2.1 Governing equations

Consider the propagation of small-amplitude waves in a stably stratified rotating fluid with constant buoyancy frequency $N$ and under $f$ -plane approximation with $f$ the Coriolis frequency. In the Boussinesq approximation and neglecting viscosity, the linearized wave equation and boundary conditions for the vertical velocity $w$ in a Cartesian domain $(x,y,z)$ can be written as (e.g. Sutherland Reference Sutherland2010)

(2.1a ) $$\begin{eqnarray}\displaystyle & \displaystyle [(\unicode[STIX]{x2202}_{t}^{2}+f^{2})\unicode[STIX]{x2202}_{z}^{2}+(\unicode[STIX]{x2202}_{t}^{2}+N^{2})\unicode[STIX]{x1D6FB}_{h}^{2}]w=0,\quad -H+b\leqslant z\leqslant 0, & \displaystyle\end{eqnarray}$$

(2.1b ) $$\begin{eqnarray}\displaystyle & \displaystyle w=0,\quad z=0, & \displaystyle\end{eqnarray}$$

(2.1c ) $$\begin{eqnarray}\displaystyle & \displaystyle w=\boldsymbol{u}_{h}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{h}b,\quad z=-H+b, & \displaystyle\end{eqnarray}$$

$\boldsymbol{u}_{h}=(u,v)$

$z$

$z=0$

$b(x,y)$

$H$

$\unicode[STIX]{x1D735}_{h}=(\unicode[STIX]{x2202}_{x},\unicode[STIX]{x2202}_{y})$

$(u,v)$

$w$

(2.2a,b ) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D6FB}_{h}^{2}u=-(\unicode[STIX]{x2202}_{t}\unicode[STIX]{x2202}_{x}+f\unicode[STIX]{x2202}_{y})\unicode[STIX]{x2202}_{z}w,\quad \unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D6FB}_{h}^{2}v=-(\unicode[STIX]{x2202}_{t}\unicode[STIX]{x2202}_{y}-f\unicode[STIX]{x2202}_{x})\unicode[STIX]{x2202}_{z}w,\end{eqnarray}$$

following a straightforward manipulation of the continuity and linearized momentum equations, with (2.2) known as the polarization relations for linear internal waves (cf. e.g. Sutherland Reference Sutherland2010).

Our goal is to investigate the effect of a finite patch of monochromatic seabed corrugations on incident internal waves coming from $x=-\infty$ at the steady state. Thus we consider a seabed topography superimposed on the mean water depth of the form

(2.3) $$\begin{eqnarray}b(x,y)=\left\{\begin{array}{@{}cc@{}}\displaystyle \frac{d}{2}(\text{e}^{\text{i}\unicode[STIX]{x1D711}}\text{e}^{\text{i}\boldsymbol{k}_{b}\boldsymbol{\cdot }\boldsymbol{x}_{h}}+\text{c.c.}), & 0\leqslant x\leqslant \ell ,-\infty \leqslant y\leqslant +\infty ,\\ 0, & \text{otherwise},\end{array}\right.\end{eqnarray}$$

with $\boldsymbol{k}_{b}$ the corrugation wavenumber vector, $\unicode[STIX]{x1D711}$ an arbitrary phase, $d$ the corrugation amplitude, $\ell$ the length of the patch, $\text{c.c.}$ denotes the complex conjugate and $\boldsymbol{x}_{h}=(x,y)$ . We look for a steady-state time-periodic solution of the form $(u,v,w)=Re[(\hat{u} ,\hat{v},{\hat{w}})\text{e}^{-\text{i}\unicode[STIX]{x1D714}t}]$ ( $Re$ denoting the real part) with $\unicode[STIX]{x1D714}$ the angular frequency of the incident waves. Therefore, $\unicode[STIX]{x2202}_{t}$ can be substituted with $-\text{i}\unicode[STIX]{x1D714}$ and $(u,v,w)$ with $Re[(\hat{u} ,\hat{v},{\hat{w}})\text{e}^{-\text{i}\unicode[STIX]{x1D714}t}]$ in (2.1)–(2.2), leading to (dropping the hat symbol)

(2.4a ) $$\begin{eqnarray}\displaystyle & \displaystyle (\unicode[STIX]{x1D707}^{2}\unicode[STIX]{x2202}_{z}^{2}-\unicode[STIX]{x1D6FB}_{h}^{2})w=0,\quad -H+b\leqslant z\leqslant 0, & \displaystyle\end{eqnarray}$$

(2.4b ) $$\begin{eqnarray}\displaystyle & \displaystyle w=0,\quad z=0, & \displaystyle\end{eqnarray}$$

(2.4c ) $$\begin{eqnarray}\displaystyle & \displaystyle w=\boldsymbol{u}_{h}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{h}b,\quad z=-H+b, & \displaystyle\end{eqnarray}$$

(2.5a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D6FB}_{h}^{2}u=-\left(\unicode[STIX]{x2202}_{x}+\frac{\text{i}f}{\unicode[STIX]{x1D714}}\unicode[STIX]{x2202}_{y}\right)\unicode[STIX]{x2202}_{z}w,\quad \unicode[STIX]{x1D6FB}_{h}^{2}v=-\left(\unicode[STIX]{x2202}_{y}-\frac{\text{i}f}{\unicode[STIX]{x1D714}}\unicode[STIX]{x2202}_{x}\right)\unicode[STIX]{x2202}_{z}w,\end{eqnarray}$$

with $\boldsymbol{u}(x,y,z)=(u,v,w)$ the time-independent complex-valued velocity field to be found and $\unicode[STIX]{x1D707}(\unicode[STIX]{x1D714})=\sqrt{(\unicode[STIX]{x1D714}^{2}-f^{2})/(N^{2}-\unicode[STIX]{x1D714}^{2})}$ the so-called internal-waves slope (cf. e.g. Bühler & Holmes-Cerfon Reference Bühler and Holmes-Cerfon2011).

Upstream ( $x\leqslant 0$ ) and downstream ( $x\geqslant \ell$ ) of the corrugated patch, the bottom is flat ( $b=0$ ) and the solution of (2.1) can be written as a sum $w=\sum _{j}w_{j}$ of internal-wave modes of the form

(2.6) $$\begin{eqnarray}w_{j}=A_{j}\sin [\unicode[STIX]{x1D6FE}_{j}(z+H)]\text{e}^{\text{i}\boldsymbol{k}_{j}\boldsymbol{\cdot }\boldsymbol{ x}_{h}},\end{eqnarray}$$

provided that $f<\unicode[STIX]{x1D714}<N$ , with $\unicode[STIX]{x1D6FE}_{j}=j\unicode[STIX]{x03C0}/H$ the vertical wavenumber and $A_{j}$ the complex wave-mode amplitude. For each wave mode (2.6), the magnitude of the horizontal wavenumber $k_{j}=|\boldsymbol{k}_{j}|$ can be related to the vertical wavenumber through the dispersion relation

(2.7) $$\begin{eqnarray}k_{j}=\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FE}_{j}=\unicode[STIX]{x1D707}\frac{j\unicode[STIX]{x03C0}}{H}.\end{eqnarray}$$

While the wave modes (2.6) propagate independently from each other over the flat bottom, they become coupled over the corrugations (i.e. when $b\neq 0$ ) through the bottom boundary condition (2.4c ). We will demonstrate that energy transfers at the steady state between the wave modes can become significant under resonance conditions.

2.2 Regular perturbation

We consider the case of bottom corrugations that are small compared to the mean water depth, i.e. we let $\unicode[STIX]{x1D716}=d/H\ll 1$ , such that the problem can be solved via asymptotic analysis. To ease the derivation, we use primed dimensionless variables given by $(u,v)=(Uu^{\prime },Uv^{\prime })$ , $w=\unicode[STIX]{x1D707}Uw^{\prime }$ , $b=db^{\prime }$ , $\boldsymbol{x}_{h}=(H/\unicode[STIX]{x1D707})\boldsymbol{x}_{h}^{\prime }$ , $z=Hz^{\prime }$ , with $U$ the magnitude of the characteristic horizontal velocity of the incident waves. The proposed scaling is suggested by the dimensional form of the wave modes over the flat bottom, i.e. (2.6). With the proposed scaling, the dimensionless form of the flat-bottom modes reads

(2.8) $$\begin{eqnarray}w_{j}^{\prime }=A_{j}^{\prime }\sin [\unicode[STIX]{x1D705}_{j}(z^{\prime }+1)]\text{e}^{\text{i}\unicode[STIX]{x1D73F}_{j}\boldsymbol{\cdot }\boldsymbol{ x}_{h}^{\prime }},\end{eqnarray}$$

where $\unicode[STIX]{x1D73F}_{j}=\boldsymbol{k}_{j}H/\unicode[STIX]{x1D707}$ is the dimensionless wavenumber ( $\unicode[STIX]{x1D705}_{j}=|\unicode[STIX]{x1D73F}_{j}|$ ), $w_{j}^{\prime }\sim \sqrt{(u_{j}^{\prime })^{2}+(v_{j}^{\prime })^{2}}\sim O(1)$ , and the bottom topography becomes

(2.9) $$\begin{eqnarray}b^{\prime }={\textstyle \frac{1}{2}}(\text{e}^{\text{i}\unicode[STIX]{x1D711}}\text{e}^{\text{i}\unicode[STIX]{x1D73F}_{b}\boldsymbol{\cdot }\boldsymbol{x}_{h}^{\prime }}+\text{c.c.}),\end{eqnarray}$$

with $\unicode[STIX]{x1D73F}_{b}=\boldsymbol{k}_{b}H/\unicode[STIX]{x1D707}$ .

Using dimensionless variables the steady-state equations are now written as (dropping the primes from here onward)

(2.10a ) $$\begin{eqnarray}\displaystyle & \displaystyle (\unicode[STIX]{x2202}_{z}^{2}-\unicode[STIX]{x1D6FB}_{h}^{2})w=0,\quad -1+\unicode[STIX]{x1D716}b\leqslant z\leqslant 0, & \displaystyle\end{eqnarray}$$

(2.10b ) $$\begin{eqnarray}\displaystyle & \displaystyle w=0,\quad z=0, & \displaystyle\end{eqnarray}$$

(2.10c ) $$\begin{eqnarray}\displaystyle & \displaystyle w=\unicode[STIX]{x1D716}\boldsymbol{u}_{h}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{h}b,\quad z=-1+\unicode[STIX]{x1D716}b, & \displaystyle\end{eqnarray}$$

(2.11a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D6FB}_{h}^{2}u=-\left(\unicode[STIX]{x2202}_{x}+\frac{\text{i}f}{\unicode[STIX]{x1D714}}\unicode[STIX]{x2202}_{y}\right)\unicode[STIX]{x2202}_{z}w,\quad \unicode[STIX]{x1D6FB}_{h}^{2}v=-\left(\unicode[STIX]{x2202}_{y}-\frac{\text{i}f}{\unicode[STIX]{x1D714}}\unicode[STIX]{x2202}_{x}\right)\unicode[STIX]{x2202}_{z}w.\end{eqnarray}$$

Let us approximate the solution by a perturbation series of the form

(2.12) $$\begin{eqnarray}\boldsymbol{u}=\boldsymbol{u}^{(0)}+\unicode[STIX]{x1D716}^{1}\boldsymbol{u}^{(1)}+\cdots\end{eqnarray}$$

with $\boldsymbol{u}^{(j)}\sim O(1)$ for $j=1,2$ . Substituting (2.12) into (2.10c ) yields

(2.13) $$\begin{eqnarray}w^{(0)}+\unicode[STIX]{x1D716}w^{(1)}+\cdots =\unicode[STIX]{x1D716}[u^{(0)}+\unicode[STIX]{x1D716}u^{(1)}+\cdots \,]\unicode[STIX]{x2202}_{x}b+\unicode[STIX]{x1D716}[v^{(0)}+\unicode[STIX]{x1D716}v^{(1)}+\cdots \,]\unicode[STIX]{x2202}_{y}b,\quad z=-1+\unicode[STIX]{x1D716}b,\end{eqnarray}$$

which, using a first-order Taylor series approximation about $z=-1$ , can be further expanded as

(2.14) $$\begin{eqnarray}\displaystyle & & \displaystyle w^{(0)}+\unicode[STIX]{x1D716}b\unicode[STIX]{x2202}_{z}w^{(0)}+\unicode[STIX]{x1D716}w^{(1)}+\unicode[STIX]{x1D716}^{2}b\unicode[STIX]{x2202}_{z}w^{(1)}+\cdots \nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x1D716}[u^{(0)}+\unicode[STIX]{x1D716}b\unicode[STIX]{x2202}_{z}u^{(0)}+\unicode[STIX]{x1D716}u^{(1)}+\unicode[STIX]{x1D716}^{2}b\unicode[STIX]{x2202}_{z}u^{(1)}+\cdots \,]\unicode[STIX]{x2202}_{x}b\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\unicode[STIX]{x1D716}[v^{(0)}+\unicode[STIX]{x1D716}b\unicode[STIX]{x2202}_{z}v^{(0)}+\unicode[STIX]{x1D716}v^{(1)}+\unicode[STIX]{x1D716}^{2}b\unicode[STIX]{x2202}_{z}v^{(1)}+\cdots \,]\unicode[STIX]{x2202}_{y}b,\quad z=-1.\end{eqnarray}$$

The zero- and first-order equations obtained from (2.14) become, respectively,

(2.15a ) $$\begin{eqnarray}\displaystyle & \displaystyle w^{(0)}=0,\quad z=-1, & \displaystyle\end{eqnarray}$$

(2.15b ) $$\begin{eqnarray}\displaystyle & \displaystyle w^{(1)}=u^{(0)}\unicode[STIX]{x2202}_{x}b+v^{(0)}\unicode[STIX]{x2202}_{y}b-b\unicode[STIX]{x2202}_{z}w^{(0)}=\unicode[STIX]{x1D735}_{h}\boldsymbol{\cdot }(\boldsymbol{u}_{h}^{(0)}b),\quad z=-1, & \displaystyle\end{eqnarray}$$

$\boldsymbol{u}^{(0)}$

$\boldsymbol{u}_{h}^{(0)}=(u^{(0)},v^{(0)})$

$\boldsymbol{u}^{(0)}$

$\boldsymbol{u}^{(1)}$

The asymptotic expansion resulting in (2.15) for the bottom boundary condition is valid provided that $\unicode[STIX]{x1D716}=d/H\ll 1$ and the right-hand side of (2.15b ) remains of order one. For monochromatic bottom corrugations given by (2.9), this means that we require (i) $\unicode[STIX]{x1D73F}_{b}\sim O(1)$ (from the terms $u^{(0)}\unicode[STIX]{x2202}_{x}b$ and $v^{(0)}\unicode[STIX]{x2202}_{y}b$ on the right-hand side of (2.15b )) and (ii) $\unicode[STIX]{x1D705}\sim O(1)$ where $\unicode[STIX]{x1D705}$ is the dimensionless vertical wavenumber of internal waves (from the term $b\unicode[STIX]{x2202}_{z}w^{(0)}$ on the right-hand side of (2.15b )). The first condition (i) implies in terms of dimensional variables $k_{b}d/\unicode[STIX]{x1D707}\sim O(\unicode[STIX]{x1D716})\ll 1$ , i.e. the bottom slope $k_{b}d$ must be much smaller than the internal-wave slope $\unicode[STIX]{x1D707}$ , such that (i) is a stronger requirement than the typical subcritical-slope assumption $k_{b}d\leqslant \unicode[STIX]{x1D707}$ (cf. Bühler & Holmes-Cerfon Reference Bühler and Holmes-Cerfon2011). The second condition (ii) implies in terms of dimensional variables $\unicode[STIX]{x1D6FE}d\sim O(\unicode[STIX]{x1D716})\ll 1$ , i.e. the corrugation height much be much smaller than the vertical wavelength ( $2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FE}$ ) of internal waves. Note that (ii) is always verified for the incident mode-one wave since the vertical wavelength is half the water depth.

It can be noted that in considering the linear form (2.1) of the wave equations, we have neglected the advection of bottom-generated waves by the incident flow. When the incident flow corresponds to semi-diurnal internal tides, which is the case of our interest, this simplification is acceptable because velocities of the internal tides are in general of order $U\sim O(1~\text{cm}~\text{s}^{-1})$ or smaller (e.g. Zhao et al. Reference Zhao, Alford, Girton, Rainville and Simmons2016) such that the advective term is much smaller than the acceleration term, i.e. in dimensional form $(U\unicode[STIX]{x2202}_{x}/\unicode[STIX]{x2202}_{t})\sim (Uk/\unicode[STIX]{x1D714})\ll 1$ , for wavelengths down to a few kilometres or smaller.

2.3 Chain resonance conditions

As expected, the zero-order solution is not affected by bottom corrugations, such that it can be written as a superposition of wave modes given by (2.8). Using the dimensionless polarization relations (2.11), the forcing term on the right-hand side of (2.15b ) due to the interaction of a leading-order wave mode (2.8) with bottom corrugations can thus be written as

(2.16) $$\begin{eqnarray}\text{i}\left[\left(\cos \unicode[STIX]{x1D703}+\frac{\text{i}f}{\unicode[STIX]{x1D714}}\sin \unicode[STIX]{x1D703}\right)\unicode[STIX]{x2202}_{x}(b\text{e}^{\text{i}\unicode[STIX]{x1D73F}_{j}\boldsymbol{\cdot }\boldsymbol{x}_{h}})+\left(\sin \unicode[STIX]{x1D703}-\frac{\text{i}f}{\unicode[STIX]{x1D714}}\cos \unicode[STIX]{x1D703}\right)\unicode[STIX]{x2202}_{y}(b\text{e}^{\text{i}\unicode[STIX]{x1D73F}_{j}\boldsymbol{\cdot }\boldsymbol{x}_{h}})\right],\end{eqnarray}$$

where $\unicode[STIX]{x1D703}$ denotes the direction of propagation of the internal waves, i.e. we note $\unicode[STIX]{x1D73F}_{j}=(\unicode[STIX]{x1D705}_{j}\cos \unicode[STIX]{x1D703},\unicode[STIX]{x1D705}_{j}\sin \unicode[STIX]{x1D703})$ .

For incident waves with wavenumber $\unicode[STIX]{x1D73F}_{j}=\unicode[STIX]{x1D73F}_{in}$ and $b$ given by (2.9), it can be seen from (2.16) that the forcing of $w^{(1)}$ is by wave-like terms with wavenumber $\unicode[STIX]{x1D73F}_{in}\pm \unicode[STIX]{x1D73F}_{b}$ . Because $w^{(0)}$ and $w^{(1)}$ satisfy the same governing equations (except for the bottom boundary condition), we thus know from the theory of wave–bottom resonance that $w^{(1)}$ can grow unbounded over the corrugated patch and potentially become much larger than $O(1)$ when $\unicode[STIX]{x1D73F}_{in}\pm \unicode[STIX]{x1D73F}_{b}$ is a natural wavenumber of the zero-order problem, i.e. when there exists an integer $m>0$ such that $|\unicode[STIX]{x1D73F}_{in}\pm \unicode[STIX]{x1D73F}_{b}|=m\unicode[STIX]{x03C0}$ (see e.g. Liu & Yue Reference Liu and Yue1998, for the case of surface waves). Regular perturbation fails to capture an accurate solution at resonance (the unbounded growth of $w^{(1)}$ is unphysical) because the wave–bottom interaction are so strong that the incident wave loses significant energy and new waves become strongly and coherently generated by the many small seabed undulations. This resonant scattering can be dealt with by the method of multiple scales, i.e. including all possibly blowing-up modes at leading order and allowing them to slowly evolve over the corrugations, as done in § 3. It is important to note here that the wave–bottom resonance is a resonance condition on the horizontal wavenumbers only because the interaction happens at a boundary between a freely propagating wave and a frozen bottom wave (i.e. of zero frequency). This is a situation different from nonlinear wave–wave interactions for which the resonance condition requires a frequency match and a wavenumber match in all spatial dimensions.

When a resonance is obtained between the incident wave and the bottom corrugations, the newly (resonantly) generated wave is added to the set of zero-order wave modes and we note the corresponding wavenumber $\unicode[STIX]{x1D73F}_{r_{\pm 1}}=\unicode[STIX]{x1D73F}_{in}\pm \unicode[STIX]{x1D73F}_{b}$ (cf. figure 1 a). In a homogeneous fluid, the interaction of an incident wave with monochromatic corrugations can resonate at most one wave (i.e. the reflected wave $\unicode[STIX]{x1D73F}_{r_{-1}}$ ). In a continuously stratified fluid, however, the initial interaction can resonate both $\unicode[STIX]{x1D73F}_{r_{+}1}$ and $\unicode[STIX]{x1D73F}_{r_{-1}}$ , and each one of these two waves can then resonate other new waves. Such a chain resonance is shown in figure 1(b), which includes the incident mode-one internal wave ( $\unicode[STIX]{x1D73F}_{in}$ ) interacting with monochromatic corrugations ( $\unicode[STIX]{x1D73F}_{b}$ ), the resulting forcing (resonant) wavenumbers (noted $\unicode[STIX]{x1D73F}_{r_{\pm 1}}$ ), as well as the forcing (again resonant) wavenumbers $\unicode[STIX]{x1D73F}_{r_{+2}}=\unicode[STIX]{x1D73F}_{r_{+1}}+\unicode[STIX]{x1D73F}_{b}$ due to the interaction of $\unicode[STIX]{x1D73F}_{r_{+1}}$ with the same monochromatic corrugations. Other forcing resonant higher-mode wavenumbers ( $\unicode[STIX]{x1D73F}_{r_{-1}},\unicode[STIX]{x1D73F}_{r_{-2}},\ldots$ , $\unicode[STIX]{x1D73F}_{r_{+2}},\unicode[STIX]{x1D73F}_{r_{+3}},\ldots$ ) also participate in the chain resonance, with the sequence being potentially infinite.

As can be expected from a purely geometric argument, the successive forcing wavenumbers cannot all be perfectly resonant unless the incident wave comes at normal incidence (i.e.  $\unicode[STIX]{x1D703}_{in}=0$ ), since in this case the spacing between the successively resonated wavenumbers is always $\unicode[STIX]{x1D705}_{1}=\unicode[STIX]{x03C0}$ in a linearly stratified fluid (Bühler & Holmes-Cerfon Reference Bühler and Holmes-Cerfon2011). In figure 1(c), we illustrate the effect of detuning due to oblique incidence by showing that the $\unicode[STIX]{x1D73F}_{r_{+1}}$ wave is in fact only near resonance with the corrugations $\unicode[STIX]{x1D73F}_{b}$ , such that the forcing and resonant wavenumbers, $\unicode[STIX]{x1D73F}_{r_{+1}}+\unicode[STIX]{x1D73F}_{b}$ and $\unicode[STIX]{x1D73F}_{r_{+2}}$ respectively, are separated by a (small) detuning wavenumber, which we note $\unicode[STIX]{x1D716}\unicode[STIX]{x1D739}_{+2}$ .

In the general case, we write the positive and negative branches of the successively resonated wavenumbers as

(2.17a ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D73F}_{r_{+m}}=\unicode[STIX]{x1D73F}_{r_{+(m-1)}}+\unicode[STIX]{x1D73F}_{b}+\unicode[STIX]{x1D716}\unicode[STIX]{x1D739}_{+m}, & \displaystyle\end{eqnarray}$$

(2.17b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D73F}_{r_{-m}}=\unicode[STIX]{x1D73F}_{r_{-(m-1)}}-\unicode[STIX]{x1D73F}_{b}+\unicode[STIX]{x1D716}\unicode[STIX]{x1D739}_{-m}, & \displaystyle\end{eqnarray}$$

$m\geqslant 1$

$\unicode[STIX]{x1D73F}_{r_{\pm 0}}=\unicode[STIX]{x1D73F}_{in}$

$\unicode[STIX]{x1D73F}_{r_{\pm m}}$

$\unicode[STIX]{x1D716}\Vert \unicode[STIX]{x1D739}_{\pm m}\Vert _{2}=\unicode[STIX]{x1D716}\unicode[STIX]{x1D6FF}_{\pm m}$

$L^{2}$

$\unicode[STIX]{x1D716}\unicode[STIX]{x1D6FF}_{\pm m}$

$\unicode[STIX]{x1D705}_{1}=\unicode[STIX]{x03C0}$

$\unicode[STIX]{x1D716}\{\unicode[STIX]{x1D6FF}_{-4},\unicode[STIX]{x1D6FF}_{-3},\unicode[STIX]{x1D6FF}_{-2},\unicode[STIX]{x1D6FF}_{-1},\unicode[STIX]{x1D6FF}_{+1},\unicode[STIX]{x1D6FF}_{+2},\unicode[STIX]{x1D6FF}_{+3}\}$

$O(1)$

$\unicode[STIX]{x1D73F}_{r_{+m}}$

$m$

$r_{\pm m}$

$r$

$\pm$

$m$

4.1 Chain resonance at normal incidence

In this section we describe the chain resonance obtained at normal incidence and for long corrugations (i.e.  $\unicode[STIX]{x1D705}_{b}\sim \unicode[STIX]{x1D705}_{in}=\unicode[STIX]{x03C0}$ and $\unicode[STIX]{x1D703}_{in}=0$ ). The incident wave can be either perfectly tuned or detuned, i.e. such that $\unicode[STIX]{x1D716}\unicode[STIX]{x1D6FF}_{+1}=0$ or $\unicode[STIX]{x1D716}\unicode[STIX]{x1D6FF}_{+1}\neq 0$ (cf. (2.17)). We consider ${\mathcal{N}}=25$ seabed bars and the corrugation amplitude is $\unicode[STIX]{x1D716}=d/H=0.008$ .

The perfectly tuned one-dimensional case was briefly discussed in Bühler & Holmes-Cerfon (Reference Bühler and Holmes-Cerfon2011), following a prior derivation by Chen (Reference Chen2009) who obtained a closed-form solution for the amplitude variations. Solving (3.13)–(3.14) with $\unicode[STIX]{x1D705}_{b}=\unicode[STIX]{x1D705}_{1}$ , we find the variations of the incident ( $A_{in}$ ) and resonated ( $A_{+m}$ ) wave amplitudes as shown in figure 2(a); there are no back-scattered $A_{-m}$ waves because the spacing between the corrugations is large and the detuning of the $\unicode[STIX]{x1D705}_{r_{-1}}$ wave is also large. Note that our results agree perfectly with the solution of Chen (Reference Chen2009), which is expected since we solve (3.13)–(3.14) with a sufficiently large number of waves (i.e.  $M^{\pm }$ in (3.15) is sufficiently large that the truncated solution is close to the infinite series solution). The chain resonance for perfectly tuned normally incident waves only includes perfect resonances, such that we observe a cascade of the incident wave energy from the longest to the smallest scales. The initial amplitude decays monotonically over the bars while newly resonated waves successively capture the incident-wave energy.

Figure 3.

Same as figure 2, except that this time we have backward-going resonated ( $A_{-m}$ ) waves in addition to the forward-going ( $A_{+m}$ ) waves as we consider shorter corrugations, i.e. for which $\unicode[STIX]{x1D705}_{b}\sim 2\unicode[STIX]{x1D705}_{in}$ . The first resonated wave $A_{+1}$ is a mode-three wave, and $A_{-1}$ is a mode-one wave. The detuning $\unicode[STIX]{x1D716}\unicode[STIX]{x1D6FF}_{+1}/\unicode[STIX]{x1D705}_{in}$ is again (a) 0, (b) 2 % and (c) 4 %.

When the first wave–bottom interaction is not perfectly resonant, we obtain the amplitude variations shown in figure 2(b–c). For $\unicode[STIX]{x1D716}\unicode[STIX]{x1D6FF}_{+1}/\unicode[STIX]{x1D705}_{in}=0.02$ , figure 2(b) shows that the energy cascade still takes place, albeit at a slower rate over the seabed bars compared to the perfectly tuned case. With increased detuning $\unicode[STIX]{x1D716}\unicode[STIX]{x1D6FF}_{+1}/\unicode[STIX]{x1D705}_{in}=0.04$ , we can see in figure 2(c) that the chain resonance is stopped at approximately the second or third resonated wave. We also observe that the incident wave even gains energy back from the resonated waves, such that downstream of the patch the incident mode-one internal wave is again close to 1. With half the number of seabed bars, the incident wave amplitude would on the contrary be at its minimum, as can be seen at $x/\unicode[STIX]{x1D706}_{b}\sim 12.5$ in figure 2(c). Indeed, the solution obtained for forward-scattered waves only is independent of the number of seabed bars, such that the results shown in figure 2 also apply for smaller patches. As the detuning is further increased (not shown), energy exchanges become even less strong and oscillate more rapidly. With $\unicode[STIX]{x1D716}\unicode[STIX]{x1D6FF}_{+1}/\unicode[STIX]{x1D705}_{in}=0.1$ , the decrease of the incident wave is negligible. It should be remarked, however, that detuning effects depend upon the interaction strength, measured as $d/H$ . From (3.13), it can be seen that the ratio of the detuning term to the interaction term is proportional to $(\unicode[STIX]{x1D716}\unicode[STIX]{x1D6E5}_{\pm m}/\unicode[STIX]{x1D705}_{r_{\pm m}})/(d/H)$ , such that detuning effects become less important as $d/H$ is increased.

Figure 4.

Variations of the incident and resonated wave envelopes for incidence angles (a) $\unicode[STIX]{x1D703}_{in}=\unicode[STIX]{x03C0}/6$ , (b) $\unicode[STIX]{x1D703}_{in}=\unicode[STIX]{x03C0}/4$ , (c) $\unicode[STIX]{x1D703}_{in}=\unicode[STIX]{x03C0}/3$ . The first resonance resulting in the mode-two wave $A_{+1}$ is perfect ( $\unicode[STIX]{x1D716}\unicode[STIX]{x1D6FF}_{+1}=0$ ). Back-resonated ( $A_{-m}$ ) waves have negligible amplitude in all three cases due to large detuning that damp energy exchanges, and therefore are not shown. As in figure 2, ${\mathcal{N}}=25$ , $f/\unicode[STIX]{x1D714}=0.6$ , $\unicode[STIX]{x1D705}_{in}=\unicode[STIX]{x03C0}$ , but the corrugation height is this time increased to $d/H=0.015$ .

We show in figure 3 the chain resonance when the normally incident waves and the corrugations first resonate a back-scattered mode-one wave $A_{-1}$ , propagating in the $-x$ direction, in addition to the forward-going mode-three wave $A_{+1}$ (which is the situation that corresponds to corrugations approximately half the size of those considered in figure 2, i.e.  $\unicode[STIX]{x1D705}_{b}\sim 2\unicode[STIX]{x1D705}_{in}$ ). The first resonance condition is perfectly satisfied with $\unicode[STIX]{x1D705}_{b}=2\unicode[STIX]{x1D705}_{in}$ , hence so are the successive resonances (cf. figure 3 a). A cascade of energy is again obtained from the longest to the smallest scales. Back-scattered waves have zero amplitude at the end of the patch, but grow monotonically to the upstream. With detuning, as in figure 2, we see that fewer waves are resonated, but overall the effect of detuning up to 4 % is relatively small such that many new waves can be resonated. Note that the effect of detuning is smaller for the case with back-scattered waves compared to the case of figure 2(b–c) partly because $\unicode[STIX]{x1D716}\unicode[STIX]{x1D6FF}_{\pm 1}/\unicode[STIX]{x1D705}_{b}$ is smaller in figure 3(b–c) due to the larger $\unicode[STIX]{x1D705}_{b}$ .

4.2 Chain resonance at oblique incidence

An oblique incidence of the mode-one waves results in non-zero detuning wavenumbers for most resonated waves, even when the incident wave is perfectly tuned (see figure 1 b): we call this the angular detuning. To isolate the effect of the angular detuning from the incident detuning (between the incident waves and the corrugations), here we first assume that the first forward-going forced wave is perfectly resonated, i.e.  $\unicode[STIX]{x1D716}\unicode[STIX]{x1D6FF}_{+1}=0$ .

For a relatively broad range of incidence angles, i.e.  $\unicode[STIX]{x1D703}_{in}<\unicode[STIX]{x03C0}/8$ , we find that the chain resonance results are very similar to the normal-incidence case with no detuning. For $\unicode[STIX]{x1D703}_{in}=\unicode[STIX]{x03C0}/6$ , figure 4(a) shows a number of new waves attaining large amplitude over the patch, suggesting that the chain resonance still takes place despite an individual detuning $\unicode[STIX]{x1D716}\unicode[STIX]{x1D6FF}_{+m}/\unicode[STIX]{x1D705}_{in}$ and cumulative detuning $\unicode[STIX]{x1D716}\unicode[STIX]{x1D6E5}_{+m}/\unicode[STIX]{x1D705}_{r_{+m}}$ increasing up to approximately 2 % for the first 10 waves. The detuning for the back-scattered waves is quite large (up to 50 % for $A_{-1}$ ), but even when we consider them, their predicted amplitude is negligible (less than 0.01), such that the results at leading order do not violate multiple scales.

For larger incidence angle $\unicode[STIX]{x1D703}_{in}=\unicode[STIX]{x03C0}/4$ , figure 4(b) shows that the chain resonance is effectively stopped at just 2 or 3 resonated waves. In this case, individual $\unicode[STIX]{x1D716}\unicode[STIX]{x1D6FF}_{+m}/\unicode[STIX]{x1D705}_{in}$ and cumulative $\unicode[STIX]{x1D716}\unicode[STIX]{x1D6E5}_{+m}/\unicode[STIX]{x1D705}_{r_{+m}}$ detuning wavenumbers go up to 15 % and 4 % for the first 4 resonated waves, and are higher for further resonances. As for the previous case (figure 4 a), the detuning of the back-scattered waves is large such that no back-scattered waves are resonated. Figure 4(c) shows that the two resonated forward-going waves ( $A_{+1}$ and $A_{+2}$ ) are of even smaller amplitudes when the incidence angle is increased to $\unicode[STIX]{x1D703}_{in}=\unicode[STIX]{x03C0}/3$ compared to $\unicode[STIX]{x1D703}_{in}=\unicode[STIX]{x03C0}/4$ .

In figure 5(a–c) we show the same case of figure 4(b), i.e. with incidence angle $\unicode[STIX]{x1D703}_{in}=\unicode[STIX]{x03C0}/4$ , but this time we consider a detuned incident wave. In figure 5(a) we show the results obtained for $\unicode[STIX]{x1D716}\unicode[STIX]{x1D739}_{+1}=-0.025\unicode[STIX]{x1D73F}_{r_{+1}}$ (i.e.  $\unicode[STIX]{x1D716}\unicode[STIX]{x1D6FF}_{+1}=0.05\unicode[STIX]{x1D705}_{in}$ ), corresponding to incident waves that are too short to perfectly resonate the $\unicode[STIX]{x1D73F}_{r_{+1}}$ waves. Fewer waves are resonated and have smaller maximum amplitude than in figure 4(b). The effect of the incident-wave detuning can thus be seen to be added to that due to the oblique incidence. The oblique angular detuning is indeed opposite to the direction of the $\unicode[STIX]{x1D73F}_{r_{+m}}$ waves (as is the case of the incident detuning), which makes the wave-bottom interactions always result in a wavenumber larger than the next closest natural resonated wavenumber $\unicode[STIX]{x1D73F}_{r_{+(m+1)}}$ . When the incident-wave detuning is $\unicode[STIX]{x1D716}\unicode[STIX]{x1D739}_{+1}=0.025\unicode[STIX]{x1D73F}_{r_{+1}}$ (cf. figure 5 b), however, it counteracts the angular detuning such that more waves with relatively large amplitudes are resonated and a chain resonance is again obtained. With a larger positive detuning of $\unicode[STIX]{x1D716}\unicode[STIX]{x1D739}_{+1}=0.05\unicode[STIX]{x1D73F}_{r_{+1}}$ , even more waves are resonated but the maximum amplitudes of the low-mode waves over the patch are smaller (figure 5 c). The trend continues as the detuning is increased, until no resonances remain.

Figure 5.

Same case as figure 4(b), except that this time the incident-wave detuning $\unicode[STIX]{x1D716}\unicode[STIX]{x1D6FF}_{+1}/\unicode[STIX]{x1D705}_{in}$ is 5 % for (a,b) and 10 % for (c). Unlike in other cases, in (a) the detuning is negative, i.e.  $\unicode[STIX]{x1D716}\unicode[STIX]{x1D739}_{+1}\boldsymbol{\cdot }\unicode[STIX]{x1D73F}_{r_{+1}}<0$ (cf. (2.17)). In (a) the effects of the incident detuning and the angular detuning are added, such that fewer waves of smaller amplitudes are obtained compared to the perfectly tuned incident-wave case of figure 4(b). In (b,c), the two effects are opposite such that more waves are obtained resulting in an oblique chain resonance of many waves with different angles of propagation.

4.3 Discussion of results

From the various results shown in figures 2–5 it is found that the resonant generation of multiple internal waves over seabed corrugations is not limited to the special case of perfectly tuned normally incident waves, although the efficiency of energy transfers does change with changing incidence angle and detuning. The sensitivity of the resonance to the incident detuning can be partly assessed from the variations with $\unicode[STIX]{x1D716}\unicode[STIX]{x1D6FF}_{+1}/\unicode[STIX]{x1D705}_{in}$ of

(4.1) $$\begin{eqnarray}\tilde{F}=\max _{x}(1-|A_{in}|^{2}),\end{eqnarray}$$

which is the maximum energy flux of the resonated waves in $x$ direction over the patch (normalized by the incident energy flux; see (3.9)). For normal incidence ( $\unicode[STIX]{x1D703}_{in}=0$ ), figure 6(a) shows that the resonated waves capture more than 50 % of the incident energy flux with zero detuning and more than 40 % for a broad range of detuning wavenumber. The peak as well as the half-width of this curve would be increased provided that the corrugation amplitude increases too; the peak would also increase with the number of corrugations. While figure 6(a) implies that strong energy exchanges occur with or without detuning, it does not yet allow us to assess whether high-wavenumber waves are generated for relatively large detuning. In fact, as shown in figure 4(c), the incident wave can cede a significant fraction of its energy to just two resonated waves.

Figure 6.

Effect of detuning on (a) the maximum of energy flux of the resonated waves $\tilde{F}$ (4.1) and (b) the minimum Richardson number $\tilde{Ri}$ (see (4.2)). The results are shown as a function of the normalized detuning wavenumber $\unicode[STIX]{x1D716}\unicode[STIX]{x1D6FF}_{+1}/\unicode[STIX]{x1D705}_{in}$ for the case of normal incidence ( $\unicode[STIX]{x1D703}_{in}=0$ ). As in figure 2, $\unicode[STIX]{x1D705}_{in}=\unicode[STIX]{x03C0}$ , ${\mathcal{N}}=25$ , $f/\unicode[STIX]{x1D714}=0.6$ and $d/H=0.008$ .

Figure 7.

Same as figure 6 but for incidence angle $\unicode[STIX]{x1D703}_{in}=\unicode[STIX]{x03C0}/6$ (——, blue), $\unicode[STIX]{x03C0}/4$ (——, red), $\unicode[STIX]{x03C0}/3$ (——, yellow) and $d/H=0.015$ . The energy flux ceded to the resonated waves is close to unity in all three cases when $\unicode[STIX]{x1D6FF}_{+1}\sim 0$ . $\tilde{Ri}$ decreases significantly for $\unicode[STIX]{x1D703}_{in}=\unicode[STIX]{x03C0}/6,\unicode[STIX]{x03C0}/4$ , although the location of the global minimum is obtained for non-zero detuning; this implies that the effect of oblique incidence and incident detuning can cancel each other, resulting in the generation of multiple higher-wavenumber resonated modes. Parts of the curves shown as dotted lines correspond to cases for which the theory is relatively stretched, and therefore may not be quantitatively accurate.

Because generating high-wavenumber waves is expected to increase the vertical shear, here we make a parallel between the efficiency of chain resonance at generating high-wavenumber waves and the decrease of the Richardson number $Ri=(HN/U)^{2}/(\unicode[STIX]{x2202}V/\unicode[STIX]{x2202}z)^{2}$ over the patch, where $V$ is the dimensionless horizontal velocity and $z$ the dimensionless vertical coordinate. The Richardson number is of great interest to the oceanographic community as it is often used as a proxy to predict the likelihood of overturning events and enhanced mixing in a stratified fluid, due to shear instability acting against the stable buoyancy field. Though the existence of a universal critical Richardson number $Ri_{c}$ at which turbulent mixing can start is still debated, the lower bound $Ri<Ri_{c}=1/4$ , originally derived for horizontal shear flows, of the order of unity is often assumed (Mack & Schoeberlein Reference Mack and Schoeberlein2004; Galperin, Sukoriansky & Anderson Reference Galperin, Sukoriansky and Anderson2007). The Richardson number for mode-one internal waves at tidal frequency can be roughly estimated from field measurements in the open ocean. Velocities of mode-one semi-diurnal internal waves have been reported to be of order $U\sim O(0.01~\text{m}~\text{s}^{-1}{-}0.1~\text{m}~\text{s}^{-1})$ and higher close to large topographic features (Alford & Zhao Reference Alford and Zhao2007; Alford et al. Reference Alford, MacKinnon, Zhao, Pinkel, Klymak and Peacock2007; Garrett & Kunze Reference Garrett and Kunze2007), such that for a depth of $H=4000~\text{m}$ and constant buoyancy of $N=2\times 10^{-4}~\text{s}^{-1}$ (which is within the range reported in some parts of the deep ocean, see e.g. St Laurent & Garrett Reference St Laurent and Garrett2002), the minimum Richardson number is of order $Ri=(NH/U)^{2}/\unicode[STIX]{x03C0}^{2}\sim O(10)-O(10^{3})\gg 1$ . Low-mode internal tides at the semi-diurnal frequency are thus unlikely to overturn because of shear instability, which explains why it is generally assumed that a nonlinear mechanism or modal transformation must take place in order to destabilize the flow and induce mixing (Staquet & Sommeria Reference Staquet and Sommeria2002).

Here with simple monochromatic corrugations we showed that an efficient modal transformation can occur for a wide range of incidence angles and detuning. The efficiency of the energy transfer from low-mode to high-mode internal waves is now related to the drop of Richardson number $\tilde{Ri}$ , i.e.

(4.2) $$\begin{eqnarray}\tilde{Ri}=\frac{\displaystyle \left(1+\frac{f^{2}}{\unicode[STIX]{x1D714}^{2}}\right)\unicode[STIX]{x1D705}_{in}^{2}}{\max _{x,z}\left[\unicode[STIX]{x2202}_{z}\sqrt{|u^{(0)}|^{2}+|v^{(0)}|^{2}}\right]^{2}},\end{eqnarray}$$

which gives the minimum of the Richardson number over the corrugations normalized by the Richardson number of the incident wave. For the case of normal incidence (see figure 6 b), we find that $\tilde{Ri}$ is close to $10^{-2}$ , with no detuning. This is a relatively large decrease of the Richardson number. However, whether it is a sufficient decrease for the waves to overturn depends on the Richardson number of the initial wave (say $Ri_{0}$ ), since the actual Richardson number is $Ri_{0}\times \tilde{Ri}$ . Interestingly, the decrease of Richardson number can be used to obtain a lower bound on the number of short internal waves generated by the chain resonance. Assuming that all of the incident energy flux is sent to the $\unicode[STIX]{x1D705}_{r_{+m}}$ wave only, the wave amplitude becomes $\unicode[STIX]{x1D705}_{r_{+m}}^{1/2}$ such that $\tilde{Ri}\sim (2\unicode[STIX]{x1D705}_{in})^{2}/(4\unicode[STIX]{x1D705}_{r_{+m}}^{5/2})$ or $\tilde{Ri}\sim 1/(m+1)^{5/2}$ if $\unicode[STIX]{x1D705}_{r_{+m}}=(m+1)\unicode[STIX]{x1D705}_{in}$ (as is the case for figure 6). Therefore, a drop of $10^{-2}$ implies the lower bound $m=5$ for the number of resonated waves; it can be checked from figure 2(a) that at least 5 waves are resonated with non-negligible amplitude.

We show in figure 7(a) the maximum energy flux of the resonated waves and in figure 7(b) the minimum Richardson number for oblique incident waves ( $\unicode[STIX]{x1D703}_{in}=\unicode[STIX]{x03C0}/6,\unicode[STIX]{x03C0}/4,\unicode[STIX]{x03C0}/3$ ). The maximum energy flux of the resonated waves is large (even higher than in the case of normal incidence, see figure 6 a), partly because the corrugation amplitude is larger, but also because the energy transfer from the incident to the resonated waves can occur more rapidly with only one wave resonated. In figure 7(b) we observe that there exists an incident detuning for which the minimum Richardson number drops by several orders of magnitude for $\unicode[STIX]{x1D703}_{in}=\unicode[STIX]{x03C0}/6,\unicode[STIX]{x03C0}/4$ , as was the case for normal incidence. Again, the occurrence of chain resonance for non-zero detuning at oblique incidence comes from the fact that the angular detuning is negative, hence can be counter-balanced with a positive detuning of the first resonance. Because the first interaction that feeds all subsequent resonances is weakened with non-zero incident detuning, the resonated waves have relatively smaller amplitudes in chain resonance at oblique incidence (for which we require a non-zero incident detuning) than in chain resonance at normal incidence (obtained with zero incident detuning). At oblique incidence, however, the projected patch length onto the internal waves’ path is longer than at normal incidence, hence resulting in the generation of more waves (this is the effect of the $\cos \unicode[STIX]{x1D703}_{\pm m}$ terms in (3.13)). These two effects compete with each other, allowing for significant decrease of the Richardson number at oblique incidence (cf. figure 7 b). The fact that the minimum Richardson number is obtained for increasing detuning as the incidence angle is increased is simply the effect of the increased angular detuning that must be counter-balanced.

4.4 Extension to polychromatic seabed

In this section we briefly discuss the extension of the present theory to the case of a polychromatic seabed, i.e. with corrugations given by (in dimensionless form)

(4.3) $$\begin{eqnarray}b(x)=\mathop{\sum }_{n=1}^{N_{c}}\frac{d_{n}}{2H}\text{e}^{\text{i}\unicode[STIX]{x1D711}_{n}}\text{e}^{\text{i}\unicode[STIX]{x1D705}_{bn}x}+\text{c.c.}\end{eqnarray}$$

In (4.3) the bottom wavenumbers $\unicode[STIX]{x1D705}_{bn}$ are ordered from smallest to largest, $d_{n}$ is real ( $d_{n}/H\ll 1$ ) and $\unicode[STIX]{x1D711}_{n}$ is the mode phase. For simplicity we restrict the analysis to one horizontal dimension, no rotation and we consider the discretized bathymetric spectrum to be narrow-banded, such that all seabed modes resonate the same sequence of internal waves $\{\unicode[STIX]{x1D705}_{r_{\pm m}}\}$ for a given incident wavenumber $\unicode[STIX]{x1D705}_{in}$ . In other words we assume that all wavenumbers in $[\unicode[STIX]{x1D705}_{b1},\unicode[STIX]{x1D705}_{bN}]$ are closest to the same perfectly resonant bottom wavenumber $\unicode[STIX]{x1D705}_{b}^{\ast }$ .

Figure 8.

Histograms of (a) $\tilde{F}$ (4.1) and (b) $\tilde{Ri}$ (4.2) obtained for $10^{6}$ distinct uniform random bottom phase distributions $\unicode[STIX]{x1D711}_{n}\in [\!0,2\unicode[STIX]{x03C0}\![$ (see (4.3)). Three bottom modes are considered with amplitudes $d_{n}/H=[0.0025,0.005,0.0025]$ and detuning $\unicode[STIX]{x1D6FC}_{n}/\unicode[STIX]{x1D705}_{in}=[-0.02,0,0.02]$ . The incident wave is normal to the ${\mathcal{N}}=25$ corrugations considered. The vertical dashed lines show the solutions when the two detuned bottom modes are discarded, and the light green solid line shows the unfilled histogram obtained when two bottom modes of amplitudes $0.0025$ and detuning $\pm 0.08$ are added to the three modes used in constructing the filled histogram. $f/\unicode[STIX]{x1D714}=0$ .

Equation (4.3) can be recast advantageously as

(4.4) $$\begin{eqnarray}b(x)=\text{e}^{\text{i}\unicode[STIX]{x1D705}_{b}^{\ast }x}\mathop{\sum }_{n=1}^{N_{c}}\frac{d_{n}}{2H}\text{e}^{\text{i}\unicode[STIX]{x1D711}_{n}}\text{e}^{-\text{i}\unicode[STIX]{x1D6FC}_{n}x}+\text{c.c.},\end{eqnarray}$$

with $\unicode[STIX]{x1D6FC}_{n}$ a bottom detuning representing the deviation of each bottom wavenumber from $\unicode[STIX]{x1D705}_{b}^{\ast }$ . For a narrow-banded topographic spectrum we have $|\unicode[STIX]{x1D6FC}_{n}|\ll \unicode[STIX]{x1D705}_{b}^{\ast }\sim 1$ , such that the sum term in (4.4) is a slowly varying function of $x$ . Therefore, considering a narrow-banded polychromatic seabed is the same as considering a slowly varying and oscillating bottom corrugation amplitude $d\equiv D(x)$ , such that the envelope equation (3.8) can be readily modified as

(4.5) $$\begin{eqnarray}\frac{\text{d}A_{\pm m}}{\text{d}x}=\text{i}\unicode[STIX]{x1D705}_{r_{\pm m}}[A_{\pm (m+1)}{\mathcal{I}}_{\pm }(x)+A_{\pm (m-1)}{\mathcal{I}}_{\mp }(x)],\end{eqnarray}$$

with the slowly varying interaction term

(4.6) $$\begin{eqnarray}{\mathcal{I}}_{\pm }(x)=\mathop{\sum }_{n=1}^{N_{c}}\frac{d_{n}}{2H}\text{e}^{\pm \text{i}\unicode[STIX]{x1D711}_{n}}\text{e}^{\mp \text{i}\unicode[STIX]{x1D6FC}_{n}x},\end{eqnarray}$$

for bottom corrugations of the form (4.3). It should be noted that (4.5)–(4.6) can be derived formally as in § 3 provided that a superposition of $N_{c}$ interactions is considered on the right-hand side of the bottom boundary condition (3.4c ); while necessary for broadband spectra, this procedure is, however, not justified for the present case.

Equations (4.5)–(4.6) demonstrate that the resonance mechanism due to multiple seabed components can be described as a superposition of resonances (from the sum term). The interference effects resulting from the superposition can be destructive or constructive depending on the bottom phases $\unicode[STIX]{x1D711}_{n}$ , detuning $\unicode[STIX]{x1D6FC}_{n}$ , and location $x$ over the corrugations. Some results on the sensitivity of the chain resonance to arbitrary bottom phase distributions are provided in figure 8. We consider 3 bottom modes and solve (4.5) for $10^{6}$ different cases of uniform random phase distributions $\unicode[STIX]{x1D711}_{n}\in [\!0,2\unicode[STIX]{x03C0}\![$ . Figure 8(a,b) shows the histograms of the maximum energy flux ceded by the incident wave to the resonated waves ( $\tilde{F}$ ), and of the minimum normalized Richardson number ( $\tilde{Ri}$ ) for the $10^{6}$ random phase distributions (cf. (4.1)–(4.2)). The most probable values of the histograms of $\tilde{F}$ and $\tilde{Ri}$ are approximately equal to what is obtained when only the perfectly tuned bottom mode is considered (shown by the dashed vertical lines). The histograms are skewed toward higher/smaller values of $\tilde{F}/\tilde{Ri}$ , such that adding bottom modes with random phases is found to lead to constructive interference (higher $\tilde{F}$ , lower $\tilde{Ri}$ ) with slightly higher probabilities than destructive interference. This suggests that the chain resonance mechanism can be expected to occur in most cases of topographic spectra even though the intensity is as expected sensitive to the phase distribution. The minimum/maximum values of $\tilde{F}$ and maximum/minimum values of $\tilde{Ri}$ are obtained when the interference lead to the minimum/maximum possible transfer of energy to the shorter waves. With two bottom modes of relatively large detuning added (light green solid lines), it is seen that the histograms do not change significantly. This is expected since the higher the detuning the lesser the contribution to the generation of new resonated waves.