2.1. Shallow water

We consider Poincaré waves propagating in rotating shallow water with flat bottom and a background geostrophic flow. The background velocity $\boldsymbol {U} = (U, V, 0)$ and height $H$ are related through the geostrophic balance

(2.3) \begin{equation} f \boldsymbol {e}_z \times \boldsymbol {U} = - g \boldsymbol {\nabla }_{\boldsymbol {x}} \Delta H, \end{equation}

where $f$ is the Coriolis frequency, $g$ the gravitational constant and $\boldsymbol {e}_z$ the unit vertical vector. Here we have introduced $\Delta H$ , the geostrophic perturbation to the mean height $\bar H$ such that $H = \bar H + \Delta H$ (see figure 1). The perturbation $\Delta H$ , and hence $\boldsymbol {U}$ , is assumed to vary slowly in time and space compared with the period and wavelength of the Poincaré waves. For completeness we verify that the action conservation (2.1) holds in this case in Appendix A.

Assuming that $\Delta H \ll \bar {H}$ , the frequency of waves with wavevector $\boldsymbol {k} = (k_1, k_2)$ can be approximated as

(2.4) \begin{equation} \omega = \big(f^2 + g (\bar {H} + \Delta H) k_h^2\big)^{1/2} + \boldsymbol {U} \boldsymbol {\cdot } \boldsymbol {k} \approx \omega _{0} + \frac {g \Delta H k_h^2}{2 \omega _{0}} + \boldsymbol {U} \boldsymbol {\cdot } \boldsymbol {k}, \end{equation}

where $k_h = (k_1^2 + k_2^2)^{1/2}$ is the wavenumber and $\omega _{0} = (f^2 + g \bar {H} k_h^2)^{1/2}$ is the intrinsic frequency for constant height $\bar {H}$ . Thus, there are two contributions to the frequency inhomogeneity,

(2.5) \begin{equation} \omega _{1} = \underbrace {\frac {g \Delta H k_h^2}{2 \omega _{0}}}_{\text{height fluctuation}} + \underbrace {\vphantom {\frac {g \Delta H k_h^2}{2 \omega _{0}}} \boldsymbol {U} \boldsymbol {\cdot } \boldsymbol {k}}_{\text{Doppler shift}}, \end{equation}

and hence two contributions to the scattering.

Figure 2.

Ratio $R_{{sw}}$ (solid lines) defined by (2.8), the estimated relative importance of height fluctuation and Doppler shift terms in the shallow-water system, plotted for different values of the Burger number $Bu$ against non-dimensionalised horizontal wavenumber $k_h/K_* \gt 1$ . Quasi-geostrophic flow is $Bu = 1$ . Burger numbers $Bu = 0.1,\ 0.25$ are associated with the planetary geostrophic regime. Dashed lines are the square of this ratio, $R^{\prime}_{{sw}}$ (3.29), which gives a more accurate relative importance ratio for the diffusion regime, as shown in § 3.1.

We now compare the relative size of the terms in (2.5) to argue that height fluctuation effects can be just as important as Doppler shift in ray tracing and induced diffusion. We introduce the characteristic background flow speed $U_*$ , characteristic horizontal wavenumber $K_*$ and the flow Burger number

(2.6) \begin{equation} Bu = \frac {g\bar {H}}{f^2} K_*^2. \end{equation}

Using the geostrophic balance (2.3) and expressing $\omega _{0}$ in terms of $Bu$ , the height fluctuation term scales like

(2.7) \begin{equation} \frac {g \Delta H k_h^2}{2 \omega _{0}} \sim \frac { U_* k_h^2 }{2 K_* (1 + Bu (k_h/K_*)^2)^{1/2}}. \end{equation}

Then, the ratio between the height fluctuation and Doppler shift terms is given by

(2.8) \begin{equation} R_{{sw}} = \frac {\text {height fluctuation}}{\text {Doppler shift}} \sim \frac {k_h/K_*}{2(1 + Bu \, (k_h/K_*)^2)^{1/2}}. \end{equation}

In figure 2, we plot the ratio $R_{{sw}}$ given by (2.8) against non-dimensionalised wavenumber $k_h/K_*$ for different realistic values of the Burger number: $Bu = O(1)$ corresponds to the quasi-geostrophic regime and $Bu \ll 1$ to the planetary geostrophic regime. We take the limit of (2.8) for large $k_h/K_*$ , a necessary condition for (2.1) to hold:

(2.9) \begin{equation} R_{{sw}} \rightarrow Bu^{-1/2}/2, \quad k_h/K_* \gg 1. \end{equation}

We see that for $Bu = 0.25$ , $R_{{sw}} \rightarrow 1$ and the height fluctuation and Doppler shift terms in (2.5) have the same magnitude. For $Bu = 1$ corresponding to quasi-geostrophic flow, $R_{{sw}} \rightarrow 0.5$ . For smaller $Bu$ associated with planetary geostrophy, height fluctuations dominate over Doppler shift. For larger $Bu$ , Doppler shift dominates over height fluctuations.

In § 3.1, we show that a better estimate for the relative importance of height fluctuation to Doppler shift effects in the diffusion regime is $R^{\prime}_{{sw}} = R_{{sw}}^2$ , which is also plotted in figure 2.

2.2. Three-dimensional Boussinesq

Chavanne et al. (Reference Chavanne, Flament, Luther and Gurgel2010) compare the effect of refraction through flow buoyancy gradients with that of Doppler shift on internal tide propagation. Our set-up is similar to theirs – we consider 3-D IGWs of wavevector $\boldsymbol {k} = (k_1, k_2, k_3)$ propagating in a geostrophic flow $\boldsymbol {U} = (U, V, 0)$ with buoyancy $B$ . Unlike the tides, our waves are not in hydrostatic balance.

We first consider waves propagating with no background flow buoyancy gradients. The dispersion relation is

(2.10) \begin{equation} \omega = \left (f^2 \cos ^2 \theta + N^2 \sin ^2 \theta \right )^{1/2} + \boldsymbol {U} \boldsymbol {\cdot } \boldsymbol {k}. \end{equation}

The intrinsic frequency is the first term, dependent on $\theta$ , the angle between $\boldsymbol {k}$ and the vertical, and $N$ is the buoyancy frequency. With uniform vertical buoyancy gradients, $N^2 = \bar {N}^2 = \text{const.}$ , the bare frequency coincides with the intrinsic frequency. To obtain (2.10), a WKB ansatz is substituted into the 3-D Boussinesq equations. We omit the full derivation, but it follows the same method as Appendix A for the shallow-water system. (See also derivations of action conservation with vertical buoyancy gradients by (i) Bretherton (Reference Bretherton1966) for gravity waves in a shear flow without the effect of rotation and (ii) Pan et al. (Reference Pan, Haley and Lermusiaux2021) for internal tides in hydrostatic balance.)

We make a rough argument for the inclusion of inhomogeneous vertical buoyancy gradients associated with the geostrophic flow in (2.10). Horizontal geostrophic balance and vertical hydrostatic balance lead to the thermal wind balance

(2.11) \begin{equation} \boldsymbol {f} \times \partial _{z}\boldsymbol {U} = \boldsymbol {\nabla }_{\boldsymbol {x},h} B. \end{equation}

Subscript $h$ indicates a purely horizontal gradient. This means that horizontal flow buoyancy gradients are induced by vertical shear. If the flow’s vertical shear is nonlinear, vertical buoyancy gradients are also induced.

In deriving (2.10), the buoyancy frequency $N^2$ only appears in the equation for wave buoyancy $b$ which is given, for negligible background flow buoyancy gradients, by

(2.12) \begin{equation} \partial _{t} b + \boldsymbol {U} \boldsymbol {\cdot } \boldsymbol {\nabla }_{\boldsymbol {x}} b + N^2 w = 0. \end{equation}

Here, $w$ is the vertical wave velocity and $N^2 = \bar {N}^2$ . Gradients in $\boldsymbol {U}$ have been neglected under the WKB ansatz (see e.g. Olbers Reference Olbers1981).

The full wave velocity is $\boldsymbol {u} = (u, v, w)$ . Introducing flow buoyancy gradients such that $B = B(\boldsymbol {x})$ , (2.12) becomes

(2.13) \begin{equation} \partial _{t} b + \boldsymbol {U} \boldsymbol {\cdot } \boldsymbol {\nabla }_{\boldsymbol {x}} b + \boldsymbol {u}_h \boldsymbol {\cdot } \boldsymbol {\nabla }_{\boldsymbol {x},h} B + N^2 w = 0, \end{equation}

where

(2.14) \begin{equation} N^2 = \bar {N}^2 + \partial _{z} B \end{equation}

and $\boldsymbol {u}_h = (u,v,0)$ is the horizontal wave velocity. This is (A5) of Appendix A of Chavanne et al. (Reference Chavanne, Flament, Luther and Gurgel2010). The vertical buoyancy gradient $\partial _{z}B$ acts as a perturbation to $\bar {N}^2$ . If horizontal gradients are neglected, then (2.10) remains the same with variable $N^2$ defined by (2.14).

Horizontal buoyancy gradients introduce $u$ and $v$ terms into the buoyancy equation. This changes the nature of the eigenvalue problem for $\omega$ . It can be shown, as discussed in Chavanne et al. (Reference Chavanne, Flament, Luther and Gurgel2010), that the perturbations to the total frequency (2.10) caused by these $u$ and $v$ terms are imaginary. This corresponds to growing and decaying wave modes which exchange energy with the background flow. This shear instability is because buoyancy is a non-conservative force in a baroclinic fluid (Jones Reference Jones2001). Following Bretherton (Reference Bretherton1966), Olbers (Reference Olbers1981) and Chavanne et al. (Reference Chavanne, Flament, Luther and Gurgel2010), we neglect these horizontal gradients which amounts to an assumption of large Richardson number and leave exploration of this instability and its interaction with wavevector diffusion to future work.

Accounting for weak flow-induced vertical buoyancy gradients, we expand the dispersion relation (2.10) with $N^2$ given by (2.14) to obtain

(2.15) \begin{equation} \omega = \underbrace {\vphantom {\frac {\partial _{z}B \sin ^2 \theta }{2\omega _0}}\left (f^2 \cos ^2 \theta + \bar {N}^2 \sin ^2 \theta \right )^{1/2}}_{\omega _{0}} + \underbrace {\frac {\partial _{z}B \sin ^2 \theta }{2\omega _0} + \boldsymbol {U} \boldsymbol {\cdot } \boldsymbol {k}}_{\omega _{1}}, \end{equation}

valid for small $\partial _{z}B/\bar {N}^2$ . We compare the size of the Doppler shift and buoyancy gradient terms in $\omega _{1}$ . We introduce the background flow aspect ratio $\alpha$ defined by

(2.16) \begin{equation} \alpha = \frac {L_*}{H_*} = \frac {K_{v*}}{K_*}, \end{equation}

for characteristic length scales $L_* = 1/K_*$ and $H_* = 1/K_{v*}$ in the horizontal and vertical, respectively. Using the thermal wind relation (2.11) and aspect ratio, the buoyancy fluctuation term scales like

(2.17) \begin{equation} \frac {\partial _{z} B \sin ^2 \theta }{2 \omega _{0}} \sim \frac {\alpha ^2 U_* K_* \sin ^2 \theta }{2 (\cos ^2 \theta + (\bar {N}^2/f^2)\sin ^2 \theta )^{1/2}}. \end{equation}

Thus, the ratio $R_{{B}}$ between the buoyancy fluctuation and Doppler shift terms is roughly

(2.18) \begin{equation} {R_{{B}}} = \frac {\text{buoyancy fluctuation}}{\text{Doppler shift}} \sim \frac {\alpha ^2 \sin \theta }{2(\cos ^2 \theta +(\bar {N}^2/f^2) \sin ^2 \theta )^{1/2}(k/K_*)}, \end{equation}

where $k = k_h/\sin \theta$ is the wavenumber magnitude. It is instructive to consider $R_{{B}}$ as a function of non-dimensionalised frequency and horizontal wavenumber, $\omega _0/f$ and $k_h/K_*$ :

(2.19) \begin{equation} {R_{{B}}} \sim \frac {1}{2} \, \frac {\alpha ^2}{\bar {N}^2/f^2 - 1} \, \frac {(\omega _{0}/f)^2 - 1}{\omega _{0}/f} \, \frac {1}{k_h/K_*} \approx \frac {1}{2} \, \left (\frac {\alpha }{\bar {N}/f}\right )^2 \, \frac {(\omega _{0}/f)^2 - 1}{\omega _{0}/f} \, \frac {1}{k_h/K_*}. \end{equation}

The second expression holds for $(\bar {N}/f)^2 \gg 1$ , which is true in both the atmosphere and ocean. At $\omega _0 = f$ , ${R_{{B}}} = 0$ . This is to be expected: buoyancy effects are absent for vertically propagating inertial waves. The ratio attains a maximum value of ${R_{{B}}} \sim \alpha ^2(2(\bar {N}/f)(k_h/K_*))^{-1}$ when $\omega _{0} = \bar {N}$ . The ratio decays as $(k_h/K_*)^{-1}$ . In the WKB regime we consider, $k_h/K_* \gg 1$ and so it appears justified to assume the Doppler shift term dominates. However, for large values of $\alpha$ or when considering the lower limit of the WKB regime, this may not be the case.

Figure 3 shows the ratio (2.19) for $(\bar {N}/f)^2 \gg 1$ and three values of $\alpha /(\bar {N}/f)$ , including $\alpha \sim \bar {N}/f$ , a realistic regime for geostrophic turbulence. Contours of ${R_{{B}}} = 0.1,\ 1$ and $10$ , corresponding to a negligible, balanced and dominant buoyancy term, respectively, are given for each value of $\alpha$ . The buoyancy term can be equal to or greater than the Doppler shift term, namely for high aspect ratio $\alpha$ , higher frequencies and lower wavenumbers.

Figure 3.

Ratio $R_{{B}}$ as given in (2.19), the relative importance of buoyancy fluctuation and Doppler shift terms in the full Boussinesq system, against non-dimensionalised frequency $\omega _0/f$ and horizontal wavenumber $k_h/K_*$ : (a) $\alpha = \bar {N}/(2f)$ , (b) $\alpha=\bar{N}/f$ and (c) $ \alpha = 3\bar {N}/f$ , with $(\bar {N}/f)^2 \gg 1$ . Contours are shown for ${R_{{B}}} = 0.1$ (dotted black), $1$ (solid black) and $10$ (dotted white).

3.1. Shallow water

To evaluate the shallow-water diffusivity, we first evaluate the correlation function (3.10), then find its Fourier transform and substitute into (3.13). We introduce the background flow stream function $\psi$ such that

(3.15a,b) \begin{equation} \boldsymbol {U} = \boldsymbol {\nabla }_{\boldsymbol {x}}^{\perp } \psi = (-\partial _{x_2} \psi , \partial _{x_1} \psi , 0) \quad \textrm {and} \quad g \Delta H = f \psi , \end{equation}

where $\boldsymbol {\nabla }_{\boldsymbol {x}}^{\perp } = (-\partial _{x_2},\partial _{x_1})$ is the skew gradient and we use geostrophic balance (2.3). Substituting this and the frequency (2.5) into (3.10) yields

(3.16) \begin{equation} \Lambda (\boldsymbol {y},\boldsymbol {k}, r)= \underbrace {\vphantom {\frac {f^2 k_h^4}{4 \omega _{0}^2}}-\big(\boldsymbol {k} \boldsymbol {\cdot } \boldsymbol {\nabla }_{\boldsymbol {y}}^{\perp }\big)^2\langle \psi (\boldsymbol {x}, t) \psi (\boldsymbol {x} + \boldsymbol {y}, t + r)\rangle }_{\text{Doppler shift}} \, + \, \underbrace {\frac {f^2 k_h^4}{4 \omega _{0}^2} \langle \psi (\boldsymbol {x}, t)\psi (\boldsymbol {x} + \boldsymbol {y}, t + r)\rangle }_{\text{height fluctuation}}. \end{equation}

For the Doppler shift term, the skew gradients have been moved through the ensemble average using integration by parts and the symmetry of $\boldsymbol {x}$ and $\boldsymbol {y}$ arguments. Notably, there are no cross terms in (3.16) and the height fluctuation and Doppler shift effects are uncoupled. This is because, using integration by parts under the ensemble average,

(3.17) \begin{align} \text {cross terms} &\propto \langle \psi (\boldsymbol {x} + \boldsymbol {y}, t + r) \boldsymbol {k} \boldsymbol {\cdot } \boldsymbol {\nabla }_{\boldsymbol {x}}^{\perp } \psi (\boldsymbol {x}, t) \rangle + \langle \psi (\boldsymbol {x}, t) \boldsymbol {k} \boldsymbol {\cdot } \boldsymbol {\nabla }_{\boldsymbol {x}}^{\perp } \psi (\boldsymbol {x} + \boldsymbol {y}, t + r) \rangle \notag \\ &\propto - \langle \psi (\boldsymbol {x}, t) \boldsymbol {k} \boldsymbol {\cdot } \boldsymbol {\nabla }_{\boldsymbol {x}}^{\perp } \psi (\boldsymbol {x} + \boldsymbol {y}, t + r) \rangle + \langle \psi (\boldsymbol {x}, t) \boldsymbol {k} \boldsymbol {\cdot } \boldsymbol {\nabla }_{\boldsymbol {x}}^{\perp } \psi (\boldsymbol {x} + \boldsymbol {y}, t + r) \rangle \notag \\ &= 0. \end{align}

The Fourier transform of the correlation function (3.16) is

(3.18) \begin{equation} \hat {\Lambda }(\boldsymbol {K}, \boldsymbol {k}, \mathit {\unicode{x1D6FA} }) = |\boldsymbol {k} \times \boldsymbol {K}|^2 E_\psi (\boldsymbol {K}, \mathit {\unicode{x1D6FA} }) \, + \, \frac {f^2 k_h^4}{4 \omega _{0}^2} E_\psi (\boldsymbol {K}, \mathit {\unicode{x1D6FA} }), \end{equation}

where

(3.19) \begin{equation} E_\psi (\boldsymbol {K}, \mathit {\unicode{x1D6FA} }) = \langle \hat {\psi }(-K, -\mathit {\unicode{x1D6FA} }) \hat {\psi }(K, \mathit {\unicode{x1D6FA} })\rangle \end{equation}

is the spectrum of the stream function. Defining the horizontal energy spectrum of the background flow as

(3.20) \begin{equation} E(\boldsymbol {K}, \mathit {\unicode{x1D6FA} }) = K_h^2 E_\psi /2, \end{equation}

we rewrite (3.18) as

(3.21) \begin{equation} \hat {\Lambda }(\boldsymbol {K}, \boldsymbol {k}, \mathit {\unicode{x1D6FA} }) = 2 k_h^2 \sin ^2 \gamma E(\boldsymbol {K}, \mathit {\unicode{x1D6FA} }) + \frac {f^2 k_h^4}{2 K_h^2 \omega _{0}^2} E(\boldsymbol {K}, \mathit {\unicode{x1D6FA} }), \end{equation}

where $K_h$ is the horizontal wavenumber of the background flow and $\gamma$ is the angle between $\boldsymbol {K}$ and $\boldsymbol {k}$ . Combined with (3.13), we have a diffusivity that accounts for height fluctuation effects for a time-dependent flow.

As proof of concept, we simplify to a time-independent flow. This (i) allows for inexpensive ray tracing simulations in § 3.2 and (ii) results in a simpler form for the diffusivity. Adding time dependence is possible, as done solely for the Doppler shift effect by Dong et al. (Reference Dong, Bühler and Smith2020) and – for the full 3-D set-up – by CKV (Reference Cox, Kafiabad and Vanneste2023). Combining the time-independent-flow limit of (3.21) with (3.14), we have

(3.22) \begin{align} {{\unicode{x1D60B}}}_{ij} = \, &2 \pi k_h^2 \int _{0}^{\infty } {\text {d}}K_h\int _{0}^{2\pi } {\text {d}}\gamma \, K_h K_i K_j \sin ^2 \gamma E(\boldsymbol {K}) \delta (\boldsymbol {K} \boldsymbol {\cdot } \boldsymbol {c})\notag \\ &+ \frac {\pi f^2 k_h^4}{2 \omega _{0}^2} \int _{0}^{\infty } {\text {d}}K_h\int _{0}^{2\pi } {\text {d}}\gamma \, \frac {K_i K_j}{K_h} E(\boldsymbol {K})\delta (\boldsymbol {K} \boldsymbol {\cdot } \boldsymbol {c}), \end{align}

in polar coordinates $\boldsymbol {K} = (K_h, \gamma )$ . Here $E(\boldsymbol {K})$ is the flow kinetic energy spectrum marginalised over frequencies. In the local polar basis $(\boldsymbol {e}_{k_h}, \boldsymbol {e}_{\phi })$ associated with $\boldsymbol {k}$ , this diffusivity has one non-zero component:

(3.23) \begin{equation} {{\unicode{x1D60B}}}_{\phi \phi } = \boldsymbol {e}_{\phi } \boldsymbol {\cdot } {\unicode{x1D63F}} \boldsymbol {\cdot } \boldsymbol {e}_{\phi }. \end{equation}

This is because the constant frequency surface in spectral space is the circle $\omega = \omega _{0}(k_h) + O(\epsilon ) = \text{const}$ . There is no diffusion of action across this circle because for a time-independent linear system, wave frequency does not change. This means there are no radial components of the diffusivity, only (3.23). We see this explicitly by expanding $\boldsymbol {K}$ as

(3.24) \begin{equation} \boldsymbol {K} = K_h \cos \gamma \boldsymbol {e}_{k_h} + K_h \sin \gamma \boldsymbol {e}_{\phi }. \end{equation}

Then, noting $\delta (\boldsymbol {K} \boldsymbol {\cdot } \boldsymbol {c}) \propto \delta (\cos \gamma )$ as $\boldsymbol {c} = \boldsymbol {\nabla }_{\boldsymbol {k}} \omega _0 (k_h) = c \boldsymbol {e}_{k_h}$ shows that any integrand containing $\cos \gamma$ integrates to zero (providing the energy spectrum $E(\boldsymbol {K})$ is well behaved). This is the case for all components of (3.22) bar (3.23), which, using (3.24), is

(3.25) \begin{align} {{\unicode{x1D60B}}}_{\phi \phi } = \, &2 \pi k_h^2 \int _{0}^{\infty } {\text {d}}K_h\int _{0}^{2\pi } {\text {d}}\gamma \, K_h^3 \sin ^4 \gamma E(\boldsymbol {K}) \delta (K_hc\cos \gamma )\notag \\ &+ \frac {\pi f^2 k_h^4}{2 \omega _{0}^2} \int _{0}^{\infty } {\text {d}}K_h\int _{0}^{2\pi } {\text {d}}\gamma \, K_h \sin ^2 \gamma E(\boldsymbol {K})\delta (K_hc \cos \gamma ). \end{align}

Assuming flow isotropy such that $E(\boldsymbol {K}) = E(K_h)$ , we integrate in $\gamma$ to get

(3.26) \begin{equation} {{\unicode{x1D60B}}}_{\phi \phi } = \underbrace {\frac {4 \pi k_h^2}{c \vphantom {\omega ^2_0}} \int _{0}^{\infty } K_h^2 E(K_h) \, {\text {d}}K_h}_{\text{Doppler shift}} \, + \, \underbrace {\frac {\pi f^2 k_h^4}{c \omega _{0}^2} \int _{0}^{\infty } E(K_h) \, {\text {d}}K_h}_{\text{height fluctuation}}. \end{equation}

With this single component, the diffusion equation (3.8) reduces to

(3.27) \begin{equation} \partial _{t} a + \boldsymbol {c} \boldsymbol {\cdot } \boldsymbol {\nabla }_{\boldsymbol {x}} a = \mu \partial _{\phi \phi } a, \end{equation}

where we define the directional diffusivity

(3.28) \begin{equation} \mu = {{\unicode{x1D60B}}}_{\phi \phi } / k_h^2. \end{equation}

To examine the relative importance of each term in (3.26), we consider the ratio between the two:

(3.29) \begin{equation} R^{\prime}_{{sw}} = \frac {\left [{{\unicode{x1D60B}}}_{\phi \phi }\right ]_{\text{height fluctuation}}}{\left [{{\unicode{x1D60B}}}_{\phi \phi }\right ]_{\text{Doppler shift}}} = \frac {f^2 k_h^2}{4 \omega _{0}^2} \frac {\int _{0}^{\infty } E(K_h) \, {\text{d}}K_h}{\int _{0}^{\infty } K_h^2 E(K_h) \, {\text {d}}K_h} \sim \frac {(k_h/K_*)^2}{4 (1 + Bu \, (k_h/K_*)^2)}. \end{equation}

Here, we approximate the integrals by

(3.30a,b) \begin{equation} 2 \pi \int _{0}^{\infty } K_h^2 E(K_h) \, {\text {d}}K_h \sim K_* U_*^2/2 \quad \text {and} \quad 2 \pi \int _{0}^{\infty } E(K_h) \, {\text {d}}K_h \sim U_*^2/(2 K_*) \end{equation}

using the characteristic speed and horizontal wavenumber of the flow. The final expression in (3.29) is the square of the scaling argument estimate $R_{{sw}}$ (2.8) which comes from the wave dispersion relation. This is sensible – the diffusivity (3.7) consists of a square frequency term. This means that in the diffusion regime, the scaling argument of § 2.1 is an underestimate for $R_{{sw}} \gt 1$ and an overestimate for $R_{{sw}} \lt 1$ . In figure 2, the adjustment from $R_{{sw}}$ to $R^{\prime}_{{sw}}$ is shown.

3.2. Shallow-water ray tracing

The 2-D diffusion equation (3.27) has an exact solution as outlined in § 4 of Villas Bôas & Young (Reference Villas Bôas and Young2020). We use this solution and ray tracing – a numerical method to find constant-wave-action trajectories – to assess the validity of the diffusion approximation and form of the diffusivity (3.26).

We start by multiplying the diffusion equation by $\cos \phi$ and integrating over the entire $(x, y)$ plane and $\phi \in (-\pi , \pi )$ to get

(3.31) \begin{equation} \frac {\text {d}}{{\text {d}}t} \iiint \cos \phi \, a \, {\text {d}}x\, {\text {d}}y\, {\text {d}}\phi = -\mu \iiint \cos \phi \, a \, {\text {d}}x\, {\text {d}}y\, {\text {d}}\phi . \end{equation}

Note that without pre-multiplying by $\cos \phi$ , the left-hand side is zero by action conservation. Here, the $\boldsymbol {c} \boldsymbol {\cdot } \boldsymbol {\nabla }_{\boldsymbol {x}} a$ term disappears under the $(x,y)$ integrals because the $a$ we consider is ensemble-averaged and we assume statistical spatial homogeneity in deriving the diffusion equation. Integrating with respect to time yields

(3.32) \begin{equation} \ln {(\langle \cos \phi \rangle _a)} = - \mu t + \ln {(\langle \cos \phi _0 \rangle _a)}, \end{equation}

where $\langle \cdot \rangle _a$ denotes the action-weighted ensemble average and $\phi _0$ is the initial angle. This is an alternative form of (4.3) in Villas Bôas & Young (Reference Villas Bôas and Young2020).

We confirm that (3.32) holds by ray tracing: solving the characteristic equations of the action conservation equation (2.1) numerically to give many constant-action trajectories and taking an ensemble average. These ray equations are

(3.33a,b) \begin{equation} \partial _t \boldsymbol {x} = \boldsymbol {\nabla }_{\boldsymbol {k}} \omega \quad \text {and} \quad \partial _t \boldsymbol {k} = -\boldsymbol {\nabla }_{\boldsymbol {x}} \omega . \end{equation}

We define non-dimensional quantities

(3.34a–d) \begin{equation} t^{\prime} = t f, \quad \boldsymbol {k}^{\prime} = \boldsymbol {k}/K_*, \quad \boldsymbol {x}^{\prime} = \boldsymbol {x} K_* \quad \text {and} \quad \boldsymbol {U}^{\prime} = \boldsymbol {U}/U_* \end{equation}

(and $\psi^{\prime} = \psi K_*/U_*$ ). Substituting the approximate frequency for shallow-water waves (2.4) into (3.33a,b ) gives

(3.35a) \begin{align} & \partial _{t^{\prime}} \boldsymbol {x}^{\prime} =\ \ \frac {Bu}{{(1 + Bu k_h^{\prime 2})}^{1/2}}\boldsymbol {k}^{\prime} + \quad Ro \boldsymbol {U}^{\prime} \quad\quad \quad \quad +\quad Ro \frac {2 + Bu k_h^{\prime 2}}{2 {(1 + Bu k_h^{\prime 2})}^{3/2}} \psi^{\prime} \boldsymbol {k}^{\prime} , \end{align}

(3.35b) \begin{align} & \partial _{t^{\prime}} \boldsymbol {k}^{\prime} = \ \ 0 \qquad\qquad \quad\quad\ \ - \quad Ro \boldsymbol {\nabla }_{\boldsymbol {x}^{\prime}}\left (\boldsymbol {U}^{\prime} \boldsymbol {\cdot } \boldsymbol {k}^{\prime}\right ) \ - \quad Ro \frac {k_h^{\prime 2}}{2 (1 + Bu k_h^{\prime 2}) }\boldsymbol {U}^{\prime \perp },\\ & \qquad \quad \ \underbrace {\hphantom {\frac {Bu}{(1 + Bu k_h^{\prime 2})^{1/2}}\boldsymbol {k}^{\prime}}}_{\text{no flow}} \quad \quad \underbrace {\hphantom {Ro \boldsymbol {\nabla }_{\boldsymbol {x}^{\prime}}\left (\boldsymbol {U}^{\prime} \boldsymbol {\cdot } \boldsymbol {k}^{\prime}\right )}}_{\text{Doppler shift}} \quad \quad \underbrace {\hphantom {Ro \frac {2 + Bu k_h^{\prime 2}}{2 {(1 + Bu k_h^{\prime 2})}^{3/2}} \psi^{\prime} \boldsymbol {k}^{\prime}}}_{\text{height fluctuation}}\nonumber \end{align}

where

(3.36) \begin{equation} Ro = \frac {U_*K_*}{f} \end{equation}

is the flow Rossby number and we introduce $\boldsymbol {U}^\perp = (V, -U, 0)$ .

The rays propagate in a flow which is a snapshot of a fully evolved 2-D quasi-geostrophic Navier–Stokes simulation, solved using a pseudo-spectral method and Crank–Nicholson time-stepping with time step ${\text {d}}t = 0.01$ . The doubly periodic domain $\boldsymbol {x} \in [0, L]$ is discretised by $1024^2$ gridpoints. Viscosity is $\nu = 10^{-5}$ . The initial wavenumber distribution of the flow is Gaussian. We prescribe nine values of the Rossby radius of deformation $L_D = Bu^{1/2}/K_*$ and three values of the initial wavenumber $K_*$ to give $3^3$ flows. This gives final snapshots with varying $Ro$ and $Bu$ . A typical flow is shown in figure 4(a).

We perform ray simulations with the inhomogeneity $\omega _{1}$ given by (i) Doppler shift only; (ii) height fluctuation only; and (iii) both Doppler shift and height fluctuation.

We initialise $50^2$ rays equally distributed across the periodic $\boldsymbol {x}$ domain with a constant horizontal wavenumber $k_h \approx 32.2 K_*$ (within the WKB regime) and random initial angle. By giving the rays different initial angles, our rays sample more of the flow and the ensemble average approaches the exact solution more rapidly as the number of rays increases than it would with a constant initial angle. For simplicity, we consider the angle $\phi$ to be the angle of deviation from the ray’s initial angle meaning $\ln {(\langle \cos \phi _0 \rangle _a)} = 0$ . This is possible because of the flow’s statistical isotropy. The angle of deviation for each ray is calculated at each time step, and $\cos \phi$ is averaged over. We choose action $a=1$ for each ray so that $\langle \cdot \rangle _a$ and the ensemble average are identical. Some of the $3^3$ flows have very similar Burger or Rossby numbers; we average over these Burger and Rossby numbers to leave $3^2$ results with distinct Burger and Rossby numbers. We estimate $\mu$ , (3.28), the negative gradient of $\ln {(\langle \cos \phi \rangle _a)}$ against time $t$ with ${{\unicode{x1D60B}}}_{\phi \phi }$ computed from (3.26). We compare this with the simulation gradient.

An example ray tracing simulation is shown in figure 4 in physical and wavevector space. The Rossby number of the flow is the small parameter in § 3, $\epsilon \sim Ro \sim O(0.01)$ , and so the flow is weak and the rays are only slightly deflected from their initial angle of propagation. In $\boldsymbol {k}$ space, the small Rossby number characterises the thickness of the constant-frequency ring. The exact solution and simulation approximation of (3.32) are shown in figure 4(c). This confirms the validity of the diffusion approximation and of the formulas (3.26). As expected from these formulas, the height fluctuations and Doppler shift make comparable contributions to the spectral diffusivity.

Figure 4.

Ray trajectories satisfying the characteristic equations (3.35a , b ), propagating in the flow described in § 3.2 with $Ro = 0.03$ , $Bu = 0.32$ and $K_* = 3.10$ . (a) A sample of 10 rays in physical space superimposed on the flow vorticity field (darker sections indicate higher-magnitude vorticity). (b) A sample of 50 rays in spectral space initialised on a constant-frequency circle given by the dotted black line. (c) The exact solution (3.32) approximated by the rays’ ensemble average (solid lines) and the diffusivity (3.26) (dashed lines). Red lines are calculated solely with the height fluctuation terms in (3.26) and (3.35a , b ); green lines are solely Doppler shift and; blue lines are the entire system with both Doppler and height fluctuation effects. The vertical dashed line indicates $\tau$ , the point at which the gradient calculation for figure 5 begins.

The $Ro$ and $Bu$ dependence of $\mu$ can be seen from (3.26). We define the non-dimensional directional diffusivity $\tilde {\mu }$ as

(3.37) \begin{equation} \tilde {\mu } = \frac {Bu^{1/2 }}{f Ro^2}\mu \sim \underbrace {\frac {(\omega _{0}/f)}{Bu^{1/2} (k_h/K_*)}}_{\text{Doppler shift}} \, + \, \underbrace {\frac { (k_h/K_*) }{4 Bu^{1/2} (\omega _{0}/f)}}_{\text{height fluctuation}} \rightarrow \underbrace {\vphantom {\frac {(\omega _{0}/f)}{Bu^{1/2} (k_h/K_*)}}1}_{\text{D.s.}} \,+\, \underbrace {\vphantom {\frac {(\omega _{0}/f)}{Bu^{1/2} (k_h/K_*)}}(4Bu)^{-1}}_{\text{h.f.}}, \end{equation}

where the final limit holds for $(k_h/K_*)^2 \gg 1$ and we have used (3.30a,b ). The scaling of $\mu$ is chosen so that the Doppler shift component of $\tilde {\mu }$ is independent of $Bu$ and $Ro$ . Note that the ratio between height fluctuation and Doppler shift effects is in agreement with $R^{\prime}_{{sw}}$ , (3.29). In figure 5, $\tilde {\mu }$ is displayed for a range of Burger numbers. The Burger dependence of (3.37) is confirmed for each contribution and, because three separate Rossby numbers are used in the plot, so too is the Rossby dependence.

The characteristic flow velocity $U_*$ is chosen such that (3.30a,b ) holds exactly and $\tilde {\mu } = 1$ is the exact limit of the non-dimensionalised, Doppler shift directional diffusivity. Then the height fluctuation $\tilde {\mu }$ shown in figure 5 coincides with the ratio between the height fluctuation and Doppler shift effects $R^{\prime}_{{sw}}$ . We see good agreement between the height fluctuation $\tilde {\mu }$ computed exactly and from simulations and thus confirm our estimate of $R^{\prime}_{{sw}}$ given by (3.29).

In deriving the diffusivity (3.7), we approximate an integral between $0$ and $t$ by one between $0$ and $\infty$ and so there is a delay between the start of the ray simulation to the point at which the exact solution (3.32) is well approximated. Therefore, our $\mu$ estimates in figure 5 begin a short time $\tau /f$ after the simulations have begun, as indicated in figure 4(c).

Figure 5.

Non-dimensional directional diffusivity $\tilde {\mu }$ , as defined by (3.28) and (3.37), plotted against flow Burger number $Bu$ . Red lines correspond to the height fluctuation component of the directional diffusivity, green lines to the Doppler shift component and blue lines to the full directional diffusivity. Crosses indicate the estimates of $\tilde {\mu }$ from the $50^2$ -ray simulations and exact solution (3.32), pluses are computed from the diffusivity expression (3.26) and dotted lines are the power laws predicted by (3.37), fitted to the first of the exact data points. The nine different Burger numbers cover three sets of Rossby number, labelled $1$ ( $Ro \approx 0.023$ ), $2$ ( $Ro \approx 0.030$ ) and $3$ ( $Ro \approx 0.057$ ). The Doppler shift directional diffusivity $\tilde {\mu } \approx 1$ and so the height fluctuation directional diffusivity corresponds to the ratio between the two, $R^{\prime}_{{sw}}$ (3.29).

We stress that the good agreement in figures 4(c) and 5 between the $\mu$ of (3.28) computed from the diffusivity (3.26) and the $\mu$ of (3.32) estimated from the ensemble average of rays validates the diffusion approximation (3.27) of the action conservation equation (2.1) with inhomogeneities of the form (2.5).

3.3. Three-dimensional Boussinesq

The method to evaluate the general diffusivity (3.13) for a Boussinesq system with vertical buoyancy gradients is similar to that for the shallow-water system. We give the result here, providing a full derivation in Appendix B.1. For a Doppler-shift-induced diffusivity, CKV (Reference Cox, Kafiabad and Vanneste2023) show that it is well justified to assume a slowly evolving flow is time-independent. The same likely applies to a buoyancy-gradient-induced diffusivity, so we focus on the time-independent case.

The two non-zero components of the diffusivities for a Boussinesq system with vertical buoyancy gradients and dispersion relation (2.15) are

(3.38) \begin{align} & \displaystyle\left.\begin{array}{ll}{{\unicode{x1D60B}}}_{kk} & = \dfrac {4\pi k^3 \omega _{0} \sin ^2 \theta }{(\bar {N}^2 - f^2) |\cos ^5 \theta |}\nonumber\\[10pt]& \qquad \displaystyle\times \int _{0}^{\infty } \int _{\theta }^{\pi - \theta } K^3 \cos ^2 \mathit {\unicode{x1D6E9} } (\cot ^2 \theta - \cot ^2 \mathit {\unicode{x1D6E9} })^{1/2} E(K, \mathit {\unicode{x1D6E9} }) \, {\text {d}}K {\text {d}}\mathit {\unicode{x1D6E9} } \end{array} \right\} {\text{ Doppler shift}}\\[2pt] & \quad \displaystyle\left.\begin{array}{ll} & \quad\ \ + \dfrac {\pi k f^2 \sin ^2 \theta }{ \omega _{0} (\bar {N}^2 - f^2)|\cos ^3 \theta |}\\[10pt] & \!\!\qquad\quad \displaystyle\times \int _{0}^{\infty } \int _{\theta }^{\pi - \theta } \dfrac {K^5 \cos ^6 \mathit {\unicode{x1D6E9} } E(K, \mathit {\unicode{x1D6E9} })}{ \sin ^2 \mathit {\unicode{x1D6E9} } (\cot ^2 \theta - \cot ^2 \mathit {\unicode{x1D6E9} })^{1/2}} \, {\text {d}}K {\text {d}}\mathit {\unicode{x1D6E9} } \end{array} \right\} \text{buoyancy fluctuation} \end{align}

and

(3.39) \begin{align} & \!\!\displaystyle\left.\begin{array}{ll}{{\unicode{x1D60B}}}_{\phi \phi } & = \dfrac {4\pi k^3 \omega _{0} \sin ^4 \theta }{(\bar {N}^2 - f^2) |\cos ^5 \theta |}\\& \quad\displaystyle\times \int _{0}^{\infty } \int _{\theta }^{\pi - \theta } K^3 \sin ^2 \unicode{x1D6E9} (\cot ^2 \theta - \cot ^2 \mathit {\unicode{x1D6E9} })^{3/2}E(K, \mathit {\unicode{x1D6E9} })\, {\text {d}}K {\text {d}}\mathit {\unicode{x1D6E9} }\end{array} \right\} {\text{ Doppler shift}} \\[3pt] & \quad \displaystyle\left.\begin{array}{ll}& \quad\ + \dfrac {\pi k f^2 \sin ^4 \theta }{ \omega _{0} (\bar {N}^2 - f^2) |\cos ^3 \theta |}\\[9pt]& \!\!\!\qquad \displaystyle\times\! \int _{0}^{\infty }\!\! \int _{\theta }^{\pi - \theta }\!\!\! K^5 \cos ^4 \mathit {\unicode{x1D6E9} } (\cot ^2 \theta {-} \cot ^2 \mathit {\unicode{x1D6E9} })^{1/2} E(K,\mathit {\unicode{x1D6E9} }) {\text {d}}K {\text {d}}\mathit {\unicode{x1D6E9} } \end{array} \!\right\}\!\text{buoyancy fluctuation}, \end{align}

where we use polar coordinates $\boldsymbol {K} = (K, \gamma , \mathit {\unicode{x1D6E9} })$ for the background flow and $E$ is the flow kinetic energy spectrum, as defined in the shallow-water case (3.19)–(3.20). Here, $\gamma = \mathit {\Phi } - \phi$ is the angle between the horizontal components of $\boldsymbol {K}$ and $\boldsymbol {k}$ , with $\mathit {\Phi }$ the azimuthal angle of $\boldsymbol {K}$ . The Doppler shift and buoyancy fluctuation diffusivities are uncoupled, as with the inhomogeneities in the shallow-water system. This is true of a time-dependent background flow also (see Appendix B.1).

Only the kinetic energy spectrum of the flow appears in (3.38)–(3.39) because we assume thermal wind balance. More generally, the diffusivity will be expressed in terms of both potential and kinetic energy spectra.

The $(\cot ^2\theta -\cot ^2\mathit {\unicode{x1D6E9} })^{-1/2}$ factor in the integrand of the buoyancy fluctuation term of ${{\unicode{x1D60B}}}_{kk}$ behaves like $\delta ^{-1/2}$ a small distance $\delta$ from the singularities at $\mathit {\unicode{x1D6E9} } = \theta , \pi -\theta$ . Therefore, it is integrable and does not cause the diffusivity to diverge.

Only the part of the flow spectrum with wavevectors of polar angle $\mathit {\unicode{x1D6E9} } \in (\theta , \pi - \theta )$ contributes to the diffusivity integrals. Inertia–gravity wave diffusion is a subregime of IGW scattering. This triadic interaction occurs between an incident wave $\boldsymbol {k}$ and the background flow $\boldsymbol {K}$ , resulting in a scattered wave $\boldsymbol {k}^{\prime}$ of the same frequency. In wavevector space, $\boldsymbol {k} + \boldsymbol {K} = \boldsymbol {k}^{\prime}$ . As the incident wave and scattered wave are of the same frequency, $\boldsymbol {K}$ connects two points on the cone of constant frequency. Thus, the range of $\boldsymbol {K}$ ’s polar angle $\mathit {\unicode{x1D6E9} }$ is the range of angles connecting any two points on the cone, i.e. $(\theta , \pi - \theta )$ . This is demonstrated in figure 6. Significantly, waves of frequency $\omega _{0}(\theta )$ will not be scattered by flow fluctuations with wavevectors outside of these angle bounds, regardless of the flow’s energy.

Figure 6.

Schematic of the scattering interaction between an incident wave $\boldsymbol {k}$ and the background flow $\boldsymbol {K}$ resulting in a scattered wave $\boldsymbol {k}^{\prime}$ . The two waves have the same frequency $\omega _{0}(\theta )$ and thus the background flow connects two points on the constant-frequency cone. For an arbitrary scattering interaction, angle $\mathit {\unicode{x1D6E9} }$ of the flow’s wavevector is bounded between $\theta$ and $\pi - \theta$ , i.e. the angular range of a vector between any two points on the cone.

These invisible flow regions make it difficult to apply the scaling argument of § 2.2. The polar angle of the characteristic wavevector of the flow $\boldsymbol {K}_*$ does not necessarily lie within $(\theta , \pi -\theta )$ , in which case $R_{{B}}$ (2.18) computed at $\boldsymbol {K}_*$ is not meaningful. Instead, a dominant wavevector – the characteristic wavevector of the portion of the flow which scatters the waves – must be used.

We define ${R^{\prime}_{{B}}}$ as the ratio between the buoyancy fluctuation and Doppler shift diffusivites for radial and azimuthal components:

(3.40a,b) \begin{equation} {R^{\prime}_{B,kk}} = \frac {[{{\unicode{x1D60B}}}_{kk}]_{\text{buoyancy fluctuation}}}{[{{\unicode{x1D60B}}}_{kk}]_{\text{Doppler shift}}} \quad \text {and} \quad R^{\prime}_{B,\phi \phi } = \frac {[{{\unicode{x1D60B}}}_{\phi \phi }]_{\text{buoyancy fluctuation}}}{[{{\unicode{x1D60B}}}_{\phi \phi }]_{\text{Doppler shift}}}. \end{equation}

Both ratios scale like $(k_h/K_*)^{-2}$ through the diffusivity prefactors’ $k$ dependence. The frequency behaviour is more complicated, requiring the diffusivity integrals to be evaluated. In Appendix B.2, we show that ${R^{\prime}_{{B},kk}} \rightarrow 0$ for $\omega _{0}/f \rightarrow 1$ and for larger frequencies and high aspect ratios $\alpha$ (2.16), ${R^{\prime}_{{B},kk}} \sim (\omega _{0}/f)^{-2}$ . This means that for waves in the WKB regime with high aspect ratios and frequencies, vertical buoyancy gradients can be neglected.

We compute the ratio of diffusivity components for a typical quasi-geostrophic flow and find good agreement with the $-2$ power laws. The geostrophic energy spectrum used is extracted from a snapshot of the full Boussinesq simulation described in CKV (Reference Cox, Kafiabad and Vanneste2023) and is pictured in figure 7. For this spectrum, $N/f = 32.0$ and the aspect ratio, (2.16), $\alpha \approx 16.0$ .

Figure 7.

The geostrophic flow component $E$ , smoothed and scaled by a maximum value $E_M$ , of the full Boussinesq simulation in CKV (Reference Cox, Kafiabad and Vanneste2023). The horizontal and vertical flow wavenumbers ( $K_h, K_v$ ) are scaled by the characteristic wavevector $(K_h, K_v) = (K_*, \alpha K_*)$ – with $\alpha$ the aspect ratio of the flow (2.16) – marked by a white cross, at which the geostrophic energy is maximum.

Both ratios of diffusivities are shown in figure 8, computed with the energy spectrum of figure 7. In figure 9, for the radial diffusivity ratio, we plot cross-sections of constant frequency and horizontal wavenumber and find good agreement with the $-2$ power laws.

Figure 8.

Radial and azimuthal ratios $(a)$ $R^{\prime}_{{B},kk}$ and $(b)$ $R^{\prime}_{{B},\phi \phi }$ as given in (3.40a,b ), the ratio of buoyancy fluctuation and Doppler shift diffusivities for a snapshot of the full Boussinesq simulation of CKV (Reference Cox, Kafiabad and Vanneste2023), against non-dimensionalised frequency $\omega _0/f$ and horizontal wavenumber $k_h/K_*$ . Contours are shown for ${R_{{B}}}' = 0.01$ (dashed black), $0.1$ (solid black) and $1$ (dashed white).

Figure 9.

Cross-sections of radial ratio $R^{\prime}_{{B},kk}$ (3.40a ), as shown in figure 8, for constant $(a)$ $\omega _{0}/f$ and $(b)$ $k_h/K_*$ . The $(\omega _{0}/f/\cos ^2\theta )^{-2}$ prediction of Appendix B.2 is, for small $\theta$ , an $(\omega _{0}/f)^{-2}$ power law.

The values of ${{R_{{B}}}}_{,kk}$ and ${{R_{{B}}}}_{,\phi \phi }$ are, for the WKB regime of $k_h/K_* \gg 1$ , $\lesssim 0.1$ . Thus, at least for the waves in CKV (Reference Cox, Kafiabad and Vanneste2023), we predict that a weak vertical buoyancy gradient induces negligible spectral diffusion in the WKB regime.

Figure 8, a comparison of diffusivity magnitudes, is markedly different from the comparison of dispersion relation terms (figure 3). Although both predict a decrease in the buoyancy fluctuation effect as horizontal wavenumber increases, the scaling argument of § 2.2 predicts that the effect dominates for higher frequencies. The $(\omega _{0}/f)^{-2}$ power law predicted from diffusivities and confirmed for an example spectrum in figure 9 contradicts this. We attribute this conflicting prediction to the large part of the flow spectrum not contributing to the diffusivity at higher wave frequencies. This makes the scaling arguments of § 2.2 unreliable and calls for the exact evaluation of the diffusivities (3.38)–(3.39).

3.4. Forced equilibrium spectrum

We solve the diffusion equation (3.8) exactly in the time-independent case corresponding to the equilibrium spectrum obtained under constant forcing. Our aim is to assess how the $k^{\pm 2}$ power-law spectra obtained by KSV (Reference Kafiabad, Savva and Vanneste2019) when diffusion is solely caused by Doppler shift are modified when accounting for vertical buoyancy gradients. As in KSV (Reference Kafiabad, Savva and Vanneste2019), we focus on radial diffusion, assuming wave statistics independent of $\phi$ such that $\partial _{\phi } a = 0$ . To concisely contrast this work to theirs, we consider only the forced, equilibrium solution to (3.8). We consider the energy density for ease of interpretation, defined as $e(k;\theta ) = 2\pi k^2 \sin \theta \omega a(k;\theta )$ , such that $e \, \text {d}k\, \text {d}\theta$ is the energy contained in the box $[k, k + \text {d}k]$ and $[\theta , \theta + \text {d} \theta ]$ . Thus, ignoring unimportant prefactors on the right-hand side, (3.8) becomes

(3.41) \begin{equation} \partial _{k} \left (k^2 {{\unicode{x1D60B}}}_{k k} \partial _{k} \frac {e}{k^2} \right ) = - \delta (k - k_*). \end{equation}

The forcing in a circle at $k = k_*$ can be generalised via integration. The minus sign ensures a positive energy spectrum. Defining

(3.42) \begin{equation} Q = k^3[{{\unicode{x1D60B}}}_{kk}]_{\text{Doppler shift}} \quad \text {and} \quad P = k[{{\unicode{x1D60B}}}_{kk}]_{\text{buoyancy fluctuation}} \end{equation}

as the $k$ -independent parts of the Doppler shift and buoyancy fluctuation diffusivities, this simplifies to

(3.43) \begin{equation} Q\partial _{k} \left (\left ( k^5 + \beta k^3\right ) \partial _{k} \frac {e}{k^2} \right ) = - \delta (k - k_*), \end{equation}

where $\beta = P/Q$ . This equation has the piece-wise solution found, for example, by partial fractions:

(3.44) \begin{align} e(k) =\left \{ \begin{array}{ll} A\left (1 - \frac {k^2}{\beta } \ln {\left (1 + \frac {\beta }{k^2}\right )}\right ) + B k^2 & \textrm {for}\ 0 \lt k \lt k_* ,\\[3pt] C\left (1 - \frac {k^2}{\beta } \ln {\left (1 + \frac {\beta }{k^2}\right )}\right ) + D k^2 & \textrm {for} \ k \gt k_* .\end{array} \right . \end{align}

We require $e(k)$ is bounded as $k \rightarrow \infty$ which means $D = 0$ . We require zero energy at $k=0$ ; therefore $A = 0$ . Continuity at $k = k_*$ gives $B = C(1/k_*^2 - \ln {(1 + \beta /k_*^2)}/\beta )$ . The jump condition $[Q (k^5 + \beta k^3 ) \partial _{k} \frac {e}{k^2}]_{k_*^-}^{k_*^+} = -1$ gives $C = 1/(2\beta Q)$ . Thus,

(3.45) \begin{align} e(k) =\frac {1}{2 \beta Q}\left \{ \begin{array}{ll} \frac {k^2}{k_*^2} \left (1 - \frac {k_*^2}{\beta } \ln {\left (1 + \frac {\beta }{k_*^2}\right )}\right )& \textrm {for}\ 0 \lt k \lt k_* ,\\ 1 - \frac {k^2}{\beta } \ln {\left (1 + \frac {\beta }{k^2}\right )} & \textrm {for} \ k \gt k_* .\end{array} \right . \end{align}

Note that $Q \rightarrow 0$ is an artificial singularity introduced by the choice of factorisation and forcing in (3.43). The Doppler shift limit of $\beta \rightarrow 0$ is

(3.46) \begin{align} e(k) =\frac {1}{4 Q k_*^2}\left \{ \begin{array}{ll} (k/k_*)^2 & \textrm {for}\ 0 \lt k \lt k_* ,\\ (k_*/k)^2 & \textrm {for} \ k \gt k_* , \end{array} \right . \end{align}

which is exactly (4.1) of KSV (Reference Kafiabad, Savva and Vanneste2019). The buoyancy fluctuation limit of $\beta \rightarrow \infty$ is

(3.47) \begin{align} e(k) =\frac {1}{2 P}\left \{ \begin{array}{ll} (k/k_*)^2 & \textrm {for}\ 0 \lt k \lt k_* ,\\ 1 & \textrm {for} \ k \gt k_* .\end{array} \right . \end{align}

In figure 10, (3.45)–(3.47) are displayed for two values of $\beta /k_*^2$ .

Figure 10.

Forced equilibrium spectra (3.45)–(3.47) against non-dimensionalised wavenumber for two different values of $\beta /k_*^2 = P/(Qk_*^2)$ (3.42), the $k/k_*$ -independent ratio of buoyancy-induced and Doppler-shift-induced diffusivities. (a) Ratio $\beta /k_*^2 = 1$ corresponds to a diffusivity with equal contributions from buoyancy fluctuations and Doppler shift and (b) ratio $\beta /k_*^2 = 100$ corresponds to a buoyancy-fluctuation-dominated diffusivity. Diffusion by both effects is given in blue (3.45), whilst red and green lines are spectra derived from only the buoyancy and Doppler shift terms, respectively, given by (3.47) and (3.46).

The finite energy at $k \rightarrow \infty$ of (3.47) is unphysical. However, this solution is unstable in the sense that a vanishingly small Doppler shift contribution will result in $e(k) \rightarrow 0$ as $k\rightarrow \infty$ . This is because the limit of (3.45) as $k \rightarrow \infty$ , $\beta = o(k)$ is $e(k) \rightarrow 1/(4Qk^2)$ , i.e. the Doppler shift limit (3.46) for $k\gt k_*$ . This can be seen in figure 10(b). If a buoyancy fluctuation is present, so too is the Doppler shift term by the thermal wind balance (2.11) which means that the $k\gt k_*$ limit of (3.47) never occurs and the energy spectrum will always decay for large $k$ under Doppler shift.

Figure 10 shows how buoyancy fluctuations affect the spectrum amplitude, mainly for $k\lt k_*$ . For a Doppler-shift-dominated diffusivity ( $\beta \rightarrow 0$ ), this amplitude change is negligible. For $k \gt k_*$ , buoyancy fluctuations shallow the Doppler-shift-induced energy spectrum of KSV (Reference Kafiabad, Savva and Vanneste2019) for a small range of intermediate wavenumbers.

The scaling argument of § 2.2 and the ratio of radial diffusivity (3.38) predict the buoyancy-induced diffusivity decays to zero for large $k/K_*$ and thus the effect of vertical buoyancy gradients is small. Our findings in this section compound this prediction because (i) a small buoyancy-induced diffusivity has negligible impact on the wave energy spectrum for any wavenumber and (ii) any buoyancy-induced diffusivity has negligible impact on the wave energy spectrum for large $k_h/K_*$ .

4.1. Rotating shallow-water system

We relax the assumption of a flat bottom made in §§ 2.1, 3.1 and 3.2 by writing

(4.1) \begin{equation} \Delta H = \Delta H_1 + \Delta H_2, \end{equation}

where $\Delta H_1$ is induced by geostrophy and satisfies (2.3) and $\Delta H_2$ is the fluctuation to mean height due to bottom topography. This is shown in figure 11.

Figure 11.

Shallow-water set-up with bottom topography, a modified version of figure 1(b). The free surface is the solid blue line, the bottom is the solid black line and the horizontal dashed black lines enclose the constant mean height $\bar {H}$ . The wave perturbations are given by $h$ and the fluctuation in height due to geostrophy, the dashed-dotted line, is given by $\Delta H_1$ . The topographic variation in height is $\Delta H_2$ , which is a negative quantity at the labelled location in this figure.

As with the height fluctuation induced by geostrophy, we assume topography varies over longer length scales than the waves so that the WKB assumption holds and action is conserved (the WKB analysis of Appendix A does not rely on the flat-bottom assumption). The amplitude of topographic variation is small relative to the mean height so that the weak interaction assumption of statistical scattering is satisfied. The perturbation (2.5) to the bare frequency has an additional term induced by topography:

(4.2) \begin{equation} \omega _{1} = \underbrace {\frac {g \Delta H_1 k_h^2}{2 \omega _{0}}}_{\text{geostrophic height fluctuation}} + \underbrace {\frac {g \Delta H_2 k_h^2}{2 \omega _{0}}}_{\text{topography}} + \underbrace {\vphantom {\frac {g \Delta H k_h^2}{2 \omega _{0}}} \boldsymbol {U} \boldsymbol {\cdot } \boldsymbol {k}}_{\text{Doppler shift}}. \end{equation}

If the background flow and topography are uncorrelated, the diffusivity (3.26) has an additional term:

(4.3) \begin{equation} {{\unicode{x1D60B}}}_{\phi \phi } = \underbrace {\frac {4 \pi k_h^2}{c \vphantom {\omega ^2_0}} \int _{0}^{\infty }\!\! K_h^2 E(K_h) \, {\text {d}}K_h}_{\text{Doppler shift}} \, + \, \underbrace {\frac {\pi f^2 k_h^4}{c \omega _{0}^2} \int _{0}^{\infty }\!\! E(K_h) \, {\text {d}}K_h}_{\text{geostrophic height fluctuation}} + \, \underbrace {\frac {\pi g^2 k_h^4}{2 c \omega _{0}^2} \int _{0}^{\infty }\!\! K_h^2\mathcal {B}(K_h) \, {\text {d}}K_h}_{\text{topography}}, \end{equation}

where $\mathcal {B}$ is the bottom topography spectrum:

(4.4) \begin{equation} \mathcal {B}(\boldsymbol {K}) = \langle \widehat {\Delta H_2}(-\boldsymbol {K}) \widehat {\Delta H_2}(\boldsymbol {K})\rangle \end{equation}

which we assume to be horizontally isotropic such that $\mathcal {B}(\boldsymbol {K}) = \mathcal {B}(K_h)$ . The topography term in (4.3) is derived by replacing $E(K_h)$ in the height fluctuation term of (3.26) with $ (gK_h/f)^2 \mathcal {B}(K_h)/2$ , which comes from expressing $E(K_h)$ as a function of $\Delta H_1$ using geostrophic balance (2.3), then swapping $\Delta H_1$ for $\Delta H_2$ .

Should the background flow and topography be correlated, the diffusivity will contain geostrophic height fluctuation–topography and Doppler shift–topography cross terms.

4.2. Surface waves

Bretherton & Garrett (Reference Bretherton and Garrett1968) prove that wave action is conserved for a wide class of physical systems. One of the examples they consider is surface gravity waves propagating on a mean flow with negligible rotation effects. In this case, the absolute frequency of the waves is

(4.5) \begin{equation} \omega = \left (g k_h \tanh {(H k_h )} \right )^{1/2} + \boldsymbol {U} \boldsymbol {\cdot } \boldsymbol {k} \end{equation}

with $\boldsymbol {U} = (U, V, 0)$ the background surface velocity. As in figure 11, $H$ is the wave-free height of the water column including both topographic variations $\Delta H_2$ and other variations in the mean height $\Delta H_1$ . Unlike in previous sections, we do not necessarily attribute these variations to geostrophy.

Providing the fluctuations to mean height $\bar {H}$ are small,

(4.6) \begin{equation} \omega _{1} \approx \frac {k_h(gk_h - \omega _{0}^2)\Delta H}{2\omega _{0}} + \boldsymbol {U} \boldsymbol {\cdot } \boldsymbol {k}. \end{equation}

For small-amplitude, horizontally isotropic height variations which vary over longer length and time scales than the waves, the azimuthal diffusivity is

(4.7) \begin{equation} {{\unicode{x1D60B}}}_{\phi \phi } = \underbrace {\frac {4 \pi k_h^2}{c \vphantom {\omega ^2_0}} \int _{0}^{\infty } K_h^2 E(K_h) \, {\text {d}}K_h}_{\text{Doppler shift}} \, + \, \underbrace {\frac {\pi k_h^2(gk_h - \omega _{0}^2)^2}{2 c \omega _{0}^2} \int _{0}^{\infty } K_h^2\mathcal {H}(K_h) \, {\text {d}}K_h}_{\text{height fluctuations}}, \end{equation}

which we find by comparing the surface wave dispersion relation (4.5) with that of shallow-water waves (4.2) and switching $g^2 k_h^4 \mathcal {B}(K_h)$ for $k_h^2(gk_h - \omega _{0}^2)^2\mathcal {H}(K_h)$ in the topography-induced diffusivity of the rotating shallow-water system (4.3). Here, $\mathcal {H}$ is the spectrum of fluctuations in total water depth. The diffusivity (4.7) is valid for a background velocity $\boldsymbol {U}$ not correlated with topography or other height-fluctuation effects.

The expressions given in this section are for diffusivities induced by an approximately time-independent $\omega _{1}$ which is clearly valid for bottom topography. Tolman (Reference Tolman1990) considers surface waves propagating through temporally and spatially varying tidal currents which induce variations in depth. We refer the interested reader to CKV (Reference Cox, Kafiabad and Vanneste2023) for work on the time-dependent case.

It should be possible to show that the topography-induced diffusivity in (4.7) is a limiting case of kinetic equations for Bragg scattering of surface gravity waves by bathymetry such as those derived by Hasselmann (Reference Hasselmann1966) and Ardhuin & Herbers (Reference Ardhuin and Herbers2002). We leave this proof to future work.