoaling is important. When the linearized shallow water equations are solved with the continental shelf modeled as a sharp discontinuity, the ratio of the amplitudes is given by the transmission coefficient. On the other hand, when the slope is very broad relative to the wavelength of the incoming wave, then amplification is governed by Green’s Law, which predicts a larger amplification than the transmission coefficient, and a much smaller amplitude reflection than given by the reflection coefficient of a sharp interface. We explore the relation between these results and elucidate the behavior in the intermediate case of a very steep continental shelf.

Appendices

Appendix 1: Transmission of Volume and of Energy

Note that Green’s Law and the analysis above applies to the amplitude of waves, not to their energy or the volume of water displaced by the wave. In this appendix we discuss how the results extend to these other quantities. Of course the total volume of water is conserved and so $\int _{-\infty }^\infty (h(x) + \eta (x,t))\,dx$ is constant in time. Moreover from this or directly from the first conservation law of (1) we see that the “wave volume” ${\bar{\eta }} \equiv \int _{-\infty }^\infty \eta (x,t)\,dx$ is constant in time and always equal to the initial wave volume. Similarly, the second conservation law of (1) shows that ${\bar{u}} \equiv \int _{-\infty }^\infty u(x,t)\,dx$ is constant in time. Suppose the initial elevation $\eta$ and velocity u are identically 0 for $x>-\epsilon$ and consists of some purely right-going wave with finite volume ${\bar{\eta }}$ travelling over the deep ocean. Then $u(x,0) = \sqrt{g/h_\ell }\,\eta (x,0)$ for all x, and so ${\bar{u}} = \sqrt{g/h_\ell }\,{\bar{\eta }}$. For large times, the perturbed elevation approaches 0 over the continental slope region, and consists of a purely left-going reflected wave and a purely right-going transmitted wave. Thus for sufficiently large t we can decompose $\eta (x,t) = \eta ^R(x,t) + \eta ^T(x,t)$, and similarly for u, where we have $\eta ^R(x,t) \equiv 0$ for $x>-\epsilon$ and $u^R(x,t) = -\sqrt{g/h_\ell }\,\eta ^R(x,t)$ for all x, while $\eta ^T(x,t) \equiv 0$ for $x<\epsilon$ and $u^T(x,t) = \sqrt{g/h_r}\,\eta ^T(x,t)$ for all x. If we define ${\bar{\eta }}^R = \int _{-\infty }^\infty \eta ^R(x,t)\,dx = \int _{-\infty }^{-\epsilon } \eta (x,t)\, dx$ and ${\bar{\eta }}^T = \int _{-\infty }^\infty \eta ^T(x,t)\,dx = \int _{\epsilon }^{\infty } \eta (x,t)\, dx$, then by conservation of $\eta$ and u, we obtain a linear system of two equations

$$\begin{aligned} \begin{aligned} {\bar{\eta }}^R + {\bar{\eta }}^T&= {\bar{\eta }},\\ -\sqrt{g/h_\ell }\,{\bar{\eta }}^R + \sqrt{g/h_r}\,{\bar{\eta }}^T&= \sqrt{g/h_\ell }\,{\bar{\eta }}, \end{aligned} \end{aligned}$$

(9)

which can be solved to find that the total reflected and transmitted volume are given respectively by

$$\begin{aligned} \begin{aligned} {\bar{\eta }}^R&= \left( \frac{\sqrt{h_\ell } - \sqrt{h_r}}{\sqrt{h_\ell } + \sqrt{h_r}}\right) {\bar{\eta }} \equiv {\bar{C}}_R {\bar{\eta }},\\ {\bar{\eta }}^T&= \left( \frac{2\sqrt{h_r}}{\sqrt{h_\ell } + \sqrt{h_r}}\right) {\bar{\eta }} \equiv {\bar{C}}_T {\bar{\eta }}. \end{aligned} \end{aligned}$$

(10)

Note that ${\bar{C}}_T = \sqrt{h_r/h_\ell }\, C_T$. As the pulse passes over a discontinuous jump in bathymetry, the amplitude increases by $C_T$ but its width decreases by $\sqrt{h_r/h_\ell }$ due to the change in wave speed, and so its volume changes by the product of these, which is ${\bar{C}}_T$. The reflected wave, on the other hand, has the same width as the incident pulse, and so ${\bar{C}}_R = C_R$.

The transmission and reflection coefficients for volume from (10) apply regardless of the width of the continental slope, and in fact are the same for any specified variation of h(x) provided it varies only over a finite region and is identically equal to $h_\ell$ and $h_r$ away from the slope. They are also independent of the shape of the pulse $\eta (x,0)$ as long as the initial wave is purely right-going, $u(x,0) = \sqrt{g/h_\ell }\eta (x,0)$, and $\eta (x,0) = 0$ for $x > -\epsilon$.

Note that the square pulse shown in fig. 2 for a very broad continental slope also appears to be transmitted as a square pulse, and the change in wave speed again suggests the width of the pulse will be reduced by $\sqrt{h_r/h_\ell }$. But if the initial amplitude is A then the amplitude of the transmitted wave is close to $C_G A$ rather than $\bar{C}_T A$, and so it seems that too much volume has been transmitted onto the shelf. However, recall that there is also a small negative trailing wave. It turns out this wave has amplitude that decreases as the width of the continental slope increases, while at the same time it spreads out farther, in such a way that its total volume is constant and roughly equal to $({\bar{C}}_T - C_G)A < 0$, which cancels the excess volume that appears in the transmitted pulse. This follows from the more general results presented in George et al. (2019a).

It is interesting to consider the case of a localized variation in bathymetry, with $h_r = h_l$ and any variations restricted to $-\,\epsilon< x < \epsilon$, e.g., an underwater ridge or sill. Then ${\bar{C}}_R=0$ and ${\bar{C}}_T = 1$, so the total volume reflected is zero while the transmitted volume is equal to the volume of the original wave. This does not mean, of course, that the transmitted wave has the same form as the incident wave, nor that there are no reflections (only that the integral of the reflected wave vanishes).

It is also interesting to note that the total energy propagated onto the shelf varies as the width of the slope or the initial pulse are varied. A wave that is purely left-going or right-going satisfies equipartition of energy between potential and kinetic energy (see e.g. Berger et al. 2011; Mei 1992), so it suffices to consider, for example, the potential energy given by $E(x,t) = \rho g \int _{-\infty }^\infty \eta ^2(x,t)\,dx.$ where $\rho$ is the density of water (which is assumed to be constant in this calculation). Consider the case of a step function continental slope and an initial pulse of height A and width w, for which $E(x,0) = \rho g wA^2$. The reflected wave has height $C_RA$ and width w, while the transmitted wave has height $C_TA$ and width $w\sqrt{h_r/h_\ell }$. So the total potential energy is $\rho g (C_R^2 w + C_T^2 w\sqrt{h_r/h_\ell })A^2 = \rho g wA^2$, illustrating conservation of energy. For a very broad slope, for which Green’s Law holds, the energy in the reflected wave goes to zero (even though its volume is constant, it becomes more spread out with smaller amplitude as the width of the continental slope increases, and the energy is quadratic in $\eta$, so this integral vanishes). The transmitted wave carries all of the energy; it has height $C_GA$ and width $w\sqrt{h_r/h_\ell }$, and hence energy $\rho g C_G^2 w\sqrt{h_r/h_\ell }A^2 = \rho g wA^2$.

Appendix 2: Single Intermediate Layer

In this appendix and the next we derive two results mentioned in the paper on the propagation of a bore through a layered medium, starting with the case of a single intermediate layer where it is easy to show that the sum of all transmitted waves approaches $C_TA$ over time while the sum of all reflected waves approaches $C_RA$. The single-layer case is shown in fig. 4(b), where the intermediate step in bathymetry has depth $h_m$ with $h_r< h_m < h_\ell$. Define the following transmission coefficients:

$$\begin{aligned} C_T^{\ell m}= & {} \frac{2\sqrt{h_\ell }}{\sqrt{h_\ell }+\sqrt{h_m}}, \quad C_T^{mr} = \frac{2\sqrt{h_m}}{\sqrt{h_m}+\sqrt{h_r}}, \quad \nonumber \\ C_T^{m\ell }= & {} \frac{2\sqrt{h_m}}{\sqrt{h_\ell }+\sqrt{h_m}}, \end{aligned}$$

(11)

where for example the superscript mr indicates transmission of a right-going wave from the middle layer to the right layer, and $m\ell$ indicates transmission of a left-going wave from the middle layer to the left layer. Similarly define the reflection coefficients for internal reflections in the intermediate layer as

$$\begin{aligned} C_R^{mr} = \frac{\sqrt{h_m} - \sqrt{h_r}}{\sqrt{h_m} + \sqrt{h_r}}, \quad C_R^{m\ell } = \frac{\sqrt{h_m} - \sqrt{h_\ell }}{\sqrt{h_m} + \sqrt{h_\ell }} \end{aligned}$$

(12)

Then an initial right-going wave consisting of a discontinuity in $\eta$ of amplitude A leads to a first transmitted wave of amplitude $C_T^{\ell m}C_T^{mr}A$. This is larger than $C_TA$, where $C_T$ is the transmission coefficient with no intermediate layer from (5), but not as large as $C_GA$, the value found in Appendix 1 for a smooth transition. This first wave is followed by transmitted waves that have undergone 2n internal reflections for $n=1,~2,~3,~\ldots$, each of which has amplitude

$$\begin{aligned} C_T^{\ell m}(C_R^{mr} C_R^{ml})^n C_T^{mr}A. \end{aligned}$$

(13)

Note that $-1< C_R^{mr} C_R^{ml} < 0$ and so the sum of all the transmitted waves, including the first wave with no reflections, is given by the geometric series

$$\begin{aligned} \sum _{n=0}^\infty C_T^{\ell m}(C_R^{mr} C_R^{ml})^n C_T^{mr}A = \left( \frac{C_T^{\ell m}C_T^{mr}}{1 - C_R^{mr} C_R^{ml}}\right) A= C_T A \end{aligned}$$

(14)

As expected, the state behind the right-going waves decays to the same state $C_TA$ obtained behind the right-going transmitted wave in the case of no interior layers.

Similarly, the sum of all left-going waves that depart into the deep ocean after $2n+1$ internal reflections (for $n = 0,~1,~2,~\ldots$) is given by

$$\begin{aligned} \sum _{n=0}^\infty C_T^{\ell m}C_R^{mr} (C_R^{ml}C_R^{mr})^n C_T^{m\ell }A = C_RA, \end{aligned}$$

(15)

where $C_R$ is the reflection coefficient in the case of no intermediate layers. But note that there is a time lag between each departing wave, so the composite reflected wave takes the form of a step function with infinitely many jumps that decay exponentially with n.

Appendix 3: Infinitely Many Layers and the First Transmitted Wave

We now consider the amplification of the wave as it is transmitted through each interface of a layered medium, ignoring all the reflected waves generated in the process. The final amplitude of this “first transmitted wave” will be denoted by $C_T^fA$, and we will show that $C_T< C_T^f< C_G$ and that this transmission coefficient asymptotes to the Green’s Law coefficient $C_G$ as the number of layers increases. A different approach to deriving this same result is taken in George et al. (2019a) and George (2018), where more general results are derived for the waves that also experience internal reflections.

The amplitude of the first transmitted wave in the case of a single intermediate layer can be represented by $A_{t_1t_2}$, signifying transmission through the first and second interface:

$$\begin{aligned} A_{t_1t_2} = C_T^{\ell m}C_T^{mr}A = \frac{2\sqrt{h_\ell }}{\sqrt{h_{m}}+\sqrt{h_\ell }} \frac{2\sqrt{h_{m}}}{\sqrt{h_{m}}+\sqrt{h_r}} A. \end{aligned}$$

(16)

In the case of two intermediate layers, the amplitude of the first transmitted wave is given by

$$\begin{aligned} A_{t_1t_2t_3} = \frac{2\sqrt{h_\ell }}{\sqrt{h_{m_1}}+\sqrt{h_\ell }}\frac{2\sqrt{h_{m_1}}}{\sqrt{h_{m_1}}+\sqrt{h_{m_2}}} \frac{2\sqrt{h_{m_2}}}{\sqrt{h_{m_2}}+\sqrt{h_r}} A. \end{aligned}$$

(17)

It can be seen that $A_{t_1t_2t_3}$ is larger than $A_{t_1t_2}$ and that increasing the number of layers increases the amplitude of the first transmitted wave.

More generally, consider piecewise-constant bathymetry with N intermediate steps having depths $h_i$ chosen with $h_i = h_\ell + i\varDelta h$ where $\varDelta h = (h_r-h_\ell )/N$ for $i=1,~2,~\ldots ,~N$. We also define $h_0 = h_\ell$ and $h_{N+1} = h_r$. Then the transmission coefficient at the interface between $h_i$ and $h_{i+1}$ is given by

$$\begin{aligned} C_{T}^i = \frac{2\sqrt{h_i}}{\sqrt{h_i}+\sqrt{h_i+\varDelta h }} = \frac{2}{1+\sqrt{1+\frac{\varDelta h}{h_i}}}. \end{aligned}$$

This allows us to find the continuous limit of the leading amplitude by considering the scenario with infinitely many layers. The amplitude of the first transmitted wave would then be given by

$$\begin{aligned} \lim _{N\rightarrow \infty }\prod _{i=0}^{N} C_{T}^i A =\lim _{N\rightarrow \infty } \prod _{i=0}^{N} \bigg (\frac{2}{1+\sqrt{1+\frac{\varDelta h}{h_i}}}\bigg ) A \equiv C_T^fA, \end{aligned}$$

(18)

Taking the log of $C_T^f$ converts the infinite product into a sum,

$$\begin{aligned} \log (C_T^f) =\lim _{N\rightarrow \infty } \sum _{i=0}^{N}\log \bigg (\frac{2}{1+\sqrt{1+\frac{\varDelta h}{h_i}}}\bigg ). \end{aligned}$$

(19)

Using the Taylor expansion then results in

$$\begin{aligned} \log (C_T^f)&= -\lim _{N\rightarrow \infty } \sum _{i=0}^{N} \bigg (\frac{\varDelta h}{4h_i}-\frac{\varDelta h^2}{16h_i^2}+\cdots \bigg ). \end{aligned}$$

(20)

Since the sum of the $\varDelta h^2$ and higher order terms vanish in the limit, we obtain

$$\begin{aligned} \log (C_T^f) = -\lim _{N\rightarrow \infty } \frac{h_r-h_\ell }{N} \sum _{i=0}^{N} \frac{1}{4h_i}. \end{aligned}$$

(21)

This sum approaches an integral in the limit, giving

$$\begin{aligned} \log (C_T^f) = - \int _{h_\ell }^{h_r} \frac{1}{4x} dx = -\frac{1}{4} \log \bigg (\frac{h_r}{h_\ell }\bigg ) \end{aligned}$$

(22)

and hence

$$\begin{aligned} C_T^f= (h_\ell /h_r)^{1/4}. \end{aligned}$$

(23)

This is exactly the Green’s Law coefficient $C_G$ from Eq. (6), showing that the first transmitted wave has amplitude $C_GA$ in the limit as the number of internal layers goes to infinity, i.e. as the step function bathymetry is smoothed out to a continuous continental slope. Note also that this argument is independent of the locations $x_i$ corresponding to each jump discontinuity from $h_i$ to $h_{i+1}$, and so the same result holds regardless of whether we discretize a narrow or wide slope, and regardless of the shape of the slope; it need not be a discretization of a linear slope of the sort shown in fig. 1.