2.1 Framework

Here we give a general framework that consists of a system of equations at $\mathit{O}(\unicode[STIX]{x1D716})$ and $\mathit{O}(\unicode[STIX]{x1D716}^{2})$ , which is obtained using perturbation analysis and the method of multiple scales for a periodic wave train. We have kept the system quite general so as to use the system of equations for the purpose of wave triad interaction (see § 2.2) and Bragg resonance (see § 2.3).

We consider a two-interface system with piecewise constant density and vorticity in each layer, see figure 1. The velocity profile is continuous, but the derivative of velocity may have a discontinuity at the density interface. The total depth of the system is $H$ and $H=h_{u}+h_{l}$ . The fluid above the surface is assumed to be a zero density fluid and $R$ is the density ratio at the interface $(R\equiv \unicode[STIX]{x1D70C}_{u}/\unicode[STIX]{x1D70C}_{l})$ . The base velocity profile is piecewise linear, and has the values $U=\{U_{u},U_{l},U_{b}\}$ at $z=\{0,-h_{u},-h_{u}-h_{l}\}$ respectively. The vertical ( $z$ ) axis points upwards, hence the gravity ( $g$ ) is along the negative $z$ -direction. The elevations of the surface and the pycnocline from their respective mean levels are $\unicode[STIX]{x1D702}_{u}(x,t)$ and $\unicode[STIX]{x1D702}_{l}(x,t)$ . Similarly, the elevation of the bottom topography is $\unicode[STIX]{x1D702}_{b}(x)$ from its mean level at $z=-h_{u}-h_{l}$ . As mentioned already, for a piecewise linear base velocity profile perturbed by irrotational initial disturbances, the vorticity generation is limited to the interfaces and the bulk flow remains irrotational. This allows us to introduce the velocity potentials $\unicode[STIX]{x1D719}_{u}$ and $\unicode[STIX]{x1D719}_{l}$ respectively in the upper and the lower layers. Hence, the continuity equation reduces to the Laplace equation

(2.8a ) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D719}_{u}=0\quad -h_{u}+\unicode[STIX]{x1D702}_{l}<z<\unicode[STIX]{x1D702}_{u}, & \displaystyle\end{eqnarray}$$

(2.8b ) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D719}_{l}=0\quad -h_{u}-h_{l}+\unicode[STIX]{x1D702}_{b}<z<-h_{u}+\unicode[STIX]{x1D702}_{l}. & \displaystyle\end{eqnarray}$$

(2.9a ) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D702}_{u,t}+(U+\unicode[STIX]{x1D719}_{u,x})\unicode[STIX]{x1D702}_{u,x}=\unicode[STIX]{x1D719}_{u,z}\quad \text{at }z=\unicode[STIX]{x1D702}_{u}, & \displaystyle\end{eqnarray}$$

(2.9b ) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D702}_{l,t}+(U+\unicode[STIX]{x1D719}_{u,x})\unicode[STIX]{x1D702}_{l,x}=\unicode[STIX]{x1D719}_{u,z}\quad \text{at }z=-h_{u}+\unicode[STIX]{x1D702}_{l}, & \displaystyle\end{eqnarray}$$

(2.9c ) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D702}_{l,t}+(U+\unicode[STIX]{x1D719}_{l,x})\unicode[STIX]{x1D702}_{l,x}=\unicode[STIX]{x1D719}_{l,z}\quad \text{at }z=-h_{u}+\unicode[STIX]{x1D702}_{l}, & \displaystyle\end{eqnarray}$$

(2.9d ) $$\begin{eqnarray}\displaystyle & (U+\unicode[STIX]{x1D719}_{l,x})\unicode[STIX]{x1D702}_{b,x}=\unicode[STIX]{x1D719}_{l,z}\quad \text{at }z=-h_{u}-h_{l}+\unicode[STIX]{x1D702}_{b}. & \displaystyle\end{eqnarray}$$

(2.9e ) $$\begin{eqnarray}\unicode[STIX]{x1D719}_{u,t}+{\textstyle \frac{1}{2}}(\unicode[STIX]{x1D719}_{u,x}^{2}+\unicode[STIX]{x1D719}_{u,z}^{2})+U\unicode[STIX]{x1D719}_{u,x}-\unicode[STIX]{x1D6FA}_{u}\unicode[STIX]{x1D713}_{u}+g\unicode[STIX]{x1D702}_{u}=0\quad \text{at }z=\unicode[STIX]{x1D702}_{u},\end{eqnarray}$$

(2.9f ) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D70C}_{u}\left[\unicode[STIX]{x1D719}_{u,t}+{\textstyle \frac{1}{2}}(\unicode[STIX]{x1D719}_{u,x}^{2}+\unicode[STIX]{x1D719}_{u,z}^{2})+U\unicode[STIX]{x1D719}_{u,x}-\unicode[STIX]{x1D6FA}_{u}\unicode[STIX]{x1D713}_{u}+g\unicode[STIX]{x1D702}_{l}\right]\nonumber\\ \displaystyle & & \displaystyle \quad -\unicode[STIX]{x1D70C}_{l}\left[\unicode[STIX]{x1D719}_{l,t}+{\textstyle \frac{1}{2}}(\unicode[STIX]{x1D719}_{l,x}^{2}+\unicode[STIX]{x1D719}_{l,z}^{2})+U\unicode[STIX]{x1D719}_{l,x}-\unicode[STIX]{x1D6FA}_{l}\unicode[STIX]{x1D713}_{l}+g\unicode[STIX]{x1D702}_{l}\right]=0\quad \text{at }z=-h_{u}+\unicode[STIX]{x1D702}_{l}.\qquad\end{eqnarray}$$

We are interested in obtaining the solutions up to the first order of nonlinearity. Hence, we perform a perturbation expansion until $\mathit{O}(\unicode[STIX]{x1D716}^{2})$ , where the expansion parameter $\unicode[STIX]{x1D716}$ measures the wave steepness. It is also assumed that the steepness of the bottom topography is $\mathit{O}(\unicode[STIX]{x1D716})$ .

(2.10a ) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D719}_{u}(x,z,t)=\unicode[STIX]{x1D716}\unicode[STIX]{x1D719}_{u}^{(1)}(x,z,t,\unicode[STIX]{x1D70F})+\unicode[STIX]{x1D716}^{2}\unicode[STIX]{x1D719}_{u}^{(2)}(x,z,t,\unicode[STIX]{x1D70F}), & \displaystyle\end{eqnarray}$$

(2.10b ) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D719}_{l}(x,z,t)=\unicode[STIX]{x1D716}\unicode[STIX]{x1D719}_{l}^{(1)}(x,z,t,\unicode[STIX]{x1D70F})+\unicode[STIX]{x1D716}^{2}\unicode[STIX]{x1D719}_{l}^{(2)}(x,z,t,\unicode[STIX]{x1D70F}), & \displaystyle\end{eqnarray}$$

(2.10c ) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D702}_{u}(x,t)=\unicode[STIX]{x1D716}\unicode[STIX]{x1D702}_{u}^{(1)}(x,t,\unicode[STIX]{x1D70F})+\unicode[STIX]{x1D716}^{2}\unicode[STIX]{x1D702}_{u}^{(2)}(x,t,\unicode[STIX]{x1D70F}), & \displaystyle\end{eqnarray}$$

(2.10d ) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D702}_{l}(x,t)=\unicode[STIX]{x1D716}\unicode[STIX]{x1D702}_{l}^{(1)}(x,t,\unicode[STIX]{x1D70F})+\unicode[STIX]{x1D716}^{2}\unicode[STIX]{x1D702}_{l}^{(2)}(x,t,\unicode[STIX]{x1D70F}). & \displaystyle\end{eqnarray}$$

$\unicode[STIX]{x1D70F}$

$\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D716}t$

$\mathit{O}(\unicode[STIX]{x1D716})$

$\mathit{O}(\unicode[STIX]{x1D716}^{2})$

$\unicode[STIX]{x1D719}$

$\unicode[STIX]{x1D713}$

$\mathit{O}(\unicode[STIX]{x1D716})$

(2.11a ) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D719}_{u,z}^{(1)}-\unicode[STIX]{x1D702}_{u,t}^{(1)}-U_{u}\unicode[STIX]{x1D702}_{u,x}^{(1)}=0\quad \text{at }z=0, & \displaystyle\end{eqnarray}$$

(2.11b ) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D719}_{u,z}^{(1)}-\unicode[STIX]{x1D702}_{l,t}^{(1)}-U_{l}\unicode[STIX]{x1D702}_{l,x}^{(1)}=0\quad \text{at }z=-h_{u}, & \displaystyle\end{eqnarray}$$

(2.11c ) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D719}_{l,z}^{(1)}-\unicode[STIX]{x1D702}_{l,t}^{(1)}-U_{l}\unicode[STIX]{x1D702}_{l,x}^{(1)}=0\quad \text{at }z=-h_{u}, & \displaystyle\end{eqnarray}$$

(2.11d ) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D719}_{u,t}^{(1)}+U_{u}\unicode[STIX]{x1D719}_{u,x}^{(1)}-\unicode[STIX]{x1D6FA}_{u}\unicode[STIX]{x1D713}_{u}^{(1)}+g\unicode[STIX]{x1D702}_{u}^{(1)}=0\quad \text{at }z=0, & \displaystyle\end{eqnarray}$$

(2.11e ) $$\begin{eqnarray}\displaystyle & R[\unicode[STIX]{x1D719}_{u,t}^{(1)}+U_{l}\unicode[STIX]{x1D719}_{u,x}^{(1)}-\unicode[STIX]{x1D6FA}_{u}\unicode[STIX]{x1D713}_{u}^{(1)}+g\unicode[STIX]{x1D702}_{l}^{(1)}] & \displaystyle\end{eqnarray}$$

(2.11f ) $$\begin{eqnarray}\displaystyle & -[\unicode[STIX]{x1D719}_{l,t}^{(1)}+U_{l}\unicode[STIX]{x1D719}_{l,x}^{(1)}-\unicode[STIX]{x1D6FA}_{l}\unicode[STIX]{x1D713}_{l}^{(1)}+g\unicode[STIX]{x1D702}_{l}^{(1)}]=0\quad \text{at }z=-h_{u}, & \displaystyle\end{eqnarray}$$

(2.11g ) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D719}_{l,z}^{(1)}=0\quad \text{at }z=-h_{u}-h_{l}. & \displaystyle\end{eqnarray}$$

$j=\{1,2,\ldots \}$

$m=\{1,2\}$

$j$

$j$

$(m)$

(2.12a ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D719}_{uj}^{(m)}=\left[A_{j}^{(m)}(\unicode[STIX]{x1D70F})\frac{\cosh k_{j}(z+h_{u})}{\cosh (k_{j}h_{u})}+B_{j}^{(m)}(\unicode[STIX]{x1D70F})\frac{\sinh k_{j}z}{\cosh (k_{j}h_{u})}\right]\text{e}^{\text{i}(k_{j}x-\unicode[STIX]{x1D714}_{j}t)}+\text{c.c.}, & \displaystyle\end{eqnarray}$$

(2.12b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D719}_{lj}^{(m)}=\left[C_{j}^{(m)}(\unicode[STIX]{x1D70F})\frac{\cosh k_{j}(z+h_{u}+h_{l})}{\cosh (k_{j}h_{l})}+D_{j}^{(m)}(\unicode[STIX]{x1D70F})\frac{\sinh k_{j}(z+h_{u}+h_{l})}{\cosh (k_{j}h_{l})}\right]\text{e}^{\text{i}(k_{j}x-\unicode[STIX]{x1D714}_{j}t)}+\text{c.c.}, & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(2.12c ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D713}_{uj}^{(m)}=\text{i}\left[A_{j}^{(m)}(\unicode[STIX]{x1D70F})\frac{\sinh k_{j}(z+h_{u})}{\cosh (k_{j}h_{u})}+B_{j}^{(m)}(\unicode[STIX]{x1D70F})\frac{\cosh k_{j}z}{\cosh (k_{j}h_{u})}\right]\text{e}^{\text{i}(k_{j}x-\unicode[STIX]{x1D714}_{j}t)}+\text{c.c.}, & \displaystyle\end{eqnarray}$$

(2.12d ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D713}_{lj}^{(m)}=\text{i}\left[C_{j}^{(m)}(\unicode[STIX]{x1D70F})\frac{\sinh k_{j}(z+h_{u}+h_{l})}{\cosh (k_{j}h_{l})}+D_{j}^{(m)}(\unicode[STIX]{x1D70F})\frac{\cosh k_{j}(z+h_{u}+h_{l})}{\cosh (k_{j}h_{l})}\right]\text{e}^{\text{i}(k_{j}x-\unicode[STIX]{x1D714}_{j}t)}+\text{c.c.}, & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(2.12e ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D702}_{uj}^{(m)}=a_{j}^{(m)}(\unicode[STIX]{x1D70F})\text{e}^{\text{i}(k_{j}x-\unicode[STIX]{x1D714}_{j}t)}+\text{c.c.}, & \displaystyle\end{eqnarray}$$

(2.12f ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D702}_{lj}^{(m)}=b_{j}^{(m)}(\unicode[STIX]{x1D70F})\text{e}^{\text{i}(k_{j}x-\unicode[STIX]{x1D714}_{j}t)}+\text{c.c.}, & \displaystyle\end{eqnarray}$$

$\mathit{O}(\unicode[STIX]{x1D716})$

$k_{j}$

$\mathit{O}(\unicode[STIX]{x1D716})$

(2.13) $$\begin{eqnarray}\unicode[STIX]{x1D63F}(\unicode[STIX]{x1D714}_{j},k_{j})\boldsymbol{x}_{j}^{(1)}=0.\end{eqnarray}$$

Here, the vector $\boldsymbol{x}_{j}^{(1)}\equiv [A_{j}^{(1)},B_{j}^{(1)},C_{j}^{(1)},D_{j}^{(1)},a_{j}^{(1)},b_{j}^{(1)}]^{\dagger }$ , and the matrix $\unicode[STIX]{x1D63F}(\unicode[STIX]{x1D714}_{j},k_{j})$ is given by

(2.14) $$\begin{eqnarray}\displaystyle \left[\begin{array}{@{}cccccc@{}}{\displaystyle \frac{k_{j}}{\coth (k_{j}h_{u})}} & {\displaystyle \frac{k_{j}}{\cosh (k_{j}h_{u})}} & 0 & 0 & \text{i}\unicode[STIX]{x1D714}_{j}^{\{1\}} & 0\\ 0 & k_{j} & 0 & 0 & 0 & \text{i}\unicode[STIX]{x1D714}_{j}^{\{2\}}\\ 0 & 0 & {\displaystyle \frac{k_{j}}{\coth (k_{j}h_{l})}} & k & 0 & \text{i}\unicode[STIX]{x1D714}_{j}^{(2)}\\ -\text{i}\unicode[STIX]{x1D714}_{j}^{(1)}-{\displaystyle \frac{\text{i}\unicode[STIX]{x1D6FA}_{u}}{\coth (k_{j}h_{u})}} & -{\displaystyle \frac{\text{i}\unicode[STIX]{x1D6FA}_{u}}{\cosh (k_{j}h_{u})}} & 0 & 0 & g & 0\\ -{\displaystyle \frac{\text{i}R\unicode[STIX]{x1D714}_{j}^{\{2\}}}{\cosh (k_{j}h_{u})}} & {\displaystyle \frac{\text{i}R\unicode[STIX]{x1D714}_{j}^{\{2\}}}{\coth (k_{j}h_{u})}}-\text{i}R\unicode[STIX]{x1D6FA}_{u} & \text{i}\unicode[STIX]{x1D714}_{j}^{\{2\}}+{\displaystyle \frac{\text{i}\unicode[STIX]{x1D6FA}_{l}}{\coth k_{j}h_{l}}} & {\displaystyle \frac{\text{i}\unicode[STIX]{x1D714}_{j}^{(2)}}{\coth k_{j}h_{l}}}+\text{i}\unicode[STIX]{x1D6FA}_{l} & 0 & g(R-1)\\ 0 & 0 & 0 & {\displaystyle \frac{k_{j}}{\cosh k_{j}h_{l}}} & 0 & 0\end{array}\right], & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D714}_{j}^{\{1\}}=\unicode[STIX]{x1D714}_{j}-U_{u}k_{j}$ ; $\unicode[STIX]{x1D714}_{j}^{\{2\}}=\unicode[STIX]{x1D714}_{j}-U_{l}k_{j}$ . The dispersion relation of the above system is obtained by setting the determinant of the above matrix to zero and is given by the equation

(2.15) $$\begin{eqnarray}\det (\unicode[STIX]{x1D63F})(\unicode[STIX]{x1D714}_{j},k_{j})=0,\end{eqnarray}$$

where $\det (\unicode[STIX]{x1D63F})(\unicode[STIX]{x1D714}_{j},k_{j})$ is the determinant of the matrix $\unicode[STIX]{x1D63F}(\unicode[STIX]{x1D714}_{j},k_{j})$ .

In addition to the homogeneous solution, we also have the particular solutions at $\mathit{O}(\unicode[STIX]{x1D716})$ due to the velocity difference between the bottom and the fluid above it. In such a case, the time-independent surface elevation, capturing the non-homogeneity introduced by the mean flow’s interaction with the bottom, is given by

(2.16) $$\begin{eqnarray}\hat{\unicode[STIX]{x1D702}}_{u}=-\frac{U_{u}U_{b}U_{l}^{2}k_{b}^{6}}{\cosh (k_{b}h_{u})\cosh ^{2}(k_{b}h_{l})\det (\unicode[STIX]{x1D63F})(0,k_{b})}\hat{\unicode[STIX]{x1D702}}_{b},\end{eqnarray}$$

and other coefficients, i.e.  $\hat{\unicode[STIX]{x1D702}}_{l}$ , $A,B,C,D$ are given in appendix B.

At $\mathit{O}(\unicode[STIX]{x1D716}^{2})$ , we obtain the following equations:

(2.17a ) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D719}_{u,z}^{(2)}-\unicode[STIX]{x1D702}_{u,t}^{(2)}-U_{u}\unicode[STIX]{x1D702}_{u,x}^{(2)}=p_{1}+q_{1}\quad \text{at }z=0, & \displaystyle\end{eqnarray}$$

(2.17b ) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D719}_{u,z}^{(2)}-\unicode[STIX]{x1D702}_{l,t}^{(2)}-U_{l}\unicode[STIX]{x1D702}_{l,x}^{(2)}=p_{2}+q_{2}\quad \text{at }z=-h_{u}, & \displaystyle\end{eqnarray}$$

(2.17c ) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D719}_{l,z}^{(2)}-\unicode[STIX]{x1D702}_{l,t}^{(2)}-U_{l}\unicode[STIX]{x1D702}_{l,x}^{(2)}=p_{3}+q_{3}\quad \text{at }z=-h_{u}, & \displaystyle\end{eqnarray}$$

(2.17d ) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D719}_{u,t}^{(2)}+U_{u}\unicode[STIX]{x1D719}_{u,x}^{(2)}-\unicode[STIX]{x1D6FA}_{u}\unicode[STIX]{x1D713}_{u}^{(2)}+g\unicode[STIX]{x1D702}_{u}^{(2)}=p_{4}+q_{4}\quad \text{at }z=0, & \displaystyle\end{eqnarray}$$

(2.17e ) $$\begin{eqnarray}\displaystyle & [\unicode[STIX]{x1D719}_{u,t}^{(2)}+U_{l}\unicode[STIX]{x1D719}_{u,x}^{(2)}-\unicode[STIX]{x1D6FA}_{u}\unicode[STIX]{x1D713}_{u}^{(2)}+g\unicode[STIX]{x1D702}_{l}^{(2)}] & \displaystyle \nonumber\\ \displaystyle & -[\unicode[STIX]{x1D719}_{l,t}^{(2)}+U_{l}\unicode[STIX]{x1D719}_{l,x}^{(2)}-\unicode[STIX]{x1D6FA}_{l}\unicode[STIX]{x1D713}_{l}^{(2)}+g\unicode[STIX]{x1D702}_{l}^{(2)}]=p_{5}+q_{5}\quad \text{at }z=-h_{u}, & \displaystyle\end{eqnarray}$$

(2.17f ) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D719}_{l,z}^{(2)}=p_{6}+q_{6}\quad \text{at }z=-h_{u}-h_{l}. & \displaystyle\end{eqnarray}$$

$p_{1},p_{2},p_{3},p_{4},p_{5}$

$p_{6}$

$\mathit{O}(\unicode[STIX]{x1D716})$

$(1)$

(2.18a ) $$\begin{eqnarray}\displaystyle & p_{1}=\unicode[STIX]{x1D702}_{u,x}(\unicode[STIX]{x1D719}_{u,x}+\unicode[STIX]{x1D6FA}_{u}\unicode[STIX]{x1D702}_{u})-\unicode[STIX]{x1D702}_{u}\unicode[STIX]{x1D719}_{u,zz}\quad z=0, & \displaystyle\end{eqnarray}$$

(2.18b ) $$\begin{eqnarray}\displaystyle & p_{2}=\unicode[STIX]{x1D702}_{l,x}(\unicode[STIX]{x1D719}_{u,x}+\unicode[STIX]{x1D6FA}_{u}\unicode[STIX]{x1D702}_{l})-\unicode[STIX]{x1D702}_{l}\unicode[STIX]{x1D719}_{u,zz}\quad z=-h_{u}, & \displaystyle\end{eqnarray}$$

(2.18c ) $$\begin{eqnarray}\displaystyle & p_{3}=\unicode[STIX]{x1D702}_{l,x}(\unicode[STIX]{x1D719}_{l,x}+\unicode[STIX]{x1D6FA}_{l}\unicode[STIX]{x1D702}_{l})-\unicode[STIX]{x1D702}_{l}\unicode[STIX]{x1D719}_{l,zz}\quad z=-h_{u}, & \displaystyle\end{eqnarray}$$

(2.18d ) $$\begin{eqnarray}\displaystyle & p_{4}=-\unicode[STIX]{x1D702}_{u}(\unicode[STIX]{x1D719}_{u,tz}+U_{u}\unicode[STIX]{x1D719}_{u,xz}+\unicode[STIX]{x1D6FA}_{u}\unicode[STIX]{x1D719}_{u,x})-{\textstyle \frac{1}{2}}\left[(\unicode[STIX]{x1D719}_{u,x})^{2}+(\unicode[STIX]{x1D719}_{u,z})^{2}\right]+\unicode[STIX]{x1D6FA}_{u}\unicode[STIX]{x1D702}_{u}\unicode[STIX]{x1D713}_{u,z}\quad z=0, & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(2.18e ) $$\begin{eqnarray}\displaystyle p_{5} & = & \displaystyle R\left[\unicode[STIX]{x1D702}_{l}(\unicode[STIX]{x1D719}_{u,tz}+U_{l}\unicode[STIX]{x1D719}_{u,xz}+\unicode[STIX]{x1D6FA}_{u}\unicode[STIX]{x1D719}_{u,x})-{\textstyle \frac{1}{2}}\left[(\unicode[STIX]{x1D719}_{u,x})^{2}+(\unicode[STIX]{x1D719}_{u,z})^{2}\right]+\unicode[STIX]{x1D6FA}_{u}\unicode[STIX]{x1D702}_{l}\unicode[STIX]{x1D713}_{u,z}\right]\nonumber\\ \displaystyle & & \displaystyle -\,\left[\unicode[STIX]{x1D702}_{l}(\unicode[STIX]{x1D719}_{l,tz}+U_{l}\unicode[STIX]{x1D719}_{l,xz}+\unicode[STIX]{x1D6FA}_{l}\unicode[STIX]{x1D719}_{l,x})-{\textstyle \frac{1}{2}}\left[(\unicode[STIX]{x1D719}_{l,x})^{2}+(\unicode[STIX]{x1D719}_{l,z})^{2}\right]+\unicode[STIX]{x1D6FA}_{l}\unicode[STIX]{x1D702}_{l}\unicode[STIX]{x1D713}_{l,z}\right]\quad z=-h_{u},\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(2.18f ) $$\begin{eqnarray}\displaystyle p_{6}=\unicode[STIX]{x1D702}_{b,x}(\unicode[STIX]{x1D719}_{l,x}+\unicode[STIX]{x1D6FA}_{l}\unicode[STIX]{x1D702}_{b})-\unicode[STIX]{x1D702}_{b}\unicode[STIX]{x1D719}_{l,zz}\quad z=-h_{u}-h_{l}. & & \displaystyle\end{eqnarray}$$

Here, the right-hand side terms comprise of terms due to nonlinearity of the boundary condition as well as due to Taylor expansion about the mean level. The right-hand side terms $q_{1},q_{2},q_{3},q_{4},q_{5}$ and $q_{6}$ are the time derivatives of the $\mathit{O}(\unicode[STIX]{x1D716})$ terms:

(2.19a ) $$\begin{eqnarray}\displaystyle & q_{1}=\unicode[STIX]{x1D702}_{u,\unicode[STIX]{x1D70F}}, & \displaystyle\end{eqnarray}$$

(2.19b ) $$\begin{eqnarray}\displaystyle & q_{2}=\unicode[STIX]{x1D702}_{l,\unicode[STIX]{x1D70F}}, & \displaystyle\end{eqnarray}$$

(2.19c ) $$\begin{eqnarray}\displaystyle & q_{3}=\unicode[STIX]{x1D702}_{l,\unicode[STIX]{x1D70F}}, & \displaystyle\end{eqnarray}$$

(2.19d ) $$\begin{eqnarray}\displaystyle & q_{4}=-\unicode[STIX]{x1D719}_{u,\unicode[STIX]{x1D70F}}, & \displaystyle\end{eqnarray}$$

(2.19e ) $$\begin{eqnarray}\displaystyle & q_{5}=-R\unicode[STIX]{x1D719}_{u,\unicode[STIX]{x1D70F}}+\unicode[STIX]{x1D719}_{l,\unicode[STIX]{x1D70F}}, & \displaystyle\end{eqnarray}$$

(2.19f ) $$\begin{eqnarray}\displaystyle & q_{6}=0. & \displaystyle\end{eqnarray}$$

2.2 Analytical solution for wave triad interaction

We assume that initially at $\mathit{O}(\unicode[STIX]{x1D716})$ , the system has only three wavenumbers $\{k_{1},k_{2},k_{3}\}$ and corresponding frequencies $\{\unicode[STIX]{x1D714}_{1},\unicode[STIX]{x1D714}_{2},\unicode[STIX]{x1D714}_{3}\}$ , satisfying the resonance condition. Without any loss of generality, the resonance condition is given by $k_{1}=k_{2}+k_{3}$ and $\unicode[STIX]{x1D714}_{1}=\unicode[STIX]{x1D714}_{2}+\unicode[STIX]{x1D714}_{3}$ . The surface elevation expressed as a sum of these three modes reads

(2.20) $$\begin{eqnarray}\unicode[STIX]{x1D702}_{u}^{(1)}(x,t,\unicode[STIX]{x1D70F})=a_{1}^{(1)}(\unicode[STIX]{x1D70F})\text{e}^{\text{i}(k_{1}x-\unicode[STIX]{x1D714}_{1}t)}+a_{2}^{(1)}(\unicode[STIX]{x1D70F})\text{e}^{\text{i}(k_{2}x-\unicode[STIX]{x1D714}_{2}t)}+a_{3}^{(1)}(\unicode[STIX]{x1D70F})\text{e}^{\text{i}(k_{3}x-\unicode[STIX]{x1D714}_{3}t)}+\text{c.c}.\end{eqnarray}$$

The other functions $\unicode[STIX]{x1D719}_{u}^{(1)},\unicode[STIX]{x1D719}_{l}^{(1)},\unicode[STIX]{x1D713}_{u}^{(1)},\unicode[STIX]{x1D713}_{l}^{(1)}$ and $\unicode[STIX]{x1D702}_{l}^{(1)}$ can also be written in a similar fashion. Substituting this in the equations at $\mathit{O}(\unicode[STIX]{x1D716})$ , we would obtain the following set of linear equations:

(2.21a-c ) $$\begin{eqnarray}\unicode[STIX]{x1D63F}(\unicode[STIX]{x1D714}_{1},k_{1})\boldsymbol{x}_{1}^{(1)}=0;\quad \unicode[STIX]{x1D63F}(\unicode[STIX]{x1D714}_{2},k_{2})\boldsymbol{x}_{2}^{(1)}=0;\quad \unicode[STIX]{x1D63F}(\unicode[STIX]{x1D714}_{3},k_{3})\boldsymbol{x}_{3}^{(1)}=0.\end{eqnarray}$$

The vector $\boldsymbol{x}_{j}^{(1)}\equiv [A_{j}^{(1)},B_{j}^{(1)},C_{j}^{(1)},D_{j}^{(1)},a_{j}^{(1)},b_{j}^{(1)}]^{\dagger }$ and the matrix $\unicode[STIX]{x1D63F}(\unicode[STIX]{x1D714},k)$ are given in § 2.1. We further proceed to substitute (2.12a )–(2.12f ) in (2.17a )–(2.17f ) at $\mathit{O}(\unicode[STIX]{x1D716}^{2})$ . Here, the left-hand sides of the equations obtained at $\mathit{O}(\unicode[STIX]{x1D716}^{2})$ are similar to those obtained at $\mathit{O}(\unicode[STIX]{x1D716})$ . On substitution, we collect the terms corresponding to each wavenumber $k_{1},k_{2}$ and $k_{3}$ after using the resonance conditions $k_{1}=k_{2}+k_{3}$ and $\unicode[STIX]{x1D714}_{1}=\unicode[STIX]{x1D714}_{2}+\unicode[STIX]{x1D714}_{3}$ . We obtain equations of the form

(2.22a ) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D63F}(\unicode[STIX]{x1D714}_{1},k_{1})\boldsymbol{x}_{1}^{(2)}=\boldsymbol{v}_{1}a_{2}^{(1)}a_{3}^{(1)}+\boldsymbol{r}_{1}a_{1,\unicode[STIX]{x1D70F}}^{(1)}, & \displaystyle\end{eqnarray}$$

(2.22b ) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D63F}(\unicode[STIX]{x1D714}_{2},k_{2})\boldsymbol{x}_{2}^{(2)}=\boldsymbol{v}_{2}\bar{a}_{3}^{(1)}a_{1}^{(1)}+\boldsymbol{r}_{2}a_{2,\unicode[STIX]{x1D70F}}^{(1)}, & \displaystyle\end{eqnarray}$$

(2.22c ) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D63F}(\unicode[STIX]{x1D714}_{3},k_{3})\boldsymbol{x}_{3}^{(2)}=\boldsymbol{v}_{3}a_{1}^{(1)}\bar{a}_{2}^{(1)}+\boldsymbol{r}_{3}a_{3,\unicode[STIX]{x1D70F}}^{(1)}, & \displaystyle\end{eqnarray}$$

$\boldsymbol{x}_{j}^{(2)}\equiv [\!A_{j}^{(2)},B_{j}^{(2)},C_{j}^{(2)},D_{j}^{(2)},a_{j}^{(2)},b_{j}^{(2)}\! ]^{\dagger }$

$\boldsymbol{v}_{j}$

$\boldsymbol{r}_{j}$

$\boldsymbol{v}_{j}$

$\exp [\text{i}(k_{j}x-\unicode[STIX]{x1D714}_{j}t)]$

$\mathit{O}(\unicode[STIX]{x1D716})$

$\boldsymbol{r}_{j}$

$\mathit{O}(\unicode[STIX]{x1D716})$

$A_{i}^{(1)}a_{j}^{(1)},B_{i}^{(1)}a_{j}^{(1)},C_{i}^{(1)}C_{j}^{(1)}$

$a_{i}^{(1)}a_{j}^{(1)}$

$\unicode[STIX]{x1D63F}(\unicode[STIX]{x1D714}_{j},k_{j})$

$a_{j,\unicode[STIX]{x1D70F}}^{(1)}$

Using the Fredholm alternative in the context of the sets of (2.21) and (2.22), we deduce that the solutions for $\boldsymbol{x}_{i}^{(2)}$ exist if and only if the vectors $\boldsymbol{v}_{i}$ are orthogonal to the null space of the transpose of the respective matrices $\unicode[STIX]{x1D63F}(\unicode[STIX]{x1D714}_{j},k_{j})$ . Denoting the null space of the transpose of the matrix $\unicode[STIX]{x1D63F}(\unicode[STIX]{x1D714}_{j},k_{j})$ by $\boldsymbol{n}_{j}$ , we finally get a set of three equations:

(2.23a ) $$\begin{eqnarray}\displaystyle & \boldsymbol{n}_{1}\boldsymbol{\cdot }(\boldsymbol{v}_{1}a_{2}^{(1)}a_{3}^{(1)}+\boldsymbol{r}_{1}a_{1,\unicode[STIX]{x1D70F}}^{(1)})=0, & \displaystyle\end{eqnarray}$$

(2.23b ) $$\begin{eqnarray}\displaystyle & \boldsymbol{n}_{2}\boldsymbol{\cdot }(\boldsymbol{v}_{2}\bar{a}_{3}^{(1)}a_{1}^{(1)}+\boldsymbol{r}_{2}a_{2,\unicode[STIX]{x1D70F}}^{(1)})=0, & \displaystyle\end{eqnarray}$$

(2.23c ) $$\begin{eqnarray}\displaystyle & \boldsymbol{n}_{3}\boldsymbol{\cdot }(\boldsymbol{v}_{3}a_{1}^{(1)}\bar{a}_{2}^{(1)}+\boldsymbol{r}_{3}a_{3,\unicode[STIX]{x1D70F}}^{(1)})=0, & \displaystyle\end{eqnarray}$$

(2.24a-c ) $$\begin{eqnarray}\displaystyle a_{1,\unicode[STIX]{x1D70F}}^{(1)}=\unicode[STIX]{x1D6FD}_{1}a_{2}^{(1)}a_{3}^{(1)};\quad a_{2,\unicode[STIX]{x1D70F}}^{(1)}=\unicode[STIX]{x1D6FD}_{2}\bar{a}_{3}^{(1)}a_{1}^{(1)};\quad a_{3,\unicode[STIX]{x1D70F}}^{(1)}=\unicode[STIX]{x1D6FD}_{3}a_{1}^{(1)}\bar{a}_{2}^{(1)}, & & \displaystyle\end{eqnarray}$$

where

(2.25) $$\begin{eqnarray}\unicode[STIX]{x1D6FD}_{j}=-\frac{\boldsymbol{n}_{j}\boldsymbol{\cdot }\boldsymbol{v}_{j}}{\boldsymbol{n}_{j}\boldsymbol{\cdot }\boldsymbol{r}_{j}}.\end{eqnarray}$$

2.3 Analytical solution for Bragg resonance

The equations for the case of Bragg resonance can also be obtained using the same framework as in § 2.2. However, in the case of Bragg resonance, only two propagating waves are involved, the third one is the bottom ripple. We assume the participating waves to have the wavenumbers $\{k_{1},k_{2}\}$ with frequencies $\{\unicode[STIX]{x1D714}_{1},\unicode[STIX]{x1D714}_{2}\}$ and the bottom to have wavenumber $k_{b}$ . Substituting the normal modes, we get a set of linear equations at $\mathit{O}(\unicode[STIX]{x1D716})$ :

(2.26a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D63F}(\unicode[STIX]{x1D714}_{1},k_{1})\boldsymbol{x}_{1}^{(1)}=0;\quad \unicode[STIX]{x1D63F}(\unicode[STIX]{x1D714}_{2},k_{2})\boldsymbol{x}_{2}^{(1)}=0.\end{eqnarray}$$

Here the vector $\boldsymbol{x}_{j}^{(1)}\equiv [A_{j}^{(1)},B_{j}^{(1)},C_{j}^{(1)},D_{j}^{(1)},a_{j}^{(1)},b_{j}^{(1)}]^{\dagger }$ and the matrix $\unicode[STIX]{x1D63F}(\unicode[STIX]{x1D714},k)$ are the same as those in § 2.2. We assume that at $\mathit{O}(\unicode[STIX]{x1D716})$ , the surface consists of only two modes, $k_{1}$ and $k_{2}$ . Hence we write $\unicode[STIX]{x1D702}_{u}^{(1)}(x,t)$ as

(2.27) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D702}_{u}^{(1)}(x,t,\unicode[STIX]{x1D70F})=a_{1}^{(1)}(\unicode[STIX]{x1D70F})\text{e}^{\text{i}(k_{1}x-\unicode[STIX]{x1D714}_{1}t)}+a_{2}^{(1)}(\unicode[STIX]{x1D70F})\text{e}^{\text{i}(k_{2}x-\unicode[STIX]{x1D714}_{2}t)}+\text{c.c.}, & \displaystyle\end{eqnarray}$$

(2.28) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D702}_{b}(x)=a_{b}\text{e}^{\text{i}k_{1}x}+\text{c.c.} & \displaystyle\end{eqnarray}$$

The other functions $\unicode[STIX]{x1D719}_{u}^{(1)},\unicode[STIX]{x1D719}_{l}^{(1)},\unicode[STIX]{x1D713}_{u}^{(1)},\unicode[STIX]{x1D713}_{l}^{(1)}$ and $\unicode[STIX]{x1D702}_{l}^{(1)}$ containing the wavenumbers ‘ $k_{1}$ ’ and ‘ $k_{2}$ ’ can also be written similarly. Substituting this in the equations at $\mathit{O}(\unicode[STIX]{x1D716})$ , we obtain a set of linear equations

(2.29a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D63F}(\unicode[STIX]{x1D714}_{1},k_{1})\boldsymbol{x}_{1}^{(1)}=0;\quad \unicode[STIX]{x1D63F}(\unicode[STIX]{x1D714}_{2},k_{2})\boldsymbol{x}_{2}^{(1)}=0.\end{eqnarray}$$

The vector $\boldsymbol{x}_{j}^{(1)}\equiv [A_{j}^{(1)},B_{j}^{(1)},C_{j}^{(1)},D_{j}^{(1)},a_{j}^{(1)},b_{j}^{(1)}]^{\dagger }$ and the matrix $\unicode[STIX]{x1D63F}(\unicode[STIX]{x1D714},k)$ are the same as those in § 2.2. We further proceed to substitute (2.12a )–(2.12f ) in (2.17a )–(2.17f ) at $\mathit{O}(\unicode[STIX]{x1D716}^{2})$ . Assuming $k_{1}+k_{2}=k_{b}$ and $\unicode[STIX]{x1D714}_{1}+\unicode[STIX]{x1D714}_{2}=0$ , we obtain

(2.30) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D63F}(\unicode[STIX]{x1D714}_{1},k_{1})\boldsymbol{x}_{1}^{(2)}=\boldsymbol{v}_{1}a_{b}\bar{a}_{2}^{(1)}+\boldsymbol{r}_{1}a_{1,\unicode[STIX]{x1D70F}}^{(1)}, & \displaystyle\end{eqnarray}$$

(2.31) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D63F}(\unicode[STIX]{x1D714}_{2},k_{2})\boldsymbol{x}_{2}^{(2)}=\boldsymbol{v}_{2}a_{b}\bar{a}_{1}^{(1)}+\boldsymbol{r}_{2}a_{2,\unicode[STIX]{x1D70F}}^{(1)}. & \displaystyle\end{eqnarray}$$

Denoting the null space of the transpose of $\unicode[STIX]{x1D63F}(k_{j},\unicode[STIX]{x1D714}_{j})$ by $\boldsymbol{n}_{j}$ and using the Fredholm alternative, we get the following set of equations:

(2.32a,b ) $$\begin{eqnarray}\displaystyle a_{1,\unicode[STIX]{x1D70F}}^{(1)}=\unicode[STIX]{x1D6FD}_{1}a_{b}\bar{a}_{2}^{(1)};\quad a_{2,\unicode[STIX]{x1D70F}}^{(1)}=\unicode[STIX]{x1D6FD}_{2}a_{b}\bar{a}_{1}^{(1)}, & & \displaystyle\end{eqnarray}$$

where

(2.33) $$\begin{eqnarray}\unicode[STIX]{x1D6FD}_{j}=-\frac{\boldsymbol{n}_{j}\boldsymbol{\cdot }\boldsymbol{v}_{j}}{\boldsymbol{n}_{j}\boldsymbol{\cdot }\boldsymbol{r}_{j}}.\end{eqnarray}$$

Furthermore, when $k_{1}-k_{2}=k_{b}$ and $\unicode[STIX]{x1D714}_{1}-\unicode[STIX]{x1D714}_{2}=0$ , we get

(2.34a,b ) $$\begin{eqnarray}\displaystyle a_{1,\unicode[STIX]{x1D70F}}^{(1)}=\unicode[STIX]{x1D6FD}_{1}a_{b}a_{2}^{(1)};\quad a_{2,\unicode[STIX]{x1D70F}}^{(1)}=\unicode[STIX]{x1D6FD}_{2}\bar{a}_{b}a_{1}^{(1)}, & & \displaystyle\end{eqnarray}$$

in which $\unicode[STIX]{x1D6FD}_{j}$ remains the same as before.

3.1 Shear in the lower layer

Here, we analyse the case when shear is present only in the lower layer and the local velocity at the bottom is zero (case 2 of figure 1). Therefore, the surface modes and the interfacial modes are Doppler shifted by an equal amount with respect to the bottom ripple. Presence of shear will also result in a change in the intrinsic frequencies of the waves. For the case of shear in the lower layer, firstly we investigate the triads formed by two surface modes, i.e. ${\mathcal{S}}{\mathcal{G}}^{+}$ and ${\mathcal{S}}{\mathcal{G}}^{-}$ . We have taken the incident wave $k_{i}$ on ${\mathcal{S}}{\mathcal{G}}^{+}$ and the resonant wave $k_{r}$ on ${\mathcal{S}}{\mathcal{G}}^{-}$ . Changing the Froude number changes the resonance condition, as is evident from figure 3(a). As mentioned earlier, in the absence of shear, all the Bragg resonance triads having $k_{i}$ on the ${\mathcal{S}}{\mathcal{G}}^{+}$ branch will resonate the waves on the ${\mathcal{S}}{\mathcal{G}}^{-}$ branch having $k_{r}=k_{i}$ . This corresponds to the straight line labelled $Fr=0.0$ in figure 3(a). Increasing $Fr\;(\equiv U_{u}/\sqrt{gH})$ will mean that the surface will be positively Doppler shifted with respect to the bottom ripples. For any given positive velocity, at some value of $k$ , the dispersion curve ${\mathcal{S}}{\mathcal{G}}^{-}$ is bound to cross the $k$ -axis; see figure 3(b). However, while plotting, we have kept the values of $k$ restricted because for higher values of $k$ , even though the resonance condition is satisfied, the rate of energy exchange falls off because the waves are unable to ‘feel’ the bottom. We see that for $Fr=0.2$ , the ${\mathcal{S}}{\mathcal{G}}^{-}$ branch shifts upwards. This is naturally reflected in the change in the resonance condition in figure 3(b), in which we have plotted the two branches of the dispersion relation. (The dispersion relation is a fourth-order polynomial in $\unicode[STIX]{x1D714}$ but we have plotted only two branches on which the resonance is being studied, i.e. ${\mathcal{S}}{\mathcal{G}}^{+}$ and ${\mathcal{S}}{\mathcal{G}}^{-}$ in this case.) If $Fr$ is further increased, then for a given $k_{i}$ on ${\mathcal{S}}{\mathcal{G}}^{+}$ , there can be up to three values of $k_{r}$ on ${\mathcal{S}}{\mathcal{G}}^{+}$ which would form the triad. This is the reason that for $Fr=0.6$ curve in figure 3, for a single $k_{i}H$ , there exist 3 values of $k_{r}H$ for which resonance condition is met. Two of these triads will be formed if the bottom’s wavenumber is $k_{b}=k_{i}+k_{r}$ (shown by the solid line). However, the third $k_{r}$ would lie on the part of ${\mathcal{S}}{\mathcal{G}}^{-}$ for which $\unicode[STIX]{x1D714}>0$ and for such a triad (shown in broken lines in figure 3 a for $Fr=0.6$ ), the bottom’s wavenumber would be $k_{r}-k_{i}$ . We note here in passing that these triads represented by the broken lines (in figure 3 a, not in 3 b) are not ‘usual’ triads but are ‘explosive’ triads. In such triads, both the incident wave and the resonant wave grow simultaneously, while the total energy of the system still remains conserved. This is due to the existence of negative energy waves (Cairns Reference Cairns1979). These ‘explosive’ triads have been explored by McHugh (Reference McHugh1992), for capillary–gravity waves, as well as by the authors of this paper (Raj & Guha Reference Raj and Guha2018).

Further, in figure 3(b) we have also plotted the change in the dispersion curves of ${\mathcal{S}}{\mathcal{G}}^{-}$ and ${\mathcal{S}}{\mathcal{G}}^{+}$ for $Fr=(0,0.2,0.6)$ for $kH<4$ . It can be seen that within this window of $kH$ , for a given $\unicode[STIX]{x1D714}_{i}$ on ${\mathcal{S}}{\mathcal{G}}^{+}$ , there can be only one $k_{i}$ (lines parallel to the $k$ -axis i.e. $\unicode[STIX]{x1D714}=\pm \unicode[STIX]{x1D714}_{0}$ intersects any given ${\mathcal{S}}{\mathcal{G}}^{+}$ at exactly one point). But for a given $|\unicode[STIX]{x1D714}_{r}|$ on ${\mathcal{S}}{\mathcal{G}}^{-}$ , for $Fr=0.6$ , there can be three values of $k_{r}$ satisfying the dispersion relation, two values are negative and one positive (lines parallel to the $k$ -axis i.e. $\unicode[STIX]{x1D714}=\pm \unicode[STIX]{x1D714}_{0}$ may intersect any given ${\mathcal{S}}{\mathcal{G}}^{-}$ at either one point or at three points).

Although we have discussed the modification in the resonance condition for a positive $Fr$ , a very similar thing happens for a negative $Fr$ . In figure 3, whereas for a positive $Fr$ , there may exist up to three $k_{r}$ on ${\mathcal{S}}{\mathcal{G}}^{-}$ for a given $k_{i}$ on ${\mathcal{S}}{\mathcal{G}}^{+}$ , for a negative $Fr$ (see $Fr=0.6$ ), three different $k_{i}$ on ${\mathcal{S}}{\mathcal{G}}^{+}$ may resonate the same wavenumber $k_{r}$ on ${\mathcal{S}}{\mathcal{G}}^{-}$ (see, $Fr=-0.6$ ). Because the dispersion curves in question, i.e. ${\mathcal{S}}{\mathcal{G}}^{+}$ and ${\mathcal{S}}{\mathcal{G}}^{-}$ are symmetric for $Fr=0$ , the symmetry is also maintained for a positive and a negative $Fr$ .

It might be noticed that the value of $Fr$ needed for any appreciable change in the resonance condition varies from moderate to large. The reason for this is that for surface gravity waves, the intrinsic frequency is quite large and to Doppler shift the intrinsic frequency, a local velocity of similar magnitude is needed. For example, to get three possible resonant waves having $k_{r}H<4$ for a given $k_{i}H$ , a Froude number of approximately 0.5 is needed. However, to Doppler shift the interfacial gravity waves on the pycnocline, a significantly smaller Froude number is sufficient because the intrinsic phase speeds of the interfacial waves are significantly low.

Figure 4.

(a) Different combinations of $k_{r}$ on ${\mathcal{I}}{\mathcal{G}}^{-}$ such that $k_{i}$ is on ${\mathcal{I}}{\mathcal{G}}^{+}$ , for various values of $Fr$ when shear is in the lower layer. Here $R=0.95$ and $h_{u}/h_{l}=1/3$ . For solid lines, $k_{b}=k_{i}+k_{r}$ and for dashed lines, $k_{b}=|k_{i}-k_{r}|$ . (b) Dispersion relations for the same case for three values of $Fr$ . Here solid lines represent ${\mathcal{I}}{\mathcal{G}}^{+}$ modes and dashed lines represent ${\mathcal{I}}{\mathcal{G}}^{-}$ .

We move on to the incident/resonant wave pairs formed by two interfacial modes, i.e. by the waves on ${\mathcal{I}}{\mathcal{G}}^{+}$ and ${\mathcal{I}}{\mathcal{G}}^{-}$ for the case of shear in the upper layer (case 3 of figure 1). The pycnocline is not only Doppler shifted with respect to the bottom, but it also has a discontinuity in shear across it. This signifies the presence of vorticity–gravity waves at the pycnocline and a significant change in the intrinsic frequency as well. A figure similar to the previous case showing combinations of $k_{i}$ (on ${\mathcal{I}}{\mathcal{G}}^{+}$ ) and $k_{r}$ (on ${\mathcal{I}}{\mathcal{G}}^{-}$ ) has been plotted in figure 4(a) restricting the non-dimensionalised wavenumber to 5. Naturally, at $Fr=0$ , the resonance condition is symmetric but the resonance condition changes greatly even for a small amount of mean flow. As we increase the $Fr$ , for a small $k_{i}$ on ${\mathcal{I}}{\mathcal{G}}^{+}$ , the resonance condition is met by a larger $k_{r}$ on ${\mathcal{I}}{\mathcal{G}}^{-}$ (see curves labelled $Fr=0.01,0.02$ of figure 4 a). On increasing $Fr$ further, again we see the existence of three $k_{r}$ values for a given $k_{i}$ , similar to the resonance between ${\mathcal{S}}{\mathcal{G}}^{+}$ and ${\mathcal{S}}{\mathcal{G}}^{-}$ ( $Fr=0.08$ , figure 4 a,b). However, if we further keep on increasing $Fr$ , then the complete ${\mathcal{I}}{\mathcal{G}}^{-}$ curve will become positive (shown in figure 4 b, $Fr=0.2$ ) and in such a case, only one resonant wave for any given $k_{i}$ on ${\mathcal{S}}{\mathcal{G}}^{-}$ will exist (dashed line labelled $Fr=0.2$ in figure 4 a). The dashed line implies that the wavenumber of the bottom ripple for such a triad is $k_{b}=k_{i}-k_{r}$ unlike the usual case $k_{b}=k_{i}+k_{r}$ for the solid lines in figure 4. Again, similar to the previous case, the triads marked by the dashed lines are the explosive triads. The positive and a negative $Fr$ result in symmetric cases as shown in figure 4.

We put this in the context of a real ocean of depth $H=100~\text{m}$ having a pycnocline at $h_{u}=25~\text{m}$ from the surface. These data are similar to those used by Alam et al. (Reference Alam, Liu and Yue2009b ). In the ‘no-flow’ situation, an interfacial wave having a wavelength $\unicode[STIX]{x1D706}_{i}\sim 200~\text{m}$ will resonate an oppositely travelling wave of wavelength $\unicode[STIX]{x1D706}_{r}\sim 200~\text{m}$ . However, in the presence of a small velocity of $U_{u}=U_{l}=0.31~\text{m}~\text{s}^{-1}$ opposite to the direction of the incident wave, the resonant wave would have a wavelength $\unicode[STIX]{x1D706}_{r}\sim 140~\text{m}$ .

The third sub-case for the case of shear in the upper layer is the resonant interaction between a surface and interfacial mode having opposite intrinsic frequency. This means that the incident/resonant pair is either ${\mathcal{I}}{\mathcal{G}}^{+}/{\mathcal{S}}{\mathcal{G}}^{-}$ or ${\mathcal{I}}{\mathcal{G}}^{-}/{\mathcal{S}}{\mathcal{G}}^{+}$ . Without a loss of generality, we will discuss only the ${\mathcal{I}}{\mathcal{G}}^{+}/{\mathcal{S}}{\mathcal{G}}^{-}$ pair; see figure 5(a,b). The results about the other pair can be obtained in a straightforward manner, simply by changing the sign of  $Fr$ from positive to negative and vice versa. What matters is that whether the sign of mean flow and that of surface/interfacial waves are in the same direction or the opposite. The positive shear in this particular case will imply that the velocity at the surface/pycnocline is in the direction of the propagation of the interfacial wave ${\mathcal{I}}{\mathcal{G}}^{+}$ . Therefore, increasing $Fr$ will lead to an increase in the frequency of ${\mathcal{I}}{\mathcal{G}}^{+}$ but a non-monotonic change in the frequency of the ${\mathcal{S}}{\mathcal{G}}^{-}$ mode, as shown in figure 5(b). Even for a small value of shear, the effect on the speed of ${\mathcal{I}}{\mathcal{G}}^{+}$ is significant but ${\mathcal{S}}{\mathcal{G}}^{-}$ is relatively less affected. However, for a large value of $Fr$ , there may exist multiple values of $k_{r}$ for a given $k_{i}$ as can be seen from figure 5(a), $Fr=0.3$ , 0.4, 0.5, 0.6. The reason is simply a non-monotonic behaviour of frequency of ${\mathcal{S}}{\mathcal{G}}^{-}$ with respect to the wavenumber as can be seen from figure 5(b). For a higher value of $Fr$ , the frequency of ${\mathcal{S}}{\mathcal{G}}^{-}$ becomes positive and the triads formed by the positive part of ${\mathcal{S}}{\mathcal{G}}^{-}$ are shown in dashed lines in figure 5(a). For these triads (again these triads are explosive), the bottom’s wavenumber is $k_{r}-k_{i}$ whereas for triads marked by solid line, the bottom’s wavenumber is $k_{r}+k_{i}$ .

Figure 5.

(a) Different combinations of $k_{r}$ on ${\mathcal{S}}{\mathcal{G}}^{-}$ such that $k_{i}$ is on ${\mathcal{I}}{\mathcal{G}}^{+}$ , for various positive $Fr$ values when shear is in the lower layer. Here $R=0.95$ and $h_{u}/h_{l}=1/3$ . For solid lines, $k_{b}=k_{i}+k_{r}$ but for dashed lines, $k_{b}=|k_{i}-k_{r}|$ . (b) Dispersion relations for the same case for different increasingly positive $Fr$ . Here solid lines represent ${\mathcal{I}}{\mathcal{G}}^{+}$ modes and dashed lines represent ${\mathcal{S}}{\mathcal{G}}^{-}$ .

Figure 6.

(a) Different combinations of $k_{r}$ on ${\mathcal{S}}{\mathcal{G}}^{-}$ such that $k_{i}$ is on ${\mathcal{I}}{\mathcal{G}}^{+}$ , for various negative $Fr$ values when shear is in the lower layer. Here $R=0.95$ and $h_{u}/h_{l}=1/3$ . For solid lines, $k_{b}=k_{i}+k_{r}$ but for dashed lines, $k_{b}=|k_{i}-k_{r}|$ . (b) Dispersion relations for the same case for different negative $Fr$ . Direction of arrows imply increasingly negative $Fr$ . Here solid lines represent ${\mathcal{I}}{\mathcal{G}}^{+}$ modes and dashed lines represent ${\mathcal{S}}{\mathcal{G}}^{-}$ .

If the Froude number is negative (see figure 6 a,b), the frequency of ${\mathcal{S}}{\mathcal{G}}^{-}$ increases monotonically but that of ${\mathcal{I}}{\mathcal{G}}^{+}$ may become non-monotonic; shown in figure 6(b) for the case $Fr=-0.08$ . Because the frequency of ${\mathcal{S}}{\mathcal{G}}^{-}$ plotted in figure 6(b) is restricted, not much difference in the dispersion curves is obtained. For a higher $Fr$ , the frequency changes sign within the chosen limit of $kH=4$ and becomes negative ( $Fr=-0.1$ ). For a further increase in the velocity, the frequency becomes completely negative for ${\mathcal{I}}{\mathcal{G}}^{-}$ ( $Fr=-0.2$ ). This change in the frequency is reflected in the change in the resonance condition and the change can be visualised in figure 6(a). For $Fr=-0.01,-0.02,-0.04$ , for a single $k_{r}$ , only one $k_{i}<4$ exists but for higher $Fr$ , a single $k_{r}$ maybe resonant by multiple $k_{i}$ ( $Fr=-0.08,-0.1$ ). For $Fr=-0.1$ , the bottom’s wavenumber is $k_{i}+k_{r}$ for the solid line part in figure 6(a) and $k_{i}-k_{r}$ for the dashed line part i.e. the explosive triads.

Finally, we deal with the case when the incident/resonant modes are in the same direction i.e. ${\mathcal{I}}{\mathcal{G}}^{+}/{\mathcal{S}}{\mathcal{G}}^{+}$ or ${\mathcal{I}}{\mathcal{G}}^{-}/{\mathcal{S}}{\mathcal{G}}^{-}$ . Again, without a loss of generality, we study only the resonance between ${\mathcal{I}}{\mathcal{G}}^{+}/{\mathcal{S}}{\mathcal{G}}^{+}$ modes and in this case, positive $Fr$ will imply a flow in the same direction of the waves; see figure 7(a,b). Because all the involved waves and the mean flow are in the same direction, there is no question of sign changing of the frequency of any wave. The frequencies of all the waves increases progressively with increasing $Fr$ (figure 7 b). In figure 7(a,b), however, we have plotted both the positive $Fr$ and the negative $Fr$ having low magnitudes. Increasing $Fr$ will mean that for a given $k_{i}$ , a higher $k_{r}$ will be needed for resonance which can be seen from figure 7(a,b) (positive $Fr$ ).

For a negative $Fr$ (see figure 8 a,b), the dispersion relation is plotted in figure 8(b). For $Fr=-0.1$ , ${\mathcal{S}}{\mathcal{G}}^{+}$ is positive throughout but a part of ${\mathcal{I}}{\mathcal{G}}^{+}$ becomes negative. For a higher negative $Fr$ (say $-0.3$ ), ${\mathcal{S}}{\mathcal{G}}^{+}$ remains positive, but ${\mathcal{I}}{\mathcal{G}}^{+}$ becomes completely negative. For a further negative $Fr$ , ${\mathcal{I}}{\mathcal{G}}^{+}$ , remains negative and a part of ${\mathcal{S}}{\mathcal{G}}^{+}$ also becomes negative. The effect on the resonance conditions for the case of small negative $Fr$ (up to $-0.25$ ) has been plotted in figure 7(a) and for higher negative $Fr$ has been plotted in figure 8(a).

Figure 7.

(a) Different combinations of $k_{r}$ on ${\mathcal{S}}{\mathcal{G}}^{+}$ such that $k_{i}$ is on ${\mathcal{I}}{\mathcal{G}}^{+}$ for various values of positive $Fr$ and for low values of negative $Fr$ for the case of shear in the lower layer; $R=0.95$ , $h_{u}/h_{l}=1/3$ . For solid lines, $k_{b}=|k_{i}-k_{r}|$ and for dashed lines, $k_{b}=k_{i}+k_{r}$ . (b) Dispersion relation for positive $Fr$ . Here solid lines represent ${\mathcal{I}}{\mathcal{G}}^{+}$ modes and dashed lines represent ${\mathcal{S}}{\mathcal{G}}^{+}$ .

Figure 8.

(a) Different combinations of $k_{r}$ on ${\mathcal{S}}{\mathcal{G}}^{+}$ such that $k_{i}$ is on ${\mathcal{I}}{\mathcal{G}}^{+}$ for high values of negative $Fr$ for the case of shear in the lower layer. Here $R=0.95$ and $h_{u}/h_{l}=1/3$ . For solid lines, $k_{b}=|k_{i}-k_{r}|$ and for dashed lines, $k_{b}=k_{i}+k_{r}$ . (b) Dispersion relation for negative $Fr$ . Here solid lines represent ${\mathcal{I}}{\mathcal{G}}^{+}$ modes and dashed lines represent ${\mathcal{S}}{\mathcal{G}}^{+}$ .

3.2 Shear in the upper layer

As we have mentioned earlier, shear in the upper layer causes a relative Doppler shift between the surface and the pycnocline, as well as the bottom ripple. Further, the change in the intrinsic frequencies of the surface modes will be minimal compared to the change in the intrinsic frequencies of the interfacial modes. In § 3.1, we have performed a detailed study on how the Doppler shift changes the resonance conditions. Here we focus on the case when the intrinsic frequency of the waves changes, and because the intrinsic frequencies of the interfacial waves are more prone to change, the case of substantial interest is the resonant interaction between the ${\mathcal{I}}{\mathcal{G}}^{+}$ and ${\mathcal{I}}{\mathcal{G}}^{-}$ modes. Since shear is only in the upper layer and the pycnocline has no local base velocity, there is no role of the Doppler shift. However, when shear in the upper layer is positive ( $Fr>0$ ), ${\mathcal{I}}{\mathcal{G}}^{+}$ is sped up but the ${\mathcal{I}}{\mathcal{G}}^{-}$ mode is slowed down (we note that the wave at the interface is a vorticity–gravity wave). Although for the $Fr=0$ case, the resonant wave is $k_{r}=k_{i}$ , the conditions change when $Fr\neq 0$ . For $Fr>0$ , we have $k_{r}<k_{i}$ , and for $Fr<0$ , we get $k_{r}>k_{i}$ . The change in the resonance condition is shown in figure 9(a) and the dispersion relation for $Fr>0$ has been plotted in figure 9(b).

Figure 9.

(a) Different combinations of $k_{r}$ on ${\mathcal{I}}{\mathcal{G}}^{-}$ such that $k_{i}$ is on ${\mathcal{I}}{\mathcal{G}}^{+}$ , for various values of $Fr\equiv U_{u}/\sqrt{gH}$ when shear is in the upper layer. Here $U_{l}=U_{b}=0$ , $R=0.95$ and $h_{u}/h_{l}=1/3$ . (b) Dispersion relations for the same case for three values of $Fr$ .

4.1 Uniform flow and consequences of shear

Wave triad interaction is an energy exchange between waves on the surface and the pycnocline and there is no direct involvement of bottom topography. Therefore, a velocity field with a uniform flow (figure 1 case 1) will Doppler shift the waves on the surface and the pycnocline by the same velocity $U$ and there will not be any consequences for the resonance condition. To illustrate this, we take three waves having wavenumbers $(k_{1},k_{2},k_{3})$ and corresponding frequencies $(\unicode[STIX]{x1D714}_{1},\unicode[STIX]{x1D714}_{2},\unicode[STIX]{x1D714}_{3})$ such that $k_{1}+k_{2}-k_{3}=0$ but $\unicode[STIX]{x1D714}_{1}+\unicode[STIX]{x1D714}_{2}-\unicode[STIX]{x1D714}_{3}\neq 0$ . In the presence of a constant base velocity $U$ , every frequency $\unicode[STIX]{x1D714}_{j}$ ( $j=1,2,3$ ) would be Doppler shifted by an amount $Uk_{j}$ . In such a case, however, the intrinsic frequencies of the waves will undergo no change. Therefore, the modified frequency condition would be

(4.1) $$\begin{eqnarray}\displaystyle & & \displaystyle (\unicode[STIX]{x1D714}_{1}+Uk_{1})+(\unicode[STIX]{x1D714}_{2}+Uk_{2})-(\unicode[STIX]{x1D714}_{3}+Uk_{3})\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x1D714}_{1}+\unicode[STIX]{x1D714}_{2}-\unicode[STIX]{x1D714}_{3}+U(k_{1}+k_{2}-k_{3})\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x1D714}_{1}+\unicode[STIX]{x1D714}_{2}-\unicode[STIX]{x1D714}_{3}\nonumber\\ \displaystyle & & \displaystyle \quad \neq 0.\end{eqnarray}$$

Thus, a mere Doppler shift by the same velocity $U$ would not change the resonance condition for the wave triad interaction. However, if the surface and the interface were to be Doppler shifted by different amounts, there can be changes in the resonance conditions and naturally the three waves satisfying the resonance condition in the absence of shear might not do so in the presence of a shear. Alternatively, three waves not satisfying a resonance condition might do so in the presence of velocity shear. This can be elucidated using a simple example: let $k_{1}+k_{2}-k_{3}=0$ but $\unicode[STIX]{x1D714}_{1}+\unicode[STIX]{x1D714}_{2}-\unicode[STIX]{x1D714}_{3}\neq 0$ , i.e. the waves do not satisfy the resonant condition in the absence of a base velocity shear. Let us assume that the waves $1$ and $2$ are at the surface, which now has a base velocity $U_{u}$ , while wave $3$ is at the interface, which travels with a velocity $U_{l}$ (different from $U_{u}$ because of shear). Then, the frequency condition reads

(4.2) $$\begin{eqnarray}\displaystyle & & \displaystyle (\unicode[STIX]{x1D714}_{1}^{\prime }+U_{u}k_{1})+(\unicode[STIX]{x1D714}_{2}^{\prime }+U_{u}k_{2})-(\unicode[STIX]{x1D714}_{3}^{\prime }+U_{l}k_{3})\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x1D714}_{1}^{\prime }+\unicode[STIX]{x1D714}_{2}^{\prime }-\unicode[STIX]{x1D714}_{3}^{\prime }+U_{u}(k_{1}+k_{2})-U_{l}k_{3}\nonumber\\ \displaystyle & & \displaystyle \quad =0\quad \text{iff }\unicode[STIX]{x1D714}_{1}^{\prime }+\unicode[STIX]{x1D714}_{2}^{\prime }-\unicode[STIX]{x1D714}_{3}^{\prime }=k_{3}(U_{l}-U_{u}).\end{eqnarray}$$

The primes in the frequencies denote that the intrinsic frequencies will be modified due to shear.

4.2 Shear in the lower layer

Figure 10.

Different combinations of $k_{1}$ , $k_{2}$ and $k_{3}$ on ${\mathcal{I}}{\mathcal{G}}^{-}$ , ${\mathcal{S}}{\mathcal{G}}^{-}$ , ${\mathcal{I}}{\mathcal{G}}^{+}$ respectively forming a resonance triad for (a) shear in bottom layer only; $U_{u}^{\ast }=U_{l}^{\ast }=(0.0,0.2,0.4,0.6)$ , $U_{b}^{\ast }=0$ , $R=0.95$ and $h_{u}/h_{l}=1/3$ . (b) Shear in top layer only; $U_{u}^{\ast }=(0.0,0.2,0.4,0.6)$ , $U_{l}^{\ast }=U_{b}^{\ast }=0$ , $R=0.95$ and $h_{u}/h_{l}=1/3$ .

When shear is present only in the lower layer, there is no Doppler shift between the two waves and the only effect of the shear is felt in modifying the intrinsic frequencies of the waves. Although the shear jump is only at the pycnocline, the effect of it at lower wavenumbers would be felt in the surface mode as well. In figure 10(a), we have shown the change in the resonance condition for three interacting waves having $k_{1},k_{2}$ and $k_{3}$ on ${\mathcal{I}}{\mathcal{G}}^{-}$ , ${\mathcal{S}}{\mathcal{G}}^{-}$ and ${\mathcal{I}}{\mathcal{G}}^{+}$ respectively. The Froude numbers are as follows: $Fr=(0.0,0.2,0.4,0.6)$ . As the shear is increased in the positive direction, for a given $k_{1}$ on ${\mathcal{I}}{\mathcal{G}}^{-}$ , $k_{2}$ on ${\mathcal{S}}{\mathcal{G}}^{-}$ decreases but $k_{3}$ on ${\mathcal{I}}{\mathcal{G}}^{+}$ increases. The figure reveals that the change is not as significant as that in the Bragg resonance case even at high values of shear.

4.3 Shear in the upper layer

The presence of a shear in the upper layer modifies the flow in two ways. Firstly, there now exists a jump in base shear both at the surface and at the interface, which changes the intrinsic frequencies of all four modes. Secondly, the presence of shear automatically means that the local mean velocities at the surface and at the pycnocline are different from each other, which implies a relative Doppler shift between the two. Although such a situation may give rise to shear instabilities, such shear instabilities tend to occur at higher wavenumbers which have very low growth rates. In figure 10(b), we have shown the resonance condition for three interacting waves having $k_{1},k_{2}$ and $k_{3}$ respectively on ${\mathcal{I}}{\mathcal{G}}^{-}$ , ${\mathcal{S}}{\mathcal{G}}^{-}$ and ${\mathcal{I}}{\mathcal{G}}^{+}$ . Yet again, the behaviour is similar to that of the previous case but the change in the resonance condition is more prominent here due to the Doppler shifting of the surface and the interface.

5.1 Validation

A comprehensive benchmarking of the HOS method for two layers without a velocity field has been performed in Alam et al. (Reference Alam, Liu and Yue2009b ). In this paper, we have extended the method to incorporate the velocity field by adding requisite terms. For validation of the code, we have simulated a case of Bragg resonance in which the surface mode interacts with the bottom to generate another surface mode having an intrinsic frequency of the opposite sign. We have compared the solution of the HOS code to the analytically obtained solution. The parameters used are mentioned in the caption of figure 11. It can be seen that the analytical solution and the numerical solution are graphically indistinguishable.

Figure 11.

Code validation for $R=0.98$ , $k_{i}H=0.086$ , $k_{r}H=0.1140$ , $k_{b}H=0.2$ , $\unicode[STIX]{x1D714}_{i}^{\ast }=0.0982$ , $\unicode[STIX]{x1D714}_{r}^{\ast }=-0.0982$ , $U_{u}^{\ast }=0.1864$ , $U_{l}^{\ast }=0.0083$ , $M=3$ , $N=2048$ , $T_{i}/\unicode[STIX]{x0394}T=512$ . The analytical and numerical solutions are indistinguishable.

5.2 Numerical results

We have simulated a resonance between the waves on the same branch ( ${\mathcal{S}}{\mathcal{G}}^{-}$ ) of the dispersion curve for case 2, i.e. shear only in the lower layer. The incident wave has the wavenumber $k_{i}H=0.83$ , while the resonant wave has the wavenumber $k_{r}H=2.27$ . These two waves have the same direction of propagation and have the same frequency of $\unicode[STIX]{x1D714}^{\ast }=-0.4770$ . Because the direction of propagation of the waves is the same, the wavenumber of the bottom is the difference of the wavenumber of the incident and the resonant waves, i.e. $k_{b}H=1.44$ . The velocities are $U_{u}^{\ast }=0.5016,U_{l}^{\ast }=0,U_{b}^{\ast }=0$ . Other relevant physical parameters are $h_{u}/h_{l}=1/3$ and $R=0.95$ . The dispersion relation is plotted in figure 12(a) and the corresponding HOS simulation is shown in figure 12(b).

Figure 12.

(a) Dispersion relation showing the location of the resonant triad. Both the incident and the resonant waves lie on the ${\mathcal{S}}{\mathcal{G}}^{-}$ curve. (b) Numerical simulation using the HOS code: $a_{i}=0.00005H$ , $a_{b}=0.02H$ , $U_{u}^{\ast }=0.5016$ , $U_{l}^{\ast }=U_{b}^{\ast }=0$ , $\unicode[STIX]{x1D714}_{i}^{\ast }=-0.4770$ , $R=0.95$ , $k_{i}H=0.83$ , $k_{r}H=2.27$ , $k_{b}H=1.44$ , $h_{u}/h_{l}=1/3$ , $M=3$ , $N=1024$ , $T_{i}/\unicode[STIX]{x0394}T=2048$ . $T_{i}$ is the time period of the incident wave.

Next, we have studied the effect of shear in the upper layer on the Bragg resonance between two oppositely travelling internal modes, i.e. ${\mathcal{I}}{\mathcal{G}}^{+}$ and ${\mathcal{I}}{\mathcal{G}}^{-}$ . Because the shear is in the upper layer only, there is no Doppler shift of the concerned waves (both on the pycnocline), and the changes are only in the intrinsic frequencies. In the absence of a base flow, it is known that the resonant wavenumber will be the same as the incident wavenumber. However, we have shown analytically in § 4.1 that shear changes the resonance condition. To illustrate this here, we take our incident wave on ${\mathcal{I}}{\mathcal{G}}^{+}$ having wavenumber $k_{i}H=0.10$ and frequency $\unicode[STIX]{x1D714}_{i}^{\ast }=0.0097$ . The bottom ripple consists of three different wavenumbers: $k_{b1}H=0.19,k_{b2}H=0.20$ and $k_{b3}H=0.21$ . Thus, the incident wave will interact with the bottom and may generate three different waves having wavenumbers $k_{r1}H=0.09$ , $k_{r2}H=0.10$ , $k_{r3}H=0.11$ and frequencies $\unicode[STIX]{x1D714}_{r1}^{\ast }=-0.0088$ , $\unicode[STIX]{x1D714}_{r2}^{\ast }=0.0097$ , $\unicode[STIX]{x1D714}_{r3}^{\ast }=-0.0107$ respectively. Only one of these wavenumbers may satisfy a resonance condition for a given velocity field and the other wavenumbers will be generated in a ‘near-resonant’ way (Craik Reference Craik1988). We plot the time evolution of amplitude of all these three wavenumbers in the absence of shear; see figure 13(a). As expected, the maximum growth is only in the wavenumber $k_{r2}H=0.10$ . The amplitude plotted is simply the spatial Fourier transform of the interface, and a rapidly changing amplitude corresponding to $kH=0.10$ signifies an oppositely travelling wave increasing in amplitude. At approximately $t/T_{0}\approx 30$ , both the positively and the negatively travelling waves have the same amplitude. We also observe a small growth in the wavenumbers $kH=0.09$ and $kH=0.11$ . These two wavenumbers do not satisfy the exact resonant condition and hence are generated only near resonantly.

Figure 13.

Amplitude versus time plot for different wavenumbers on the interface for $R=0.95,h_{u}/h_{l}=1/3$ , $a_{i}=0.00005H_{0}$ , $a_{b}=0.05H_{0}$ and $U_{l}^{\ast }=U_{b}^{\ast }=0$ . (a) $U_{u}^{\ast }=0$ , (b) $U_{u}^{\ast }=-0.0136$ and (c) $U_{u}^{\ast }=0.0123$ . Parameters for the simulation are $M=3,N=512$ , $T_{0}/\unicode[STIX]{x0394}T=512$ . Here, $T_{0}$ is the time period of the wave $k_{i}$ in the absence of any background velocity.

Next, we make the shear negative in the upper layer to yield $U_{u}^{\ast }=-0.0136$ , while keeping $U_{l}^{\ast }=U_{b}^{\ast }=0$ . Due to the velocity field, the incident wave’s frequency gets modified to $\unicode[STIX]{x1D714}_{i}^{\ast }=0.0092$ . The frequencies of the three possible resonant waves respectively become $\unicode[STIX]{x1D714}_{r1}^{\ast }=-0.0092,\unicode[STIX]{x1D714}_{r2}^{\ast }=-0.0103$ and $\unicode[STIX]{x1D714}_{r3}^{\ast }=-0.0103$ . In this case, we observe that the incident wave frequency is equal to $\unicode[STIX]{x1D714}_{r1}$ , therefore the dominant resonating wavenumber is $k_{r1}H=0.09$ . The amplitude evolution has been plotted in figure 13(b). Wiggles in the plot indicate near-resonant generation of oppositely travelling waves.

Likewise, we make the shear positive in the upper layer to yield $U_{u}^{\ast }=0.0123$ , while keeping $U_{l}^{\ast }=U_{b}^{\ast }=0$ . The incident wave’s frequency changes to $\unicode[STIX]{x1D714}_{i}^{\ast }=0.0102$ . The frequency of the three possible resonant waves become $\unicode[STIX]{x1D714}_{r1}^{\ast }=-0.0083$ , $\unicode[STIX]{x1D714}_{r2}^{\ast }=-0.0093$ and $\unicode[STIX]{x1D714}_{r3}^{\ast }=-0.0102$ . We observe that $\unicode[STIX]{x1D714}_{i}=\unicode[STIX]{x1D714}_{r3}$ and hence, the dominant wavenumber generated is $k_{r3}H=0.11$ . This also corroborates figure 9, where it can be seen that for a $k_{i}$ on the ${\mathcal{I}}{\mathcal{G}}^{+}$ mode, increasing the shear results in an increase in $k_{r}$ on the ${\mathcal{I}}{\mathcal{G}}^{-}$ mode. The amplitude evolution has been plotted in figure 13(c). Again, wiggles indicate near-resonant generation of oppositely travelling waves. Thus, we see that the exclusion of shear may substantially change the condition for resonant triads. Hence, practical applications, such as broadband cloaking (Alam Reference Alam2012), in which bottom corrugations are designed in a particular way to ‘cloak’ the offshore structures, may need to account for oceanic shear for an optimum design.