2.1. Korteweg–de Vries equation

The usual KdV equation for propagation of weakly nonlinear long internal waves over variable topography with depth $h(x)$ is, in standard notation,

(2.1)\begin{equation} \frac{\partial A}{\partial t} + c\frac{\partial A}{\partial x} + \frac{1}{2Q}\frac{\partial Q}{\partial x}cA + \mu A\frac{\partial A}{\partial x} + \delta \frac{\partial^3 A}{\partial x^3} =0 \end{equation}

(see Grimshaw (Reference Grimshaw1981), and the reviews by Helfrich & Melville (Reference Helfrich and Melville2006), Grimshaw (Reference Grimshaw2007) and Grimshaw et al. (Reference Grimshaw, Pelinovsky, Talipova and Kurkina2010)). Here $\zeta = A(x,t)\phi (z; h)$ is the leading-order expression for the vertical particle displacement. The modal function $\phi (z; h)$ is determined, using the Boussinesq approximation, by a rigid upper lid approximation, and in the absence of a background shear flow,

(2.2)\begin{equation} \frac{\partial^2 \phi}{\partial z^2} + \frac{N^2 }{c^2 }\phi = 0; \quad \phi = 0, \quad z = 0, -h(x). \end{equation}

Here $N^2 = -({g}/{\rho _0 })({\mathrm {d} \rho _0}/{\mathrm {d} z})$ is the background density field. This modal equation determines the modal function $\phi (z; h)$ and the linear long-wave speed $c(h)$ where the $h$-dependence is parametric, since it is assumed that the depth $h(x)$ is slowly varying. The coefficients in (2.1) are given by, in the same approximation used in (2.2),

(2.3)\begin{gather} I \mu = 3 c^2 \int_{-h}^{0} \rho_0 \left(\frac{\partial \phi}{\partial z}\right)^3 \, \textrm{d}z, \end{gather}

(2.4)\begin{gather}I \delta = c^2 \int_{-h}^{0} \rho_0 \phi^2 \, \textrm{d}z, \end{gather}

(2.5)\begin{gather}I = 2 c \int_{-h}^{0}\, \rho_0 \left(\frac{\partial \phi}{\partial z}\right)^2\,\textrm{d}z, \quad Q =c^2 I. \end{gather}

In general, the modal equation (2.2) determines an infinite set of modes $\phi _n$, $n=1, 2, \ldots$, ordered so that $c_1 > c_2 > \cdots$. Usually only mode-1, $\phi _1$, with the fastest linear phase speed $c_1$ is considered. Then the KdV equation (2.1) describes the evolution of the amplitude of this mode. But importantly we note that the same KdV equation (2.1) can be used for a mode-2 wave, with coefficients determined by (2.3), (2.4), (2.5) using the mode-2 modal function $\phi _2$ and speed $c_2 < c_1$.

2.2. Coupled modes: linearised system

The theory of Griffiths & Grimshaw (Reference Griffiths and Grimshaw2007) uses linear long-wave theory to decompose the wave field into a sum of vertical modes, noting that the full set of modal functions defined by (2.2) are complete. This was used by Liu et al. (Reference Liu, Grimshaw and Johnson2019a) to examine the generation of a mode-2 wave by a mode-1 wave incident on the topography. As in Liu et al. (Reference Liu, Grimshaw and Johnson2019a), the theory of Griffiths & Grimshaw (Reference Griffiths and Grimshaw2007) is restricted to just two modes, so that the wave field is given by

(2.6)\begin{equation} \left. \begin{gathered} \zeta = \sum_{n=1}^{n=2} A_{n} (x, t)\phi_{n} (z; h), \\ u = \sum_{n=1}^{n=2} A_{n} (x, t ) c_n (h)\frac{\partial \phi_n (z; h)}{\partial z}, \\ p = \rho_{0}\sum_{n=1}^{n=2} A_{n} (x, t ) c_{n}^2 (h) \frac{\partial \phi_n (z; h)}{\partial z}. \end{gathered} \right\} \end{equation}

Here $\zeta$, $u$ and $p$ are the vertical particle displacement, the horizontal velocity and the dynamic pressure, respectively. The vertical velocity is $w = {\partial \zeta }/{\partial t}$ and the density perturbation is $\rho = N^2 \zeta /g$. From Griffiths & Grimshaw (Reference Griffiths and Grimshaw2007) (see (24)) the linear long-wave coupling between mode-1 and mode-2 with linear long-wave speeds $c_{1,2}$ is given, for slowly varying $h(x)$, by

(2.7)\begin{gather} \frac{1}{c_{1}^2} \frac{\partial^2 U_1}{\partial t^2} - \frac{\partial^2 U_1}{\partial x^2} = -\gamma \frac{\partial h}{\partial x} \frac{\partial U_2}{\partial x}, \end{gather}

(2.8)\begin{gather} \frac{1}{c_{2}^2} \frac{\partial^2 U_2}{\partial t^2} - \frac{\partial^2 U_2}{\partial x^2} = \gamma \frac{\partial h}{\partial x} \frac{\partial U_1}{\partial x}, \end{gather}

(2.9)\begin{gather} \gamma = \frac{2}{\tilde{\lambda}_1 - \tilde{\lambda}_2 }\left(\frac{\partial \tilde{\lambda}_1} {\partial h} \frac{\partial \tilde{\lambda}_2}{\partial h}\right)^{1/2} = \frac{4}{c_{2}^2 - c_{1}^2 }\left(c_{1}c_{2}\frac{\partial c_1}{\partial h} \frac{\partial c_2}{\partial h}\right)^{1/2}, \quad \tilde{\lambda}_1= \frac{1}{c_{1}^2 }, \quad \tilde{\lambda}_2= \frac{1}{c_{2}^2 }. \end{gather}

As noted above, this is a reduction from the full set in (24) of Griffiths & Grimshaw (Reference Griffiths and Grimshaw2007) by restriction to just two modes, and retention of only the leading-order topographic coupling terms when $h(x)$ is slowly varying; that is, terms such as ${\partial ^2 h}/{\partial x^2}$ or $({\partial h}/{\partial x})^2$ are omitted. However, the theory of Griffiths & Grimshaw (Reference Griffiths and Grimshaw2007) was developed for the horizontal velocity field in the form $u =U ({\partial \psi (z;h)}/{\partial z})$, whereas in the usual KdV theory described above by (2.6) the theory is developed for the vertical particle displacement $A \phi (z; h)$, where the modal function $\psi \propto \phi$. In the absence of a background shear flow, these are related by, temporarily omitting the modal index,

(2.10a–c)\begin{equation} u = U \frac{\partial \psi}{\partial z} = c A \frac{\partial \phi}{\partial z}, \quad \psi (z) = K \phi (z), \quad K^2 = \frac{1}{I c} = \frac{c}{Q}. \end{equation}

Hence we find that

(2.11a,b)\begin{equation} U = c^{1/2}B, \quad B= Q^{1/2} A, \end{equation}

where $Q$ and $I$ are the linear magnification and normalisation factors in the usual KdV theory; see (2.5).

Since $h(x)$ is slowly varying, we use the change of variables

(2.12)\begin{gather} T= \int^{x}_0 \frac{\textrm{d}x}{c_1}, \quad X = T -t , \end{gather}

(2.13)\begin{gather} \left. \begin{gathered} \frac{\partial U}{\partial x} = \frac{1}{c_1}\left(\frac{\partial U}{\partial X} + \frac{\partial U}{\partial T}\right), \quad \frac{\partial^2 U}{\partial t^2} = \frac{\partial^2 U}{\partial X^2}, \\ \frac{\partial^2 U}{\partial x^2} = \frac{1}{c_{1}^2}\left(\frac{\partial^2 U}{\partial X^2} + 2\frac{\partial^2 U}{\partial X \partial T } + \frac{\partial^2 U}{\partial T^2}\right) - \frac{1}{c_{1}^3 }\frac{\partial c_{1}}{\partial T} \left(\frac{\partial U}{\partial X} + \frac{\partial U}{\partial T}\right). \end{gathered} \right\} \end{gather}

In Liu et al. (Reference Liu, Grimshaw and Johnson2019a) we used (2.12) for mode-1 and an analogous change of variable for mode-2 with $c_1$ replaced with $c_2$. Here we assume that $c_1 \approx c_2$, and so we can use (2.12) for both modes. Then in each modal equation, the operator on the left-hand side becomes

(2.14)\begin{equation} \frac{1}{c_{1}^2 }\frac{\partial^2 U}{\partial t^2} - \frac{\partial^2 U}{\partial x^2}= - \frac{2}{c_{1}^2}\frac{\partial^2 U}{\partial X \partial T} -\frac{1}{c_{1}^{2} }\frac{\partial^2 U}{\partial T^2}+ \frac{1}{c_{1}^3 }\frac{\partial c_{1}}{\partial T} \left(\frac{\partial U}{\partial X} + \frac{\partial U}{\partial T}\right). \end{equation}

Next we use the transformation (2.11a,b) and assume that the $T$-derivatives vary slowly relative to $X$-derivatives. Thus we can neglect slowly varying terms such as ${\partial ^2 U}/{\partial T^2}$ and $({\partial c_{1}}/{\partial T})({\partial U}/{\partial T})$, and the coupled system ((2.7), (2.8)) becomes

(2.15)\begin{gather} \frac{\partial B_{1}}{\partial T} = \frac{\gamma c_{2}^{1/2}}{2 c_{1}^{1/2}}\frac{\partial h}{\partial T}B_2, \end{gather}

(2.16a,b)\begin{gather} \frac{\partial B_{2}}{\partial T} - \varDelta \frac{\partial B_{2}}{\partial X} = -\frac{\gamma c_{1}^{1/2}}{2 c_{2}^{1/2}}\frac{\partial h}{\partial T}B_1, \quad \varDelta = \left\{\frac{1}{c_{2}^2 } - \frac{1}{c_{1}^2 }\right\}\frac{c_{1}^{2}}{2} \approx (c_1 - c_2)/c_2 . \end{gather}

In the original $x$, $t$ variable there is a choice of either an initial condition, when $B_1, B_2$ are specified at $t=0$, or a boundary condition when $B_{1}, B_{2}$ are at $x=0$. Here, we choose $B_1$ to be a mode-1 solitary wave, either located in $x < 0$ at $t=0$ or specified at $x=0$, and in both cases $B_2 = 0$. In the initial-value problem, these initial conditions are located at $X=T$, or as boundary conditions at $T=0$. These can be shown to be asymptotically equivalent with respect to the small long-wave parameter used to derive ((2.7), (2.8)), and also the KdV equation (2.1) (see the next subsection). But, nevertheless they produce different solutions in general. Here we choose the boundary condition formulation, inter alia being more convenient for numerical simulations. Thus the ‘initial’ condition at $T=0$ is a mode-1 solitary wave, defined in $T< 0$ by $B_{1} = B_{1sol} (X- VT)$ in $T < 0$ where $h=h_b$, and a zero mode-2 wave, $B_2 = 0$. Here $V$ is the solitary wave speed (see (2.25)), and in the linear long-wave limit we set $V =0$. Importantly, note that $\varDelta > 0$ since $c_1 > c_2$, and we have used the approximation that $c_2 \approx c_1$ to simplify $\varDelta$.

2.3. The linear response

The coupled system ((2.15), (2.16a,b)) forms the bases of the investigation here. An estimate of the size of a typical solution of this system can be found using the method in Liu et al. (Reference Liu, Grimshaw and Johnson2019a) to solve the linear system approximately when $\varDelta$ is of order unity. Since the coupling term $\gamma ({\partial h}/{\partial T})$ is slowly varying and hence small, it can be neglected in (2.15) when the incident wave is a mode-1 wave. It then follows that to leading order $B_{1} = B_{1sol} (X)$ is independent of $T$ and determined from the initial condition of a mode-1 solitary wave a priori. The second equation (2.16a,b) can then be solved for the mode-2 wave:

(2.17)\begin{equation} \left. \begin{gathered} B_2 \approx -\int_{0}^{\tau} \, C (\tau^{\prime}) B_{1sol}(X+ \tau - \tau^{\prime})\, \textrm{d}\tau^{\prime }, \\ \tau (T) = \int_{0}^{T} \, \varDelta (T^{\prime} ) \, \textrm{d}T^{\prime }, \quad C(\tau) = \frac{\gamma c_{1}^{1/2}}{2 c_{2}^{1/2}\varDelta } \frac{\partial h}{\partial T}. \end{gathered} \right\} \end{equation}

Note that the evolution variable $T$ is replaced with $\tau$ and the ‘initial’ condition that ${B_2 = 0, T= 0}$, that is $B_2 = 0, \tau = 0$ has been imposed. This can be written in the alternative form, putting $Y = X+ \tau - \tau ^{\prime }$, so that $X < Y < X + \tau$ when $\tau > \tau ^{\prime } > 0$:

(2.18)\begin{equation} B_2 \approx -\int_{X}^{X + \tau } \, C (X + \tau - Y) B_{1sol}(Y)\, \textrm{d}Y. \end{equation}

This form indicates that this approximate solution is a combination of a slaved mode-2 wave depending on $X$ and a free mode-2 wave depending on $X + \tau$. Note that because $C(\tau ) \ge 0$, $B_2$ has the opposite sign to $B_{1sol}$.

Since $C$ is slowly varying compared to $B_{1sol}$, $B_{1sol} (X)$ can be regarded as localised around $X=0$, and then the expression (2.18) for $B_2$ is effectively confined to the region ${-\tau < X < 0}$, where the two boundaries correspond to a free wave and a slaved wave, respectively. For instance if we approximate $B_1 (X)$ with a $\delta$-function, $B_1 (X) = a\,\delta (KX)$ with mass $a/K$, then (2.18) becomes

(2.19)\begin{equation} B_2 \approx -\frac{a}{K}C (X+ \tau)\{\mathcal{H}(X + \tau ) - \mathcal{H} (X) \}, \end{equation}

where $\mathcal {H}(\cdot )$ is the Heaviside function. For each fixed $\tau > 0$, as $X$ increases over the range $-\tau < X < 0$, $X + \tau$ increases from $0$ to $\tau$, and this approximate solution (2.19) is non-zero only for $-\tau < X < 0$. The maximum amplitude of $B_2$ is $C_M a/ K$, where $C_M = \hbox {max} [C]$. Further from (2.17), $C(\tau (T))$ is non-zero only on the slope, that is, $0 < \tau (T) < \tau (L)$. Hence (2.19) is non-zero only for $-\tau (T) < X < 0$, and for $-\tau (T) < X < \tau (T)- \tau (L)$. In these combined regions, (2.19) is a function of $X + \tau$ and so is a free wave. On the other hand, if $C$ is rapidly varying relative to $B$, as would be the case if $h(T)$ is close to being a step, then we might approximate $C$ with a $\delta$-function $C (\tau ) = C_M \delta (L(\tau - \tau _0))$. Then the solution (2.17) becomes

(2.20)\begin{equation} B_2 \approx - \frac{C_M}{L} B_{1sol} ( X + \tau - \tau_0 ). \end{equation}

This is a free wave and importantly note that compared to (2.19) the amplitude varies as $a C_M /L$ rather than $a C_M /K$.

These approximate solutions of ((2.15), (2.16a,b)) can be compared with those presented in Liu et al. (Reference Liu, Grimshaw and Johnson2019a) where it was assumed that $\varDelta$ is of order unity, and the $T$-derivative in (2.16a,b) was omitted on the basis that the forcing term $B_{1sol} (X)$ depended only on $X$. Also, crucially, in Liu et al. (Reference Liu, Grimshaw and Johnson2019a) the initial condition that $B_2 =0$ was imposed at $t=0$, $X=T$ instead of at $T=0$. As already noted, although these initial conditions are asymptotically equivalent, they lead in practice to different outcomes. In the present notation the solution found by Liu et al. (Reference Liu, Grimshaw and Johnson2019a) is

(2.21)\begin{equation} \left. \begin{gathered} B_1 \approx B_{1sol} (X), \quad B_2 \approx B_{2P}(X, \tau ) + B_{2F} (X+ \tau ), \\ B_{2P} = - C(\tau ) \int^{\infty}_{X} \, B_{1sol} (X^{\prime}) \, \textrm{d}X^{\prime }, \\ B_{2F} (Z) = -B_{2P}(X, \tau(X)), \quad Z (X) = X + \tau (X) . \end{gathered} \right\} \end{equation}

Here in the expression $\tau = \tau (T)$ in (2.17), we put $X=T$, and then an inversion is needed to get $X = X(Z)$, so that the functional form of $B_{2F} (Z)$ can be determined. Although there is an obvious similarity between (2.18) and (2.21) they are not the same. However, the underlying assumption here is that the $T$-scale is slowly varying compared to the $X$-scale, and in that limit the two expressions agree. Indeed, if $C(\tau )$ is taken to be a constant, then they are identical, indicating that if $C(\tau )$ is sufficiently slowly varying, then the two expression are close. The linear response of the present solution when $\varDelta$ is not small thus merges smoothly with the linear results of Liu et al. (Reference Liu, Grimshaw and Johnson2019a).

2.4. Coupled modes: Korteweg–de Vries system

The present study concerns the behaviour when $c_2 \approx c_1$ so that $\varDelta \ll 1$, and also then $\gamma \propto \varDelta ^{-1}$ becomes large; see (2.9). In this limit the coupled system ((2.15), (2.16a,b)) yields high-frequency oscillations with a frequency proportional to $\varDelta ^{-1}$ and a wavenumber proportional to $\varDelta ^{-2}$. This resonance leads to the requirement that weakly nonlinear terms need to be invoked, together with weak linear dispersion, requiring the KdV extension of the coupled system, examined here.

In the absence of topographic coupling we can add a KdV extension for each mode, that is

(2.22)\begin{equation} \frac{\partial B}{\partial T} \to \frac{\partial B}{\partial T} + \nu B\frac{\partial B}{\partial X} + \lambda \frac{\partial^3 B}{\partial X^3}, \quad \nu = \frac{\mu }{cQ^{1/2}}, \quad \lambda = \frac{\delta }{c^3}, \end{equation}

after applying the transformations (2.12) to the KdV equation, and (2.1), and $\mu , \lambda$ are defined by ((2.3), (2.4)) for each mode. Hence we propose that the coupled KdV system for this resonance will be

(2.23)\begin{gather} \frac{\partial B_{1}}{\partial T} + \nu_1 B_1\frac{\partial B_{1}}{\partial X} + \lambda_1 \frac{\partial^3 B_{1}}{\partial X^3} = \frac{\gamma c_{2}^{1/2}}{2 c_{1}^{1/2}}\frac{\partial h}{\partial T}B_2, \end{gather}

(2.24)\begin{gather} \frac{\partial B_{2}}{\partial T} - \varDelta \frac{\partial B_{2}}{\partial X} + \nu_2 B_2 \frac{\partial B_{2}}{\partial X} + \lambda_2 \frac{\partial^3 B_{2}}{\partial X^3} = -\frac{\gamma c_{1}^{1/2}}{2 c_{2}^{1/2}}\frac{\partial h}{\partial T}B_1. \end{gather}

The usual KdV scaling requires that $B_{1,2} \sim \epsilon ^2$, $\partial /\partial X \sim \epsilon$ and $\partial /\partial T \sim \epsilon ^3$. Here in addition we require a resonance so that $\varDelta \sim \epsilon ^2$ and so then $\gamma ({\partial h}/{\partial T}) \sim \epsilon ^3$. From (2.9) $\gamma \propto \varDelta ^{-1}$, it follows that formally ${\partial h}/{\partial T} \sim \epsilon ^5$. Note that although we are assuming that $c_2 \approx c_1$, the modal functions $\phi _1$ and $\phi _2$ remain distinct. It is consistent asymptotically to replace $\varDelta$ in (2.16a,b) with $\varDelta \approx (c_1 - c_2)/c_2$, and also put $c_2/c_1 \approx 1$ in the coupling terms on the right-hand side. Also it is useful to note that in this $X$–$T$ reference frame, the linear phase speeds for the mode-1 and mode-2 waves are $0$ and $-\varDelta$, respectively.

The ‘initial’ condition at $T= 0$ is that the mode-2 is zero, $B_2 = 0$, and there is only a mode-1 solitary wave in $T \le 0$, that is,

(2.25)\begin{equation} B_1 = a\, \textrm{sech}^2 (K(X - VT)), \quad V = \frac{a\nu_{1b}}{3} = 4K^2 \lambda_{1b}, \quad B_2 =0. \end{equation}

3.1. Case $A$: weak resonant coupling (initial depression wave)

In case $A$ the middle layer thickness ($h_2$) is thin and the lower layer ($h_3$) is thicker than half of the middle layer ($h_2/2$) in the whole slope–shelf region. The incident mode-1 wave is thus a depression solitary wave. Figure 2 shows the coefficients of the coupled KdV ((2.23), (2.24)) model, with the upper and lower rows corresponding to the set-up $h_{gentle}$ and $h_{steep}$ (figure 1$a{,}d$), respectively. The ratio $c_2/c_1$ is consistently small (covering the range from $0.42$ (ocean edge) to $0.48$ (shelf edge)). All the parameters are in the same range except for $h_T$, which is $4$ times larger in the steep slope set-up so the forced terms in ((2.23), (2.24)) are proportionally strengthened as in Liu et al. (Reference Liu, Grimshaw and Johnson2019a).

Figure 2.

Case $A$. Coefficients of the coupled KdV ((2.23), (2.24)) model: $(a{,}c{,}e{,}g{,}i{,}k)$ gentle slope and $(b{,}d{,}\,f{,}h{,}j{,}l)$ steep slope.

Figures 3 and 4 show the wave evolution under the gentle and steep slope set-ups, respectively, using the coupled KdV system for both mode-1 and mode-2 and the KdV model for mode-1 alone. When the incident mode-1 wave encounters the slope, a mode-2 wave is generated propagating with a speed approximately $c_2$ that is much smaller than the mode-1 linear wave speed $c_1$, and so in the mode-1 coordinate system this generated mode-2 wave travels to the left. Figures 3$(d)$ and 4$(d)$ show the final state in this simulation where both the waves arrive at the shallow-water region. For the mode-1 wave, wave fission is starting to occur. There is no significant distinction between the two different topographic gradients. The feedback of the generated mode-2 wave on the incident mode-1 wave is too small to be seen. The generated mode-2 waves are convex, with an amplitude of 0.3 m in the gentle case and 1.2 m in the steep case, directly proportional to the change in the slope gradient.

Figure 3.

Case $A$. Wave evolution under the KdV (2.1) and coupled KdV ((2.23), (2.24)) model for gentle slope gradient with an initial depression mode-1 solitary wave. $(a{,}c)$ Mode-1 and mode-2 wave evolution under the coupled KdV system. $(b)$ Mode-1 wave evolution under the KdV equation. $(d)$ The final state under simulation for these waves.

Figure 4.

Case $A$. As in figure 3 except that the topographic slope is steeper.

3.2. Case $B$: moderate resonant coupling (polarity change for mode-1 wave)

In case $B$ (see figure 1$b{,}e$), the middle layer thickness is larger, and the upper and lower pycnoclines are much closer to the surface and bottom. The lower layer $(h_3)$ before the slope is thicker than a half of the middle layer $(h_2/2)$, but thinner after the slope, and so a polarity change occurs for the incident mode-1 wave that is initially a depression wave. Figure 5 show the parameters and coefficients of the coupled KdV system. For this thicker middle layer the range of the speed ratio $c_2/c_1$ is from $0.58$ (ocean) to $0.72$ (shelf). The nonlinear coefficient $\nu _1$ of the mode-1 wave passes through the critical value $0$ at $x=0$ on the slope, and $\nu _2$ is now negative in contrast to case $A$. The relative mode-2 wave speed $\varDelta$ is smaller, so the distance between the incident mode-1 wave and the generated mode-2 wave is less over the same time interval. The coupling parameter $\gamma$ increases around $10$ times while ${\partial h}/{\partial T}$ remains almost the same as in case $A$, and so coupling is considerably enhanced in this case $B$.

Figure 5.

Case $B$. Coefficients of coupled KdV ((2.23), (2.24)) model: $(a{,}c{,}e{,}g{,}i{,}k)$ gentle slope and $(b{,}d{,}\,f{,}h{,}j{,}l)$ steep slope.

Figures 6 and 7 show the wave evolution for the gentle and steep slope set-ups (figure 1$b{,}e$), respectively. The incident mode-1 changes polarity on the slope region, and in the KdV model we see an emerging rarefaction wave in the front followed by an undular bore (see figures 6$b{,}d$ and 7$b{,}d$). However, in the coupled KdV system, the mode-1 wave evolution is now significantly affected by the generated mode-2 wave. It changes polarity much faster with an elevation wave in the front and a depression wave at the back. The generated mode-2 wave is now a concave shape, because of the opposite sigh of $\nu _2$ compared to case $A$. When the topographic gradient increases, the amplitude of the generated mode-2 wave increases, which leads to a greater influence on the mode-1 wave. We see that the elevated (depressed) part in figure 7 is much smaller (larger) than in figure 6. Here, the mode-2 wave propagates faster, so it takes longer to reach the same distance from the mode-1 wave compared to case $A$.

Figure 6.

Case $B$. As in figure 3 except that the layer thicknesses are different.

Figure 7.

Case $B$. As in figure 6 except that the topographic slope is steeper.

3.3. Case $C$: strong resonant coupling (initial elevation wave)

In case $C$ (figure 1$c{,}\,f$), the middle layer thickness increases further, so the upper and lower pycnoclines are very close to the surface and bottom, with the lower layer thickness $(h_3)$ thinner than a half of the middle layer $(h_2/2)$ all through. The incident mode-1 solitary wave is now an elevation wave. Figure 8 shows the parameters and coefficients of the coupled KdV equations ((2.23), (2.24)). The mode-2 and mode-1 linear phase speeds are very close after the slope, the shelf edge value of the ratio is greater than $0.9$ and so $c_2 \approx c_1$. The nonlinear coefficients $\nu _1$ and $\nu _2$ stay positive and negative, respectively, indicating an elevation mode-1 wave and a concave mode-2 wave. The relative speed $\varDelta$ of the generated mode-2 wave is approximately $0$ after the slope, so the gap between mode-1 and mode-2 remains almost unchanged after crossing the slope region. The value of $\gamma$ grows significantly by $100$ times compared to case $A$, while ${\partial h}/{\partial T}$ does not change significantly, and so the coupling term is much stronger giving strong resonant coupling.

Figure 8.

Case $C$. Coefficients of coupled KdV ((2.23), (2.24)) model: $(a{,}c{,}e{,}g{,}i{,}k)$ gentle slope and $(b{,}d{,}\,f{,}h{,}j{,}l)$ steep slope.

Figures 9 and 10 show the corresponding wave evolution under these gentle (3.1) and steep (3.2) set-ups with now a very thick middle layer. The incident elevation mode-1 wave fissions in the KdV model (see figures 9$b{,}d$ and 10$b{,}d$) and the number of the generated solitary waves in the steep slope case are fewer than in the gentle slope case. For resonant coupling, amplitudes of the generated mode-2 wave are much larger, comparable to the incident mode-1 wave or even larger (see figures 9$c{,}d$ and 10$c{,}d$), while the mode-1 wave evolution is suppressed by the generated mode-2 wave with a much smaller leading wave followed by a downward wave tail (see figures 9$a{,}d$ and 10$a{,}d$).

Figure 9.

Case $C$. As in figure 3 except that the layer thicknesses are different.

Figure 10.

Case $C$. As in figure 9 except that the topographic slope is steeper.