2.1 The two-layer weakly nonlinear (TWN) model

We begin with Euler equations for two immiscible layers of potential fluids with unequal densities. The fluids in the two layers are assumed to be inviscid, irrotational and incompressible, with unequal densities, $\unicode[STIX]{x1D70C}_{1}$ in the upper layer and $\unicode[STIX]{x1D70C}_{2}$ ( ${>}\unicode[STIX]{x1D70C}_{1}$ for the stable case) in the lower layer. The horizontal and vertical coordinates are denoted by $x$ and $z$ , respectively. The interface and the overlaying free-surface displacements are denoted, respectively, by $\unicode[STIX]{x1D709}_{2}$ and $\unicode[STIX]{x1D709}_{1}$ (see figure 1). The pressure exerted at the surface is denoted by $P_{1}$ . Then, the governing equations for the two-layer Euler system are given as follows

where $\unicode[STIX]{x1D719}_{1}$ ( $\unicode[STIX]{x1D719}_{2}$ ) is the velocity potential of the upper (lower) fluid layer, $h_{1}$ ( $h_{2}$ ) is the undisturbed thickness of the upper (lower) fluid layer, respectively, and $g$ is the gravitational acceleration.

Figure 1.

Sketch of the two-layer fluid system. The surface ripples are located at the leading edge of the IW. Near field corresponds to the region near the central IW and far field corresponds to the region far from the central IW.

For the small-amplitude approximation, we assume that the characteristic amplitude, $a$ , of the IWs and SWs is much smaller than the thickness of the upper fluid layer,

For the long-wave approximation, we assume that the thickness of each fluid layer is much smaller than the characteristic wavelength, $l$ , of the IWs and SWs,

The two small parameters, $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ , control the nonlinear and dispersive effects, respectively. Based on the scaling (2.9)–(2.10), we non-dimensionalize all the physical variables in the Euler equations (2.1)–(2.8). Next, we apply asymptotic analysis to these dimensionless equations, similar to that applied in the derivation of the Korteweg–de Vries (KdV) equation (Whitham Reference Whitham1974). Retaining the terms in the first power of $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ , we obtain the set of equations in the dimensional form for the variables $(\unicode[STIX]{x1D709}_{1},\overline{u}_{1},\unicode[STIX]{x1D709}_{2},\overline{u}_{2})$ (which we refer to as the TWN model),

where $\unicode[STIX]{x1D70C}_{r}$ is the density ratio $\unicode[STIX]{x1D70C}_{1}/\unicode[STIX]{x1D70C}_{2}$ , and

are the depth-averaged horizontal velocities of the upper and lower fluid layers, respectively. This weakly nonlinear model can also be obtained via a direct reduction from a fully nonlinear model given in Choi & Camassa (Reference Choi and Camassa1996), Barros & Gavrilyuk (Reference Barros and Gavrilyuk2007), and Barros & Choi (Reference Barros and Choi2009). We refer to the fully nonlinear model as the Miyata–Choi–Camassa (MCC) model. We provide the dimensionless form of the TWN model in appendix A. Next, we study the basic properties of the TWN model, including the dispersion relation and the class 3 triad resonance condition.

The TWN system admits a linear dispersion relation corresponding to a monochromatic wavetrain with infinitesimal amplitude. By substituting the monochromatic solutions $(\unicode[STIX]{x1D709}_{j},\overline{u}_{j})\sim \exp [\text{i}(kx-\unicode[STIX]{x1D707}_{k}t)]$ , $j=1,2$ , into the TWN system, we obtain the pure linear dispersion relation between the frequency $\unicode[STIX]{x1D707}_{k}$ and the wavenumber $k$ as

The same dispersion relation can be found in Barros & Gavrilyuk (Reference Barros and Gavrilyuk2007). Equation (2.16) always has four real roots when the problem is considered in the oceanic regime, i.e. when the density ratio $\unicode[STIX]{x1D70C}_{r}$ is close to $1$ . Then, at the leading order in $1-\unicode[STIX]{x1D70C}_{r}$ , we can approximate the dispersion relation of the two-mode waves, denoted by $\unicode[STIX]{x1D6FA}_{k}$ and $\unicode[STIX]{x1D714}_{k}$ (figure 2 a), as

and

In the following, the two kinds of waves, corresponding to the dispersion relations $\unicode[STIX]{x1D6FA}_{k}$ (2.17) and $\unicode[STIX]{x1D714}_{k}$ (2.18), are referred to as the long-mode (baroclinic) waves and the short-mode (barotropic) waves, respectively.

Figure 2.

(a) Pure linear dispersion relations $\unicode[STIX]{x1D714}_{k}$ and $\unicode[STIX]{x1D6FA}_{k}$ of short-mode waves and long-mode waves, respectively. (b) Comparison of the phase velocity for the short-mode waves $c_{p}$ (solid line ——), the group velocity for the short-mode waves $c_{g}$ (dashed line ——), and the phase velocity for the long-mode waves $C_{p}$ (dashed-dotted line – ⋅ – ⋅ –). The resonant wavenumber $k_{res}$ satisfies the triad resonance condition (2.20). For our simulations in figures 2–9, the parameters are $(h_{1},h_{2},g,\unicode[STIX]{x1D70C}_{1},\unicode[STIX]{x1D70C}_{2})=(1,3,1,1,1.003)$ .

Resonant interactions among different modes are an important process of energy exchange. For the two-mode waves in equations (2.17) and (2.18), we can numerically verify the existence of solutions to the class 3 triad resonance condition. If the dispersion relations allow the wavenumbers $k_{j}(j=1,2,3)$ and the frequencies $\unicode[STIX]{x1D714}_{k}(k_{1})$ , $\unicode[STIX]{x1D714}_{k}(k_{2})$ and $\unicode[STIX]{x1D6FA}_{k}(k_{3})$ to satisfy the condition

the corresponding three waves constitute a class 3 resonant triad. Moreover, if the wavenumbers have the specific form $k_{1}=k_{res}+\unicode[STIX]{x0394}k/2$ , $k_{2}=k_{res}-\unicode[STIX]{x0394}k/2$ , $k_{3}=\unicode[STIX]{x0394}k$ , where $\unicode[STIX]{x0394}k\ll k_{res}$ and $\unicode[STIX]{x0394}k\rightarrow 0$ , then the resonance condition reduces to

where the group velocity $c_{g}$ and the phase velocity $C_{p}$ are given by the equations

Here, $k_{res}$ is referred to as the resonant wavenumber. Many early results (Phillips Reference Phillips1974; Hashizume Reference Hashizume1980; Craig, Guyenne & Sulem Reference Craig, Guyenne and Sulem2011; Craig et al. Reference Craig, Guyenne and Sulem2012) have confirmed that there exists a unique resonant wavenumber $k_{res}$  satisfying equation (2.20) in the two-layer system. The resonance condition (2.20) shows that, for resonantly interacting waves, the group velocity of short-mode waves and the phase velocity of long-mode waves are equal (Phillips Reference Phillips1974; Hashizume Reference Hashizume1980; Craig et al. Reference Craig, Guyenne and Sulem2011, Reference Craig, Guyenne and Sulem2012). Figure 2(b) shows the phase velocity of the short-mode waves $c_{p}=\unicode[STIX]{x1D714}_{k}/k$ , the group velocity of the short-mode waves $c_{g}$ and the phase velocity of the long-mode waves $C_{p}=\unicode[STIX]{x1D6FA}_{k}/k$ . It can be seen from the inset of figure 2(b) that there indeed exists a resonant wavenumber $k_{res}$ , satisfying the resonance condition (2.20), $c_{g}(k_{res})=C_{p}(0)$ . Thus, class 3 resonant triads exist among two short-mode waves and one long-mode wave for the TWN model. Note that the class 3 triad resonance condition (2.19) or (2.20) is not assumed during the derivation of the TWN model. Nevertheless, the resonance condition (2.19) from Alam (Reference Alam2012) can be satisfied using the dispersion relations (2.17) and (2.18) in the TWN model. As discussed in § 3.2, the class 3 triad resonance phenomena will be numerically investigated in our model in detail.

In our numerical simulations, throughout the text, the variables are all dimensionless. That is, numerically, the characteristic length in both height and wavelength is $h_{1}$ , the characteristic speed is $\sqrt{gh_{1}}$ and the characteristic time is $\sqrt{h_{1}/g}$ (see similar dimensionless forms used in numerical simulations in Choi & Camassa (Reference Choi and Camassa1999) and Choi, Barros & Jo (Reference Choi, Barros and Jo2009)). In this work, two parameter regimes are used in our simulations, one being $(h_{1},h_{2},g,\unicode[STIX]{x1D70C}_{1},\unicode[STIX]{x1D70C}_{2})=(1,3,1,1,1.003)$ and the other being $(h_{1},h_{2},g,\unicode[STIX]{x1D70C}_{1},\unicode[STIX]{x1D70C}_{2})=(1,5,1,0.856,0.996)$ .

2.2 Modified two-layer weakly nonlinear (MTWN) model

Next, we propose a modified version of the TWN model, which takes into account the effects of a BBF. The basic idea is that we are given a slowly varying, sufficiently long-wavelength BBF, and then study the behaviour of short-mode surface waves on top of this BBF (more details concerning using the BBF can be found in appendix A). We construct the modified model for the effects of the BBF on the propagation of short-mode waves, based on the following assumptions: the long-wave approximation (2.10) is only applied to the short-mode (barotropic) waves of IWs and SWs; the characteristic wavelength of short-mode waves, $l$ , is much shorter than the wavelength of the BBF, $L$ ,

The displacements associated with the BBF at the interface and at the surface are denoted by $\unicode[STIX]{x1D6EF}_{2}^{\ast }$ and $\unicode[STIX]{x1D6EF}_{1}^{\ast }$ , respectively. The depth-averaged horizontal currents of the upper and lower layers are denoted by $\overline{U}_{1}^{\ast }$ and $\overline{U}_{2}^{\ast }$ , respectively. In the rest of the paper, the vector $(\unicode[STIX]{x1D6EF}_{j}^{\ast },\overline{U}_{j}^{\ast })$ is referred to as the BBF. The slowly varying long-wavelength BBF disperses weakly, and thereafter the displacements and currents associated with the BBF, $\unicode[STIX]{x1D6EF}_{j}^{\ast }$ and $\overline{U}_{j}^{\ast }$ , can be expressed as the right-moving travelling wave $\unicode[STIX]{x1D6EF}_{j}^{\ast }(\cdot )=\unicode[STIX]{x1D6EF}_{j}^{\ast }(K(x-C_{s}t))$ and $\overline{U}_{j}^{\ast }(\cdot )=\overline{U}_{j}^{\ast }(K(x-C_{s}t))$ , where $K=2\unicode[STIX]{x03C0}/L$ is the characteristic wavenumber and $C_{s}$ is the constant phase velocity of the BBF. The BBF, $(\unicode[STIX]{x1D6EF}_{j}^{\ast },\overline{U}_{j}^{\ast })$ , satisfies the sign constraint $\unicode[STIX]{x1D6EF}_{1}^{\ast }>0$ , $\unicode[STIX]{x1D6EF}_{2}^{\ast }<0$ , $\overline{U}_{1}^{\ast }>0$ , and $\overline{U}_{2}^{\ast }<0$ . The variables $(\unicode[STIX]{x1D709}_{j},\overline{u}_{j})$ in the TWN model eventually will be represented as a sum of two components, one being the BBF, $(\unicode[STIX]{x1D6EF}_{j}^{\ast },\overline{U}_{j}^{\ast })$ , and the other the perturbations of the waves, $(\unicode[STIX]{x1D709}_{j}^{\ast },\overline{u}_{j}^{\ast })$ . We study the motion of the perturbation waves $(\unicode[STIX]{x1D709}_{j}^{\ast },\overline{u}_{j}^{\ast })$ , in particular, we focus on the dynamical behaviour of IWs and SWs on the BBF. The assumption (2.22) adopted here is similar to the one adopted in Hwung et al. (Reference Hwung, Yang and Shugan2009). By substituting $(\unicode[STIX]{x1D709}_{j},\overline{u}_{j})=(\unicode[STIX]{x1D6EF}_{j}^{\ast },\overline{U}_{j}^{\ast })+(\unicode[STIX]{x1D709}_{j}^{\ast },\overline{u}_{j}^{\ast })$ into the TWN system, applying asymptotic analysis (see appendix A for details), and casting the system in the conservation form in the right-moving frame with velocity $C_{s}$ , we obtain the effective evolution equations for the perturbations of waves $(\unicode[STIX]{x1D709}_{j}^{\ast },\overline{u}_{j}^{\ast })$ on the BBF in dimensional form as follows (referred to as the MTWN model)

Here

and the right-moving frame corresponds to the variable transformation $T=t$ and $X=x-C_{s}t$ , where $C_{s}$ is the phase velocity of the BBF $(\unicode[STIX]{x1D6EF}_{j}^{\ast },\overline{U}_{j}^{\ast })$ and $P_{1}$ is the normal pressure exerted at the surface. The parameter $\unicode[STIX]{x1D717}^{\ast }$ takes two values, 1 or 0, where $\unicode[STIX]{x1D717}^{\ast }=0$ corresponds to the linear regime and $\unicode[STIX]{x1D717}^{\ast }=1$ corresponds to the nonlinear regime for the perturbation waves $(\unicode[STIX]{x1D709}_{j}^{\ast },\overline{u}_{j}^{\ast })$ . In the right-moving frame, the BBF $(\unicode[STIX]{x1D6EF}_{j}^{\ast },\overline{U}_{j}^{\ast })$ is steady with respect to time $T$ . More details concerning the dimensionless form of the MTWN model can be found in appendix A.

3.1 Linear modulation, or ray-based, theory

In this section, we review the linear modulation, or ray-based, theory for short-mode SWs on a BBF. We take the components of the long-wavelength BBF, $(\unicode[STIX]{x1D6EF}_{1}^{\ast },\unicode[STIX]{x1D6EF}_{2}^{\ast },\overline{U}_{1}^{\ast },\overline{U}_{2}^{\ast })$ , to be proportional to

with the maximal amplitudes of $(\unicode[STIX]{x1D6EF}_{1}^{\ast },\unicode[STIX]{x1D6EF}_{2}^{\ast },\overline{U}_{1}^{\ast },\overline{U}_{2}^{\ast })$ being $(0.0003,-0.1,0.007,-0.006)$ , the slope $s_{\unicode[STIX]{x1D6EF}^{\ast }}=0.2$ , the wavelength $X_{\unicode[STIX]{x1D6EF}^{\ast }}=120$ and the phase velocity $C_{s}=0.0485$ . Here, the maximal amplitudes and the phase velocity are chosen to be close to those of the interfacial solitary wave solution of the TWN model for $\unicode[STIX]{x1D6EF}_{2}^{\ast }(0)=-0.1$ (see appendix B). The parameter $X_{\unicode[STIX]{x1D6EF}^{\ast }}=120$ characterizes the wavelength of the BBF and is chosen to be large enough to satisfy the assumption (2.22). The parameter $s_{\unicode[STIX]{x1D6EF}^{\ast }}$ characterizes the slope of the BBF. Later on in § 4.2, we will vary these parameters for a parametric study and see how their values affect the dynamical behaviour of short-mode SWs.

In the presence of the baroclinic flow in the right-moving frame, the modulated dispersion relation $\overline{\unicode[STIX]{x1D708}}_{k}$ for the perturbation waves $(\unicode[STIX]{x1D709}_{j}^{\ast },\overline{u}_{j}^{\ast })$ can be obtained by substituting $(\unicode[STIX]{x1D709}_{j},\overline{u}_{j})\sim (\unicode[STIX]{x1D6EF}_{j}^{\ast },\overline{U}_{j}^{\ast })+\exp [\text{i}(kX-\overline{\unicode[STIX]{x1D714}}_{k}T)]$ with $\overline{\unicode[STIX]{x1D714}}_{k}=\overline{\unicode[STIX]{x1D708}}_{k}+C_{s}k$ into the TWN system . Here, $\overline{\unicode[STIX]{x1D714}}_{k}$ is the dispersion relation in the resting frame, whereas $\overline{\unicode[STIX]{x1D708}}_{k}$  is the dispersion relation in the right-moving frame. The resulting equation is

where $|\cdot |$ denotes the determinant of the enclosed matrix. Moreover, we also obtain three linear eigenfunction relations among the amplitudes of $(\unicode[STIX]{x1D709}_{j}^{\ast },\overline{u}_{j}^{\ast })$ . We note that, first, $(\unicode[STIX]{x1D6EF}_{j}^{\ast },\overline{U}_{j}^{\ast })$ are steady in time $T$ in the right-moving frame, so that the dispersion relation $\overline{\unicode[STIX]{x1D708}}_{k}$  is independent of time $T$ . Second, the wavelengths of $(\unicode[STIX]{x1D6EF}_{j}^{\ast },\overline{U}_{j}^{\ast })$ are relatively long compared to the characteristic length of the short-mode waves, so $(\unicode[STIX]{x1D6EF}_{j}^{\ast },\overline{U}_{j}^{\ast })$ in the dispersion relation (3.2) can be locally treated as constant in $X$ space. Third, if $(\unicode[STIX]{x1D6EF}_{j}^{\ast },\overline{U}_{j}^{\ast })=0$ , the modulated dispersion relation $\overline{\unicode[STIX]{x1D714}}_{k}$ in equation (3.2) reduces to the pure linear dispersion relation $\unicode[STIX]{x1D707}_{k}$ in equation (2.16).

Due to the slow variation in space and time of the phase of the short-mode waves, the governing equations of the space-time rays at the location $X$ and wavenumber $k$ are given by the ray-based theory (Whitham Reference Whitham1974; Bakhanov & Ostrovsky Reference Bakhanov and Ostrovsky2002; Kodaira et al. Reference Kodaira, Waseda, Miyata and Choi2016),

where $\overline{\unicode[STIX]{x1D708}}_{k}$ is the modulated dispersion relation in equation (3.2). Since the dispersion relation $\overline{\unicode[STIX]{x1D708}}_{k}$ does not explicitly depend on time $T$ , equations (3.3) and (3.4) constitute a Hamiltonian system with $\overline{\unicode[STIX]{x1D708}}_{k}$ as the Hamiltonian, $X$ displacement and $k$ momentum. Equation (3.3) states that the wave packet propagates with the group velocity.

Figure 3.

(a) The phase portrait of the right-moving wave-packet motion in the variables $X$ and $k$ . Along each cyan-coloured curve, the frequency $\overline{\unicode[STIX]{x1D708}}_{k}$ remains constant. Inside the region enclosed by the blue level-curve, the right-moving short-mode waves are trapped. The red level-curve corresponds to the simulation in panel (c). Along the two green level-curves, the minimal wavenumbers are $k=3$ and $k=4$ , respectively (corresponding to the initial perturbation (4.1) in the next § 4). The arrows indicate the direction of wave-packet movement. (b) Plotted is the group velocity $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D708}}_{k}/\unicode[STIX]{x2202}k$ with its magnitude colour-coded (copper–black) on the background. The yellow curve corresponds to zero group velocity. (c) The maximal amplitude $A$ versus the peak location $X$ for the evolution of a narrow wave packet with a narrow band of wavenumbers. The initial value is marked by the green cross. The red curve corresponds to the first period and the blue curve corresponds to the second period. The main observation is that SWs become high at the leading edge, $100<X<120$ , and low at the trailing edge, $-120<X<-100$ (asymmetric behaviour (1c)). This main observation of asymmetric behaviour (1c) persists for both periods (red and blue curves). On the flat BBF ( $-100<X<100$ ), as a result of the wave dispersion, the initially narrow SW packet becomes smaller in maximal amplitude and wider in length. Also, due to the wave dispersion, the maximal amplitude in the second period (blue curve) becomes smaller than that in the first period (red curve). In fact, eventually the waves will be almost uniformly distributed on the flat BBF after several periods (not shown here) due to the wave dispersion and the periodic motion of SWs.

We first study the dynamical behaviour of right-moving (positive phase velocity) short-mode SWs. In figure 3(a), the phase portrait of the right-moving wave-packet motion in the variables $X$ and $k$ is constructed from the modulated dispersion relation $\overline{\unicode[STIX]{x1D708}}_{k}$ using the ray-based theory. It can be seen from figure 3(a) that right-moving short-mode waves become short in wavelength at the leading edge ( $X>0$ ), whereas they become long at the trailing edge ( $X<0$ ). Figure 3(b) displays the group velocity of these short-mode waves, $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D708}}_{k}/\unicode[STIX]{x2202}k$ , (in variables $X$ and $k$ ) with its magnitude colour-coded on top of the background. It can be seen from figure 3(b) that, above the yellow curve, $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D708}}_{k}/\unicode[STIX]{x2202}k<0$ and its absolute value is relatively small, whereas below the yellow curve, $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D708}}_{k}/\unicode[STIX]{x2202}k>0$ and its absolute value is relatively large. This suggests that the right-moving short-mode wave packets propagate towards the trailing edge (in the direction of decreasing $X$ when $C_{s}>0$ ) with a relatively small group velocity, and towards the leading edge (in the direction of increasing $X$ when $C_{s}>0$ ) with a relatively large group velocity.

We now study the relation between the maximal amplitude $A$ and the peak location $X$ of the right-moving short-mode waves. We numerically simulate the MTWN model with $\unicode[STIX]{x1D717}^{\ast }=1$ and $P_{1}=0$ in the weakly nonlinear regime. Here, the weakly nonlinear regime means that the amplitude of initial perturbation $(\unicode[STIX]{x1D709}_{j}^{\ast },\overline{u}_{j}^{\ast })$ is very small. The initial perturbation for short-mode waves is a narrow wave packet with a disturbance of a narrow band of wavenumbers. The initial perturbation of $\unicode[STIX]{x1D709}_{2}^{\ast }$ is taken to be

with $A_{\unicode[STIX]{x1D700}}=2\times 10^{-4}$ , $X_{0}=123$ , $x_{0}=3$ and $k_{0}=5$ . This initial perturbation $(X_{0},k_{0})$ is located on the red-level curve in figure 3(a). To allow for only right-moving short-mode waves in the system, the other three variables, $(\unicode[STIX]{x1D709}_{1}^{\ast },\overline{u}_{1}^{\ast },\overline{u}_{2}^{\ast })$ , have the same form as the perturbation (3.5), with their amplitudes satisfying the linear eigenfunction relations following from the matrix in equation (3.2). For example, we can obtain from the matrix in equation (3.2) that $\overline{u}_{2}^{\ast }=(\overline{\unicode[STIX]{x1D714}}_{k}-k\overline{U}_{2}^{\ast })\unicode[STIX]{x1D709}_{2}^{\ast }/(kh_{2}+k\unicode[STIX]{x1D6EF}_{2}^{\ast })$ . More details about the settings of the numerical simulations, including the numerical scheme, dealiasing strategy, boundary conditions and Kelvin–Helmholtz instability are presented in appendix C.

Figure 3(c) displays the maximal amplitude $A$ versus the peak location $X$ for the evolution of the perturbation for the right-moving short-mode waves. It can be seen that SWs become high in amplitude at the leading edge ( $X>0$ ) and low at the trailing edge ( $X<0$ ). Notice that this assertion is also true for the Euler system of equations (Lewis et al. Reference Lewis, Lake and Ko1974). In particular, this SW amplitude phenomenon can be inferred from the energy conservation equation of the Euler system. The energy conservation equation can be obtained using the concept of radiation stress for the Euler system (Lewis et al. Reference Lewis, Lake and Ko1974)

where $\widetilde{C}_{g}$ is the group velocity in the pure linear dispersion relation, $\widetilde{A}$ is the amplitude of SW packets and $\widetilde{U}$ is the surface-current elevation. At the leading edge, a negative strain rate $\unicode[STIX]{x2202}\widetilde{U}/\unicode[STIX]{x2202}x$ corresponds to the growth of the amplitude (surface convergence), whereas at the trailing edge, a positive strain rate $\unicode[STIX]{x2202}\widetilde{U}/\unicode[STIX]{x2202}x$ corresponds to the decay of the amplitude (surface divergence).

Figure 4.

(a) Snapshots of the BBF and the spatiotemporal evolution of SWs $\unicode[STIX]{x1D709}_{1}$ . The red and blue lines correspond to the group velocities of the left-moving SWs and right-moving SWs, respectively. (b) Zoomed-in version of (a). The red and blue lines are the same as those in (a). The black and magenta lines correspond to the phase velocities of the left-moving SWs and right-moving SWs, respectively. Note that the phase velocity of the left-moving SWs (black line) is negative whereas the phase velocity of the right-moving SWs (magenta line) is positive. (c) Snapshots of SWs $\unicode[STIX]{x1D709}_{1}$ . Initially at $T=0$ (red curve), there is only a narrow wave packet of SWs. At $T=1000$ , the smaller-amplitude left-moving SW packet (blue curve for $X<50$ ) and the larger-amplitude right-moving SW packet (blue curve for $X>50$ ) can be seen. Note that the left-moving SW packet, corresponding to a negative phase velocity, is not trapped in the near field. On the other hand, the right-moving SW packet, corresponding to a positive phase velocity, is trapped in the near field. At $T=3500$ , the left-moving waves (black curve for $X<-150$ ) are partially absorbed by the boundary condition (see appendix C). At $T=5000$ , the left-moving waves (green curve for $X<-150$ ) are completely absorbed by the boundary. For the right-moving waves, as a result of the wave dispersion, the initially narrow SW packet (red curve for $X>100$ ) becomes smaller in maximal amplitude and wider in length (e.g. green curve for $X>-50$ ).

Next, we study the dynamical behaviour of left-moving (negative phase velocity) short-mode SWs. Figure 4(a) displays the spatiotemporal evolution of SWs $\unicode[STIX]{x1D709}_{1}$ . The initial perturbation of $\unicode[STIX]{x1D709}_{1}^{\ast }$ is the same as that of $\unicode[STIX]{x1D709}_{1}^{\ast }$ used in figure 3(c). However, the initial perturbations of the other three variables, $\unicode[STIX]{x1D709}_{2}^{\ast },u_{1}^{\ast },u_{2}^{\ast }$ , are all taken to be zero. It can be seen that the initial narrow wave packet splits into two wave packets, one corresponding to the right-moving SW packet (blue line) and the other corresponding to the left-moving SW packet (red line). The right-moving waves are trapped in the near field as discussed above and shown in figure 3. On the other hand, the left-moving waves quickly leave the near field (figure 4), and then these left-moving waves are absorbed by the boundary (figure 4 a,c).

To summarize, the behaviour of the short-mode waves is asymmetric at the leading edge versus the trailing edge when there is an underlying BBF:

1. (1a) Right-moving SW packets propagate towards the trailing edge with a relatively small group velocity, and towards the leading edge with a relatively large group velocity (figure 3 b).

2. (1b) Right-moving SWs become short in wavelength at the leading edge and long at the trailing edge (figure 3 a).

3. (1c) Right-moving SWs become high in amplitude at the leading edge and low at the trailing edge (figure 3 c).

4. (1d) Left-moving SWs quickly leave the near field, and then only the right-moving SWs that propagate in the same direction as the BBF remain trapped in the near field (figure 4).

To the best of our knowledge, the asymmetric behaviour (1b) and (1d) was earlier discovered in references (Bakhanov & Ostrovsky Reference Bakhanov and Ostrovsky2002; Kodaira et al. Reference Kodaira, Waseda, Miyata and Choi2016), the asymmetric behaviour (1c) was earlier discovered in reference (Lewis et al. Reference Lewis, Lake and Ko1974), but the asymmetric behaviour (1a) was not pointed out earlier. Here, we review these asymmetric types of behaviour predicted by the ray-based theory using the MTWN model in the weakly nonlinear regime. These asymmetric types of behaviour are important for demonstrating the modulation-resonance mechanism, as will be discussed in § 4 below.

In the rest of the paper, we always use short-mode SWs to denote right-moving short-mode SWs that are trapped in the near field. Whenever we discuss left-moving short-mode SWs, we will directly use the phrase left-moving short-mode SWs.

3.2 Nonlinear class 3 triad resonance

Figure 5.

(a) Modulated dispersion relations (3.2) for the short-mode waves, $\overline{\unicode[STIX]{x1D714}}_{k}$ , and the long-mode waves, $\overline{\unicode[STIX]{x1D6FA}}_{k}$ . (b) The wavenumbers of two short-mode waves satisfying the class 3 triad resonance condition as in equation (2.19) but using the modulated dispersion relations. Panels (c) and (d) show the evolution of the wavenumber spectra of the SWs $\unicode[STIX]{x1D709}_{1}^{\ast }$ and the IWs $\unicode[STIX]{x1D709}_{2}^{\ast }$ . The resonant wavenumber is $k_{res}=5.0$ .

In this section, we review the phenomena associated with the nonlinear class 3 triad resonance for short-mode SWs. Class 3 triad resonance is considered responsible for the surface signature of the underlying internal waves. The resonant excitation of surface waves whose group velocity is close to the phase velocity of long internal waves has been reported in various observations (Lewis et al. Reference Lewis, Lake and Ko1974; Osborne & Burch Reference Osborne and Burch1980). Many works have been dedicated to the effective model describing class 3 triad resonance coupling phenomena between interfacial and surface waves using multi-scale analysis (Lewis et al. Reference Lewis, Lake and Ko1974; Kawahara et al. Reference Kawahara, Sugimoto and Kakutani1975; Hashizume Reference Hashizume1980; Funakoshi & Oikawa Reference Funakoshi and Oikawa1983; Sepulveda Reference Sepulveda1987; Lee et al. Reference Lee, Shugan and An2007; Hwung et al. Reference Hwung, Yang and Shugan2009; Craig et al. Reference Craig, Guyenne and Sulem2012). Alam (Reference Alam2012) showed that class 3 triad resonance can occur among two short-mode waves and one long-mode wave in which all waves propagate in the same direction in a two-layer system. Tanaka & Wakayama (Reference Tanaka and Wakayama2015) showed that in a two-layer Euler system, the spectra of SWs and IWs can change significantly due to the class 3 triad resonance. Moreover, there is an inverse energy cascade from larger wavenumbers to smaller wavenumbers, and finally a steep peak forms in the spectrum of SWs around the resonant wavenumber $k_{res}$  (Tanaka & Wakayama Reference Tanaka and Wakayama2015).

In our set-up, we will reproduce the above phenomena observed by Tanaka & Wakayama (Reference Tanaka and Wakayama2015) using our MTWN model with $\unicode[STIX]{x1D717}^{\ast }=1$ and $P_{1}=0$ in the presence of the BBF in the strongly nonlinear regime. Here, the strongly nonlinear regime means that the amplitude of the initial perturbation of $(\unicode[STIX]{x1D709}_{j}^{\ast },\overline{u}_{j}^{\ast })$ is relatively large. The BBF is constant in $X$ space, $(\unicode[STIX]{x1D6EF}_{1}^{\ast },\unicode[STIX]{x1D6EF}_{2}^{\ast },\overline{U}_{1}^{\ast },\overline{U}_{2}^{\ast })=(0.0003,-0.1,0.007,-0.006)$ , with the velocity $C_{s}=0.0485$ . Periodic boundary conditions are applied in this numerical implementation (note that, for other numerical implementations in §§ 3.1 and 4, absorbing boundary conditions are applied as discussed in appendix C). The spectrum of IW amplitude $\unicode[STIX]{x1D709}_{2}^{\ast }$ , at the initial time $T=0$ , is given by

where $a_{m}=0.25$ , $c_{w}=0.5$ , $k_{c}=8$ , $k_{m}=4.5$ and $H_{e}(k-k_{m})$ is a Heaviside step function with $H_{e}=0$ for $k\leqslant k_{m}$ and $H_{e}=1$ for $k>k_{m}$ . Here, the wavenumber $k_{c}>k_{res}$ , and the Heaviside step function $H_{e}$ is used to ensure that low wavenumbers $(k\leqslant k_{m})$ are not included in the initial disturbance of the IW spectrum, and furthermore to identify the subsequent growth of the IW spectrum for $k\leqslant k_{m}$ due to class 3 triad resonance. To ensure that the initial wave field consists only of the short-mode waves propagating in the positive $X$ direction, again, the amplitudes of the other three variables $(\unicode[STIX]{x1D709}_{1}^{\ast },\overline{u}_{1}^{\ast },\overline{u}_{2}^{\ast })$ at $T=0$ satisfy the linear eigenfunction relations stemming from the matrix in equation (3.2). The phase of the variable $\widehat{\unicode[STIX]{x1D709}}_{2}^{\ast }(k,0)$ for each wavenumber is uniformly distributed on $[0,2\unicode[STIX]{x03C0}]$ .

Figure 5(a) displays the modulated dispersion relation $\overline{\unicode[STIX]{x1D714}}_{k}$ in equation (3.2) for the short-mode waves, and $\overline{\unicode[STIX]{x1D6FA}}_{k}$ in equation (3.2) for the long-mode waves, in the presence of the constant BBF. Figure 5(b) displays the wavenumbers $k_{1}$ and $k_{2}$ of the two short-mode waves which satisfy the class 3 triad resonance condition as in equation (2.19), but using the modulated dispersion relations $\overline{\unicode[STIX]{x1D714}}_{k}$  in equation (3.2) for two short-mode waves, and $\overline{\unicode[STIX]{x1D6FA}}_{k}$ in equation (3.2) for one long-mode wave. Figure 5(c) and (d) display several snapshots of the wave spectra for SWs and IWs, respectively.

It can be clearly seen that

1. (2a) A significant amount of energy is transferred from SWs towards IWs due to the class 3 triad resonance.

2. (2b) There is an inverse energy cascade for the spectrum of SWs from larger wavenumbers above $k_{res}$ towards smaller wavenumbers, and finally a steep peak forms in the spectrum of SWs around $k_{res}$ .

Here, we have reproduced the class 3 triad resonance phenomena (as observed by Tanaka & Wakayama (Reference Tanaka and Wakayama2015)) using our MTWN model. These two resonance phenomena (2a) and (2b) are also important for demonstrating the modulation-resonance mechanism in § 4 below.

4.1 Modulation-resonance mechanism

Figure 6.

Comparison of spatiotemporal manifestation of SWs in the linear regime ( $\unicode[STIX]{x1D717}^{\ast }=0,\unicode[STIX]{x1D70E}_{m}=5\times 10^{-4}$ ) (first column), in the weakly nonlinear regime ( $\unicode[STIX]{x1D717}^{\ast }=1,\unicode[STIX]{x1D70E}_{m}=5\times 10^{-4}$ ) (second column), and in the strongly nonlinear regime ( $\unicode[STIX]{x1D717}^{\ast }=1,\unicode[STIX]{x1D70E}_{m}=2.5\times 10^{-2}$ ) (third column). The first row (a–c) displays the snapshot of the BBF and the spatiotemporal evolution of SWs $\unicode[STIX]{x1D709}_{1}=\unicode[STIX]{x1D6EF}_{1}^{\ast }+\unicode[STIX]{x1D709}_{1}^{\ast }$ . The parameters of the BBFs are given in table 1. The second row (d–f) and the third row (g–i) display the zoomed-in versions of SW profiles. The red line corresponds to the positive phase velocity of the right-moving short-mode SWs. The white line (blue line) corresponds to the group velocity of right-moving short-mode SWs when SW packets propagate from the trailing (leading) edge towards the leading (trailing) edge. The ranges of colour bars in panels (a–f) are automatically tuned. To observe the small-amplitude SW packets that propagate from the trailing edge towards the leading edge (white lines), the ranges of colour bars in panels (g–i) are manually tuned to be smaller. The fourth row (j–l) displays the snapshots of SWs and IWs at $T=15\,000$ . It can be seen that spatiotemporal manifestation of SWs differs greatly between the linear (or weakly nonlinear) regime and the strongly nonlinear regime. In the strongly nonlinear regime, SW packets are located at the leading edge due to the class 3 triad resonance (see text).

Figure 7.

Comparison of wavenumber spectra of SWs and IWs in the linear regime ( $\unicode[STIX]{x1D717}^{\ast }=0,\unicode[STIX]{x1D70E}_{m}=5\times 10^{-4}$ ) (first column), in the weakly nonlinear regime ( $\unicode[STIX]{x1D717}^{\ast }=1,\unicode[STIX]{x1D70E}_{m}=5\times 10^{-4}$ ) (second column), and in the strongly nonlinear regime ( $\unicode[STIX]{x1D717}^{\ast }=1,\unicode[STIX]{x1D70E}_{m}=2.5\times 10^{-2}$ ) (third column). The first row (a–c) displays the evolution of the wavenumber spectra of SWs $\unicode[STIX]{x1D709}_{1}^{\ast }$ . The bracket $\langle \cdot \rangle$ is interpreted as an ensemble average over $48$ realizations. The second row (d–f) displays the evolution of the wavenumber spectra of IWs $\unicode[STIX]{x1D709}_{2}^{\ast }$ . All the wavenumber spectra vanish at the initial $T=0$ , and these spectra grow as a result of the external pressure $P_{1}$ . In the linear and weakly nonlinear regimes, the SW spectrum transits between the intervals $[3,5]$ and $[6.5,9]$ as predicted by the ray-based theory (two green closed curves in figure 3 a) and thereafter there is a dip at $k_{res}$ in the SW spectrum. In contrast, in the strongly nonlinear regime, there is a peak growing at $k_{res}$ in the SW spectrum due to the class 3 triad resonance. Moreover, the IW spectrum grows at $k\approx 0.3$ also due to the class 3 triad resonance.

In this section, we will show that in the strongly nonlinear regime of the MTWN model short-mode SWs on the BBF in equation (3.1) can be located at the leading edge with the resonant wavenumber $k_{res}\approx 5$ (see figure 5 b for a description of the resonant wavenumber). For comparison, we also compute the short-mode SW behaviour in the weakly nonlinear and linear regimes (see appendix A for more discussion of these regimes). As introduced previously, $\unicode[STIX]{x1D717}^{\ast }=0$ corresponds to the linear regime and $\unicode[STIX]{x1D717}^{\ast }=1$ corresponds to the nonlinear regime. For $\unicode[STIX]{x1D717}^{\ast }=1$ , unlike in § 3, we here use the amplitude of the surface pressure $P_{1}$ to control the strength of the nonlinearity. Initially, all wave variables $(\unicode[STIX]{x1D709}_{j}^{\ast },\overline{u}_{j}^{\ast })$ are set to zero. Specifically, we trigger the MTWN system by exerting random pressure $P_{1X}$ only at the initial $T=0$ as follows

where $\unicode[STIX]{x1D70E}_{m}$ controls the strength of the initial perturbation, $\unicode[STIX]{x1D6FF}(T)$ is the Kronecker delta function which is 1 when $T=0$ and 0 otherwise, $s_{L}=2.5$ , $k_{L}=3$ , $s_{R}=1.5$ , $k_{R}=4$ and $\unicode[STIX]{x1D719}_{k}$ is a random phase, which is uniformly distributed on $[0,2\unicode[STIX]{x03C0}]$ for each wavenumber $k$ . Here, the parameters $s_{L}$ , $k_{L}$ , $s_{R}$ and $k_{R}$ are empirically chosen such that the initial perturbation waves possess wavenumbers between the two critical wavenumbers $k_{IW}$ and $k_{res}$ , where $k_{IW}=0.3$ corresponds to the subsequently growing peak location of the IW spectrum (figure 7) and $k_{res}=5$ is the resonant wavenumber of SWs (figure 7). Next, we focus on the dynamical behaviour of the short-mode SWs in both the physical $X$ space and the wavenumber $k$ space for all regimes.

Figure 6 displays the comparison of wave behaviour among the linear, weakly nonlinear and strongly nonlinear regimes. For the linear regime ( $\unicode[STIX]{x1D717}^{\ast }=0$ ), it can be seen from figure 6(a,d,g) that the SW packets propagate in the direction of decreasing $X$ with a relatively small group velocity (obvious dark stripes parallel to the blue line during the time period about $0\leqslant T\leqslant 10\,000$ ), and in the direction of increasing $X$ with a relatively large group velocity (faint dark stripes parallel to the white line during the time period about $10\,000\leqslant T\leqslant 14\,000$ ). This zigzag pattern corresponds to the asymmetric behaviour (1a) of the short-mode SWs, as can be predicted by the linear modulation theory. In fact, the obvious and faint dark stripes in figure 6(a,g) reflect the large amplitude and the small amplitude of the short-mode SWs, respectively. This implies the asymmetric behaviour (1c), i.e. that SWs become high at the leading edge and low at the trailing edge.

Moreover, it can be seen from figures 7(a) and 7(d) that both SW spectra and IW spectra are concentrated around the intervals $[3,5]$ and $[6.5,9]$ . Notice that the spectrum of the initial perturbation $P_{1X}$ in equation (4.1) is concentrated around the interval $[3,4]$ , so that, based on the linear modulation theory, the wave spectra should be concentrated around the interval $[6.5,9]$ when SW packets propagate towards the trailing edge (see the two green-level curves in figure 3 a). This numerical result implies the asymmetric behaviour (1b), i.e. that SWs become short at the leading edge and long at the trailing edge. Note that no IW spectrum grows for $k\leqslant 2$ (figure 7 d) and no steep peak forms in the SW spectrum around $k_{res}$ (figure 7 a). Thus, the triad resonance phenomena (2a) and (2b) are not observed in this linear regime. In summary, for the linear regime ( $\unicode[STIX]{x1D717}^{\ast }=0$ ), as expected, the asymmetric behaviour (1a)–(1c) is observed, while the triad resonance phenomena (2a) and (2b) are not observed. Based on the ray-based theory, the short-mode waves are linearly modulated by the BBF $(\unicode[STIX]{x1D6EF}_{j}^{\ast },\overline{U}_{j}^{\ast })$ .

In the weakly nonlinear regime ( $\unicode[STIX]{x1D717}^{\ast }=1,\unicode[STIX]{x1D70E}_{m}=5\times 10^{-4}$ ), we see that the spatiotemporal manifestations of SWs and IWs (figure 6 b,e,h) resemble those of SWs and IWs in the linear regime (figure 6 a,d,g). In the weakly nonlinear regime, class 3 triad resonance can occur in the near field. It can be seen from figure 7(b) that even the SW spectra resemble those in the linear regime (figure 7 a). The only difference arises from the IW spectrum. We see from figure 7(e) that the IW spectrum grows for $k\approx 0.3$ as time increases. This reflects the triad resonance phenomenon (2a), i.e. that the energy is transferred from SWs towards IWs.

For the strongly nonlinear regime ( $\unicode[STIX]{x1D717}^{\ast }=1,\unicode[STIX]{x1D70E}_{m}=2.5\times 10^{-2}$ ), the behaviour of SWs and IWs, in both the $X$ and $k$ spaces, becomes quite different from that in the linear and weakly nonlinear regimes. In the physical $X$ space, surprisingly, one can see from figure 6(c, f) that SW packets are trapped at the leading edge of the BBF after some initial time evolution. In the wavenumber $k$ space, both the triad resonance phenomena (2a) and (2b) are observed in figure 7(f) and figure 7(c), respectively. It can be seen from figure 7(f) that IW spectrum grows significantly for $k\approx 0.3$ as time increases, which implies that a significant amount of energy is transferred from SWs towards IWs due to the class 3 triad resonance. It can be seen from figure 7(c) that there is an inverse energy cascade for the spectrum of SWs from larger wavenumbers above $k_{res}$ towards smaller wavenumbers, and finally a steep peak forms in the spectrum of SWs around $k_{res}$ .

We now propose a possible mechanism, referred to as the modulation-resonance mechanism ((3a) and (3a ^′ ) correspond to the modulation mechanism and (3b) corresponds to the resonance mechanism), underlying this phenomenon, namely, that SWs whose typical wavenumber is the resonant wavenumber $k_{res}$ are located at the leading edge of the BBF. For convenience of discussing the wave direction, $C_{s}>0$ is assumed for the velocity of the BBF in the rest of the discussion.

1. (3a) First, after the initial transients, the large-scale BBF separates the left-moving and right-moving short-mode waves. The left-moving radiation quickly leaves the near field, and then only the right-moving waves that propagate in the same direction as the BBF remain trapped in the near field (figure 4).

1. (3a’) Next, based on the modulation, ray-based, theory, if the right-moving short-mode perturbation waves possess wavenumbers below $k_{res}$ , they are transported towards the leading edge with a relatively large group velocity (long black arrow in figure 3 a and white line in figure 6 i). Also, for right-moving short-mode perturbation waves located in front of the leading edge with wavenumbers larger than $k_{res}$ (three rightmost short black arrows in figure 3 a), they are transported towards the leading edge with a relatively slow group velocity. Thus, SWs are accumulated at the leading edge. During the sloping process at the leading edge, these waves become short in wavelength (wavenumber $k$ becomes $k>k_{res}$ ) and high in amplitude. Here, the sloping process means the tendency of short-mode waves to shorten and heighten.

1. (3b) Finally, since SW packets propagate towards the trailing edge with a relatively small group velocity during the sloping process (three short black arrows at $X\sim 80$ in figure 3 a and blue line in figure 6 f), energy can be accumulated, and meanwhile the amplitude of SWs grows locally at the leading edge as a consequence of the asymmetric behaviour (1a)–(1c). The growing amplitude increases the local nonlinearity significantly and consequently triggers the occurrence of a class 3 triad resonance locally at the leading edge. Then, an inverse energy cascade for the spectrum of SWs emerges, streaming from the larger wavenumbers above $k_{res}$ towards the smaller wavenumbers, and finally the spectrum of SWs focuses around the $k_{res}$ (figure 7 c). This spectrum evolution indicates that the class 3 triad resonance condition (2.19) reduces to the resonance condition, $\overline{c}_{g}(k_{res})=C_{s}$ , that is, SW packets propagate at the same velocity as the BBF. Here, $\overline{c}_{g}$ is the modulated group velocity given by $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D714}}_{k}/\unicode[STIX]{x2202}k$ . Therefore, SWs with the resonant wavenumber $k_{res}$ can be located at the leading edge of the BBF as shown in figure 6(c) and figure 7(c).

In the following, § 4.2 and § 4.3, we will examine the robustness of our model results by investigating the behaviour of SWs for various underlying BBFs in a parametric study. From the parametric study results, we will supplement an additional resonance mechanism (3b ^′ ) to the modulation-resonance mechanism in § 4.3.

4.2 Parametric study for parameter regime I

Table 1.

The wavelengths $X_{\unicode[STIX]{x1D6EF}^{\ast }}$ , slopes $s_{\unicode[STIX]{x1D6EF}^{\ast }}$ , maximal amplitudes of displacements $\unicode[STIX]{x1D6EF}_{j}^{\ast }$ , maximal amplitudes of currents $\overline{U}_{j}^{\ast }$ , and phase velocities $C_{s}$ of the BBF in equation (3.1) used for regime I, $(h_{1},h_{2},g,\unicode[STIX]{x1D70C}_{1},\unicode[STIX]{x1D70C}_{2})=(1,3,1,1,1.003)$ . In figure 6, the maximal amplitudes of $(\unicode[STIX]{x1D6EF}_{j}^{\ast },\overline{U}_{j}^{\ast })$ and phase velocities are chosen to be close to those of the interfacial solitary wave solutions of the TWN model for $\unicode[STIX]{x1D6EF}_{2}^{\ast }(0)=-0.1$ (see appendix B). In figures 8 and 9, the maximal amplitudes and phase velocities are chosen to be the same as those of the interfacial solitary wave solutions of the TWN model (see appendix B).

Figure 8.

Parameter regime I. (a) Snapshots of SWs $\unicode[STIX]{x1D709}_{1}$ at $T=10\,000$ for different wavelength $X_{\unicode[STIX]{x1D6EF}^{\ast }}$ of the BBF. The parameters of the BBFs are given in table 1. (b) The wavenumber spectra of SWs $\unicode[STIX]{x1D709}_{1}^{\ast }$ . The bracket $\langle \cdot \rangle$ is interpreted as the time average over $5000\leqslant T\leqslant 10\,000$ .

In this section, we investigate how the dynamical behaviour of the short-mode SWs is affected by the wavelength $X_{\unicode[STIX]{x1D6EF}^{\ast }}$ , slope $s_{\unicode[STIX]{x1D6EF}^{\ast }}$ and maximal amplitude of $(\unicode[STIX]{x1D6EF}_{j}^{\ast },\overline{U}_{j}^{\ast })$ associated with the BBF in equation (3.1) (table 1) in the strongly nonlinear regime of the parameter regime I, $(h_{1},h_{2},g,\unicode[STIX]{x1D70C}_{1},\unicode[STIX]{x1D70C}_{2})=(1,3,1,1,1.003)$ . After this parametric study, we will further discuss the modulation-resonance mechanism in § 4.3. The initial driving force $P_{1X}$  adopted here is exactly the same as the force corresponding to the strongly nonlinear regime in the previous section (4.1). First, we study the effects of BBF’s wavelength $X_{\unicode[STIX]{x1D6EF}^{\ast }}$ on the dynamical behaviour of short-mode SWs. It can be seen from figure 8(a) that the longer the wavelength of the BBF, the larger the number of SW packets that can be trapped at the leading edge in $X$ space. From figure 8(b), it can be seen that the wavenumber spectra of SWs focus around the resonant wavenumber $k_{res}$ . It can be further observed that the longer the wavelength of the BBF, the higher the peaks of the SW spectra around the resonant $k_{res}$ .

Figure 9.

Parameter regime I. The snapshot of the BBF and the corresponding spatiotemporal evolution of SWs $\unicode[STIX]{x1D709}_{1}$ for the different maximal amplitude of the BBF being (a)  $\unicode[STIX]{x1D6EF}_{2}^{\ast }(0)=-0.1$ , (b)  $\unicode[STIX]{x1D6EF}_{2}^{\ast }(0)=-0.2$ , and (c) $\unicode[STIX]{x1D6EF}_{2}^{\ast }(0)=-0.3$ . The parameters of the BBFs are given in table 1. (d) Zoomed-in version of (c) for $11\,500\leqslant T\leqslant 14\,500$ for the maximal amplitude of the BBF being $\unicode[STIX]{x1D6EF}_{2}^{\ast }(0)=-0.3$ . (e) Snapshots of SWs $\unicode[STIX]{x1D709}_{1}$ at $T=15\,000$ for different maximal amplitudes of the BBFs. (f) The temporal evolution of the maximal amplitude, $A$ , of the SWs. (g) The wavenumber spectra of SWs $\unicode[STIX]{x1D709}_{1}^{\ast }$ . The bracket $\langle \cdot \rangle$ is interpreted as the time average over $5000\leqslant T\leqslant 20\,000$ . The resonant wavenumber $k_{res}=5.0$ is depicted by the magenta line.

Next, we study the effects of BBF’s slope $s_{\unicode[STIX]{x1D6EF}^{\ast }}$ on the dynamical behaviour of short-mode SWs. Note that figures 6–8 correspond to the relatively large slope ( $s_{\unicode[STIX]{x1D6EF}^{\ast }}=0.2$ ), whereas figure 9 corresponds to the relatively small slope ( $s_{\unicode[STIX]{x1D6EF}^{\ast }}=0.022$ ) of the BBF. In $k$ space, wavenumber spectra of SWs behave similarly, that is, wave spectra focus around the resonant $k_{res}$ . In $X$ space, however, the spatiotemporal manifestations of SWs differ between large and small slopes of BBF. For a large slope (figures 6 and 8), SWs are generated in several size-ordered packets (about 6 packets in figure 6 l) with the larger amplitudes appearing at the leading edge of the BBF, which then slowly decay to smaller amplitudes at the trailing edge. However, for the small slope, SWs are generated in only one to three dominant wave packets at the leading edge (figure 9). For the largest dominant wave packet, both of its maximal amplitude and group velocity increase after the interaction with other smaller-amplitude wave packets. For example, it can be seen from figure 9(d) and (f) that after the dominant wave packet interacts with a smaller-amplitude wave packet at $T=13\,000$ , its amplitude becomes slightly larger (figure 9 f) and its group velocity becomes slightly faster (figure 9 d). At the same time, the small-amplitude wave packet becomes even smaller after the interaction (figure 9 d). During this process, energy may be transferred from the smaller-amplitude wave packet to the larger-amplitude one.

Last, we study the effects of BBF’s maximal amplitude, $(\unicode[STIX]{x1D6EF}_{j}^{\ast }(0),\overline{U}_{j}^{\ast }(0))$ , on the dynamical behaviour of short-mode SWs. It can be seen from figure 9(f) that, for larger maximal-amplitude of the BBF, the dominant SW packet with a larger amplitude can be generated at the leading edge. Also, it can be seen from figure 9(g) that, for larger maximal-amplitude of the BBF, the peak of the SW spectrum is higher in amplitude around the resonant $k_{res}$ .

4.3 Parametric study for parameter regime II

To confirm the modulation-resonance mechanism, we choose a different parameter regime II, $(h_{1},h_{2},g,\unicode[STIX]{x1D70C}_{1},\unicode[STIX]{x1D70C}_{2})=(1,5,1,0.856,0.996)$ , for studying the dynamical behaviour of the short-mode SWs. This parameter regime was studied in Ref. (Kodaira et al. Reference Kodaira, Waseda, Miyata and Choi2016). The initial driving force $P_{1X}$  adopted here is in the form of equation (4.1) with $s_{L}=2.5$ , $k_{L}=0.3$ , $s_{R}=1.5$ , $k_{R}=1$ and $\unicode[STIX]{x1D70E}_{m}=2.5\times 10^{-2}$ . Again, we investigate the effects of the wavelength $X_{\unicode[STIX]{x1D6EF}^{\ast }}$ , slope $s_{\unicode[STIX]{x1D6EF}^{\ast }}$ and maximal amplitude of $(\unicode[STIX]{x1D6EF}_{j}^{\ast },\overline{U}_{j}^{\ast })$ associated with the BBF in equation (3.1) (table 2) on the short-mode SWs in the strongly nonlinear regime.

First, for a relatively small-wavelength BBF ( $X_{\unicode[STIX]{x1D6EF}^{\ast }}=100$ ), SW packets at the leading edge have almost the same amplitudes as those at the trailing edge (black curve in figure 10 a). However, for a relatively large wavelength BBF ( $X_{\unicode[STIX]{x1D6EF}^{\ast }}=150$ and $200$ ), SW packets at the leading edge have larger amplitudes than those at the trailing edge (figure 10 a, blue curve and red curve). Thus, SW packets can be observed at the leading edge when the wavelength of the BBF is sufficiently large. In $k$ space, the spectra of SWs focus around the resonant wavenumber $k_{res}$ (magenta line in figure 10 b), where the modulated dispersion relation (3.2) ( $(\unicode[STIX]{x1D6EF}_{1}^{\ast },\unicode[STIX]{x1D6EF}_{2}^{\ast },\overline{U}_{1}^{\ast },\overline{U}_{2}^{\ast })=(0.025,-0.24,0.08,-0.02)$ ) is applied in the resonance condition (2.20). For comparison, we also plot the the resonant wavenumber $k_{res}$ (green line) where the pure linear dispersion relation $\unicode[STIX]{x1D707}_{k}$ (2.16) is used in the resonance condition (2.20). One can see that the peak locations of the SW spectra are in good agreement with the resonant $k_{res}$ using the modulated dispersion relation (3.2) as shown in figure 10(b). Thus, it is the modulated dispersive waves for short-modes, instead of the pure linear dispersive waves, that interact with the long-mode wave through a class 3 triad resonance. (For parameter regime I, we have not plotted the resonant $k_{res}$ using the pure linear dispersion relation for comparison since both resonant wavenumbers $k_{res}$ are very close to each other. That is, the resonant $k_{res}=4.9$ using the pure linear dispersion relation whereas the resonant $k_{res}=5.0$ using the modulated dispersion relation.) Note that it has been earlier reported by Kodaira et al. (Reference Kodaira, Waseda, Miyata and Choi2016) that the modulated dispersion relation should be used when there is an underlying internal wave.

Table 2.

The wavelengths $X_{\unicode[STIX]{x1D6EF}^{\ast }}$ , slopes $s_{\unicode[STIX]{x1D6EF}^{\ast }}$ , maximal amplitudes $(\unicode[STIX]{x1D6EF}_{j}^{\ast }(0),\overline{U}_{j}^{\ast }(0))$ , and velocities $C_{s}$ of the BBF in equation (3.1) used for regime II $(h_{1},h_{2},g,\unicode[STIX]{x1D70C}_{1},\unicode[STIX]{x1D70C}_{2})=(1,5,1,0.856,0.996)$ (see this parameter regime in Kodaira et al. (Reference Kodaira, Waseda, Miyata and Choi2016)). In figures 10 and 11, the maximal amplitudes and phase velocities are chosen to be the same as those of the interfacial solitary wave solutions of the TWN model (see appendix B).

Figure 10.

Parameter regime II. (a) Snapshots of SWs $\unicode[STIX]{x1D709}_{1}$ at $T=10\,000$ for different wavelength $X_{\unicode[STIX]{x1D6EF}^{\ast }}$ of the BBF. The parameters of the BBFs are given in table 2. (b) The wavenumber spectra of SWs $\unicode[STIX]{x1D709}_{1}^{\ast }$ . The bracket $\langle \cdot \rangle$ is interpreted as the time average over $5000\leqslant T\leqslant 10\,000$ . The resonant wavenumber $k_{res}$ depicted by the green line is $1.6$ , where the pure linear dispersion relation $\unicode[STIX]{x1D707}_{k}$ (2.16) is used in the resonance condition (2.20). The resonant wavenumber $k_{res}$ depicted by the magenta line is $2.1$ , where the modulated dispersion relation (3.2) with $(\unicode[STIX]{x1D6EF}_{1}^{\ast },\unicode[STIX]{x1D6EF}_{2}^{\ast },\overline{U}_{1}^{\ast },\overline{U}_{2}^{\ast })=(0.025,-0.24,0.08,-0.02)$ is used in the resonance condition (2.20).

Next, by comparing the results in figures 10 and 11, we can see that SWs are generated in several wave packets for the large-slope BBF ( $s_{\unicode[STIX]{x1D6EF}^{\ast }}=0.2$ ) (figure 10), whereas SWs are generated in only one dominant wave packet at the front followed by a small-amplitude dispersive tail for the small-slope BBF ( $s_{\unicode[STIX]{x1D6EF}^{\ast }}=0.0167$ ) (figure 11). Again, the dominant wave packet becomes higher in amplitude and faster in group velocity after the interaction with many small-amplitude modulated dispersive waves (figure 11 a–c,e). One can observe that once the amplitude of the dominant wave packet is large enough, its speed can exceed the speed of the BBF and then move even beyond the leading edge.

Figure 11.

Parameter regime II. Snapshots of the BBF and the spatiotemporal evolution of SWs $\unicode[STIX]{x1D709}_{1}$ with the maximal amplitude of the BBF being (a)  $\unicode[STIX]{x1D6EF}_{2}^{\ast }(0)=-0.24$ , (b)  $\unicode[STIX]{x1D6EF}_{2}^{\ast }(0)=-0.77$ , and (c)  $\unicode[STIX]{x1D6EF}_{2}^{\ast }(0)=-1.21$ . The parameters of the BBFs are given in table 2. (d) Snapshots of SWs $\unicode[STIX]{x1D709}_{1}$ at $T=15\,000$ for different maximal amplitudes of the BBF. (e) The temporal evolution of the maximal amplitude, $A$ , of the SWs. (f) The wavenumber spectra of SWs $\unicode[STIX]{x1D709}_{1}^{\ast }$ . The bracket $\langle \cdot \rangle$ is interpreted as the time average over $5000\leqslant T\leqslant 15\,000$ . The resonant wavenumber $k_{res}=2.1$ as depicted by the magenta line.

Finally, for the larger-amplitude of the BBF, the dominant SW packet can grow higher in amplitude at the leading edge (figure 11 e). In figure 11(f), we still plot the resonant wavenumber $k_{res}=2.1$ for reference. One can see that for the larger-amplitude BBF, the peak of the SW spectrum becomes higher in amplitude and also the resonant wavenumber becomes slightly larger.

In summary, the effects of BBF’s wavelength, slope, and amplitude on the dynamical behaviour of short-mode SWs can be summarized as follows:

1. (4a) Surface waves can be trapped at the leading edge only when the wavelength of the BBF is sufficiently long (figure 8 or figure 10).

2. (4b) Fewer numbers of SW packets are generated at the leading edge when the slope of the BBF is smaller (comparing figures 8 and 9 or comparing figures 10 and 11).

3. (4c) Surface wave packets are larger in amplitude at the leading edge when the amplitude of the BBF is larger (figure 9 or figure 11).

Based on the above parametric study of the BBF on the dynamical behaviour of short-mode SWs, an additional statement can be supplemented in the description of the modulation-resonance mechanism as discussed in § 4.1, namely:

1. (3b’) If the slope of the BBF is relatively small, only one or very few dominant ( $X$ -space localized large-amplitude) wave packets can be generated at the leading edge. These localized wave packets become high in amplitude and large in group velocity after the interaction with other smaller-amplitude wave packets (figure 9) or with background perturbation waves (figure 11).

Our understanding of this numerical result is provided as follows: Modulation mechanism (3a ^′ ) cannot contribute to the growth of the dominant SW packet’s amplitude and group velocity since, in the linear modulation theory, the amplitude of a SW packet grows only when it propagates towards the trailing edge (figure 3). Also, dispersive effects cannot contribute to this growth since dispersion can only reduce the maximal amplitude of the dominant wave packet (figures 3 and 4 c). Thus, only nonlinear effects can contribute to the growing amplitude of the dominant SW packet. Due to the modulation mechanism (3a), left-moving and right-moving waves are initially separated and only right-moving waves remain at the leading edge after long-time evolution. Therefore, only the nonlinear class 3 triad resonance can contribute to the growing amplitude of these co-propagating SWs. This possibly results from the inverse energy cascade flowing from the spectrum of surrounding perturbation waves with larger wavenumbers above $k_{res}$  towards the dominant SW packets with characteristic wavenumber $k_{res}$ . In summary, when the slope of the BBF is small, one SW packet or very few SW packets, localized in both $X$ and $k$ spaces, can be generated at the leading edge of the underlying BBF.