2.1 Equations in physical space

We use a Cartesian coordinate system $(\boldsymbol{x},z)$, with the $xy$ plane being the mean free surface and the $z$ axis being directed upward. We consider a two-layer model with the free surface on top and the interface between the layers. We denote the respective depths of the upper and lower layers by $h_{u},h_{l}$ and their densities by $\unicode[STIX]{x1D70C}_{u}$, $\unicode[STIX]{x1D70C}_{l}$, with the difference in density between layers being $\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D70C}_{l}-\unicode[STIX]{x1D70C}_{u}$, where subscript ‘$u$’ refers to the upper layer and subscript ‘$l$’ to the lower layer. We assume the fluid is homogeneous, incompressible, immersible, inviscid and irrotational in both layers.

To derive the closed set of coupled equations for the surface velocity potential $\unicode[STIX]{x1D6F9}^{(u)}(x,y,t)$ and displacement $\unicode[STIX]{x1D701}^{(u)}(x,y,t)$ and interfacial velocity potential $\unicode[STIX]{x1D6F9}^{(l)}(x,y,t)$ and displacement $\unicode[STIX]{x1D701}^{(l)}(x,y,t)$, we start from the Euler equations, incompressibility condition and kinematic boundary conditions for velocity and pressure continuity along the surface/interface. We then introduce a nonlinearity parameter $\unicode[STIX]{x1D716}$, the slope of the waves, and make a formal assumption that $\unicode[STIX]{x1D716}\ll 1$. This allows us to iterate the resulting equations for a solution representing a wavetrain with wavenumber $k$. This procedure was recently executed in Choi (Reference Choi2019), and leads to the following system of equations, truncated at the second order of the nonlinearity parameter:

(2.1a)$$\begin{eqnarray}\displaystyle \dot{\unicode[STIX]{x1D701}}^{(u)} & = & \displaystyle \unicode[STIX]{x1D6FE}_{11}\unicode[STIX]{x1D6F9}^{(u)}+\unicode[STIX]{x1D6FE}_{12}\unicode[STIX]{x1D6F9}^{(l)}-\unicode[STIX]{x1D70C}_{u}\unicode[STIX]{x1D6FE}_{11}[\unicode[STIX]{x1D701}^{(u)}(\unicode[STIX]{x1D6FE}_{11}\unicode[STIX]{x1D6F9}^{(u)}+\unicode[STIX]{x1D6FE}_{12}\unicode[STIX]{x1D6F9}^{(l)})]\nonumber\\ \displaystyle & & \displaystyle -\,\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D6FE}_{21}[\unicode[STIX]{x1D701}^{(l)}(\unicode[STIX]{x1D6FE}_{21}\unicode[STIX]{x1D6F9}^{(u)}+\unicode[STIX]{x1D6FE}_{22}\unicode[STIX]{x1D6F9}^{(l)})]\nonumber\\ \displaystyle & & \displaystyle -\,\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D701}^{(u)}\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F9}^{(u)})/\unicode[STIX]{x1D70C}_{u}+\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D70C}_{l}/\unicode[STIX]{x1D70C}_{u})\unicode[STIX]{x1D6FE}_{31}\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D701}^{(l)}\unicode[STIX]{x1D6FE}_{31}\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F9}^{(u)})\nonumber\\ \displaystyle & & \displaystyle -\,\unicode[STIX]{x1D70C}_{l}\unicode[STIX]{x1D6FE}_{31}\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D701}^{(l)}\unicode[STIX]{x1D6FE}_{33}\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F9}^{(l)}),\end{eqnarray}$$

(2.1b)$$\begin{eqnarray}\displaystyle \dot{\unicode[STIX]{x1D701}}^{(l)} & = & \displaystyle \unicode[STIX]{x1D6FE}_{21}\unicode[STIX]{x1D6F9}^{(u)}+\unicode[STIX]{x1D6FE}_{22}\unicode[STIX]{x1D6F9}^{(l)}-\unicode[STIX]{x1D70C}_{u}\unicode[STIX]{x1D6FE}_{12}[\unicode[STIX]{x1D701}^{(u)}(\unicode[STIX]{x1D6FE}_{11}\unicode[STIX]{x1D6F9}^{(u)}+\unicode[STIX]{x1D6FE}_{12}\unicode[STIX]{x1D6F9}^{(l)})]\nonumber\\ \displaystyle & & \displaystyle -\,\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D6FE}_{22}[\unicode[STIX]{x1D701}^{(l)}(\unicode[STIX]{x1D6FE}_{21}\unicode[STIX]{x1D6F9}^{(u)}+\unicode[STIX]{x1D6FE}_{22}\unicode[STIX]{x1D6F9}^{(l)})]\nonumber\\ \displaystyle & & \displaystyle -\,\unicode[STIX]{x1D70C}_{l}\unicode[STIX]{x1D6FE}_{33}\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D701}^{(l)}\unicode[STIX]{x1D6FE}_{31}\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F9}^{(u)})-\unicode[STIX]{x1D70C}_{l}J\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D701}^{(l)}J\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F9}^{(l)})\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D70C}_{u}\unicode[STIX]{x1D6FE}_{32}\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D701}^{(l)}\unicode[STIX]{x1D6FE}_{32}\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F9}^{(l)}),\end{eqnarray}$$

(2.1c)$$\begin{eqnarray}\displaystyle \dot{\unicode[STIX]{x1D6F9}}^{(u)} & = & \displaystyle -\unicode[STIX]{x1D70C}_{u}g\unicode[STIX]{x1D701}^{(u)}+{\textstyle \frac{1}{2}}\unicode[STIX]{x1D70C}_{u}(\unicode[STIX]{x1D6FE}_{11}\unicode[STIX]{x1D6F9}^{(u)}+\unicode[STIX]{x1D6FE}_{12}\unicode[STIX]{x1D6F9}^{(l)})^{2}-{\textstyle \frac{1}{2}}(\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F9}^{(u)})\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F9}^{(u)})/\unicode[STIX]{x1D70C}_{u},\end{eqnarray}$$

(2.1d)$$\begin{eqnarray}\displaystyle \dot{\unicode[STIX]{x1D6F9}}^{(l)} & = & \displaystyle -\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}g\unicode[STIX]{x1D701}^{(l)}+{\textstyle \frac{1}{2}}\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D6FE}_{21}\unicode[STIX]{x1D6F9}^{(u)}+\unicode[STIX]{x1D6FE}_{22}\unicode[STIX]{x1D6F9}^{(l)})^{2}+{\textstyle \frac{1}{2}}\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D70C}_{l}/\unicode[STIX]{x1D70C}_{u})(\unicode[STIX]{x1D6FE}_{31}\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F9}^{(u)})\boldsymbol{\cdot }(\unicode[STIX]{x1D6FE}_{31}\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F9}^{(u)})\nonumber\\ \displaystyle & & \displaystyle -\,{\textstyle \frac{1}{2}}\unicode[STIX]{x1D70C}_{l}(J\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F9}^{(l)})\boldsymbol{\cdot }(J\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F9}^{(l)})+{\textstyle \frac{1}{2}}\unicode[STIX]{x1D70C}_{u}(\unicode[STIX]{x1D6FE}_{32}\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F9}^{(l)})\boldsymbol{\cdot }(\unicode[STIX]{x1D6FE}_{32}\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F9}^{(l)})\nonumber\\ \displaystyle & & \displaystyle -\,\unicode[STIX]{x1D70C}_{l}(\unicode[STIX]{x1D6FE}_{31}\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F9}^{(u)})\boldsymbol{\cdot }(\unicode[STIX]{x1D6FE}_{33}\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F9}^{(l)}),\end{eqnarray}$$

$\unicode[STIX]{x1D6FE}_{ij}$

$J$

2.2 Hamiltonian

We use the Fourier transformations of the interface variables for the two-layer system depicted in figure 1:

(2.2)$$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D701}^{(j)}(\boldsymbol{x},t)=\int \hat{\unicode[STIX]{x1D701}}^{(j)}(\boldsymbol{k},t)\text{e}^{-\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}}\,\text{d}\boldsymbol{k},\\ \displaystyle \unicode[STIX]{x1D6F9}^{(j)}(\boldsymbol{x},t)=\int \hat{\unicode[STIX]{x1D6F9}}^{(j)}(\boldsymbol{k},t)\text{e}^{-\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}}\,\text{d}\boldsymbol{k}\quad \text{for }j\in \{u,l\}.\end{array}\right\}\end{eqnarray}$$

The equations of motion (2.1) can then be represented by canonically conjugated Hamilton’s equations for the Hamiltonian $H$, given by Choi (Reference Choi2019):

(2.3)$$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D701}}^{(j)}}{\unicode[STIX]{x2202}t}=\frac{\unicode[STIX]{x1D6FF}H}{{\unicode[STIX]{x1D6FF}\hat{\unicode[STIX]{x1D6F9}}^{(j)}}^{\ast }},\quad \frac{\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D6F9}}^{(j)}}{\unicode[STIX]{x2202}t}=-\frac{\unicode[STIX]{x1D6FF}H}{{\unicode[STIX]{x1D6FF}\hat{\unicode[STIX]{x1D701}}^{(j)}}^{\ast }},\quad j\in \{u,l\}. & & \displaystyle\end{eqnarray}$$

This is a generalization for two layers of the Hamiltonian formulation described in Zakharov (Reference Zakharov1968) for surface waves. Here the Hamiltonian, $H$, is a sum of a quadratic Hamiltonian, describing linear non-interacting waves, and a cubic Hamiltonian, describing wave–wave interactions, $H=H_{2}+H_{3}$, where

(2.4)$$\begin{eqnarray}\displaystyle H_{2} & = & \displaystyle \frac{1}{2}\iint [\!h_{1}^{(1a)}\hat{\unicode[STIX]{x1D701}}_{1}^{(u)}\hat{\unicode[STIX]{x1D701}}_{2}^{(u)}+h_{1}^{(2a)}\hat{\unicode[STIX]{x1D6F9}}_{1}^{(u)}\hat{\unicode[STIX]{x1D6F9}}_{2}^{(u)}\nonumber\\ \displaystyle & & \displaystyle +\,h_{1}^{(3a)}\hat{\unicode[STIX]{x1D701}}_{1}^{(l)}\hat{\unicode[STIX]{x1D701}}_{2}^{(l)}+h_{1}^{(4a)}\hat{\unicode[STIX]{x1D6F9}}_{1}^{(l)}\hat{\unicode[STIX]{x1D6F9}}_{2}^{(l)}+h_{1,2}^{(5a)}\hat{\unicode[STIX]{x1D6F9}}_{1}^{(u)}\hat{\unicode[STIX]{x1D6F9}}_{2}^{(l)}\! ]\unicode[STIX]{x1D6FF}(\boldsymbol{k}_{1}+\boldsymbol{k}_{2})\,\text{d}\boldsymbol{k}_{1}\,\text{d}\boldsymbol{k}_{2},\end{eqnarray}$$

(2.5)$$\begin{eqnarray}\displaystyle H_{3} & = & \displaystyle \iiint [\!h_{123}^{(1)}\hat{\unicode[STIX]{x1D6F9}}_{1}^{(u)}\hat{\unicode[STIX]{x1D6F9}}_{2}^{(u)}\hat{\unicode[STIX]{x1D701}}_{3}^{(u)}+h_{123}^{(2)}\hat{\unicode[STIX]{x1D6F9}}_{1}^{(u)}\hat{\unicode[STIX]{x1D6F9}}_{2}^{(l)}\hat{\unicode[STIX]{x1D701}}_{3}^{(u)}+h_{123}^{(3)}\hat{\unicode[STIX]{x1D6F9}}_{1}^{(l)}\hat{\unicode[STIX]{x1D6F9}}_{2}^{(l)}\hat{\unicode[STIX]{x1D701}}_{3}^{(u)}\nonumber\\ \displaystyle & & \displaystyle \!+\,h_{123}^{(4)}\hat{\unicode[STIX]{x1D6F9}}_{1}^{(u)}\hat{\unicode[STIX]{x1D6F9}}_{2}^{(u)}\hat{\unicode[STIX]{x1D701}}_{3}^{(l)}+h_{123}^{(5)}\hat{\unicode[STIX]{x1D6F9}}_{1}^{(u)}\hat{\unicode[STIX]{x1D6F9}}_{2}^{(l)}\hat{\unicode[STIX]{x1D701}}_{3}^{(l)}+h_{123}^{(6)}\hat{\unicode[STIX]{x1D6F9}}_{1}^{(l)}\hat{\unicode[STIX]{x1D6F9}}_{2}^{(l)}\hat{\unicode[STIX]{x1D701}}_{3}^{(l)}\! ]\unicode[STIX]{x1D6FF}(\boldsymbol{k}_{1}+\boldsymbol{k}_{2}+\boldsymbol{k}_{3})\,\text{d}\boldsymbol{k}_{1}\,\text{d}\boldsymbol{k}_{2}\,\text{d}\boldsymbol{k}_{3},\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where the coupling coefficients $h_{j}^{i}$ are given in appendix B. Here $k=|\boldsymbol{k}|$ denotes the wavenumber and we use the notation that subscripts represent vector arguments, i.e. $h_{ijl}\equiv h(\boldsymbol{k}_{i},\boldsymbol{k}_{j},\boldsymbol{k}_{l})$.

This Hamiltonian is expressed explicitly in terms of the variables at the surfaces of the fluids, and is a significant step forward over the Hamiltonian structure of the two-layer system derived in Ambrosi (Reference Ambrosi2000), where the implicit form of the Hamiltonian was obtained.

The Hamiltonian provides a firm theoretical foundation to develop the theory of weak nonlinear interactions of surface and interfacial waves. However, to describe the time evolution of the spectral energy density of the waves, the quadratic part of the Hamiltonian of the system (2.4) needs to be diagonalized, so that the linear part of the equations of motion corresponds to distinct non-interacting linear waves. In other words, we need to calculate the normal modes of the system. This task is done in the next section.

3.1 Transformation to complex field variable

We start from the surface variables for the Fourier image of displacement of the upper and lower layers $\hat{\unicode[STIX]{x1D701}}_{\boldsymbol{k}}^{(i)}$ and the Fourier image of the velocity potential on upper and lower surfaces $\hat{\unicode[STIX]{x1D6F9}}_{\boldsymbol{k}}^{(i)}$, where $(i)$ denotes the layer, with $(u)$ being the upper layer and $(l)$ being the lower layer. We then perform a canonical transformation to complex action variables describing the complex amplitude of wave with wavenumber $\boldsymbol{k}$:

$$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle \hat{\unicode[STIX]{x1D701}}_{\boldsymbol{k}}^{(u)}=\left(\frac{h_{\boldsymbol{k}}^{(2a)}}{4h_{\boldsymbol{k}}^{(1a)}}\right)^{1/4}(a_{\boldsymbol{k}}^{(u)}+a_{-\boldsymbol{k}}^{(u)\ast }),\quad \hat{\unicode[STIX]{x1D6F9}}_{\boldsymbol{k}}^{(u)}=\text{i}\left(\frac{h_{\boldsymbol{k}}^{(1a)}}{4h_{\boldsymbol{k}}^{(2a)}}\right)^{1/4}(a_{\boldsymbol{k}}^{(u)}-a_{-\boldsymbol{k}}^{(u)\ast }),\\ \displaystyle \hat{\unicode[STIX]{x1D701}}_{\boldsymbol{k}}^{(l)}=\left(\frac{h_{\boldsymbol{k}}^{(4a)}}{4h_{\boldsymbol{k}}^{(3a)}}\right)^{1/4}(a_{\boldsymbol{k}}^{(l)}+a_{-\boldsymbol{k}}^{(l)\ast }),\quad \hat{\unicode[STIX]{x1D6F9}}_{\boldsymbol{k}}^{(l)}=\text{i}\left(\frac{h_{\boldsymbol{k}}^{(3a)}}{4h_{\boldsymbol{k}}^{(4a)}}\right)^{1/4}(a_{\boldsymbol{k}}^{(l)}-a_{-\boldsymbol{k}}^{(l)\ast }).\end{array}\right\} & & \displaystyle \nonumber\end{eqnarray}$$

In these variables the Hamiltonian takes the form

(3.1)$$\begin{eqnarray}\displaystyle & \displaystyle H_{2}=\int [F_{\boldsymbol{k}}^{(1)}|a_{\boldsymbol{k}}^{(U)}|^{2}+F_{\boldsymbol{k}}^{(2)}|a_{\boldsymbol{k}}^{(L)}|^{2}+F_{\boldsymbol{k}}^{(3)}[(a_{\boldsymbol{k}}^{(U)}a_{-\boldsymbol{k}}^{(L)}-a_{\boldsymbol{k}}^{(U)}a_{\boldsymbol{k}}^{(L)\ast })+\text{c.c.}]]\,\text{d}\boldsymbol{k}, & \displaystyle\end{eqnarray}$$

(3.2)$$\begin{eqnarray}\displaystyle & \displaystyle H_{3}=\mathop{\sum }_{S_{1},S_{2},S_{3}\in \{U,L\}}\iiint \,\text{d}\boldsymbol{k}_{1}\,\text{d}\boldsymbol{k}_{2}\,\text{d}\boldsymbol{k}_{3}\hphantom{\unicode[STIX]{x1D60E}_{123}^{(S_{1}S_{2}S_{3})}a_{1}^{(S_{1})}a_{2}^{(S_{2})}a_{3}^{(S_{3})}(\unicode[STIX]{x1D6FF}_{1+2+3})+\text{c.c.}} & \displaystyle \nonumber\\ \displaystyle & \displaystyle \quad \times \,[(\unicode[STIX]{x1D61D}_{123}^{(S_{1}S_{2}S_{3})}a_{1}^{(S_{1})\ast }a_{2}^{(S_{2})\ast }a_{3}^{(S_{3})}\unicode[STIX]{x1D6FF}_{1+2-3}+\unicode[STIX]{x1D60E}_{123}^{(S_{1}S_{2}S_{3})}a_{1}^{(S_{1})}a_{2}^{(S_{2})}a_{3}^{(S_{3})}\unicode[STIX]{x1D6FF}_{1+2+3})+\text{c.c.}]. & \displaystyle\end{eqnarray}$$

This is the standard form of the wave turbulence Hamiltonian of the spatially homogeneous nonlinear system with two types of waves and with the quadratic nonlinearity. The corresponding canonical equations of motion assume standard canonical form (Zakharov et al. Reference Zakharov, L’vov and Falkovich1992):

(3.3)$$\begin{eqnarray}\displaystyle \text{i}\dot{a_{\boldsymbol{k}}}^{(U)}=\frac{\unicode[STIX]{x1D6FF}H}{\unicode[STIX]{x1D6FF}a_{\boldsymbol{k}}^{(U)\ast }},\quad \text{i}\dot{a_{\boldsymbol{k}}}^{(L)}=\frac{\unicode[STIX]{x1D6FF}H}{\unicode[STIX]{x1D6FF}a_{\boldsymbol{k}}^{(L)\ast }}. & & \displaystyle\end{eqnarray}$$

Here the coefficient functions are given by

$$\begin{eqnarray}\displaystyle F_{\boldsymbol{k}}^{(1)}=\sqrt{h_{\boldsymbol{k}}^{(1a)}h_{\boldsymbol{k}}^{(2a)}},\quad F_{\boldsymbol{k}}^{(2)}=\sqrt{h_{\boldsymbol{k}}^{(3a)}h_{\boldsymbol{k}}^{(4a)}},\quad F_{\boldsymbol{k}}^{(3)}=-\frac{h_{\boldsymbol{k},-\boldsymbol{k}}^{(5)}}{4}\left[\frac{h_{\boldsymbol{k}}^{(1a)}h_{\boldsymbol{ k}}^{(3a)}}{h_{\boldsymbol{k}}^{(2a)}h_{\boldsymbol{k}}^{(4a)}}\right]^{1/4},\quad & & \displaystyle \nonumber\end{eqnarray}$$

with the matrix elements $\unicode[STIX]{x1D61D}_{123}^{(S_{1}S_{2}S_{3})}$ and $\unicode[STIX]{x1D60E}_{123}^{(S_{1}S_{2}S_{3})}$ given in appendix C.

3.2 Hamiltonian in terms of normal mode amplitudes

We now need to diagonalize the quadratic part of the resulting Hamiltonian. We perform this task by finding a canonical transformation that would decouple linear waves of the Hamiltonian (3.1). In other words, we are seeking a canonical transformation to remove the term $F_{\boldsymbol{k}}^{(3)}[(a_{\boldsymbol{k}}^{(U)}a_{-\boldsymbol{k}}^{(L)}-a_{\boldsymbol{k}}^{(U)}a_{\boldsymbol{k}}^{(L)\ast })+\text{c.c.}]$ from the Hamiltonian (3.1). The transformation is given by equation (C 5) of appendix C. As a result we obtain the normal modes of the system while maintaining the canonical structure of the equations of motion. Finding such a transformation to determine the normal modes of the system appears to be a non-trivial task, since we needed to solve an overdetermined system of nonlinear algebraic equations. Details of this procedure are explained in appendix C. Applying the transformation (C 5) to (2.4), the quadratic part of the Hamiltonian assumes the desired form

$$\begin{eqnarray}\displaystyle H_{2}=\int [\tilde{\unicode[STIX]{x1D714}}_{\boldsymbol{k}}^{(S)}|c_{\boldsymbol{k}}^{(S)}|^{2}+\tilde{\unicode[STIX]{x1D714}}_{\boldsymbol{k}}^{(I)}|c_{\boldsymbol{k}}^{(I)}|^{2}]\,\text{d}\boldsymbol{k}, & & \displaystyle \nonumber\end{eqnarray}$$

where the superscripts $S$ and $I$ correspond to the respective surface or interfacial normal modes.

The linear dispersion relationships of the surface and interfacial normal modes are given by

(3.4a)$$\begin{eqnarray}\displaystyle & \displaystyle \tilde{\unicode[STIX]{x1D714}}_{\boldsymbol{k}}^{(I)}=\unicode[STIX]{x1D6FC}_{\boldsymbol{k}}\cosh 2\unicode[STIX]{x1D719}_{\boldsymbol{k}}+2\unicode[STIX]{x1D6FE}_{\boldsymbol{k}}\sinh 2\unicode[STIX]{x1D719}_{\boldsymbol{k}}, & \displaystyle\end{eqnarray}$$

(3.4b)$$\begin{eqnarray}\displaystyle & \displaystyle \tilde{\unicode[STIX]{x1D714}}_{\boldsymbol{k}}^{(S)}=\unicode[STIX]{x1D6FD}_{\boldsymbol{k}}\cosh 2\unicode[STIX]{x1D713}_{\boldsymbol{k}}+2\unicode[STIX]{x1D701}_{\boldsymbol{k}}\sinh 2\unicode[STIX]{x1D713}_{\boldsymbol{k}}, & \displaystyle\end{eqnarray}$$

$$\begin{eqnarray}\displaystyle H_{3} & = & \displaystyle \mathop{\sum }_{S_{1},S_{2},S_{3}\in \{S,I\}}\iiint \,\text{d}\boldsymbol{k}_{1}\,\text{d}\boldsymbol{k}_{2}\,\text{d}\boldsymbol{k}_{3}\nonumber\\ \displaystyle & & \displaystyle \times \,[(\unicode[STIX]{x1D611}_{123}^{(S_{1}S_{2}S_{3})}c_{1}^{(S_{1})\ast }c_{2}^{(S_{2})\ast }c_{3}^{(S_{3})}\unicode[STIX]{x1D6FF}_{1+2-3}+\unicode[STIX]{x1D613}_{123}^{(S_{1}S_{2}S_{3})}c_{1}^{(S_{1})}c_{2}^{(S_{2})}c_{3}^{(S_{3})}\unicode[STIX]{x1D6FF}_{1+2+3})+\text{c.c.}].\nonumber\end{eqnarray}$$

Here $\unicode[STIX]{x1D611}_{123}^{(S_{1}S_{2}S_{3})}$ and $\unicode[STIX]{x1D613}_{123}^{(ijk)}$ are the interaction matrix elements, also called scattering cross-sections. These matrix elements describe the strength of the nonlinear coupling between wavenumbers of $\boldsymbol{k}_{1}$, $\boldsymbol{k}_{2}$ and $\boldsymbol{k}_{3}$ of the normal modes of types $S_{1}$, $S_{2}$ and $S_{3}$. Calculation of these matrix elements is a tedious but straightforward task completed in appendix D. An alternative, but equivalent, formulation is described in detail for both the resonant and near-resonant cases in Choi (Reference Choi2019). Knowledge of these matrix elements and the linear dispersion relations allows us to use the wave turbulence formalism to derive the kinetic equations describing the time evolution of the spectral energy density of interacting waves. This is done in the next section.

4.1 Kinetic equations

The kinetic equation is the classical analogue of the Boltzmann collision integral. The basic ideas for writing down the kinetic equation to describe how weakly interacting waves share their energies go back to Peierls. The modern theory has its origin in the works of Hasselmann, Benney and Saffmann, Kadomtsev, Zakharov, and Benney and Newell.

There are many ways of deriving the kinetic equation which are well understood and well studied (Benney & Saffmann Reference Benney and Saffmann1966; Zakharov et al. Reference Zakharov, L’vov and Falkovich1992; Choi, Lvov & Nazarenko Reference Choi, Lvov and Nazarenko2004, Reference Choi, Lvov and Nazarenko2005; Lvov & Nazarenko Reference Lvov and Nazarenko2004; Nazarenko Reference Nazarenko2011; Choi et al. Reference Choi, Lvov, Nazarenko and Pokorni2005). Here we use the slightly more general approach for the derivation of the kinetic equations, which allows not only for resonant, but also for near-resonant interactions, as was done in Lvov et al. (Reference Lvov, Lvov, Newell and Zakharov1997) and Lvov, Polzin & Yokoyama (Reference Lvov, Polzin and Yokoyama2012). We generalize Lvov et al. (Reference Lvov, Lvov, Newell and Zakharov1997, Reference Lvov, Polzin and Yokoyama2012) for the case of a system of two types of interacting waves, namely surface and interfacial waves. The resulting system of kinetic equations is

(4.2)$$\begin{eqnarray}\displaystyle {\dot{n}}^{(S_{0})}(\boldsymbol{k},t) & = & \displaystyle \mathop{\sum }_{S_{1},S_{2}\in \{S,I\}}\iint \,\text{d}\boldsymbol{k}_{1}\,\text{d}\boldsymbol{k}_{2}\times (\!|\unicode[STIX]{x1D611}_{012}^{(S_{0}S_{1}S_{2})}|^{2}f_{012}^{(S_{0}S_{1}S_{2})}\unicode[STIX]{x1D6FF}(\boldsymbol{k}-\boldsymbol{k}_{1}-\boldsymbol{k}_{2}){\mathcal{L}}_{\boldsymbol{k},\boldsymbol{k}_{1},\boldsymbol{k}_{2}}^{(S_{0}S_{1}S_{2})}\nonumber\\ \displaystyle & & \displaystyle -\,2|\unicode[STIX]{x1D611}_{102}^{(S_{1}S_{0}S_{2})}|^{2}f_{102}^{(S_{1}S_{0}S_{2})}\unicode[STIX]{x1D6FF}(\boldsymbol{k}_{1}-\boldsymbol{k}-\boldsymbol{k}_{2}){\mathcal{L}}_{\boldsymbol{k}_{1},\boldsymbol{k},\boldsymbol{k}_{2}}^{(S_{1}S_{0}S_{2})}\! ),\quad S_{0}\in \{S,I\},\end{eqnarray}$$

where $f^{(S_{1}S_{2}S_{3})}$ is the three-wave kinetic equation kernel for two types of waves:

$$\begin{eqnarray}\displaystyle f_{123}^{(S_{1}S_{2}S_{3})}=n_{1}^{(S_{1})}n_{2}^{(S_{2})}n_{3}^{(S_{3})}\left(\frac{1}{n_{1}^{(S_{1})}}-\frac{1}{n_{2}^{(S_{2})}}-\frac{1}{n_{3}^{(S_{3})}}\right). & & \displaystyle \nonumber\end{eqnarray}$$

The frequency-conserving Dirac delta function is replaced by its broadened version, the Lorenzian ${\mathcal{L}}$:

(4.3)$$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle {\mathcal{L}}_{\boldsymbol{k},\boldsymbol{k}_{1},\boldsymbol{k}_{2}}^{(S_{1}S_{2}S_{3})}=\frac{\unicode[STIX]{x1D6E4}_{\boldsymbol{k}_{1},\boldsymbol{k}_{2},\boldsymbol{k}_{3}}^{(S_{1}S_{2}S_{3})}}{\left(\unicode[STIX]{x1D714}_{\boldsymbol{k}}^{(S_{1})}-\unicode[STIX]{x1D714}_{\boldsymbol{k}_{2}}^{(S_{2})}-\unicode[STIX]{x1D714}_{\boldsymbol{k}_{2}}^{(S_{2})}\right)^{2}+\left(\unicode[STIX]{x1D6E4}_{\boldsymbol{k},\boldsymbol{k}_{1},\boldsymbol{k}_{2}}^{(S_{1}S_{2}S_{3})}\right)^{2}},\\ \displaystyle \unicode[STIX]{x1D6E4}_{\boldsymbol{k}_{1},\boldsymbol{k}_{2},\boldsymbol{k}_{3}}^{(S_{1}S_{2}S_{3})}=\unicode[STIX]{x1D6FE}_{1}^{(S_{i})}+\unicode[STIX]{x1D6FE}_{2}^{(S_{i})}+\unicode[STIX]{x1D6FE}_{3}^{(S_{i})},\\ \displaystyle \unicode[STIX]{x1D6FE}_{\boldsymbol{k}}^{(S_{0})}=\mathop{\sum }_{S_{1},S_{2}\in \{S,I\}}\iint \,\text{d}\boldsymbol{k}_{1}\,\text{d}\boldsymbol{k}_{2} (\!|\unicode[STIX]{x1D611}_{012}^{(S_{0}S_{1}S_{2})}|^{2}(n_{1}^{(S_{1})}+n_{2}^{(S_{2})})\unicode[STIX]{x1D6FF}(\boldsymbol{k}-\boldsymbol{k}_{1}-\boldsymbol{k}_{2}){\mathcal{L}}_{\boldsymbol{k},\boldsymbol{k}_{1},\boldsymbol{k}_{2}}^{(S_{0}S_{1}S_{2})}\\ \displaystyle \quad -2|\unicode[STIX]{x1D611}_{102}^{(S_{1}S_{0}S_{2})}|^{2}(n_{1}^{(S_{1})}-n_{2}^{(S_{2})})\unicode[STIX]{x1D6FF}(\boldsymbol{k}_{1}-\boldsymbol{k}-\boldsymbol{k}_{2}){\mathcal{L}}_{\boldsymbol{k}_{1},\boldsymbol{k},\boldsymbol{k}_{2}}^{(S_{1}S_{0}S_{2})} ),\quad S_{0}\in \{S,I\}.\end{array}\right\} & & \displaystyle\end{eqnarray}$$

Here $\unicode[STIX]{x1D6E4}_{\boldsymbol{k}_{1},\boldsymbol{k}_{2},\boldsymbol{k}_{3}}^{(j)}$ is the total broadening of a given resonance between wavenumbers $\boldsymbol{k}_{1},\boldsymbol{k}_{2},\boldsymbol{k}_{3}$, and $\unicode[STIX]{x1D6FE}_{i}^{(S_{i})}$ is the Boltzmann rate for wavevector $(\boldsymbol{k}_{i},S_{i})$.

The principal new feature of this system of kinetic equations is that instead of the resonant interactions taking place along Dirac delta functions, the near-resonances appear acting along the broadened resonant manifold that includes not only resonant, but also near-resonant interactions, as was done in Lvov et al. (Reference Lvov, Lvov, Newell and Zakharov1997, Reference Lvov, Polzin and Yokoyama2012).

The interpretation of this formula is the following. Nonlinear wave–wave interactions lead to a change of wave amplitude, which in turn makes the lifetime of the waves to be finite. Consequently, interactions can be near-resonant. A self-consistent evaluation of $\unicode[STIX]{x1D6FE}_{\boldsymbol{k}}$ requires the iterative solution of (4.2) and (4.3) over the entire field. Indeed, one can see from (4.3) that the width of the resonance depends on the lifetime of an individual wave, which in turn depends on the resonance width over which wave interactions occur.

We define our characteristic time for interfacial wave growth to be $\unicode[STIX]{x1D70F}_{i}^{(S_{i})}=-1/\unicode[STIX]{x1D6FE}_{i}^{(S_{i})}$, i.e. the $\text{e}$-scaling rate of the action density variable $n_{i}^{(S_{i})}$. Together (4.2)–(4.3) form a closed set of equations which can be iteratively solved to obtain the time evolution of the energy spectrum of surface and interfacial waves.

4.2 Three-wave resonances

Figure 2.

The resonant set of interfacial wavenumbers for fixed surface wavenumber $\unicode[STIX]{x1D705}=(2\unicode[STIX]{x03C0}/10)~\text{m}^{-1}$. (a) Class III and surrounding resonances; (b) class I and surrounding resonances.

Figure 3.

Schematic construction of three-wave resonances as determined by a fixed surface wave $\unicode[STIX]{x1D73F}$ and the corresponding set of resonant interfacial waves, as done in Ball (Reference Ball1964). Two of such triads are depicted.

Wave turbulence theory considers resonant wave–wave interactions. The rationale for this is that out of all possible interactions of three wavenumbers it is only resonant and near-resonant interactions that lead to the effective irreversible energy exchange between wavenumbers. Consequently, we seek wavenumbers that satisfy resonances of the form

(4.4)$$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{k}_{1}\pm \boldsymbol{k}_{2}\pm \boldsymbol{k}_{3}=0, & \displaystyle\end{eqnarray}$$

(4.5)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D714}_{1}^{(S_{1})}\pm \unicode[STIX]{x1D714}_{2}^{(S_{2})}\pm \unicode[STIX]{x1D714}_{3}^{(S_{3})}=0,\quad S_{1},S_{2},S_{3}\in \{S,I\}. & \displaystyle\end{eqnarray}$$

In figure 2 we plot an example of the two-dimensional resonant set as described in Ball (Reference Ball1964) and Thorpe (Reference Thorpe1966). We first fix the surface wavenumber $\boldsymbol{k}_{1}^{S}$ to be a fixed, given number. We arbitrarily choose $\boldsymbol{k}_{1}^{S}=2\unicode[STIX]{x03C0}/10$, and calculate wavenumbers $\boldsymbol{k}_{2}^{I}$ and $\boldsymbol{k}_{3}^{S}$ so that the conditions $\boldsymbol{k}_{1}^{S}=\boldsymbol{k}_{2}^{I}+\boldsymbol{k}_{3}^{S}$ and $\unicode[STIX]{x1D714}_{1}^{(S)}=\unicode[STIX]{x1D714}_{2}^{(I)}+\unicode[STIX]{x1D714}_{3}^{(S)}$ are satisfied. We then plot the corresponding values of the interfacial wavenumber $\boldsymbol{k}_{2}^{I}$. Schematically, this construction process is depicted in figure 3.

The intercept in figure 2(a) corresponds to the case when the interfacial wave co-propagates in the direction of surface waves; this precisely corresponds to the class III resonances studied in Alam (Reference Alam2012). We note that in this case the interfacial waves excited are much longer and slower than the surface wave, because comparatively it takes much less energy to distort the interface between the layers than it does to lift and disturb the upper layer. In the interfacial wave case the heavier fluid is lifted in slightly less dense fluid, while for the surface waves the water is lifted into the air.

Similarly, the intercept in figure 2(b) corresponds to counter-propagating waves in class I resonance. Notably, the resonance curves are not scale-invariant; figure 4 depicts the resonance curves for the cases of fixed surface wavenumber $\unicode[STIX]{x1D705}=(2j\unicode[STIX]{x03C0}/25)~\text{m}^{-1}$, $j=1,3,5$, with the dashed lines depicting the first resonance curve scaled by the respective factors of three and five. While the resonant manifold is approximately scale-invariant for large wavenumbers, scale invariance is particularly violated near the class III collinear resonance. This change in structure results in a different regime of dynamics for longer surface wavelengths, as we will see in § 5.1.

Figure 4.

Resonances are not scale-invariant: the solid curve depicts interfacial wavenumbers in resonance when one surface wavenumber is fixed to be (a) $\unicode[STIX]{x1D705}=(2\unicode[STIX]{x03C0}/25)~\text{m}^{-1}$, (b) $\unicode[STIX]{x1D705}=(3\times 2\unicode[STIX]{x03C0}/25)~\text{m}^{-1}$ and (c) $\unicode[STIX]{x1D705}=(5\times 2\unicode[STIX]{x03C0}/25)~\text{m}^{-1}$. The dashed curves correspond to scaling curve (a) by factors of 3 and 5.

5.1 Surface plane wave

5.1.1 Formulation of the problem

Let us now consider a model problem of a single plane wave on the surface of the top layer and an interfacial layer that is at rest initially at $t=0$. While this problem may seem to be oversimplified, it is motivated by real oceanographic scenarios. Indeed, when wind blows on top of the ocean, the spectral energy density of the generated surface gravity waves has a prominent narrow peak for a specific wavenumber. We denote this wavevector of the initial surface wave distribution by $\unicode[STIX]{x1D73F}$, and assume that the lower interface is undisturbed except for small-amplitude noise $\unicode[STIX]{x1D716}_{n}$. Such a choice corresponds to the initial conditions

(5.1)$$\begin{eqnarray}\displaystyle & \displaystyle n^{(S)}(\boldsymbol{k},t=0)=\tilde{A}\unicode[STIX]{x1D6FF}(\boldsymbol{k}-\unicode[STIX]{x1D73F}), & \displaystyle\end{eqnarray}$$

(5.2)$$\begin{eqnarray}\displaystyle & \displaystyle n^{(I)}(\boldsymbol{k},t=0)=O(\unicode[STIX]{x1D716}_{n}). & \displaystyle\end{eqnarray}$$

In physical space such choice corresponds to

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D701}_{1}(\boldsymbol{x},t=0)=A\cos (\unicode[STIX]{x1D73F}\boldsymbol{\cdot }\boldsymbol{x}-\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D73F}}t),\quad \unicode[STIX]{x1D701}_{1}(\boldsymbol{x},t=0)=\unicode[STIX]{x1D716}_{n}. & & \displaystyle \nonumber\end{eqnarray}$$

In the calculations below, we take the surface wavenumber $\unicode[STIX]{x1D73F}$ to correspond to the peak wavenumber of the JONSWAP spectrum (Hassleman et al. Reference Hassleman, Barnett, Bouws and Carlson1973). Here, the one-dimensional JONSWAP spectrum can be expressed in terms of $U_{19.5}$, the wind speed 19.5 m above the ocean surface, and is given by

(5.3)$$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle S(\unicode[STIX]{x1D714})=\frac{\unicode[STIX]{x1D6FC}g^{2}}{\unicode[STIX]{x1D714}^{5}}\exp \left(-\frac{5}{4}\left(\frac{\unicode[STIX]{x1D714}_{0}}{\unicode[STIX]{x1D714}}\right)^{4}\right)\unicode[STIX]{x1D6FE}^{r},\\ \displaystyle r=\exp \left(-\frac{(\unicode[STIX]{x1D714}-\unicode[STIX]{x1D714}_{0})^{2}}{2\unicode[STIX]{x1D70E}^{2}\unicode[STIX]{x1D714}_{0}^{2}}\right),\quad \unicode[STIX]{x1D714}_{0}=g/U_{19.5},\quad \unicode[STIX]{x1D70E}=\left\{\begin{array}{@{}ll@{}}0.07,\quad & \text{if }\unicode[STIX]{x1D714}\leqslant \unicode[STIX]{x1D714}_{0},\\ 0.09,\quad & \text{if }\unicode[STIX]{x1D714}>\unicode[STIX]{x1D714}_{0}.\end{array}\right.\end{array}\right\} & & \displaystyle\end{eqnarray}$$

Consequently, the peak wavenumber $\unicode[STIX]{x1D73F}$ and wave amplitude $A$ that we use are determined solely by the speed of the wind. Due to the surface spectrum being unidirectional, the dominant energy exchange from the surface to interfacial spectrum occurs on and near the class III resonance (Alam Reference Alam2012).

Substituting (5.1) into (4.3), dropping terms of order $\unicode[STIX]{x1D716}_{n}$ and keeping only the resonant and near-resonant terms, we obtain the following algebraic equations for the Boltzmann rates of the interfacial and surface wave fields at $t=0$:

(5.4)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FE}_{\boldsymbol{k}}^{(I)}(t=0)=-2\unicode[STIX]{x03C0}\tilde{A}|\unicode[STIX]{x1D611}^{(SIS)}(\unicode[STIX]{x1D73F},\boldsymbol{k},\unicode[STIX]{x1D73F}-\boldsymbol{k})|^{2}{\mathcal{L}}(\tilde{\unicode[STIX]{x1D714}}^{(S)}(\unicode[STIX]{x1D73F})-\tilde{\unicode[STIX]{x1D714}}^{(I)}(\boldsymbol{k})-\tilde{\unicode[STIX]{x1D714}}^{(S)}(\unicode[STIX]{x1D73F}-\boldsymbol{k})), & \displaystyle \nonumber\\ \displaystyle & \displaystyle \unicode[STIX]{x1D6FE}_{\boldsymbol{k}}^{(S)}(t=0)=2\unicode[STIX]{x03C0}\tilde{A}|\unicode[STIX]{x1D611}^{(SIS)}(\unicode[STIX]{x1D73F},\unicode[STIX]{x1D73F}-\boldsymbol{k},\boldsymbol{k})|^{2}{\mathcal{L}}(\tilde{\unicode[STIX]{x1D714}}^{(S)}(\unicode[STIX]{x1D73F})-\tilde{\unicode[STIX]{x1D714}}^{(I)}(\unicode[STIX]{x1D73F}-\boldsymbol{k})-\tilde{\unicode[STIX]{x1D714}}^{(S)}(\boldsymbol{k})),\quad \boldsymbol{k}\in R_{III}. & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

5.1.2 Unidirectional wave propagation

We now consider unidirectional wave propagation. We find the self-consistent value for $\unicode[STIX]{x1D6FE}_{\boldsymbol{k}}^{(I)}$ by numerically solving the one-dimensional version of the algebraic equation (5.4). The numerical solution for the growth rate of interfacial waves collinear to the surface wave is shown in figure 5(a) for a surface wavenumber of $\unicode[STIX]{x1D73F}=(2\unicode[STIX]{x03C0}/19)~\text{m}^{-1}$. Here the results were obtained by iterating (5.4). The value of $\unicode[STIX]{x1D6FE}_{\boldsymbol{k}}^{(I)}$ appears to be narrowly peaked around the resonant frequency $\unicode[STIX]{x1D714}_{\boldsymbol{k}}^{(I)}=\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D73F}}^{(S)}-\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D73F}-\boldsymbol{k}}^{(S)}$. We now vary the value of the wind speed and plot the magnitude of the value of the peak of $\unicode[STIX]{x1D6FE}^{(I)}(\boldsymbol{k})$ as a function of the wind speed. Results are shown in figure 5(b). We see that for a wind speed of roughly $8~\text{m}~\text{s}^{-1}$ there is a much more effective transfer of energy to the interfacial waves than at lower or higher wind speeds. Interestingly, this wind speed is around that at which white capping starts to occur. Here the parameters used are $\unicode[STIX]{x1D70C}_{u}=1027~\text{kg}~\text{m}^{-3}$, $\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}=1~\text{kg}~\text{m}^{-3}$, $h_{u}=800~\text{m}$ and $h_{l}=4000~\text{m}$. We can actually analytically estimate the amplitude of the peak using the following arguments. If we choose the interfacial wavenumber $\boldsymbol{k}_{0}$ so that the resonance condition is satisfied, i.e. $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D73F}}^{(S)}-\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D73F}-\boldsymbol{k}_{0}}^{(S)}-\unicode[STIX]{x1D714}_{\boldsymbol{k}_{0}}^{(I)}=0$, and assume that $\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D73F}}^{(S)},\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D73F}-\boldsymbol{k}}^{(S)}\ll \unicode[STIX]{x1D6FE}_{\boldsymbol{k}}^{(I)}$, we obtain the following estimate for the growth rate of the resonant interfacial wavenumber $\boldsymbol{k}_{0}$:

(5.5)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FE}_{\boldsymbol{k}_{0}}^{(I)}(t=0)=\sqrt{2\unicode[STIX]{x03C0}\tilde{A}}|\unicode[STIX]{x1D611}^{(SIS)}(\unicode[STIX]{x1D73F},\boldsymbol{k}_{0},\unicode[STIX]{x1D73F}-\boldsymbol{k}_{0})|. & & \displaystyle\end{eqnarray}$$

The amplitude of $\unicode[STIX]{x1D6FE}$ found by this equation is shown as red dots in figure 5; the result agrees with the peak values given by iterating (5.4), plotted in blue.

Figure 5.

(a) Comparison of numerical solution of (5.4) and the analytical value of the peak growth rate using (5.5) for a surface spectrum with wind speed of $10~\text{m}~\text{s}^{-1}$. (b) Peak growth rate of interfacial waves versus the wind speed for various values of wind speed according to both formulas.

Now that we have numerically evaluated the resonance width function $\unicode[STIX]{x1D6E4}$, we can visualize the broadening of the resonant manifold. So we broaden the resonant manifold (4.5) by allowing not only resonant, but also near-resonant interactions. We therefore replace resonant condition (4.5) by a more general condition:

(5.6)$$\begin{eqnarray}\displaystyle R_{III}=\{\boldsymbol{k}_{I}:|\tilde{\unicode[STIX]{x1D714}}^{(S)}(\unicode[STIX]{x1D73F})-\tilde{\unicode[STIX]{x1D714}}^{(I)}(\boldsymbol{k}_{I})-\tilde{\unicode[STIX]{x1D714}}^{(S)}(\unicode[STIX]{x1D73F}-\boldsymbol{k}_{I}))|<\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D73F},\boldsymbol{k}_{I},\unicode[STIX]{x1D73F}-\boldsymbol{k}_{I}}^{(SIS)}\}. & & \displaystyle\end{eqnarray}$$

The results are depicted in in figure 6, for the cases of the surface being a plane wave of wavelength 20 and 80 m. This figure replaces figure 2(a) inset and figure 4. Here the amplitude of the plane wave is determined from the JONSWAP spectrum. We observe that the greatest broadening of the resonant curves occurs at and around the class III collinear resonance.

Figure 6.

Resonance curves as plotted in figure 4 including physical width for (a) $\unicode[STIX]{x1D706}=20~\text{m}$ surface spectrum and (b) $\unicode[STIX]{x1D706}=80~\text{m}$ surface spectrum.

5.1.3 Unidirectional surface wave can generate oblique interfacial waves

We now remove the constraint of surface and interfacial waves being collinear and consider the more general case of arbitrary angle between them. Indeed, we observe that (5.5) can be evaluated for resonant two-dimensional interfacial wavenumbers which are not necessarily collinear to $\unicode[STIX]{x1D73F}$. Consequently, we calculate the peak growth rate for interfacial waves all along the resonance curve, of which the general shape is depicted in figure 2.

Figure 7.

The ratio between the resonance width $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D73F},\unicode[STIX]{x1D73F}-\boldsymbol{k}_{I},\boldsymbol{k}_{I}}^{(2)}$ and the frequency of the excited interfacial wave $\unicode[STIX]{x1D714}_{\boldsymbol{k}^{I}}^{(I)}$ for two different wind speeds, as a function of wavenumber and angle along the resonance curve.

In figure 7 we plot the ratio of the resonance width and the frequency of the interfacial wave which is excited, $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D73F},\boldsymbol{k}^{I},\unicode[STIX]{x1D73F}-\boldsymbol{k}_{I}}^{(SIS)}/\unicode[STIX]{x1D714}_{\boldsymbol{k}^{I}}^{(I)}$, along the entire resonance curve for wind speeds of $7$ and $10.7~\text{m}~\text{s}^{-1}$. We observe that for both wind speeds the low collinear wavenumbers experience the most resonance broadening, and that the resonance width decreases roughly exponentially as a function of wavenumber. Similarly, in figure 7 we also plot the ratio of resonance width and frequency as a function of the angle between the excited interfacial wavenumber and the fixed surface wavenumber, verifying that it is largest when the angle is small, though non-zero for other angles.

In figure 8 we plot the Boltzmann growth rate of interfacial wavenumbers along the two-dimensional resonance curve as a function of the angle between the interfacial wave and the fixed surface wave. We see that a band of angles is excited on a time scale similar to that of the peak collinear wave. Furthermore, as the wind speed increases, the band of excited angles becomes broader, so in addition to the collinear waves in the direction of the wind speed, oblique waves are generated as well, making the mechanism of transfer of energy from the wind to the internal waves much more effective.

Figure 8.

The excitation rate, $\unicode[STIX]{x1D6FE}_{\boldsymbol{k}}^{(I)}$, plotted as a function of $\unicode[STIX]{x1D703}$, the angle between interfacial wave $\boldsymbol{k}$ and surface wave $\unicode[STIX]{x1D73F}$.

5.2 Analysis of matrix element

Figure 9.

The matrix element $\unicode[STIX]{x1D611}^{(SSI)}(\unicode[STIX]{x1D73F},\unicode[STIX]{x1D73F}+\boldsymbol{k}_{I},\boldsymbol{k}_{I})$ with the resonant values of $k_{I}$ overlaid on top. Surface wavelength is (a) $\unicode[STIX]{x1D706}=8~\text{m}$, (b) $\unicode[STIX]{x1D706}=14.5~\text{m}$ and (c) $\unicode[STIX]{x1D706}=150~\text{m}$. We see a transition into a regime in which collinear resonances are no longer dominant, i.e. as the surface waves grow in wavelength the resonant curve crosses a peak spatial frequency in which energy transfer is most efficient.

The principal new feature of this paper is that we provide a first-principles derivation of the magnitude of the strength of interactions between surface and interfacial waves. The value of such an interaction is called the matrix element, or interaction cross-section in wave turbulence theory. In figure 9 we plot the interaction coefficient $\unicode[STIX]{x1D611}^{(SIS)}$ which governs the interaction strength along resonances of the form depicted in figure 2 for the cases of surface wavelengths $\unicode[STIX]{x1D706}=8$, 14.5 and 150 m. Here we fix the second surface wavenumber so that the resonance condition on wavenumber (4.4) is satisfied, plotting $\unicode[STIX]{x1D611}^{(SIS)}(\unicode[STIX]{x1D73F}+\boldsymbol{k}_{I},\boldsymbol{k}_{I},\unicode[STIX]{x1D73F})$ as a function of the free wavenumber $\boldsymbol{k}_{I}$. The overlaid white curve shows the resonant values of $\boldsymbol{k}_{I}$ determined by the resonance condition on frequency. Combined, the contour plot shows the interaction strength, with the resonance curve showing where interactions are restricted to occur. Notably, the interaction coefficient is approximately scale-invariant, as the structure remains nearly the same for the case of surface waves with wavelength 9 m and surface swell waves with wavelength 150 m. In contrast, the shape of the resonance curve experiences changes dependent on the magnitude of the surface wavenumbers, meaning that varying regimes occur depending on where the resonance curve lies with respect to the interaction coefficient. For a surface wavelength of approximately $\unicode[STIX]{x1D706}=14.5$  m, the resonance curve aligns with the maximum value of the matrix element, as seen in figure 9(b). In this regime, interactions as a whole seem most efficient. Further beyond this range the collinear resonance becomes less dominant, and for long surface wavelengths, as in the case when $\unicode[STIX]{x1D706}=150$  m, the maximum growth rate is along non-collinear interfacial waves, as seen in figure 9(c).

5.3 Excitation of interfacial waves by the JONSWAP surface wave spectrum, kinetic approach

5.3.1 Collinear waves

The main motivation for this paper is to predict the transfer of energy from wind-generated surface waves to the depth of the ocean. Here we fix the spectrum of the surface waves to be the JONSWAP spectrum in the $x$ direction and homogeneous in the $y$ direction for a wind speed of $15~\text{m}~\text{s}^{-1}$. We assume that initially there are no interfacial waves, i.e. $n^{(I)}(\boldsymbol{k},t=0)=0$. To calculate the growth rates, we iterate formula (4.3) until a self-consistent value is obtained. We perform iterations until an iterate is within $10^{-8}$ of the previous iterate. After obtaining self-consistent values for the growth rates of each wavenumber, we then evolve in time equations (4.2) and plot the resulting spectrum of the interfacial waves as a function of time in figure 10. Here the interfacial waves with wavelengths of less than 10 m are damped, the typical 10 m cutoff for internal wave breaking. We see that the surface wave spectrum excites interfacial waves on a time scale of days. Furthermore, the relative growth of the interfacial wave spectrum slows down over a period of 20 weeks, with no visual difference between week 19 and week 20 other than near the peak frequency.

Figure 10.

(a) The fixed surface wave JONSWAP spectrum. (b) The interfacial wave spectrum over the course of 20 weeks, where $n_{\boldsymbol{k}}^{(I)}(t=0)=0$ initially.

It appears that the spectral energy density of interfacial waves is a ‘resonant reflection’ of the spectra of the surface waves. Indeed, the spectrum in figure 10 is defined solely by class III resonances with the surface waves, and contributions from the interfacial wave interactions are sub-dominant. It therefore appears that the interactions between the interfacial waves are not an effective mechanism for the redistribution of energy in the interfacial waves, at least for the parameters chosen here. Spectral energy transfers in the field of internal waves have been studied extensively (Muller, Henyey & Pomphrey Reference Muller, Henyey and Pomphrey1986; Lvov & Tabak Reference Lvov and Tabak2001, Reference Lvov and Tabak2004; Lvov, Polzin & Tabak Reference Lvov, Polzin and Tabak2004; Lvov & Yokoyama Reference Lvov and Yokoyama2009; Lvov et al. Reference Lvov, Tabak, Polzin and Yokoyama2010; Polzin & Lvov Reference Polzin and Lvov2011, Reference Polzin and Lvov2017). It is now understood that spectral energy transfer in internal waves is dominated by the special class of non-local wave–wave interactions, called induced diffusion. This mechanism is absent in our model, since it is a two-layer system.

5.4 Simulation of growth rates for continuous two-dimensional surface spectrum

For the case when the surface spectrum is a more general two-dimensional spectrum, we resort to numerically iterating (4.3). Here we consider the two-dimensional version of (5.3), the JONSWAP spectrum. Introducing simple angular dependence via a directional spreading function as described in Janssen (Reference Janssen2004), we consider

(5.7)$$\begin{eqnarray}n^{(S)}(k,\unicode[STIX]{x1D703})=A\cos ^{2}\unicode[STIX]{x1D703}\times S(\unicode[STIX]{x1D714}_{\boldsymbol{ k}}^{(S)})\frac{\text{d}\unicode[STIX]{x1D714}_{\boldsymbol{k}}}{\text{d}\boldsymbol{k}},\quad -\frac{\unicode[STIX]{x03C0}}{2}\leqslant \unicode[STIX]{x1D703}\leqslant \frac{\unicode[STIX]{x03C0}}{2},\end{eqnarray}$$

where the spectrum is renormalized so that the total energy is the same as in the one-dimensional case, i.e. $A=\iint \cos ^{2}\unicode[STIX]{x1D703}S(\unicode[STIX]{x1D714})\,\text{d}\unicode[STIX]{x1D714}\,\text{d}\unicode[STIX]{x1D703}/\!\int S(\unicode[STIX]{x1D714})\,\text{d}\unicode[STIX]{x1D714}$. We now substitute (5.8) into (4.3) and find the self-consistent solution for $\unicode[STIX]{x1D6FE}^{(I)}(k,\unicode[STIX]{x1D714})$; the results are shown in figure 11.

Figure 11.

(a) Fixed surface spectrum and (b) simulated growth rate of interfacial wave spectrum for a wind speed of $5~\text{m}~\text{s}^{-1}$.

We obtain self-consistent values for the growth rate of interfacial waves excited for the case of a two-dimensional Pierson–Moskowitz spectrum corresponding to a wind speed of $5~\text{m}~\text{s}^{-1}$ in figure 11 and $50~\text{m}~\text{s}^{-1}$ in figure 12. Note that while these two representations may look similar, the scales of the figures are different. Higher wind speed generates a much larger band width of excited wavenumbers, and the growth rates are higher. Notably, the structure of the growth rate of interfacial waves does not match the structure of the surface spectrum. There is a peak interfacial wave analogous to the peak of the surface spectrum, but there are three lobes corresponding to interactions where long surface waves are in resonance with oblique interfacial waves.

Figure 12.

(a) Fixed surface spectrum and (b) simulated growth rate of interfacial wave spectrum for a hurricane wind speed of $50~\text{m}~\text{s}^{-1}$.

Having calculated matrix element $\unicode[STIX]{x1D611}_{123}^{(SIS)}$, and the growth rates of internal waves as a function of their wavenumber $k$ and direction $\unicode[STIX]{x1D703}$, we are now in a position to simulate the full evolving two-dimensional interfacial wave spectrum governed by (4.2). This is the subject of future work.