2.1. Posing the problem

In a 2-D Cartesian coordinate system $Oxz$, with the vertical axis $z$ pointing upward, we consider a horizontally infinite layer of a density-stratified fluid bounded from above by the ocean surface, modelled as a ‘rigid lid’ $\{z=0\}$, and from below by the impermeable bottom $\{z=-h(x)\}$ with $h>0$. We assume that the topography is asymptotically flat, that is, its slope vanishes at infinity, $\lim _{x\to \pm \infty }[h_x] =0$, and we define $\lim _{x\to \pm \infty }h = h_\pm$. The latter requirement allows us to take into account fluid domains with different depths at infinity. The parameters associated with the topography are the characteristic depth $h_0$, the characteristic height $\varLambda$ and the characteristic horizontal scaling length $L$. We do not limit ourselves to small heights; i.e. $\varLambda$ does not have to be much smaller than $h_0$ or any other characteristic vertical length scale.

In static equilibrium, the fluid velocity is zero and the background density profile $\rho _{eq}(z)$ weakly departs from a constant reference density $\rho _0$ so that the Boussinesq approximation applies. We further assume that $\rho _{eq}(z)$ decreases linearly with $z$ such that the Brunt–Väisälä frequency $N = \sqrt {(-g/\rho _0)\,\text {d}\rho _{eq}/\text {d}z}$ is constant. The hydrostatic pressure $p_0(z)$ is then defined by its vertical gradient $p_{0,z} \equiv \text {d} p_0/\text {d}z= -\rho _0 g$. The hydrodynamic problem is posed on the $f$-plane, with $f$ the Coriolis parameter. Our aim is to find perturbations of this state driven by the interaction of a background barotropic tidal flow with the bottom topography. The barotropic flow oscillates with an angular frequency $\omega \in (f, N)$ for $N>f$ and is associated with a volume flux of constant amplitude $Q$ corresponding to a uniform current $Q/h_{\pm }\cos (\omega t)$ at $x\rightarrow \pm \infty$.

Under the Boussinesq approximation, our set-up is characterised by eight dimensional parameters ($h_{\pm }$, $L$, $\varLambda$, $f$, $N$, $\omega$ and $Q$), which we summarise in figure 1, measured in combinations of metres and seconds. In the case of an isolated ridge, $h_0$ is the depth as $x\rightarrow \pm \infty$, and $\varLambda$ is the maximum height of the ridge, $\varLambda =\max \{h_0-h\}$. For a shelf, we assume that $h_- > h_+$ ($h_-$ is the oceanward depth) so that the maximum height is $\varLambda =\max \{h_0-h\}$, where we have also chosen $h_0=h_-$ as the characteristic depth. A complete dynamical description of our system therefore requires five non-dimensional numbers. First, we introduce a ‘funnelling ratio’ that measures the reduction in cross-section of the flow,

(2.1)\begin{equation} \delta = \frac{\max\{h_0-h\}}{h_0} \sim \frac\varLambda{h_0}. \end{equation}

The second and third parameters are the non-dimensional frequency $\omega /f$ and the characteristic slope of the internal wave,

(2.2)\begin{equation} \mu^{{-}1} = \sqrt{\frac{\omega^2 - f^2}{N^2 - \omega^2}} = \tan\theta, \end{equation}

$\theta$ being the angle of the free internal wave group velocity with respect to the horizontal plane (see (2.8a–c)). Note that internal waves are hydrostatic to a good approximation when $\omega \ll N$, or equivalently, $\mu \gg 1$. Our fourth and fifth parameters are the relative steepness $\varepsilon$,

(2.3)\begin{equation} \varepsilon = \mu\max\{|\partial_x h|\} \sim \frac{\mu\varLambda}L, \end{equation}

and the tidal excursion $\tau$ defined by

(2.4)\begin{equation} \tau = \frac{Q}{(h_0 - \varLambda)\omega L}. \end{equation}

The parameter $\varepsilon$ represents the ratio $\tan \alpha /\tan \theta$, where $\alpha$ is the maximum inclination of the topography with respect to the horizontal plane. Its purpose is to measure the criticality of the topography, with $\varepsilon < 1$ ($>1$) corresponding to the subcritical (supercritical) regime. The parameter $\tau$ compares the typical displacement amplitude of a water parcel above topography, $Q/[(h_0-\varLambda )\omega ]$, with the horizontal scale $L$. If $\tau$ is finite, the curvature of the particle trajectories at the bottom generates internal waves with frequencies other than $\omega$ (Bell Reference Bell1975). To ensure monochromatic disturbances, we therefore assume $\tau \ll 1$. In the ocean, for the lunar semi-diurnal tide $M_2$, the tidal excursion over flat bottom $Q/(\omega h_0)$ is $O(100\,\text {m})$ (Bell Reference Bell1975), and therefore even a moderate topographic width $L=O(10\,\text {km})$ would satisfy this so-called ‘acoustic limit’.

Figure 1. Sketch of the set-up and summary of several parameters used in this article. We use a supercritical topography ($\varepsilon >1$) for purposes of illustration, although our model can also be applied to subcritical ones.

Of the five parameters defined above, $\omega /f= O(1)$ corresponds to a tidal component at mid-latitudes, and we will keep $\mu ^{-1}$ fixed to a small value (for illustration purposes, $f=10^{-4}$ s$^{-1}$ is the value around latitude 45 $^\circ$N, $\omega /f = 1.4$ for the M$_2$ component and we will use $N = 1.5\times 10^{-3}$ s$^{-1}$, which implies $\mu \approx 15$). Starting in § 4, $\varepsilon$ and $\delta$ are the parameters that we vary primarily. Finally, we adopt a standard fixed value for $Q = 120$ m$^2$ s$^{-1}$ corresponding e.g. to a barotropic velocity amplitude at $x\to -\infty$ of $U_0 =Q/h_0 = 4$ cm s$^{-1}$ and a depth $h_0 = 3$ km. As $\varepsilon$ and $\delta$ vary, the value of $\tau$ therefore varies between calculations while remaining small.

With the parameters defined above, we can discuss the linearity of our equations. A first way to define it is to estimate the susceptibility of radiated internal waves to undergo instabilities, which, in the $\tau \ll 1$, $\varepsilon \ll 1$ regime, is small when $\varepsilon \tau \ll 1$ (e.g. Balmforth, Ierley & Young Reference Balmforth, Ierley and Young2002; Garrett & Kunze Reference Garrett and Kunze2007). In other words, the $\tau \ll 1$, $\varepsilon \ll 1$ regime is linear by construction. However, regardless of the value of $\tau$, linearity breaks down as $\varepsilon$ increases, implying that this parameter is a better measure of linearity (Bühler & Muller Reference Bühler and Muller2007; Garrett & Kunze Reference Garrett and Kunze2007; Grisouard & Bühler Reference Grisouard and Bühler2012). In this article, we adopt the common approach of letting $\varepsilon$ be significantly supercritical in some cases while always solving a linear set of equations (Pétrélis et al. Reference Pétrélis, Llewellyn Smith and Young2006; Griffiths & Grimshaw Reference Griffiths and Grimshaw2007; Balmforth & Peacock Reference Balmforth and Peacock2009; Echeverri & Peacock Reference Echeverri and Peacock2010; Mathur et al. Reference Mathur, Carter and Peacock2016). Indeed, we are primarily interested in predicting conversion from a given large-scale barotropic forcing to a topography-scale response. The subsequent nonlinear evolution of this response, which could take the form of different instabilities (see Dauxois et al. (Reference Dauxois, Joubaud, Odier and Venaille2018) for a review), could be addressed by separate parametrised procedures such as that of Muller & Bühler (Reference Muller and Bühler2009), but would be beyond the scope of this article.

Based on the discussion above, we neglect nonlinear effects and diffusion of momentum and buoyancy, and we model the flow by the linearised, inviscid Boussinesq equations. Introducing the buoyancy $b= -g(\rho /\rho _0 - 1)$, where $\rho$ is the total density field, the governing equations are

(2.5a)\begin{gather} u_t - f v ={-}p_x,\quad v_t + f u = 0, \end{gather}

(2.5b)\begin{gather}w_t ={-}p_z + b, \end{gather}

(2.5c)\begin{gather}b_t + N^2 w = 0, \end{gather}

(2.5d)\begin{gather}u_x + w_z = 0, \end{gather}

where $(u,v,w)$ are the velocity components, $p$ is the associated pressure perturbation divided by $\rho _0$ and subscripts denote partial derivatives. On the boundaries $z=-h$ and $z=0$, we have the impermeability conditions

(2.6a,b)\begin{equation} h_x u(x,-h) + w(x,-h) = 0\quad\text{and}\quad w(x,0) = 0,\end{equation}

and we also assume that the total flux through a vertical cross-section is oscillating with frequency $\omega$ and constant amplitude $Q$,

(2.7)\begin{equation} \int_{{-}h}^0u \,\,\mathrm{d} z = Q \cos\omega t. \end{equation}

Introducing the time harmonic stream function $\psi = \text {Re}\{\phi (x,z)\exp ({-\text {i}\omega t})\}$, with $u = -\psi _z$, $w = \psi _x$, where $\phi$ is a complex amplitude and $\text {Re}$ stands for the real part, (2.5)–(2.7) lead to a hyperbolic boundary value problem (BVP) on $\phi$,

(2.8a–c)\begin{equation} \mathcal{L}_{\mu}\phi := \left(\partial_x^2 - \mu^{{-}2}\partial_z^2\right) \phi= 0,\quad \phi(x,0) =0\quad\text{and}\quad \phi(x,-h) = Q, \end{equation}

where $\mu$ is defined in (2.2) (Vlasenko et al. Reference Vlasenko, Stashchuk and Hutter2005; Garrett & Gerkema Reference Garrett and Gerkema2007). Equation (2.8a–c), also known as the boundary forcing formulation, is completed by the requirement that $\phi$ has the form of a barotropic flow plus a flow radiating away from the topography in the form of internal waves. We introduce the barotropic flow in the following subsection.

2.2. Barotropic flow

The barotropic velocity components $U$, $V$, $W$ and scaled pressure $P$ satisfy (2.5)–(2.6a,b), where the buoyancy in the vertical momentum equation (2.5b) is absent and (2.5c) acts as a diagnostic equation for the induced buoyancy $B$, see (A1) in Appendix A. Introducing a barotropic stream function $\varPsi = \text {Re}\{\varPhi \exp ({-\text {i} \omega t})\}$ for some time independent function $\varPhi$ with $U = -\varPsi _z$ and $W = \varPsi _x$, we obtain from (A1),

(2.9a–c)\begin{equation} \varPhi_{xx}+ \mu_0^{{-}2}\varPhi_{zz}=0,\quad \varPhi(x,0)=0\quad\text{and}\quad \varPhi(x,-h)=Q, \end{equation}

with $\mu _0^{-2}=1 - f^2/\omega ^2$ (Garrett & Gerkema Reference Garrett and Gerkema2007), where the boundary conditions ensure that the total mass transport through any vertical cross-section is solely due to the barotropic flow. Note that the PDE (2.9a) is elliptic and is also obtained by (2.8a) with $N\equiv 0$. The solution of (2.9a–c) can be written as

(2.10)\begin{equation} \varPhi = \varPhi^{(0)}+\varPhi^{r},\quad\text{with}\ \varPhi^{(0)} ={-}Qz/h, \end{equation}

where $\varPhi ^{(0)}$ represents the hydrostatic part of the barotropic flow. It is obtained by neglecting $W_t$ in (A1b) and, consequently, $\varPhi _{xx}$ in (2.9a); see Appendix A for a detailed derivation. Here, $\varPhi ^{r}$ represents the residual non-hydrostatic part which solves

(2.11a–c)\begin{equation} \varPhi^{r}_{xx}+ \mu_0^{{-}2}\varPhi^{r}_{zz}={-}\varPhi^{(0)}_{xx},\quad \varPhi^{r}(x,0)=0\quad\text{and}\quad \varPhi^{r}(x,-h)=0. \end{equation}

Note that $\varPhi$ is spatially non-uniform, that is, it depends on $z$ and $h(x)$. Also note that $\varPhi ^{r}$ vanishes at $x\rightarrow \pm \infty$. We obtain a general semi-analytical solution of (2.11a–c) that is valid for arbitrary smooth $h$ by means of a modal decomposition (§ 3.2). In Appendix A, we derive a perturbative solution that is valid if $h_0/L\ll 1$. In fact, $\varPhi ^{(0)}$ coincides with the leading order part of this solution. The corresponding velocities are given by $[U^{(0)},W^{(0)}]=[Q/h, Qz(1/h)_x]\cos (\omega t)$. As $x\rightarrow \pm \infty$, $W^{(0)}=0$ and $U^{(0)}=Q /h_{\pm }\cos (\omega t)$ coincide with the spatially uniform (depth-averaged) barotropic currents far from the topography. In the IT model of Griffiths & Grimshaw (Reference Griffiths and Grimshaw2007), $U^{(0)}$ is also used as a forcing in the case of a shelf without coastline.

2.3. The internal tide generation problem

Introducing $\phi ^{\#}=\phi -\varPhi$, we obtain from (2.8a–c) the BVP

(2.12a–c)\begin{equation} \mathcal{L}_{\mu} \phi^{\#} ={-}\mathcal{L}_{\mu}\varPhi,\quad \phi^{\#}(x,0)=0\quad\text{and}\quad \phi^{\#}(x,-h)= 0, \end{equation}

which shows that the barotropic flow $\varPhi$ forces a purely baroclinic response $\phi ^{\#}$. Alternatively, introducing $\phi ^{{\dagger} }=\phi ^{\#} +\varPhi ^{r}$ and exploiting the linearity of $\mathcal {L}_\mu$, the BVP (2.12a–c) becomes

(2.13a–c)\begin{equation} \mathcal{L}_{\mu} \phi^{{{\dagger}}} ={-}\mathcal{L}_{\mu}\varPhi^{(0)} ={-}\varPhi^{(0)}_{xx},\quad \phi^{{{\dagger}}}(x,0)=0\quad\text{and}\quad \phi^{{{\dagger}}}(x,-h)= 0, \end{equation}

which shows that the hydrostatic part of barotropic flow, $\varPhi ^{(0)}$, forces a baroclinic response plus a non-hydrostatic barotropic one (Garrett & Gerkema Reference Garrett and Gerkema2007). This formulation is referred to as the body-forcing formulation since the forcing appears only in the wave equation (Garrett & Gerkema Reference Garrett and Gerkema2007; Garrett & Kunze Reference Garrett and Kunze2007). An advantage of working with such a formulation is that the unknown field satisfies homogeneous Dirichlet conditions that make the application of a coupled-mode approach straightforward (§ 3). Here, we shall proceed with (2.13a–c) mainly because $\varPhi ^{(0)}$ is given by the simple explicit expression (2.10); $\varPhi ^{\text r}$ can be computed independently to extract the purely baroclinic field $\phi ^{\#}=\phi ^{{\dagger} }-\varPhi ^{r}$ if needed. Also, $\varPhi ^{\text r}$ results in a spatially trapped correction to the flow; namely, it vanishes as $x\rightarrow \pm \infty$ and thus does not influence the far-field energy. Equation (2.13a–c) is supplemented with radiation conditions ensuring that waves generated in the interior of the domain propagate outward as plane waves. Thus, we have

(2.14)\begin{equation} \phi^{{{\dagger}}} = \sum_{n=1}^{\infty}c^\pm_n\exp({\pm \text{i} k^{{\pm}}_n x})\sin \left(\frac{n{\rm \pi} z}{h_{{\pm}}}\right),\quad \text{with}\ k^{{\pm}}_n = \frac{n{\rm \pi}}{\mu h_{{\pm}}},\enspace c^{{\pm}}_n\in\mathbb{C},\quad \text{as}\ x\rightarrow\pm\infty. \end{equation}

It is useful to write down expressions for the flow fields in terms of the amplitudes of the stream functions. From the above analysis, $\phi = \varPhi ^{(0)}+\varPhi ^{r}+\phi ^{\#} =\varPhi ^{(0)} + \phi ^{{\dagger} }$ and we may introduce the corresponding definitions

(2.15)\begin{equation} \xi = \varXi^{(0)} + \varXi^{\text r} + \xi^\# = \varXi^{(0)} + \xi^{\dagger}, \end{equation}

where $\xi$ is a placeholder for any of $u$, $v$, $w$, $b$ or $p$, and $\varXi$ is a placeholder for any of $U$, $V$, $W$, $B$ or $P$. The flow components appearing in (2.15) satisfy the relations

(2.16)\begin{equation} \begin{pmatrix} U^{{\diamond}} & V^{{\diamond}} & W^{{\diamond}} & B^{{\diamond}}\\ u^\star & v^\star & w^\star & b^\star \end{pmatrix}= \text{Re}\left\{\begin{pmatrix} -\varPhi^{{\diamond}}_z & \dfrac{\text{i} f}\omega\varPhi^{{\diamond}}_z & \varPhi^{{\diamond}}_x & -\dfrac{\text{i} N^2}{\omega}\varPhi^{{\diamond}}_x\\ -\phi^\star_z & \dfrac{\text{i} f}\omega\phi_z^\star & \phi_x^\star & -\dfrac{\text{i} N^2}{\omega}\phi^\star_x \end{pmatrix}\exp({-\text{i} \omega t})\right\}, \end{equation}

where the superscript $\star$ stands for either ${\dagger}$ or $\#$, and the superscript $\diamond$ stands for either $(0)$ or ${r}$. The relations for $v^{\star },V^{\diamond }$ (respectively $b^{\star },B^{\diamond }$) follow from the second equation in (2.5a) (respectively (2.5c)) and the assumption that all fields have the same time periodicity. A similar expression can be derived for the pressure containing additionally boundary terms. For easy reference, we summarise the notation for the different flow fields in table 1. Plugging the second equality of (2.15) into (2.5)–(2.6a,b) and taking into account (A15)–(A18a,b) with $i=0$, we obtain

(2.17a)\begin{gather} u^{{{\dagger}}}_t - f v^{{{\dagger}}} ={-}p^{{{\dagger}}}_x, \quad v^{{{\dagger}}}_t + f u^{{{\dagger}}} = 0, \end{gather}

(2.17b)\begin{gather}w^{{{\dagger}}}_t ={-}p^{{{\dagger}}}_z + b^{{{\dagger}}} + \left(1-\frac{\omega^2}{N^2}\right)B^{(0)}, \end{gather}

(2.18)\begin{gather} b^{{{\dagger}}}_t + N^2 w^{{{\dagger}}} = 0, \end{gather}

(2.19)\begin{gather}u^{{{\dagger}}}_x + w^{{{\dagger}}}_z = 0, \end{gather}

(2.20a,b)\begin{gather}h_x u^{{{\dagger}}}(x,-h)+w^{{{\dagger}}}(x,-h) = 0,\quad w^{{{\dagger}}}(x,0) = 0. \end{gather}

In deriving (2.17b), we used the relation $B^{(0)} = N^2 W^{(0)}_t/ \omega ^2$, which itself is derived from (2.16). In terms of $\xi ^{\#}$, the above equations stay the same except for (2.17b), which becomes

(2.21)\begin{equation} w^{\#}_t ={-}p^{\#}_z + b^{\#} + B, \end{equation}

showing that the baroclinic flow is forced by the buoyancy force created by the barotropic flow (Garrett & Gerkema Reference Garrett and Gerkema2007).

Table 1. The notation for the fields we use. The symbol $\xi$ stands for $u$, $v$, $w$, $b$, $p$ and $\varXi$ for their capitalized versions. The relations in the third column hold for $\xi$ and $\varXi$ replaced by $\phi$ and $\varPhi$.

2.4. Energy equation and conversion rate

We derive here the energy equation for the above IT generation problem and use it to define the energy conversion rates.

The dot product of (2.17)–(2.18) with $(\boldsymbol {u}^{{\dagger} },b^{{\dagger} }/N^2) \equiv (u^{{\dagger} }, v^{{\dagger} }, w^{{\dagger} },b^{{\dagger} }/N^2)$ gives

(2.22)\begin{equation} \mathcal{E}^{\dagger}_t + \boldsymbol{\nabla}\boldsymbol{\cdot}\left(p^{\dagger}\boldsymbol{u}^{\dagger}\right) = \left(1-\frac{\omega^2}{N^2}\right)B^{(0)} w^{\dagger},\quad \text{with}\ \mathcal{E}^{\dagger}= \frac{1}{2}(\boldsymbol{u}^{{{\dagger}}})^2 +\frac{1}{2}\frac{(b^{{{\dagger}}})^2}{N^2},\end{equation}

where we have used (2.19) and where $\boldsymbol {\nabla } \equiv (\partial _x, 0,\partial _z )$. Integrating (2.22) over the domain $\varOmega = [-\infty,+\infty ]\times [-h,0]$, using the divergence theorem and (2.20a,b), we obtain

(2.23)\begin{equation} \left(\int_\varOmega \mathcal{E}^{\dagger} \,\mathrm{d} \varOmega\right)_t + \left[\int_{{-}h}^0 p^{{{\dagger}}} u^{{{\dagger}}} \,\mathrm{d} z\right]^{+\infty}_{-\infty} = \left(1-\frac{\omega^2}{N^2}\right)\int_{\varOmega}B^{(0)} w^{{{\dagger}}} \,\,\mathrm{d} \varOmega, \end{equation}

with $[\,{\cdot }\,]^{+\infty }_{-\infty }=\lim _{x\to \infty }({\cdot }) - \lim _{x\to -\infty }({\cdot })$.

Applying the time-average operator $\langle \,{\cdot }\,\rangle = 1/T \int _0^{\rm T} {\cdot } \,\mathrm {d} t$, with $T = 2{\rm \pi} /\omega$, and taking into account the periodicity of $\mathcal {E}^{{\dagger} }(t)$, we find

(2.24)\begin{equation} C_+-C_{-}\stackrel{\text{def}}=\left[\int_{{-}h}^0 \langle p^{{{\dagger}}} u^{{{\dagger}}}\rangle \,\mathrm{d} z\right]^{+\infty}_{-\infty}=\left(1-\frac{\omega^2}{N^2}\right)\int_{\varOmega} \langle B^{(0)} w^{{{\dagger}}}\rangle \,\mathrm{d} \varOmega \stackrel{\text{def}}=C_{int}, \end{equation}

where we have defined the energy conversion rates $C_{\pm }$ that represent the rates at which energy is radiated at $\pm \infty$, and the total energy convergence rate $C_{int}$ given as a volume integral. Note that the non-hydrostatic barotropic flow does not contribute in (2.24) because $u^{{\dagger} }=u^{\#}+U^r \rightarrow u^{\#}$ as $x\rightarrow \pm \infty$ and $\langle B^{(0)} w^{{\dagger} }\rangle = \langle B^{(0)} w^{\#}\rangle + \langle B^{(0)} W^r\rangle = \langle B^{(0)} w^{\#}\rangle$ since $B^{(0)}$ and $W^r$ are out of phase by ${\rm \pi} /2$ (2.16). Thus, (2.24) remains valid with ${\dagger}$ replaced by $\#$.

We proceed by expressing $C_{\pm }$ and $C_{int}$ in terms of $\phi ^{{\dagger} }$ and $\varPhi ^{(0)}$. Using integration by parts and the fact that $\psi ^{{\dagger} }(x,0) =\psi ^{{\dagger} }(x,-h)=0$, we obtain $\int _{-h}^0 \langle p^{{\dagger} } u^{{\dagger} }\rangle {\rm d} z = \int _{-h}^0 \langle p_z^{{\dagger} } \psi ^{{\dagger} }\rangle {\rm d} z$. Then, expressing $p_z^{{\dagger} }$ from (2.17b) using (2.16), we find

(2.25)\begin{equation} \langle p_z^{{{\dagger}}}\psi^{{{\dagger}}}\rangle = \frac{\omega^2-N^2}{\omega} \langle\text{Re}\{\text{i}\phi^{{{\dagger}}}_x \exp({-\text{i}\omega t})\}\text{Re}\{\phi^{{{\dagger}}} \exp({-\text{i}\omega t})\}\rangle\quad \text{as}\ x\rightarrow\pm\infty. \end{equation}

Writing $\exp ({-\text {i}\omega t}) = \cos \omega t - \text {i} \sin \omega t$ and noting that $\langle \cos \omega t\sin \omega t\rangle = 0$, $\langle \cos ^2\omega t\rangle = \langle \sin ^2\omega t\rangle = 1/2$, we obtain

(2.26)\begin{equation} C_{{\pm}} = \frac{\omega^2-N^2}{2\omega}\int_{{-}h}^0 \text{Im}\{\phi^{{{\dagger}}} \overline{\phi_x^{{{\dagger}}}}\}\,\,\mathrm{d} z,\quad \text{as}\ x\rightarrow\pm\infty, \end{equation}

where the overline denotes the complex conjugate and $\text {Im}$ the imaginary part. Similarly,

(2.27)\begin{equation} C_{int} = \left(1-\frac{\omega^2}{N^2}\right)\frac{N^2}{2\omega} \int_{\varOmega} \varPhi^{(0)}_x\text{Im}\{\overline{\phi^{{{\dagger}}}_x}\}\,\,\mathrm{d} \varOmega. \end{equation}

Using the radiation conditions (2.14) in (2.26), we see that $C_+\geq 0$ (respectively $C_-\leq 0$) as $x\rightarrow +\infty$ (respectively $x\rightarrow -\infty$). Equations (2.26) and (2.27) provide us with two ways of calculating the total conversion rate, either by using the the far-field baroclinic flow or by using a barotropic–baroclinic interaction term defined in the entire fluid domain. This fact will be used in § 4 for validation purposes.

3.1. Stream function modal representation

We reformulate the IT generation problem (2.13a–c)–(2.14) by representing $\phi ^{{\dagger} }$ as

(3.1)\begin{equation} \phi^{{{\dagger}}}(x,z) = \sum_{n=1}^{\infty}\phi_n(x)Z_n(z;x),\end{equation}

where $\{Z_n(z;x)\}_{n=0}^{\infty }$ are prescribed vertical basis functions with a parametric dependence on $x$ and $\{\phi _n(x)\}_{n=0}^{\infty }$ are unknown complex modal amplitudes to be determined. For the expansion (3.1) to be exact, the set $\{Z_n(z;x)\}_{n=0}^{\infty }$ must be complete. In the present constant stratification case, this set is obtained as the set of eigenfunctions of a Sturm–Liouville problem parametrised by $x$, also called the ‘reference waveguide’ (Brekhovskikh & Godin Reference Brekhovskikh and Godin1992),

(3.2a–c)\begin{equation} Z_{n,zz} + \frac{n^2{\rm \pi}^2}{h^2} Z_n = 0,\quad Z_n(0 ;x) = 0,\quad Z_n({-}h ;x)= 0. \end{equation}

The eigenfunctions $Z_n$ are given by

(3.3)\begin{equation} Z_n(z;x) = \sin\left(\frac{n{\rm \pi} z}{h}\right),\end{equation}

and satisfy the orthogonality relation $\int _{-h}^{0}Z_n Z_m \,{\rm d} z = h\delta _{nm}/2$, where $\delta _{nm}$ is the Kronecker delta. Note that (3.1) satisfies exactly and term by term the boundary conditions in (2.13a–c). It follows from (3.1) and (3.3) that the $\phi _n$ are defined by

(3.4)\begin{equation} \phi_n = \frac{2}{h}\int_{{-}h}^0\phi^{{{\dagger}}} Z_n \,{\rm d} z = \frac{2h^2}{{\rm \pi}^3 n^3}\left( \left[\phi^{{{\dagger}}}_{zz}Y_n\right]_{{-}h}^0 - \int_{{-}h}^0 \phi^{{{\dagger}}}_{z z z}Y_n \,\,\mathrm{d} z\right), \end{equation}

where $Y_n=\cos (n{\rm \pi} z / h)$ and the second equality is obtained after integrating by parts three times, which is allowed provided $\phi ^{{\dagger} }$ is sufficiently smooth, and using the boundary conditions in (2.13a–c). This shows that $\|\phi _n\|_{\infty }:=\max |\phi _n|=O(n^{-3})$ and that (3.1) converges uniformly in this case. Similar estimates are obtained for $\|\phi _{n,x}\|_{\infty }$ and $\|\phi _{n,xx}\|_{\infty }$ by adapting the procedure developed in Athanassoulis & Papoutsellis (Reference Athanassoulis and Papoutsellis2017, § 4) and suffice to establish the term-wise differentiability of (3.1) required for the exact modal reformulation of (2.13a–c).

Griffiths & Grimshaw (Reference Griffiths and Grimshaw2007) use a similar expansion for $u$ and derive a series representation for $w$ by using the incompressibility and the bottom-boundary conditions. The maximum decay rate in this case is $O(n^{-2})$ due to the non-vanishing of $u$ on the boundaries. Note also that in contrast to (3.1), the truncated version of this expansion does not satisfy term by term the bottom boundary condition. Kelly (Reference Kelly2016) uses two sets of eigenfunctions: one for $u$ and $p$, and the other for $w$ and $b$. In this approach, $w$ vanishes identically on the bottom which is not the case for arbitrary $h$. Consequently, this approach should be regarded as an approximation, see the discussion by Kelly & Lermusiaux (Reference Kelly and Lermusiaux2016) for more details. Despite (3.1) being limited to 2-D flows, its major advantage is that it satisfies the bottom boundary condition in (2.13c) exactly and term by term, and exhibits faster convergence in comparison with existing approaches.

3.2. The coupled-mode system

We proceed by projecting (2.13a–c)–(2.14) onto (3.3). Substituting (3.1) into (2.13a), multiplying with $Z_n$ and integrating over the interval $[-h(x),0]$, we find that $\{\phi _n(x)\}_{n=1}^{\infty }$ solves the following CMS, for $m\geq 1$:

(3.5)\begin{equation} \phi_{m,xx}+\frac{m^2{\rm \pi}^2}{\mu^2h^2}\phi_m+\sum_{n=1}^{\infty} \left[\frac{b_{mn}h_x}{h}\phi_{n,x} + \left(\frac{c_{mn}h_x^2}{h^2}+ \frac{d_{mn}h_{xx}}{h}\right)\phi_n\right]= 2 g_m h\left(\frac{1}{h}\right)_{xx}, \end{equation}

where $b_{mn}$, $c_{mn}$, $d_{mn}$ are $x$-independent coefficients given in Appendix B, and $g_m = Q (-1)^{m+1}/(m{\rm \pi} )$. Substituting (3.1) into (2.14), we obtain

(3.6)\begin{equation} \sum_{n=1}^{\infty}\phi_n(x)Z_n(x;z) = \sum_{n=1}^{\infty}c_n^\pm\exp({\pm} \text{i} k^{{\pm}}_n x)\sin \left(\frac{n{\rm \pi} z}{h_{{\pm}}}\right)\quad\text{as}\ x\rightarrow \pm\infty. \end{equation}

Multiplying both sides by $Z_q$, $q\geq 1$, integrating over $[-h,0]$ and taking into account that $Z_n\rightarrow \sin (n{\rm \pi} z/ h_{\pm })$ as $x\rightarrow \pm \infty$, we obtain $\phi _n = c_n^\pm \exp (\pm \text {i} k^{\pm }_n x)$, and therefore

(3.7)\begin{equation} \phi_{n,x} \mp\text{i} k^{{\pm}}_n\phi_n= 0,\quad\text{as}\ x\rightarrow {\pm}\infty. \end{equation}

Thus, (2.13a–c)–(2.14) are exactly reformulated as the CMS (3.5)–(3.7). Solving the latter, we may reconstruct the solution of the former by using (3.1). The energy conversion rates $C_{\pm }$ are evaluated by (2.26):

(3.8)\begin{equation} C_{{\pm}} = \frac{\omega^2-N^2}{2\omega}\frac{h}{2} \sum_{n=1}^{\infty}\text{Im} \{\phi_n^{{{\dagger}}}\overline{\phi^{{{\dagger}}}}_{\!\!\!n,x}\} ={\pm} \frac{N^2-\omega^2}{4\omega\mu}{\rm \pi} \sum_{n=1}^{\infty}n \phi_n^{{{\dagger}}}\overline{\phi^{{{\dagger}}}}_{\!\!\!n}\quad\text{as}\ x\rightarrow {\pm}\infty, \end{equation}

where (3.7) and the definition of $k_n^{\pm }$ in (2.14) have been used to obtain the second equality. The purely baroclinic field $\phi ^{\#}$ is computed using $\phi ^{\#} = \phi ^{{\dagger} }-\varPhi ^{r}$. For the computation of $\varPhi ^{r}$, we apply the same modal decomposition to (2.11a–c). The resulting CMS is given by (3.5) with $\mu ^2$ replaced by $-\mu _0^2$ and vanishing conditions at infinity, instead of (3.7); if $h_0/L\ll 1$, one could use instead the approximate asymptotic expression in (A8a,b). In the following subsection, we derive a perturbative solution of the CMS (3.5)–(3.7) for infinitesimal topography. For arbitrary topographies, we solve the CMS numerically (see § 3.4).

3.3. Infinitesimal topography solution

Let $h(x) = h_0 -\epsilon r(x)$ with $|\epsilon |\ll 1$ and $r=O(1)$, for some characteristic depth $h_0$. Introducing the asymptotic expansion $\phi _m =\sum _{i=0}^K\epsilon ^i\phi _m^{(i)}$ and Taylor-expanding $1/h$ in (3.5), we deduce that $\phi _m^{(0)}=0$ and that $\phi _m^{(1)}$ solves

(3.9)\begin{equation} \phi^{(1)}_{m,xx}+\ell_m^2\phi^{(1)}_m = 2 g_m \frac{r_{xx}}{h_0}\quad\text{for}\ m\geq 1, \end{equation}

with $\ell _m=m{\rm \pi} /(h_0\mu )$ and the radiation conditions in (3.7) with $k_m^+ = k_m^- =\ell _m$. Substituting the solution (C3) in (2.26), we obtain the conversion rate for weak topography,

(3.10)\begin{equation} C^{WTA} = \frac{F_0}{h_0^{2}}\sum_{n=1}^{\infty} |\hat{r}(\ell_n)|^2 \frac{n{\rm \pi}^2}{(\mu h_0)^2}\quad \text{with}\ F_0 = \frac{\left[(N^2-\omega^2)(\omega^2-f^2)\right]^{1/2}}{2{\rm \pi}\omega}U_0^2h_0^2, \end{equation}

where $\hat {r}(\xi ) = \int _{-\infty }^{+\infty }\exp (-\text {i} x\xi )r(s)\,{\rm d} s$ is the Fourier transform of $r$ (Appendix C). This is the formula given by Laurent et al. (Reference St. Laurent, Stringer, Garrett and Perrault-Joncas2003) (up to multiplication with $\rho _0$), which was first derived by Llewellyn Smith & Young (Reference Llewellyn Smith and Young2002) in the case of hydrostatic internal waves. Khatiwala (Reference Khatiwala2003) derived a different formula starting from a multi-frequency representation of the response fields in terms of Bessel functions. In the acoustic limit (recall discussion following (2.4)), the dominant contribution to the far-field energy comes from the fundamental frequency, say, $\omega _0$ (Bell Reference Bell1975). The quantity $J_1(U_0 k_{j1}/\omega _0)$ appearing in Khatiwala (Reference Khatiwala2003, (28)), where $J_1$ is the order-one Bessel function of the first kind, and $U_0$ and $k_{j1}$ are his barotropic tidal amplitude and wavenumber for the $j$th mode of the fundamental frequency, respectively, may be replaced by its asymptotic value $U_0k_{j1}/(2\omega _0)$ leading to our (3.10) with $\omega =\omega _0$.

3.4. Numerical solution for arbitrary topography

We truncate the infinite CMS (3.5)–(3.7) by keeping the first $M$ equations and replacing the infinite domain by a finite interval $X=[x_L,x_R]$ of length $L_X = x_R-x_L$. We discretise $X$ with a uniform spacing $\delta x$, $\{x_i$, $i=1,N_X\}$ and we approximate the derivatives using fourth-order finite differences up to the boundary points $x=x_L,x_R$. The corresponding formulae can be found in Papoutsellis et al. (Reference Papoutsellis, Yates, Simon and Benoit2019, Appendix C). We thus obtain a sparse square linear system of dimension $(N_X M)^2$ for the grid values of each modal amplitude $\phi _n(x_i)$, $i=1,\ldots,N_X$, $n=1,\ldots,M$, which we solve by means of a $LU$ decomposition. Using this solution, we reconstruct the field $\phi ^{{\dagger} }$ by means of a truncated version of (3.1). We then compute the baroclinic fields and conversion rates using (2.16) and (2.26). We also provide an estimation of the free-surface elevation due to the IT motion given by $\eta = [p^{\#}]_{z=0}/ g$, where $[p^{\#}]_{z=0}$ is the pressure induced on $z=0$ by the baroclinic flow (Appendix D).

4.1. Topographic profiles

We consider two ridge profiles, namely the ‘Gaussian’ $h=h_0-h_{G}$ and the ‘bump’ $h=h_0-h_{B}$, where

(4.1a,b)\begin{equation} h_{G}=\varLambda\exp\left(-\frac{x^2}{2L^2}\right)\quad\text{and}\quad h_{\text{B}}=\varLambda\exp\left(1-\frac{1}{1-x^2/L^2}\right)\mathbb{1}_{({-}L,L)}, \end{equation}

with $\mathbb {1}_{(-L,L)}=1\ \text {in}\ (-L,L)$ and $\mathbb {1}_{(-L,L)}=0$ otherwise. Note that the ‘bump’ profile has a compact support, in contrast with the Gaussian, and all its derivatives are continuous at $x = \pm L$; see figure 2. We also consider the case of a shelf connecting two different constant depths. The shelf profile is the same as that in Griffiths & Grimshaw (Reference Griffiths and Grimshaw2007, § 5), namely, $h=h_{S}$ with

(4.2)\begin{equation} h_{S} = \left\{\begin{array}{@{}l} h_-\quad\text{for}\quad x\leq 0,\\ h_-+ (h_+-h_-) \sin^2({\rm \pi} x/2L) \quad\text{for}\ 0\leq x\leq L,\quad \text{and}\\ h_+\quad\text{for}\ x\geq L. \end{array}\right. \end{equation}

We recall from § 2.1 that two important parameters will be considered, namely, the relative topography height $\delta$ and the criticality $\varepsilon$ of the bottom slope. We also recall that, for illustration purposes, $(\omega,U_0)$ corresponds to a typical $M_2$ tide, i.e. $(\omega,U_0) = (1.4\times 10^{-4}\,\text {s}^{-1}, 0.04\,\text {m}\,\text {s}^{-1})$, and that the Brunt–Väisälä and Coriolis frequencies are $N = 1.5\times 10^{-3}\,\text {s}^{-1}$ and $f = 10^{-4}\,\text {s}^{-1}$, respectively. For all ridges, the depth far from the topography is $h_0 = 3000$ m.

Figure 2. The three topographic profiles we consider (in the shelf case, we have used $h_- = 0$, $h_+ = \varLambda$).

5.1. Gaussian ridge

Here, we consider solutions of the CMS for a subcritical ($\varepsilon =0.8$) and a supercritical ($\varepsilon =1.2$) Gaussian ridge. In figure 4, we show the purely baroclinic stream function $\phi ^{\#}$, the energy density $\mathcal {E}^{\#}= (\boldsymbol {u}^{\#})^2/2 +(b^{\#})^2/2/N^2$, the reconstructed free-surface elevation and the body-forcing term $\varPhi ^{(0)}_{xx}$.

Figure 4. Gaussian ridges with $\delta =0.5$; (a–c) $\varepsilon =0.8$ and (d–f) $\varepsilon =1.2$. From top to bottom, non-dimensional baroclinic stream function $\psi ^{\#}/Q$, energy density $\mathcal {E}^{\#}/U_0^2$ and body-forcing term $\varepsilon ^2b^2\varPhi _{xx}^{(0)}/Q$. Free-surface elevation $3\times 10^4\eta /h_0$ is also shown in light blue. Black lines correspond to iso-energy curves of $\mathcal {E}^{\#}/U_0^2 = 0.36$. Magenta dashed lines correspond to characteristic lines of slope $\pm \mu ^{-1}$. Vertical dashed lines correspond to the critical points of the topography. Insets show the field $\phi ^{\#}_x+\phi ^{\#}_z/\mu$ in the region $[-30,+7] \times [-1.05,0]$. The colour maps used in this and the subsequent figures are from Thyng et al. (Reference Thyng, Greene, Hetland, Zimmerle and DiMarco2016).

We clearly observe the beam-like structure of ITs, which is finer and more intense in the supercritical case. In this case, the solution approximates a field that is continuous along the beams, where the energy density attains very large values (theoretically infinite). The free surface also transforms from smooth to cusp-like as $\varepsilon$ exceeds $1$. Such fine-scale large-amplitude features are regularised in the ocean via nonlinearity and viscosity, which are beyond the present theory. Nevertheless, such flow-field calculations are readily obtained using the CMS and give us useful information for observable quantities. For example, in the supercritical case, the global maximum on the horizontal velocity is attained at the point of intersection of the beams above the ridge. The following maxima are attained at the first points of reflection at the free surface. Right at these points, the free surface attains its maximum slope.

Interestingly, the solution varies considerably between the sloping regions and the flatter regions. This is due to the body-forcing term in (2.13a), which couples the baroclinic motion with the non-uniform barotropic current and scales with $(1/h)_{xx}$ (figure 4c,f). This effect becomes more apparent if we write the PDE (2.13a) as a first-order system,

(5.1)\begin{equation} \left(\phi^{{{\dagger}}}_x\mp\frac{1}{\mu}\phi^{{{\dagger}}}_z\right)_x\pm \frac{1}{\mu}\left(\phi^{{{\dagger}}}_x\mp\frac{1}{\mu}\phi^{{{\dagger}}}_z\right)_z ={-}\varPhi^{(0)}_{xx}. \end{equation}

Note that the above equation also holds with $\phi ^{{\dagger} }$ replaced by $\phi ^{\#}$ and $\varPhi ^{(0)}$ by $\varPhi$. Equation (5.1) shows that the solution is not constant along a given characteristic unless $\varPhi ^{(0)}_{xx}=0$. This is captured by our solution as an adjustment of the iso-energy curves near the ridge crest (figure 4b,e). In the insets of figure 4(b,e), we also plot the field $\phi ^{\#}_x+ \phi ^{\#}_z/\mu$ to highlight the variation of the beams. This effect seems to have gone unnoticed in the literature, although it is clearly present in the calculations of Stashchuk & Cherkesov (Reference Stashchuk and Cherkesov1991). It can also be inferred from the calculations of Baines (Reference Baines1973), even though he does not visualise flow fields in the physical domain.

5.2. A non-radiating ridge

For further validation, we reproduce a case investigated by Maas (Reference Maas2011, § 4.2, ‘Ridge’). Maas considered the dimensionless version of (2.13a–c), obtained by the scaling $(x,z,\phi ^{{\dagger} },h) = (L\tilde {x},L/\mu \tilde {z},Q\widetilde {\phi ^{{\dagger} }},L/\mu \tilde {h})$, and showed that it has an exact solution, denoted here by $\phi ^{ex}$, that radiates no energy at infinity for a specially constructed ridge; see also Wunsch & Wunsch (Reference Wunsch and Wunsch2022) for an alternative construction of non-radiating topographies. For the case in Maas (Reference Maas2011, (4.15)), which is very similar to a Gaussian ridge with $\varLambda =0.19$ and $L=e/\sqrt {2}$ in our (4.1a,b), the exact non-radiating solution $\phi ^{ex}$ is shown in figure 5(a). Our calculation, $\phi ^{{\dagger} }$, is in agreement with $\phi ^{ex}$ as shown in figure 5(b) where the difference $\phi ^{ex}-\phi ^{{\dagger} }$ is plotted. The field $\phi ^{ex}$ (or $\phi ^{{\dagger} }$) is a combination of a baroclinic and a non-hydrostatic barotropic response trapped near the ridge. The purely baroclinic response is shown in figure 5(c). The hydrostatic body-forcing term is shown in figure 5(d). The calculated non-dimensional energy conversion rate is of the order of $10^{-17}$.

Figure 5. The non-radiating ridge of Maas (Reference Maas2011); Non-dimensional (a) exact baroclinic response $\phi ^{ex}$ and (b) difference $\phi ^{ex}-\phi ^{{\dagger} }$, where $\phi ^{{\dagger} }$ is calculated using the CMS, (c) purely baroclinic response $\phi ^{\#}$ calculated with the CMS and (d) body-forcing term $\varPhi ^{(0)}_{xx}$. All variables are dimensionless.