Coupling between internal and surface waves

Abstract

In a fluid system in which two immiscible layers are separated by a sharp free interface, there can be strong coupling between large amplitude nonlinear waves on the interface and waves in the overlying free surface. We study the regime where long waves propagate in the interfacial mode, which are coupled to a modulational regime for the free-surface mode. This is a system of Boussinesq equations for the internal mode, coupled to the linear Schrödinger equations for wave propagation on the free surface, and respectively a version of the Korteweg-de Vries equation for the internal mode in case of unidirectional motions. The perturbation methods are based on the Hamiltonian formulation for the original system of irrotational Euler’s equations, as described in (Benjamin and Bridges, J Fluid Mech 333:301–325, 1997, Craig et al., Comm Pure Appl Math 58:1587–1641, 2005a, Zakharov, J Appl Mech Tech Phys 9:190–194, 1968), using the perturbation theory for the modulational regime that is given in (Craig et al. to appear). We focus in particular on the situation in which the internal wave gives rise to localized bound states for the Schrödinger equation, which are interpreted as surface wave patterns that give a characteristic signature of the presence of an internal wave soliton. We also comment on the discrepancies between the free interface-free surface cases and the approximation of the upper boundary condition by a rigid lid.

Appendix

We present here a recursion formula for the higher-order terms in the Taylor series expansion for $G_{j\ell}^{(m)}(\eta,\eta_1).$ We distinguish two cases. The first is the special case where m = (m ₀, 0) or (0, m ₁), and the second is the more general case, where m = (m ₀, m ₁) where neither m ₀, m ₁ = 0. In the first case, let m = (m ₀, 0). Then we can read from the matrix Eqs. 11, using 30, 31, 32 and 33, the following expressions for the matrix coefficients: the (11)-coefficient is

$$ \begin{aligned} G_{11}^{(m_0,0)}(\eta) & = \frac{1}{m_0!} D \eta^{m_0}(x) D^{m_0} \left( \frac{e^{-h_1D}}{e^{h_1D} - e^{-h_1D}}+\frac{(-1)^{m_0}e^{h_1D}} {e^{h_1D} - e^{-h_1D}}\right)\\ & + \sum\limits_{\begin{subarray}{c} p_0 \geq 1 \\ q_0+p_0=m_0\\ q_1 = 0 = p_1\end{subarray}} G_{11}^{(q_0,0)}(\eta) \frac{1}{p_0!} \eta^{p_0}(x) D^{p_0} \left( \frac{e^{-h_1D}}{e^{h_1D} - e^{-h_1D}}+ \frac{(-1)^{p_0+1}e^{h_1D}}{e^{h_1D} - e^{-h_1D}}\right), \end{aligned} $$

(64)

the (21)-coefficient is

$$ G_{21}^{(m_0,0)}(\eta) = \sum\limits_{\begin{subarray}{c} p_0 \geq 1 \\ q_0+p_0=m_0 \\ q_1 = 0 = p_1 \end{subarray}} G_{21}^{(q_0,0)}(\eta) \frac{1}{p_0!} \eta^{p_0}(x) D^{p_0} \left( \frac{e^{-h_1D}} {e^{h_1D} - e^{-h_1D}} + \frac{(-1)^{p_0+1}e^{h_1D}}{e^{h_1D} - e^{-h_1D}}\right), $$

(65)

the (12)-coefficient is

$$ \begin{aligned} G_{12}^{(m_0,0)}(\eta) & = -\frac{1}{m_0!} D \eta^{m_0}(x) D^{m_0} \left(\frac{1}{e^{h_1D} - e^{-h_1D}} + \frac{(-1)^{m_0}}{e^{h_1D} - e^{-h_1D}} \right) \\ & -\sum\limits_{\begin{subarray}{c} p_0 \geq 1 \\ q_0+p_0=m_0 \\ q_1 = 0 = p_1 \end{subarray}} G_{11}^{(q_0,0)}(\eta) \frac{1}{p_0!} \eta^{p_0}(x) D^{p_0} \left( \frac{1}{e^{h_1D} - e^{-h_1D}} + \frac{(-1)^{p_0+1}} {e^{h_1D} - e^{-h_1D}}\right), \end{aligned} $$

(66)

and the (22)-coefficient is

$$ G_{22}^{(m_0,0)}(\eta) = - \sum\limits_{\begin{subarray}{c} p_0 \geq 1 \\ q_0+p_0=m_0 \\ q_1 = 0 = p_1 \end{subarray}} G_{21}^{(q_0,0)}(\eta) \frac{1}{p_0!} \eta^{p_0}(x) D^{p_0} \left( \frac{1}{e^{h_1D} - e^{-h_1D}} + \frac{(-1)^{p_0+1}}{e^{h_1D} - e^{-h_1D}} \right). $$

(67)

A recursive computation of $G_{j\ell}^{(m_0, 0)}(\eta)$ can be based upon formula (64) for $G_{11}^{(m_0, 0)}(\eta), m_0 > 0$ and formula (65) for $G_{21}^{(m_0, 0)}(\eta), m_0 > 0.$ This is sufficient information in order to calculate $G_{12}^{(m_0, 0)}(\eta)$ and $G_{22}^{(m_0, 0)}(\eta)$ from respectively (66) and (67).

It is a general fact that

$$ G_{j\ell}^{(m_0,m_1)}(\eta,\eta_1) = G_{\ell j}^{(m_1,m_0)}(-\eta_1,-\eta) $$

(68)

which allows us to deduce the form of $G_{j\ell}^{(0,m_1)}(\eta_1),$ with j, ℓ = 1, 2 from the above expressions. As well, each matrix operator $G_{j\ell}^{(m)}$ is self adjoint, which is not necessarily self-evident from the above formulae. Thus in particular $\left(G_{12}^{(m)}\right)^{\ast} = G_{21}^{(m)}.$ Therefore, the latter can be obtained from (66), which itself depends upon the recursion (64) alone.

The second case consists of those multi indices m = (m ₀, m ₁) where neither m ₀ nor m ₁ vanish. The m-order terms on the right-hand side of the relation (11) are zero, as is seen in (32, 33). Working as in the first case, we find an expression for the (11)-coefficient to be

$$ \begin{aligned} G_{11}^{(m_0,m_1)}(\eta,\eta_1) & = \sum\limits_{\begin{subarray}{c}1\leq p_0\leq m_0 \\q_0+p_0=m_0\\ p_1 = 0\end{subarray}} G_{11}^{(q_0,m_1)}(\eta,\eta_1) \frac{1}{p_0!} \eta^{p_0}(x) D^{p_0} \left( \frac{e^{-h_1D}}{e^{h_1D} - e^{-h_1D}} + \frac{(-1)^{p_0+1}e^{h_1D}} {e^{h_1D} - e^{-h_1D}}\right) \\ & + \sum\limits_{\begin{subarray}{c} p_0=0 \\ 1\leq p_1\leq m_1 \\ q_1+p_1=m_1\end{subarray}} G_{12}^{(m_0,q_1)}(\eta,\eta_1) \frac{1}{p_1!} \eta_1^{p_1}(x) D^{p_1} \left( \frac{1}{e^{h_1D} - e^{-h_1D}} + \frac{(-1)^{p_1+1}}{e^{h_1D} - e^{-h_1D}}\right). \end{aligned} $$

The (21)-coefficient is

$$ \begin{aligned} G_{21}^{(m_0,m_1)}(\eta,\eta_1) & = \sum\limits_{\begin{subarray}{c} 1\leq p_0\leq m_0 \\q_0+p_0=m_0\\ p_1 = 0\end{subarray}} G_{21}^{(q_0,m_1)}(\eta,\eta_1) \frac{1}{p_0!} \eta^{p_0}(x) D^{p_0} \left( \frac{e^{-h_1D}}{e^{h_1D} - e^{-h_1D}} + \frac{(-1)^{p_0+1}e^{h_1D}} {e^{h_1D} - e^{-h_1D}}\right) \\ & + \sum\limits_{\begin{subarray}{c} p_0=0 \\ 1\leq p_1\leq m_1 \\ q_1+p_1=m_1\end{subarray}} G_{22}^{(m_0,q_1)}(\eta,\eta_1) \frac{1}{p_1!} \eta_1^{p_1}(x) D^{p_1} \left( \frac{1}{e^{h_1D} - e^{-h_1D}} + \frac{(-1)^{p_1+1}}{e^{h_1D} - e^{-h_1D}}\right). \end{aligned} $$

The (12)-coefficient is the operator

$$ \begin{aligned} G_{12}^{(m_0,m_1)}(\eta,\eta_1) & = -\sum\limits_{\begin{subarray}{c}1\leq p_0\leq m_0 \\ q_0+p_0=m_0\\ p_1 = 0\end{subarray}} G_{11}^{(q_0,m_1)}(\eta,\eta_1) \frac{1}{p_0!} \eta^{p_0}(x) D^{p_0} \left( \frac{1}{e^{h_1D} - e^{-h_1D}} + \frac{(-1)^{p_0+1}}{e^{h_1D} - e^{-h_1D}}\right)\\ & -\sum\limits_{\begin{subarray}{c}p_0=0 \\ 1\leq p_1\leq m_1 \\ q_1+p_1=m_1\end{subarray}} G_{12}^{(m_0,q_1)}(\eta,\eta_1) \frac{1}{p_1!} \eta_1^{p_1}(x) D^{p_1} \left( \frac{e^{h_1D}}{e^{h_1D} - e^{-h_1D}} + \frac{(-1)^{p_1+1}e^{-h_1D}}{e^{h_1D} - e^{-h_1D}}\right). \end{aligned} $$

Finally, the (22)-coefficient is

$$ \begin{aligned} G_{22}^{(m_0,m_1)}(\eta,\eta_1) & = -\sum\limits_{\begin{subarray}{c}1\leq p_0\leq m_0 \\ q_0+p_0=m_0\\ p_1 = 0\end{subarray}} G_{21}^{(q_0,m_1)}(\eta,\eta_1) \frac{1}{p_0!} \eta^{p_0}(x) D^{p_0} \left( \frac{1}{e^{h_1D} - e^{-h_1D}} + \frac{(-1)^{p_0+1}}{e^{h_1D} - e^{-h_1D}}\right) \\ & -\sum\limits_{\begin{subarray}{c} p_0=0 \\ 1\leq p_1\leq m_1 \\ q_1+p_1=m_1\end{subarray}} G_{22}^{(m_0,q_1)}(\eta,\eta_1) \frac{1}{p_1!} \eta_1^{p_1}(x) D^{p_1} \left( \frac{e^{h_1D}} {e^{h_1D} - e^{-h_1D}} + \frac{(-1)^{p_1+1}e^{-h_1D}}{e^{h_1D} - e^{-h_1D}}\right). \end{aligned} $$