2 Equations used

To verify the far-field wave solution (1.2) we need the linearized equations outside the vortex core. The irrotational background velocity field $\boldsymbol{u}_{0}$ is $\boldsymbol{u}_{0}(r)\,=\,Ur_{0}r^{-1}\hat{\unicode[STIX]{x1D73D}}\,=\,\unicode[STIX]{x1D716}c_{0}r_{0}r^{-1}\hat{\unicode[STIX]{x1D73D}}$ where $\hat{\unicode[STIX]{x1D73D}}$ is the unit vector in the $\unicode[STIX]{x1D703}$ direction. The equations are most succinctly written in their Bernoulli form

(2.1) $$\begin{eqnarray}\displaystyle & \displaystyle \left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+\boldsymbol{u}_{0}\!\cdot \unicode[STIX]{x1D735}\right)\unicode[STIX]{x1D719}^{\prime }=-c^{2}\unicode[STIX]{x1D702}^{\prime }+\unicode[STIX]{x1D712}^{\prime }, & \displaystyle\end{eqnarray}$$

(2.2) $$\begin{eqnarray}\displaystyle & \displaystyle \left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+\boldsymbol{u}_{0}\!\cdot \unicode[STIX]{x1D735}\right)\unicode[STIX]{x1D702}^{\prime }=-\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D719}^{\prime }, & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D719}^{\prime }$ is the velocity potential for the irrotational wavemotion, $\boldsymbol{u}^{\prime }=\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}^{\prime }$ , say, while $\unicode[STIX]{x1D702}^{\prime }$ is the fractional layer-thickness or density disturbance, in the shallow water or acoustical interpretation respectively, and $\unicode[STIX]{x1D712}^{\prime }$ is a prescribed oscillatory forcing potential. Such forcing is a convenient way of representing wave sources and sinks, as used in Reference Bühler and McIntyreBM03 and Reference BühlerB14. In the limiting cases of problem (ii) these sources and sinks will recede to infinity, leaving $\unicode[STIX]{x1D712}^{\prime }=0$ for all finite $(x,y)$ . Equation (2.2) is the linearized mass-conservation equation. Eliminating $\unicode[STIX]{x1D702}^{\prime }$ and noting that $(\unicode[STIX]{x2202}_{t}+\boldsymbol{u}_{0}\!\cdot \unicode[STIX]{x1D735})c=0$ , because $c$ is a function of $r$ alone, we have

(2.3) $$\begin{eqnarray}\left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+\boldsymbol{u}_{0}\!\cdot \unicode[STIX]{x1D735}\right)^{\!2}\unicode[STIX]{x1D719}^{\prime }-c^{2}\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D719}^{\prime }=\left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+\boldsymbol{u}_{0}\!\cdot \unicode[STIX]{x1D735}\right)\unicode[STIX]{x1D712}^{\prime }.\end{eqnarray}$$

Now if $\unicode[STIX]{x1D719}^{\prime }\propto \exp (\text{i}\unicode[STIX]{x1D6F7})$ with $\unicode[STIX]{x1D6F7}$ as in (1.2)–(1.3), we have a local wavenumber vector

(2.4) $$\begin{eqnarray}\boldsymbol{k}\;=\;\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F7}\;=\;k_{0}\hat{\boldsymbol{x}}-\unicode[STIX]{x1D6FC}r^{-1}\hat{\unicode[STIX]{x1D73D}}+O(\unicode[STIX]{x1D716}^{2}k_{0}r_{0}^{2}r^{-2})\;=\;k_{0}\{\hat{\boldsymbol{x}}-\unicode[STIX]{x1D716}r_{0}r^{-1}\hat{\unicode[STIX]{x1D73D}}+O(\unicode[STIX]{x1D716}^{2}r_{0}^{2}r^{-2})\},\end{eqnarray}$$

where $\hat{\boldsymbol{x}}$ is the unit vector in the $x$ direction and where the error term has been assumed to have length-scale ${\gtrsim}k_{0}^{-1}$ . We note that $\boldsymbol{k}$ is slightly rotated away from the $x$ direction, pointing slightly into the background flow, as already seen in figure 2 where the wavecrests are rotated away from the $y$ direction. So

(2.5) $$\begin{eqnarray}\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}^{\prime }\;=\;\text{i}\boldsymbol{k}\unicode[STIX]{x1D719}^{\prime }\;=\;\text{i}k_{0}\left\{\hat{\boldsymbol{x}}-\unicode[STIX]{x1D716}r_{0}r^{-1}\hat{\unicode[STIX]{x1D73D}}+O(\unicode[STIX]{x1D716}^{2}r_{0}^{2}r^{-2})\right\}\unicode[STIX]{x1D719}^{\prime },\end{eqnarray}$$

and with $\boldsymbol{u}_{0}(r)\,=\,\unicode[STIX]{x1D716}c_{0}r_{0}r^{-1}\hat{\unicode[STIX]{x1D73D}}$ we have, noting that $\hat{\unicode[STIX]{x1D73D}}\!\cdot \hat{\boldsymbol{x}}\,=\,\sin \unicode[STIX]{x1D703}$ ,

(2.6) $$\begin{eqnarray}\left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+\boldsymbol{u}_{0}\!\cdot \unicode[STIX]{x1D735}\right)\unicode[STIX]{x1D719}^{\prime }\;=\;\text{i}k_{0}c_{0}\left\{-1+\unicode[STIX]{x1D716}r_{0}r^{-1}\sin \unicode[STIX]{x1D703}+O(\unicode[STIX]{x1D716}^{2}r_{0}^{2}r^{-2})\right\}\unicode[STIX]{x1D719}^{\prime }\end{eqnarray}$$

and

(2.7) $$\begin{eqnarray}\phantom{MMN}\left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+\boldsymbol{u}_{0}\!\cdot \unicode[STIX]{x1D735}\right)^{\!2}\unicode[STIX]{x1D719}^{\prime }\;=\;-k_{0}^{2}c_{0}^{2}\left\{1-2\unicode[STIX]{x1D716}r_{0}r^{-1}\sin \unicode[STIX]{x1D703}+O(\unicode[STIX]{x1D716}^{2}r_{0}^{2}r^{-2})\right\}\unicode[STIX]{x1D719}^{\prime },\end{eqnarray}$$

which equals $c_{0}^{2}\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D719}^{\prime }\,=\,-|\boldsymbol{k}|^{2}c_{0}^{2}\unicode[STIX]{x1D719}^{\prime }$ to the same accuracy and therefore satisfies (2.3) with $\unicode[STIX]{x1D712}^{\prime }\,=\,0$ , to that accuracy. The next order $O(\unicode[STIX]{x1D716}^{2}r_{0}^{2}r^{-2})$ fails to satisfy (2.3), because a contribution $-2k_{0}^{2}c_{0}^{2}\unicode[STIX]{x1D716}^{2}r_{0}^{2}r^{-2}\unicode[STIX]{x1D719}^{\prime }$ on the right of (2.7) disagrees with a contribution $-k_{0}^{2}c_{0}^{2}\unicode[STIX]{x1D716}^{2}r_{0}^{2}r^{-2}\unicode[STIX]{x1D719}^{\prime }$ to $c_{0}^{2}\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D719}^{\prime }$ , with no coefficient 2. At higher orders there are contributions from $\unicode[STIX]{x1D735}(\sin \unicode[STIX]{x1D703})$ that also disagree. We note in passing that, by contrast, $\exp (\text{i}\unicode[STIX]{x1D6F7})$ with no error term is an exact, and not merely a far-field asymptotic, solution to the Schrödinger equation of the original Aharonov–Bohm problem (details in appendix A). The Schrödinger equation (A 1) differs from (2.3) except in the limit $\unicode[STIX]{x1D716}\rightarrow 0$ (e.g. Stone Reference Stone2000a ).

The expression (2.4) for $\boldsymbol{k}$ is consistent with ray theory, as can be checked from Reference Bühler and McIntyreBM03 (4.12) or Reference BühlerB14 (14.5). Also of interest is the direction of the absolute group velocity

(2.8) $$\begin{eqnarray}\boldsymbol{C}^{\text{abs}}=\frac{c\boldsymbol{k}}{|\boldsymbol{k}|}+\boldsymbol{u}_{0}(r)=\frac{c_{0}\boldsymbol{k}}{|\boldsymbol{k}|}+\boldsymbol{u}_{0}(r)+O(\unicode[STIX]{x1D716}^{2}c_{0}r_{0}^{2}r^{-2}).\end{eqnarray}$$

Correct to $O(\unicode[STIX]{x1D716}c_{0}r_{0}r^{-1})$ , $\boldsymbol{C}^{\text{abs}}$ is parallel to $\hat{\boldsymbol{x}}$ , as can be checked by taking the $y$ components of (2.4) and of $\boldsymbol{u}_{0}(r)\,=\,\unicode[STIX]{x1D716}c_{0}r_{0}r^{-1}\hat{\unicode[STIX]{x1D73D}}$ . Propagation due to the $y$ component of $\boldsymbol{k}$ cancels advection due to the $y$ component of $\boldsymbol{u}_{0}$ . The cancellation follows also from the irrotationality of the background flow outside the vortex core, in virtue of the curl-curvature formula of ray theory (Reference BühlerB14 p. 86).

We avoided relying on ray theory here, when verifying (1.2), because integrating the ray equations over large distances might, conceivably, accumulate significant errors in $\unicode[STIX]{x1D6F7}$ , giving incorrect values for the Aharonov–Bohm phase jump, whereas the error $O(\unicode[STIX]{x1D716}^{2}r_{0}^{2}/r^{2})$ in (1.2) is small enough to rule out any such accumulation. A convenient corollary is that (1.2) can be used in problem (i) as well as in problem (ii), because when ray theory is valid it is permissible, correct to $O(\unicode[STIX]{x1D716})$ , to replace the constant amplitude $A$ by a $y$ -dependent amplitude envelope that restricts the wavetrain appropriately, as sketched in figure 1(a), again using the $O(\unicode[STIX]{x1D716})$  property $\boldsymbol{C}^{\text{abs}}\Vert \hat{\boldsymbol{x}}$ just noted (as contrasted with the $O(\unicode[STIX]{x1D716}^{2})$ bending of ray paths in figure 3).

Ray theory will, on the other hand, be sufficient for our treatment of problem (iii), in which the Aharonov–Bohm phase jump has no role. Details are postponed until § 8.

For the mean flow, a natural and efficient framework for solving problems (i)–(iii) is that of generalized Lagrangian-mean (GLM) theory, as laid out for instance in Reference BühlerB14. However, except where stated otherwise the reader unfamiliar with GLM theory can read the equations as involving, to sufficient accuracy, only the Eulerian-mean velocity $\overline{\boldsymbol{u}}$ and the Stokes drift $\overline{\boldsymbol{u}}^{\text{S}}$ , or phonon current per unit mass. Whenever the $O(a)$  wavemotions are irrotational and describable by ray theory, the exact GLM pseudomomentum  $\unicode[STIX]{x1D5FD}$ per unit mass can be replaced by $\overline{\boldsymbol{u}}^{\text{S}}$ and the exact Lagrangian-mean velocity $\overline{\boldsymbol{u}}^{\text{L}}$ by $\overline{\boldsymbol{u}}+\overline{\boldsymbol{u}}^{\text{S}}$ , with error $O(a^{2}\unicode[STIX]{x1D716}^{2})$ ; see e.g. Reference BühlerB14, equations (10.15) and (10.17). Then the combination $\overline{\boldsymbol{u}}^{\text{L}}-\unicode[STIX]{x1D5FD}$ , which occurs frequently in the exact theory, can be read simply as  $\overline{\boldsymbol{u}}$ , and the exact mean vorticity $\widetilde{\unicode[STIX]{x1D74E}}$ defined by

(2.9) $$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D74E}}\;=\;\unicode[STIX]{x1D735}\!\times (\overline{\boldsymbol{u}}^{\text{L}}-\unicode[STIX]{x1D5FD})\end{eqnarray}$$

can be read simply as  $\overline{\unicode[STIX]{x1D74E}}$ , the Eulerian-mean vorticity. (The quantity $\widetilde{\unicode[STIX]{x1D74E}}$ is the simplest exact measure of mean vorticity. It arises from frozen-field distortions of the three-dimensional vorticity field by the wave-induced displacement field.) The relation $\unicode[STIX]{x1D5FD}=\overline{\boldsymbol{u}}^{\text{S}}$ is always valid sufficiently far from the vortex core, in all three problems, where ray theory is always valid. In problems (i) and (ii) we need only the vertical or $z$ component of (2.9).

The power and economy of the GLM formalism comes from Kelvin’s circulation theorem and its consequence, e.g. Reference BühlerB14 § 10.2.9, that $\overline{\boldsymbol{u}}^{\text{L}}$ exactly advects mean vorticities $\widetilde{\unicode[STIX]{x1D74E}}$ , or $\widetilde{\unicode[STIX]{x1D74E}}+\boldsymbol{f}$ in problem (iii), with $\boldsymbol{f}$ the vector Coriolis parameter. This will prove useful throughout our analyses. In problem (iii) it expresses Ursell’s insight into the anti-Stokes flow, in a succinct and natural way to be pointed out in § 8. The advection property is neatly summarized by the exact three-dimensional form of the non-dissipative equation for $\widetilde{\unicode[STIX]{x1D74E}}$ ; see e.g. Reference BühlerB14, equations (10.99) and (10.153). It is

(2.10) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D74E}}}{\unicode[STIX]{x2202}t}~-~\unicode[STIX]{x1D735}\!\times \{\overline{\boldsymbol{u}}^{\text{L}}\!\times (\,\boldsymbol{f}+\widetilde{\unicode[STIX]{x1D74E}})\}~=~0\end{eqnarray}$$

or alternatively

(2.11) $$\begin{eqnarray}\frac{\overline{\text{D}}^{\text{L}}\widetilde{\unicode[STIX]{x1D74E}}}{\text{D}t}~+~(\,\boldsymbol{f}+\widetilde{\unicode[STIX]{x1D74E}})\unicode[STIX]{x1D735}\!\cdot \overline{\boldsymbol{u}}^{\text{L}}~=~(\,\boldsymbol{f}+\widetilde{\unicode[STIX]{x1D74E}})\!\cdot \unicode[STIX]{x1D735}\overline{\boldsymbol{u}}^{\text{L}},\end{eqnarray}$$

where $\overline{\text{D}}^{\text{L}}/\text{D}t=\unicode[STIX]{x2202}/\unicode[STIX]{x2202}t+\overline{\boldsymbol{u}}^{\text{L}}\!\cdot \unicode[STIX]{x1D735}$ . If we now set $\boldsymbol{f}=0$ and apply the foregoing recipe to (2.10), replacing $\widetilde{\unicode[STIX]{x1D74E}}$ by  $\overline{\unicode[STIX]{x1D74E}}$ and $\overline{\boldsymbol{u}}^{\text{L}}$ by $\overline{\boldsymbol{u}}+\overline{\boldsymbol{u}}^{\text{S}}$ , then we get the approximate version of (2.10) known as the Craik–Leibovich equation:

(2.12) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D74E}}}{\unicode[STIX]{x2202}t}-\unicode[STIX]{x1D735}\!\times (\overline{\boldsymbol{u}}\!\times \overline{\unicode[STIX]{x1D74E}})=\unicode[STIX]{x1D735}\!\times (\overline{\boldsymbol{u}}^{\text{S}}\!\times \overline{\unicode[STIX]{x1D74E}}).\end{eqnarray}$$

The right-hand side of (2.12) is the curl of the Craik–Leibovich vortex force  $\overline{\boldsymbol{u}}^{\text{S}}\!\times \overline{\unicode[STIX]{x1D74E}}$ , which as mentioned earlier makes a local contribution $\overline{\boldsymbol{u}}^{\text{S}}\!\times \unicode[STIX]{x1D74E}_{0}$ to the effective force on the vortex in problem (ii), where $\unicode[STIX]{x1D74E}_{0}=\unicode[STIX]{x1D735}\!\times \boldsymbol{u}_{0}$ . Equation (2.12) was originally derived by Craik & Leibovich (Reference Craik and Leibovich1976), via a much longer route, to study another problem – the dynamics of Langmuir vortices – assuming incompressible flow $\unicode[STIX]{x1D735}\!\cdot \boldsymbol{u}=0$ and steady wave fields, and under the strong parameter restriction $a^{2}\sim \unicode[STIX]{x1D716}\ll 1$ , i.e. that all mean flows, whether wave-induced or pre-existing, have order of magnitude $O(a^{2})$ . The route via GLM just recalled, which is not only much shorter but also has wider validity, was first pointed out by Leibovich (Reference Leibovich1980).

To complete the mean-flow equations we need a mass-conservation equation. As usual in GLM, we define a mean two-dimensional density or layer depth $\tilde{h}$ such that the areal mass element $\propto \tilde{h}\text{d}x\,\text{d}y$ exactly; see equations (10.42)–(10.47) of Reference BühlerB14. Then mass conservation is expressed by

(2.13) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\tilde{h}}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D735}_{\!\text{H}}^{\phantom{L}}\!\cdot (\tilde{h}\overline{\boldsymbol{u}}_{\text{H}}^{\text{L}})=0,\end{eqnarray}$$

where suffix H denotes horizontal projection, on to the $xy$ plane, superfluous in the two-dimensional acoustical setting but needed in the shallow-water setting and in problem (iii). In all three problems, however, we shall find that (2.13) can be simplified to

(2.14) $$\begin{eqnarray}\unicode[STIX]{x1D735}_{\!\text{H}}^{\phantom{L}}\!\cdot \overline{\boldsymbol{u}}_{\text{H}}^{\text{L}}=0\end{eqnarray}$$

if we are willing to work to the lowest significant accuracy – the lowest that captures the remote-recoil effects to be discussed. This will keep the analysis extraordinarily simple yet able to illustrate the main points. Equation (2.14) will be justified shortly for problems (i) and (ii) and in § 8 for problem (iii), with error estimates. We can then define a streamfunction, $\tilde{\unicode[STIX]{x1D713}}$ say, such that the horizontal velocity components can be written as

(2.15) $$\begin{eqnarray}\overline{u}_{\text{H}}^{\text{L}}~=~-\frac{\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D713}}}{\unicode[STIX]{x2202}y}\quad \text{and}\quad \overline{v}_{\text{H}}^{\text{L}}~=~\frac{\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D713}}}{\unicode[STIX]{x2202}x}.\end{eqnarray}$$

The Bretherton flow has streamfunction

(2.16) $$\begin{eqnarray}\tilde{\unicode[STIX]{x1D713}}_{\text{B}}^{\phantom{i}}~=~\tilde{\unicode[STIX]{x1D713}}~-~\tilde{\unicode[STIX]{x1D713}}_{0},\end{eqnarray}$$

where $\tilde{\unicode[STIX]{x1D713}}_{0}$ is the streamfunction for the non-divergent velocity field $\boldsymbol{u}_{0}$ of the vortex flow.

To compute the Bretherton flows in problems (i) and (ii) to sufficient accuracy (see § 4), we need only (2.15)–(2.16) and the vertical component of (2.9). We have $\widetilde{\unicode[STIX]{x1D74E}}=\unicode[STIX]{x1D735}\!\times (\overline{\boldsymbol{u}}^{\text{L}}-\unicode[STIX]{x1D5FD})=\unicode[STIX]{x1D735}\!\times \boldsymbol{u}_{0}=\unicode[STIX]{x1D74E}_{0}$ , expressing irrotationality outside the vortex core. The vertical component of  $\unicode[STIX]{x1D735}\!\times (\overline{\boldsymbol{u}}^{\text{L}}-\boldsymbol{u}_{0})$ is just $\unicode[STIX]{x1D6FB}_{\!\text{H}}^{2}\tilde{\unicode[STIX]{x1D713}}_{\text{B}}^{\phantom{i}}$ . We therefore have

(2.17) $$\begin{eqnarray}\unicode[STIX]{x1D6FB}_{\!\text{H}}^{2}\tilde{\unicode[STIX]{x1D713}}_{\text{B}}^{\phantom{i}}~=~\hat{\boldsymbol{z}}\!\cdot \unicode[STIX]{x1D735}\!\times \unicode[STIX]{x1D5FD},\end{eqnarray}$$

where $\hat{\boldsymbol{z}}$ is the vertical unit vector. The right-hand side of (2.17) is known as soon as we know the wave pseudomomentum field $\unicode[STIX]{x1D5FD}$ , which as mentioned earlier can be identified with the $\overline{\boldsymbol{u}}^{\text{S}}$ field whenever the wavemotion is irrotational and ray theory applies.

For problem (iii) it will be shown in § 8 that we need only two modifications. First, the vorticity $\unicode[STIX]{x1D6FB}_{\!\text{H}}^{2}\tilde{\unicode[STIX]{x1D713}}_{\text{B}}^{\phantom{i}}$ must be replaced in the standard way by $(\unicode[STIX]{x1D6FB}_{\!\text{H}}^{2}-L_{\text{D}}^{-2})\tilde{\unicode[STIX]{x1D713}}_{\text{B}}^{\phantom{i}}$ , the quasigeostrophic potential vorticity (PV), of the Bretherton flow, where $L_{\text{D}}$ is the Rossby deformation length-scale $L_{\text{D}}=f^{-1}(gH)^{1/2}$ where $g$ is gravity and $f=|\,\boldsymbol{f}|$ . Second, we must replace $\unicode[STIX]{x1D5FD}$ , which for deep-water surface gravity waves is strongly $z$ -dependent, by its vertical average $\langle \unicode[STIX]{x1D5FD}\rangle$ . So in place of (2.17) we have simply

(2.18) $$\begin{eqnarray}(\unicode[STIX]{x1D6FB}_{\!\text{H}}^{2}-L_{\text{D}}^{-2})\tilde{\unicode[STIX]{x1D713}}_{\text{B}}^{\phantom{i}}~=~\hat{\boldsymbol{z}}\!\cdot \unicode[STIX]{x1D735}\!\times \langle \unicode[STIX]{x1D5FD}\rangle .\end{eqnarray}$$

The derivation of (2.18) involves some delicate arguments about the asymptotics and will be postponed until § 8 and appendix C. The elliptic operators in (2.17) and (2.18) illustrate, by implication, a generic property of Bretherton flows, that they can extend well outside the wavetrain where $\unicode[STIX]{x1D5FD}\neq 0$ . That is one way of seeing the generic nature of remote recoil.

We now justify replacing (2.13) by (2.14) for problems (i) and (ii). Among the errors thus incurred, the largest is $O(a^{2}\unicode[STIX]{x1D716}^{1})$ . It arises in problem (i), from the variation of wave amplitude $A$ across the wavetrain and illustrating, incidentally, a need to avoid textbook arguments for the near-incompressibility of low-Mach-number or low-Froude-number flows. Those arguments do not take into account the kinds of spatial heterogeneity that are possible here, especially in problem (i).

For our asymptotic regime we need to let $a\rightarrow 0$ and $\unicode[STIX]{x1D716}\rightarrow 0$ keeping $a\ll \unicode[STIX]{x1D716}$ , for a given geometry of the background flow and incident wave field. Where the vortex flow $\boldsymbol{u}_{0}(r)$ crosses the wavetrain, in problem (i), it encounters $\tilde{h}$  values that are reduced by $O(a^{2}\unicode[STIX]{x1D716}^{0})$  because of the Brillouin radiation stress in the wavetrain. And since $\unicode[STIX]{x1D735}_{\!\text{H}}^{\phantom{L}}\!\cdot \boldsymbol{u}_{0}=0$ , the resulting contribution to $\unicode[STIX]{x1D735}_{\!\text{H}}^{\phantom{L}}\!\cdot (\tilde{h}\overline{\boldsymbol{u}}_{\text{H}}^{\text{L}})$ in (2.13), which is neglected in going to (2.14), is just $\boldsymbol{u}_{0}\!\cdot \unicode[STIX]{x1D735}_{\!\text{H}}^{\phantom{L}}\tilde{h}$ to leading order, with magnitude $O(a^{2}\unicode[STIX]{x1D716}^{1})$  since $\boldsymbol{u}_{0}=$ $O(\unicode[STIX]{x1D716})$ . (This contribution is significant, however, at the greater accuracy required in the case of figure 3, as can be seen from (5.1) of Reference Bühler and McIntyreBM03, even though it will not be required in the present analyses.)

The $O(a^{2}\unicode[STIX]{x1D716}^{0})$  local reduction in $\tilde{h}$ within the wavetrain (set-down, in the shallow-water setting) is necessary to ensure that the $O(a^{2})$  sideways mean fluid acceleration vanishes. The sideways gradients of Brillouin radiation stress and $O(a^{2})$  mean pressure must cancel. Only the isotropic part of the Brillouin radiation stress is involved, the so-called ‘hard-spring’ contribution, unrelated to pseudomomentum fluxes, and equal to $\unicode[STIX]{x2202}\ln c/\unicode[STIX]{x2202}\ln h$ times wave-energy per unit area, where $h$ is layer depth or two-dimensional mass density (e.g. Brillouin Reference Brillouin1936; Reference BühlerB14 § 10.5.1).

3 Impulse and pseudomomentum

The impulse–pseudomomentum theorem applies to problems (i)–(iii) as well as to a more general set of problems involving multiple vortices and more complicated wave fields, with arbitrary wave source and sink regions. The theorem provides an elegant way of showing that the pseudomomentum rule is satisfied in all these problems. There is, however, a severe limitation. The theorem relies crucially on horizontal non-divergence, (2.14). So it applies only at the lowest significant order of accuracy. There is a challenge here since the limitation puts the case of figure 3 outside the scope of the theorem. As just pointed out, (5.1) of Reference Bühler and McIntyreBM03 shows that (2.14) is not accurate enough in that case; in fact (2.14) must then be replaced by the anelastic equation

(3.1) $$\begin{eqnarray}\unicode[STIX]{x1D735}_{\!\text{H}}^{\phantom{L}}\!\cdot (\tilde{h}\overline{\boldsymbol{u}}_{\text{H}}^{\text{L}})=0.\end{eqnarray}$$

Yet Reference Bühler and McIntyreBM03’s analysis shows that the pseudomomentum rule still holds, a point to which we will return.

Before proceeding, we revisit the thought-experiment in which the waves are removed and an artificial external force field $\boldsymbol{F}$ exerted on the vortex core $\unicode[STIX]{x1D714}_{0}(r)$ , producing a rate of change of Kelvin impulse. In order to make the vortex translate bodily without change of shape, with velocity $\boldsymbol{u}_{\text{tr}}$ , say, we need $\boldsymbol{F}=-\unicode[STIX]{x1D714}_{0}\hat{\boldsymbol{z}}\!\times \boldsymbol{u}_{\text{tr}}$ . (The curl of this force field is just that required to move the vortex core through the fluid at velocity $\boldsymbol{u}_{\text{tr}}$ , while the divergence sets up the dipolar pressure field required to produce the corresponding changes outside the core, where the velocity field is irrotational.) Being transverse to the vortex motion, the resultant force $\boldsymbol{R}$ has the character of a Magnus force:

(3.2) $$\begin{eqnarray}\boldsymbol{R}\;=\;\int \!\!\int \!\boldsymbol{F}\text{d}x\,\text{d}y\;=\;-\hat{\boldsymbol{z}}\!\times \boldsymbol{u}_{\text{tr}}\int \!\!\int \!\unicode[STIX]{x1D714}_{0}\text{d}x\,\text{d}y\;=\;-\unicode[STIX]{x1D6E4}\hat{\boldsymbol{z}}\!\times \boldsymbol{u}_{\text{tr}}.\end{eqnarray}$$

The Kelvin impulse is defined for our two-dimensional shallow water or acoustical domain as

(3.3) $$\begin{eqnarray}\boldsymbol{I}~=\;\int \!\!\int (\,y,-x)Q\text{d}x\,\text{d}y~=\;\int \!\!\int -\hat{\boldsymbol{z}}\!\times \boldsymbol{x}Q\text{d}x\,\text{d}y,\end{eqnarray}$$

(e.g. Batchelor Reference Batchelor1967, equation (7.3.7)) where $\boldsymbol{x}=(x,y)$ and where $Q$ is the vorticity, $Q=\unicode[STIX]{x1D714}_{0}$ in this case. When the vortex translates in response to $\boldsymbol{F}$ , the rate of change of $\boldsymbol{I}$ is just $\boldsymbol{R}$ ; cf. (3.7) below. In the corresponding thought-experiment for problem (iii), the same statements hold if $Q$ is redefined as the quasigeostrophic PV.

The impulse–pseudomomentum theorem makes the following assumptions, in addition to (2.14) and its consequences (2.15)–(2.18). The wave field together with its sources and sinks is taken to have finite extent, prior to taking any infinite-wavetrain limits that might be of interest, while the domain of integration is taken infinite so as to enclose within it the vortex core, or cores, as well as all the waves and their source and sink regions. It is assumed that the pseudomomentum field satisfies a two-dimensional equation of the form (see appendix D)

(3.4) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D5FD}}{\unicode[STIX]{x2202}t}~+~\unicode[STIX]{x1D735}_{\!\text{H}}^{\phantom{L}}\!\cdot \unicode[STIX]{x1D5D5}~=\;-\,(\unicode[STIX]{x1D735}_{\!\text{H}}^{\phantom{L}}\overline{\boldsymbol{u}}^{\text{L}})\!\cdot \unicode[STIX]{x1D5FD}~+~\pmb{{\mathcal{F}}},\end{eqnarray}$$

with vertical averaging understood in problem (iii). The first term on the right comes from wave refraction and scattering by the mean flow, and $\pmb{{\mathcal{F}}}$ is the rate of generation or absorption of pseudomomentum in the wave source and sink regions, per unit area. In the refraction term, $\unicode[STIX]{x1D5FD}$ contracts with $\overline{\boldsymbol{u}}^{\text{L}}$ and not with $\unicode[STIX]{x1D735}_{\!\text{H}}^{\phantom{L}}$ . On the left, the precise form of the pseudomomentum flux tensor $\unicode[STIX]{x1D5D5}$ is immaterial but we note for later reference that, wherever ray theory holds, we shall have, in Cartesian tensor notation, with $i$ and $j$ running from 1 to 2, the standard group-velocity property

(3.5) $$\begin{eqnarray}\mathsf{B}_{ij}~=~\mathsf{p}_{i}\,C_{j}^{\text{abs}},\end{eqnarray}$$

where $\boldsymbol{C}^{\text{abs}}$ is the absolute group velocity, $\boldsymbol{u}_{0}$ plus the intrinsic group velocity, $c\boldsymbol{k}/|\boldsymbol{k}|$ in problems (i) and (ii) as in (2.8), and $c\boldsymbol{k}/2|\boldsymbol{k}|$ in problem (iii). The divergence operator contracts with $\boldsymbol{C}^{\text{abs}}$ so that, in Cartesians, the $i$ th component of $\unicode[STIX]{x1D735}_{\!\text{H}}^{\phantom{L}}\!\cdot \unicode[STIX]{x1D5D5}$ is $\mathsf{B}_{ij,j}$ . The theorem states that

(3.6) $$\begin{eqnarray}\displaystyle \frac{\text{d}}{\text{d}t}\left(\boldsymbol{I}~+~\unicode[STIX]{x1D5E3}\right)~=\;\int \!\!\int \pmb{{\mathcal{F}}}\;\text{d}x\,\text{d}y,\end{eqnarray}$$

where $\unicode[STIX]{x1D5E3}=\int \!\int \unicode[STIX]{x1D5FD}\text{d}x\,\text{d}y$ , the total pseudomomentum, again with vertical averaging understood in problem (iii).

The proof begins by noting that $Q\text{d}x\,\text{d}y$ is materially invariant, so that

(3.7) $$\begin{eqnarray}\frac{\text{d}\boldsymbol{I}}{\text{d}t}~=~\!\!\int \!\!\int \frac{\overline{\text{D}}^{\text{L}}(\,y,-x)}{\text{D}t}Q\text{d}x\,\text{d}y~=~\!\!\int \!\!\int (\overline{v}_{\text{H}}^{\text{L}},-\overline{u}_{\text{H}}^{\text{L}})Q\text{d}x\,\text{d}y.\end{eqnarray}$$

From here on, with everything in two dimensions $(x,y)$ , we drop the suffixes H so that, for instance, $\unicode[STIX]{x1D735}_{\!\text{H}}^{\phantom{L}}$ will be denoted by $\unicode[STIX]{x1D735}$ . Then, recalling (2.9) and (2.15)–(2.18), we have

(3.8) $$\begin{eqnarray}\frac{\text{d}\boldsymbol{I}}{\text{d}t}~=~\!\!\int \!\!\int Q\unicode[STIX]{x1D735}\tilde{\unicode[STIX]{x1D713}}\,\text{d}x\,\text{d}y~=~\!\!\int \!\!\int \left\{(\unicode[STIX]{x1D6FB}^{2}-L_{\text{D}}^{-2})\tilde{\unicode[STIX]{x1D713}}-\,\hat{\boldsymbol{z}}\!\cdot \unicode[STIX]{x1D735}\!\times \unicode[STIX]{x1D5FD}\right\}\unicode[STIX]{x1D735}\tilde{\unicode[STIX]{x1D713}}\,\text{d}x\,\text{d}y,\end{eqnarray}$$

with $L_{\text{D}}$ finite in problem (iii) but infinite in problems (i)–(ii). Now $\unicode[STIX]{x1D6FB}^{2}\tilde{\unicode[STIX]{x1D713}}\unicode[STIX]{x1D735}\tilde{\unicode[STIX]{x1D713}}$ contributes nothing because, in Cartesians, its $i$ th component is $\tilde{\unicode[STIX]{x1D713}}_{,jj}\tilde{\unicode[STIX]{x1D713}}_{,i}=(\tilde{\unicode[STIX]{x1D713}}_{,j}\tilde{\unicode[STIX]{x1D713}}_{,i})_{,j}-{\textstyle \frac{1}{2}}(\tilde{\unicode[STIX]{x1D713}}_{,j}\tilde{\unicode[STIX]{x1D713}}_{,j})_{,i}$ , which integrates to zero. The integrated terms at infinity vanish because, if we consider a domain of integration having radius $r\rightarrow \infty$ , the integrated terms have integrands $O(r^{-2})$ in problems (i) and (ii) and $O(\exp (-2r/L_{\text{D}}))$ in problem (iii), from the vortex-only contributions. The Bretherton flows, being dipolar because of the $\unicode[STIX]{x1D735}$ on the right-hand sides of (2.17) and (2.18), decay at the same rate or faster as $r\rightarrow \infty$ . In problem (iii) we have the additional contribution $-L_{\text{D}}^{-2}\tilde{\unicode[STIX]{x1D713}}\unicode[STIX]{x1D735}\tilde{\unicode[STIX]{x1D713}}\;\propto \;{\textstyle \frac{1}{2}}\unicode[STIX]{x1D735}(\tilde{\unicode[STIX]{x1D713}}^{2})$ , which also integrates to zero because $\tilde{\unicode[STIX]{x1D713}}^{2}=O(\exp (-2r/L_{\text{D}}))$ . Therefore (3.8) reduces, in all three problems, to

(3.9) $$\begin{eqnarray}\frac{\text{d}\boldsymbol{I}}{\text{d}t}~=\,-\int \!\!\int (\hat{\boldsymbol{z}}\!\cdot \unicode[STIX]{x1D735}\!\times \unicode[STIX]{x1D5FD})\unicode[STIX]{x1D735}\tilde{\unicode[STIX]{x1D713}}\,\text{d}x\,\text{d}y.\end{eqnarray}$$

Upon exchanging the dot with the cross and then integrating by parts, using the finite extent of the wave field, we see that the right-hand side is equal to

(3.10) $$\begin{eqnarray}-\!\!\int \!\!\int \{(\hat{\boldsymbol{z}}\!\times \unicode[STIX]{x1D735})\!\cdot \unicode[STIX]{x1D5FD}\}\unicode[STIX]{x1D735}\tilde{\unicode[STIX]{x1D713}}\,\text{d}x\,\text{d}y~=~\!\!\int \!\!\int \{\unicode[STIX]{x1D5FD}\!\cdot (\hat{\boldsymbol{z}}\!\times \unicode[STIX]{x1D735})\}\unicode[STIX]{x1D735}\tilde{\unicode[STIX]{x1D713}}\,\text{d}x\,\text{d}y~=~\!\!\int \!\!\int (\unicode[STIX]{x1D735}\overline{\boldsymbol{u}}^{\text{L}})\!\cdot \unicode[STIX]{x1D5FD}\,\text{d}x\,\text{d}y\quad\end{eqnarray}$$

since $\hat{\boldsymbol{z}}\!\times \unicode[STIX]{x1D735}$ commutes with $\unicode[STIX]{x1D735}$ , and $\hat{\boldsymbol{z}}\!\times \unicode[STIX]{x1D735}\tilde{\unicode[STIX]{x1D713}}=\overline{\boldsymbol{u}}^{\text{L}}$ by (2.15), so that the integrand on the right is minus the refraction term in (3.4). On eliminating that term between (3.4) and (3.10), and noting that $\unicode[STIX]{x1D735}\!\cdot \unicode[STIX]{x1D5D5}$ integrates to zero, again because of the finite extent of the wave field, we arrive at (3.6).

The proof has no dependence on whether or not ray theory applies. It depends only on (2.14)–(2.18) and on the form of (3.4), not on any particular formulae for $\unicode[STIX]{x1D5FD}$ , $\unicode[STIX]{x1D5D5}$ and $\pmb{{\mathcal{F}}}$ . The group-velocity property (3.5) will, however, be useful when considering pseudomomentum budgets in detail, for problems (i)–(iii), because along with ray theory it always holds far from the vortex core, together with the approximation $\unicode[STIX]{x1D5FD}=\overline{\boldsymbol{u}}^{\text{S}}$ .

The case of figure 3 prompts the question: Can we replace (2.14) by (3.1) and still prove the theorem? After considerable effort, the author has been forced to the conclusion that we cannot. At higher orders in $\unicode[STIX]{x1D716}$ there is an incompatibility between the per-unit-mass basis of vorticity – being the curl of velocity rather than of mass transport – and the per-unit-volume, or per-unit-area, basis of conservation relations, and their refractive extensions such as (3.4), in which $\unicode[STIX]{x1D5FD}$ would need to be replaced by $\tilde{h}\unicode[STIX]{x1D5FD}$ in order to attain enough accuracy to be compatible with (3.1) (again see appendix D). But one cannot simply insert a factor $\tilde{h}$ into the integrand of (3.3), because $\tilde{h}Q\text{d}x\,\text{d}y$ is not materially invariant and the subsequent steps from (3.7) onward are invalidated.

It seems likely that this limitation is not a limitation of the pseudomomentum rule as such, but only a limitation of the Kelvin impulse concept. As is well known, the ability to replace momentum budgets by impulse budgets depends on incompressibility. Incompressibility is needed in order to banish to infinity the large-scale $O(a^{2})$  pressure-field adjustments in transient situations, including transient versions of thought-experiments like that associated with the Magnus relation (3.2).

Indeed, there is a variant of the case of figure 3 in which the pseudomomentum rule holds to all orders in $\unicode[STIX]{x1D716}$ , as was shown in Reference Bühler and McIntyreBM03 § 5.2. However, that result depended on keeping the mean flow exactly steady – over an infinite domain – by applying an artificial ‘holding force’ to the vortex core as described in § 1, and then taking account of the full momentum budget in the far field correct to $O(a^{2})$ . Conditions in the far field are greatly simplified by assuming exact steadiness everywhere. The result depends on the generic relation between the fluxes of momentum and pseudomomentum. As shown in GLM theory they differ only by an isotropic term (see appendix D), which can be balanced by changes in mean pressure. Whether that can lead to further generalization remains to be explored.

4 Bretherton flows and recoil forces for problem (i)

In this section we review Reference Bühler and McIntyreBM03’s leading-order results on recoil forces in problem (i), as a preliminary to the subsequent analyses of problem (ii). We confine attention to recoil forces computed from Bretherton flows, omitting Reference Bühler and McIntyreBM03’s ray-theoretic calculations of the $O(a^{2}\unicode[STIX]{x1D716}^{1})$  net pseudomomentum flux. The impulse–pseudomomentum theorem tells us that such calculations must give the same results, at this lowest order, as indeed they were found to do. However, as pointed out in Reference Bühler and McIntyreBM03, the Bretherton flows provide the simplest route to the results. Just as $\boldsymbol{R}$ in (3.2) can be computed correct to $O(a^{2}\unicode[STIX]{x1D716}^{1})$  from $\boldsymbol{u}_{\text{tr}}$ correct to $O(a^{2}\unicode[STIX]{x1D716}^{0})$ , because $\unicode[STIX]{x1D6E4}$ is an $O(\unicode[STIX]{x1D716})$  quantity, a wave-induced recoil can be computed correct to $O(a^{2}\unicode[STIX]{x1D716}^{1})$  from a Bretherton flow correct to $O(a^{2}\unicode[STIX]{x1D716}^{0})$ , that is, from a Bretherton flow computed for an unrefracted [sic] wavetrain.

To take advantage of this simplification, in problem (i), we need to consider a wavetrain of finite length. That is because of the curl-curvature formula mentioned below (2.8), with its implication that the absolute group velocity $\boldsymbol{C}^{\text{abs}}$ remains parallel to $\hat{\boldsymbol{x}}$ when wave refraction is computed correct to $O(\unicode[STIX]{x1D716})$ . It remains parallel despite the $O(\unicode[STIX]{x1D716})$  refraction effects illustrated in figure 2; recall the cancellation noted below (2.8). The $O(a^{2}\unicode[STIX]{x1D716}^{1})$  net pseudomomentum flux and recoil force, which depend entirely on $O(\unicode[STIX]{x1D716})$  wavecrest rotations of the kind illustrated in figure 2, therefore vanish when the wave source and sink are allowed to recede to infinity.

So to obtain the simplest, leading-order version of problem (i), following Reference Bühler and McIntyreBM03, we let the unrefracted wavetrain extend between wave source and sink regions centred at finite locations $(x,y)=(-X,Y)$ and $(+X,Y)$ , say, as sketched in figure 4 (heavy straight line). For consistency with Reference Bühler and McIntyreBM03’s use of ray theory we take $X,Y\gg W$ , and $W\gg k_{0}^{-1}$ , where $W$ is a width-scale for the wavetrain. In order to generate and absorb an approximately monochromatic wavetrain, the wave source and sink are prescribed, just as in Reference Bühler and McIntyreBM03, by taking the oscillatory forcing potential $\unicode[STIX]{x1D712}^{\prime }$ to be $\exp (\text{i}\boldsymbol{k}\!\cdot \boldsymbol{x}-\text{i}kct)$ times a slowly-varying forcing envelope whose length-scale is at least of the same order of magnitude as $W$ , and where the real part is understood. However, the forcing envelope scale is kept $\ll X,Y$ , allowing us to think of the source and sink regions as approximately localized. It is convenient to take $\boldsymbol{k}\!\cdot \boldsymbol{x}=k_{0}x$ in the source along with $kct=k_{0}ct$ , with constant $k_{0}$ .

The $O(a^{2}\unicode[STIX]{x1D716}^{0})$  Bretherton flow for the unrefracted wavetrain satisfies (2.17), with evanescence at infinity. It consists of the Stokes drift $\overline{\boldsymbol{u}}^{\text{S}}$ straight along the wavetrain, parallel to $\hat{\boldsymbol{x}}$ , together with irrotational return flows symmetrically on both sides, as sketched in figure 4. All the vorticity of this Bretherton flow comes from $\hat{\boldsymbol{z}}\!\cdot \unicode[STIX]{x1D735}\!\times \unicode[STIX]{x1D5FD}$ on the right-hand side of (2.17). Within the wavetrain but outside the wave source and sink, $\hat{\boldsymbol{z}}\!\cdot \unicode[STIX]{x1D735}\!\times \unicode[STIX]{x1D5FD}$ is simply minus the horizontal shear of the Stokes drift.

Figure 4.

Schematic of Bretherton-flow streamlines in problem (i), as analysed in Reference Bühler and McIntyreBM03 correct to lowest order $O(a^{2}\unicode[STIX]{x1D716}^{0})$  for the finite wavetrain whose ray path is shown by the heavy straight line. At this order the Stokes drift is non-divergent except within the wave source and sink regions. The waves propagate from a source on the left to a sink on the right.

In virtue of the incompressibility expressed by (2.14)–(2.16), the irrotational return flow outside the wavetrain, which advects the vortex core, is the same as if it were induced by a two-dimensional mass sink at the left end of the wavetrain and a mass source at the right end whose strengths are equal to the mass flow in the Stokes drift within the wavetrain. Omitting factors $\unicode[STIX]{x1D70C}$ , where $\unicode[STIX]{x1D70C}$ is fluid mass density, we define the source and sink strengths $\pm S$ as the volume fluxes in a layer of unit depth; thus

(4.1) $$\begin{eqnarray}S=\int \overline{u}^{\text{S}}(\,y)\text{d}y=\int \mathsf{p}_{1}(\,y)\text{d}y,\end{eqnarray}$$

with the integral taken across the wavetrain. The pseudomomentum within the wavetrain has been written correct to $O(a^{2}\unicode[STIX]{x1D716}^{0})$  as $\mathsf{p}_{1}(\,y)\,\hat{\boldsymbol{x}}$ . Using $W\ll X$ and $W\ll Y$ , again following Reference Bühler and McIntyreBM03, we can approximate the mass-source flow in problem (i) as radially outward from $(x,y)=(X,Y)$ at speed $S/[2\unicode[STIX]{x1D70B}\{(x-X)^{2}+{(\,y-Y)^{2}\}}^{1/2}]$ , and similarly the mass-sink flow as radially inward toward $(x,y)=(-X,Y)$ . When these flows are added vectorially we obtain the flow pattern sketched in figure 4; and the net velocity advecting the vortex core at $(x,y)=(0,0)$ is

(4.2) $$\begin{eqnarray}\overline{\boldsymbol{u}}^{\text{L}}(0,0)~=~\frac{S}{\unicode[STIX]{x1D70B}}\frac{X}{X^{2}+Y^{2}}(-\hat{\boldsymbol{x}})\end{eqnarray}$$

correct to $O(a^{2}\unicode[STIX]{x1D716}^{0})$ . Because of the Magnus relation this corresponds to a resultant recoil force $\boldsymbol{R}=-\unicode[STIX]{x1D6E4}\hat{\boldsymbol{z}}\!\times \overline{\boldsymbol{u}}^{\text{L}}(0,0)$ , i.e.

(4.3) $$\begin{eqnarray}\boldsymbol{R}~=~\frac{\unicode[STIX]{x1D6E4}S}{\unicode[STIX]{x1D70B}}\frac{X}{X^{2}+Y^{2}}(+\hat{\boldsymbol{y}})\end{eqnarray}$$

correct to $O(a^{2}\unicode[STIX]{x1D716}^{1})$ , where $\hat{\boldsymbol{y}}$ is the unit vector in the $y$ direction.

Notice that $\overline{\boldsymbol{u}}^{\text{L}}(0,0)$ and $\boldsymbol{R}$ tend toward zero in the formal limit $X\rightarrow \infty$ , cross-checking what was deduced from the direction of $\boldsymbol{C}^{\text{abs}}$ and consequent vanishing of the $O(a^{2}\unicode[STIX]{x1D716}^{1})$  net pseudomomentum flux and recoil force in that limit. As the wavetrain gets longer, the irrotational return part of the Bretherton flow becomes increasingly spread out in the $y$ direction, diluting its effect at $(x,y)=(0,0)$ .

5 Bretherton flows and recoil forces for problem (ii)

The dilution effect just pointed out is the easiest way of seeing the non-interchangeability of limits in problem (ii). The same dilution effect occurs for any wavetrain whose width $W$ is held fixed while its length $L=2X\rightarrow \infty$ . In that formal limit the magnitude of the return flow at any fixed point tends toward zero; and it remains zero if the formal limit $W\rightarrow \infty$ is taken subsequently. The vortex core is then advected solely by the Stokes drift $\overline{\boldsymbol{u}}^{\text{S}}(0,0)=\overline{\boldsymbol{u}}^{\text{S}}=\mathsf{p}_{1}\hat{\boldsymbol{x}}$ , which is now constant across the wavetrain, so that with $L\rightarrow \infty$ first in problem (ii), (4.2) and (4.3) are replaced by

(5.1) $$\begin{eqnarray}\overline{\boldsymbol{u}}^{\text{L}}(0,0)~=~\overline{\boldsymbol{u}}^{\text{S}}~=~\mathsf{p}_{1}\hat{\boldsymbol{x}}\end{eqnarray}$$

and

(5.2) $$\begin{eqnarray}\boldsymbol{R}~=~-\unicode[STIX]{x1D6E4}\mathsf{p}_{1}\hat{\boldsymbol{y}}.\end{eqnarray}$$

Not only the magnitudes but also the signs have changed. These results hold for arbitrary $k_{0}r_{0}$ , because in the unrefracted wavetrain we always have $\overline{\boldsymbol{u}}^{\text{S}}=\unicode[STIX]{x1D5FD}=\mathsf{p}_{1}\hat{\boldsymbol{x}}$ .  Notice that (5.2) is equal to the Craik–Leibovich vortex force  $\overline{\boldsymbol{u}}^{\text{S}}\!\times \overline{\unicode[STIX]{x1D74E}}=\overline{\boldsymbol{u}}^{\text{S}}\!\times \unicode[STIX]{x1D74E}_{0}$ integrated over the vortex core, corresponding to the Iordanskii force in the quantum fluids literature (e.g. Sonin Reference Sonin1997), with $\overline{\boldsymbol{u}}^{\text{S}}=\unicode[STIX]{x1D5FD}$ corresponding to the phonon current per unit mass. The relation to the Aharonov–Bohm effect is discussed in the next two sections.

In any other version of problem (ii) there will be two contributions, one from the Craik–Leibovich vortex force and the other from the return part of the Bretherton flow. In the opposite formal limit, with the width $W$ of the wavetrain going to infinity first, the two contributions cancel. The return flow is then uniform, and equal and opposite to the Stokes drift, being diluted only near the extremities $y\sim \pm {\textstyle \frac{1}{2}}W$ . In that formal limit, therefore, the recoil force $\boldsymbol{R}$ vanishes.

In all other versions of problem (ii) there is always some dilution, making the return flow weaker than the Stokes drift and keeping the sign of the recoil opposite to that in problem (i). For instance, consider the ‘square’ limit $L=W\rightarrow \infty$ . Then it is readily shown that

(5.3) $$\begin{eqnarray}\overline{\boldsymbol{u}}^{\text{L}}(0,0)~=~{\textstyle \frac{1}{2}}\overline{\boldsymbol{u}}^{\text{S}}(0,0)~=~{\textstyle \frac{1}{2}}\mathsf{p}_{1}\hat{\boldsymbol{x}}\end{eqnarray}$$

so that in place of (5.2) we have

(5.4) $$\begin{eqnarray}\boldsymbol{R}~=~-{\textstyle \frac{1}{2}}\unicode[STIX]{x1D6E4}\mathsf{p}_{1}\hat{\boldsymbol{y}}.\end{eqnarray}$$

More generally, the factor ${\textstyle \frac{1}{2}}$ is replaced by $\{1-\,2\unicode[STIX]{x1D70B}^{-1}\lim \arctan (W/L)\}$ .

To derive this last result, one can regard the wide wavetrain as made up of narrow wavetrains each with $S=\mathsf{p}_{1}\text{d}y$ , for constant $\mathsf{p}_{1}$ , and then integrate (4.3), with $Y$ replaced by $y$ , over the whole wavetrain to get the contribution to $\boldsymbol{R}$ from the return flow. That contribution, for $L\rightarrow \infty$ and $W\rightarrow \infty$ in various ways, is therefore

(5.5) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D6E4}\mathsf{p}_{1}}{\unicode[STIX]{x1D70B}}\lim \int _{-{\textstyle \frac{1}{2}}W}^{{\textstyle \frac{1}{2}}W}\frac{X\text{d}y}{X^{2}+y^{2}}(+\hat{\boldsymbol{y}})\;=\;\frac{2\unicode[STIX]{x1D6E4}\mathsf{p}_{1}}{\unicode[STIX]{x1D70B}}\lim \arctan (W/L)\,\hat{\boldsymbol{y}}.\end{eqnarray}$$

In the next two sections, the foregoing results for problem (ii) will be cross-checked against computations of far-field pseudomomentum fluxes correct to $O(a^{2}\unicode[STIX]{x1D716}^{1})$ , taking account of the refracted wavecrest shapes illustrated in figure 2 and their mathematical description (1.2). The impulse–pseudomomentum theorem tells us that the results must agree; but it is interesting, nevertheless, not only to carry out the cross-checks but also to see how the refraction works in more detail, thereby gaining mechanistic insight, and another view of the non-interchangeability of limits.

Perhaps surprisingly, the quantum fluids literature – going back over the past fifty years or so – tends to ignore many of the points under discussion, including the $O(a^{2})$  mean flow problem, the distinction between momentum and pseudomomentum, and the non-interchangeability of limits. The author has, however, found one big quantum fluids paper (Sonin Reference Sonin1997) in which the non-interchangeability of limits is mentioned toward the end of the paper, almost as an afterthought; see below equation (83) therein. Another paper (Wexler & Thouless Reference Wexler and Thouless1998) takes a different path but flags up the dangers of manipulating divergent infinite series. Some but not all of the $O(a^{2})$  effects are discussed in Stone (Reference Stone2000b ), while all are consistently dealt with in Guo & Bühler (Reference Guo and Bühler2014), within the Gross–Pitaevskii superfluid model, but only for problem (i). The issues still seem to be surrounded by controversy, perhaps involving unconscious assumptions (e.g. McIntyre Reference McIntyre2017) about, for instance, the distinction between particles and quasiparticles.

6 Wave refraction in problem (ii): the far field outside the wake

The impulse-pseudomomentum theorem says that $\boldsymbol{R}$ can be computed as $-\oint \unicode[STIX]{x1D5D5}\!\cdot \hat{\boldsymbol{n}}\text{d}s$ , correct to $O(a^{2}\unicode[STIX]{x1D716}^{1})$ , for a steady wave field whose sources and sinks lie outside the contour of integration. The unit normal $\hat{\boldsymbol{n}}$ is directed outward, and $\unicode[STIX]{x1D5D5}$ is the pseudomomentum flux tensor appearing in (3.4). We take advantage of the simplicity of the refracted far-field solution (1.2), and its compatibility with ray theory and the group-velocity property (3.5), by letting the contour expand appropriately as $L\rightarrow \infty$ and $W\rightarrow \infty$ . It is convenient to take the contour to be a rectangle with dimensions $L$ by $W$ , where $L$ is now to be read as the length of the wavetrain excluding its source and sink regions, as they recede to infinity. Then, correct to $O(a^{2}\unicode[STIX]{x1D716}^{1})$ ,

(6.1) $$\begin{eqnarray}\boldsymbol{R}\,=\,-\lim \oint \unicode[STIX]{x1D5D5}\!\cdot \hat{\boldsymbol{n}}\text{d}s\;=\,\lim \left(\mathop{\int }_{-\frac{1}{2}W}^{\frac{1}{2}W}\unicode[STIX]{x1D5D5}\!\cdot \hat{\boldsymbol{x}}\text{d}y\biggm\vert_{x=-\frac{1}{2}L}-\int _{-\frac{1}{2}W}^{\frac{1}{2}W}\unicode[STIX]{x1D5D5}\!\cdot \hat{\boldsymbol{x}}\text{d}y\biggm\vert_{x=\frac{1}{2}L}\right).\end{eqnarray}$$

We have used (1.2) and (3.5) to neglect the contributions from the sides of the rectangle parallel to $\hat{\boldsymbol{x}}$ , as follows. For the transverse, $y$ component, the only relevant component of (3.5) on the sides parallel to $\hat{\boldsymbol{x}}$ is $\mathsf{B}_{22}=\hat{\boldsymbol{y}}\!\cdot \unicode[STIX]{x1D5FD}\,\boldsymbol{C}^{\text{abs}}\!\cdot \hat{\boldsymbol{y}}$ . In the far field, again thanks to ray theory, we have $\unicode[STIX]{x1D5FD}\,=\,\overline{\boldsymbol{u}}^{\text{S}}\,=\,{\mathcal{A}}\boldsymbol{k}$ where ${\mathcal{A}}$ is the wave-action per unit mass and where $\boldsymbol{k}\,=\,\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F7}\,=\,k_{0}\{\hat{\boldsymbol{x}}-\unicode[STIX]{x1D716}r_{0}r^{-1}\hat{\unicode[STIX]{x1D73D}}+O(\unicode[STIX]{x1D716}^{2}r_{0}^{2}r^{-2})\}$ , from (1.2) and (1.3) or from (2.4). We also have $c=c(r)=c_{0}\{1+O(\unicode[STIX]{x1D716}^{2}r_{0}^{2}r^{-2})\}$ , as noted in § 1. Denoting $\unicode[STIX]{x1D5FD}\!\cdot \hat{\boldsymbol{x}}$ by $\mathsf{p}_{1}$ as before, and $\unicode[STIX]{x1D5FD}\!\cdot \hat{\boldsymbol{y}}$ by $\mathsf{p}_{2}$ , we have $\mathsf{p}_{2}/\mathsf{p}_{1}=\boldsymbol{k}\!\cdot \hat{\boldsymbol{y}}/\boldsymbol{k}\!\cdot \hat{\boldsymbol{x}}=O(\unicode[STIX]{x1D716}r_{0}r^{-1})$ . From the formula (2.8), again noting the cancellation of leading-order $y$ components, we have $\boldsymbol{C}^{\text{abs}}\!\cdot \hat{\boldsymbol{y}}=O(\unicode[STIX]{x1D716}^{2}c_{0}r_{0}^{2}r^{-2})$ . Hence $\mathsf{B}_{22}=\mathsf{p}_{1}c_{0}$ times $O(\unicode[STIX]{x1D716}^{3}r_{0}^{3}r^{-3})$ . For the longitudinal, $x$  component, we have $\mathsf{B}_{12}=\mathsf{p}_{1}\boldsymbol{C}^{\text{abs}}\!\cdot \hat{\boldsymbol{y}}$ $=\mathsf{p}_{1}c_{0}$ times $O(\unicode[STIX]{x1D716}^{2}r_{0}^{2}r^{-2})$ . Both make negligible contributions as the rectangle expands to infinity.

Now it is clear from § 5 that $\boldsymbol{R}\Vert \hat{\boldsymbol{y}}$ , so that $\boldsymbol{R}=R\hat{\boldsymbol{y}}$ , say, correct to $O(a^{2}\unicode[STIX]{x1D716}^{1})$ . For the sake of brevity, therefore, we restrict attention from now on to evaluating $R$ from the $y$ component of (6.1). We do this in two stages, to be described in this and the next section. The first stage is to compute the contribution $R_{\text{o}}$ from outside the wake. The non-interchangeability of limits comes from that contribution. Outside the wake, we can again use (1.2), (2.4), (2.8) and (3.5). At the second stage, in the next section, we compute $R=R_{\text{o}}+R_{\text{w}}$ , where $R_{\text{w}}$ is the contribution from within the wake. It will be found that $R_{\text{w}}$ agrees with (5.2) and that it is proportional to the Aharonov–Bohm phase jump. The first contribution $R_{\text{o}}$ will be found to agree with (5.5) and to depend solely on the other relevant refraction effect, the $O(\unicode[STIX]{x1D716})$  rotation of wavecrests seen in figure 2 and expressed by the $\hat{\unicode[STIX]{x1D73D}}$ term in $\boldsymbol{k}$ . From here on we denote $\unicode[STIX]{x1D5D5}$ evaluated from (1.2), (2.4), (2.8) and (3.5) by $\unicode[STIX]{x1D5D5}_{\text{o}}$ , so that

(6.2) $$\begin{eqnarray}R_{\text{o}}=\lim \hat{\boldsymbol{y}}\!\cdot \!\left(\int _{-\frac{1}{2}W}^{\frac{1}{2}W}\unicode[STIX]{x1D5D5}_{\text{o}}\!\cdot \hat{\boldsymbol{x}}\text{d}y\biggm\vert_{x=-\frac{1}{2}L}\!\!-\!\!\int _{-\frac{1}{2}W}^{0-}\unicode[STIX]{x1D5D5}_{\text{o}}\!\cdot \hat{\boldsymbol{x}}\text{d}y\biggm\vert_{x=\frac{1}{2}L}\!\!-\!\!\int _{0+}^{\frac{1}{2}W}\unicode[STIX]{x1D5D5}_{\text{o}}\!\cdot \hat{\boldsymbol{x}}\text{d}y\biggm\vert_{x=\frac{1}{2}L}\right),\end{eqnarray}$$

which, in the limit, correctly excludes the wake contribution because of the relative narrowness of the wake, whose width $w\ll W$ , as detailed in the next section.

We can evaluate $\hat{\boldsymbol{y}}\!\cdot \unicode[STIX]{x1D5D5}_{\text{o}}\!\cdot \hat{\boldsymbol{x}}=\mathsf{p}_{2}\boldsymbol{C}^{\text{abs}}\!\cdot \hat{\boldsymbol{x}}$ from $\boldsymbol{C}^{\text{abs}}\!\cdot \hat{\boldsymbol{x}}=c_{0}\{1+O(\unicode[STIX]{x1D716}r_{0}r^{-1})\}$ and $\mathsf{p}_{2}=\mathsf{p}_{1}\boldsymbol{k}\!\cdot \hat{\boldsymbol{y}}/\boldsymbol{k}\!\cdot \hat{\boldsymbol{x}}=-\mathsf{p}_{1}\{\unicode[STIX]{x1D716}r_{0}r^{-1}\hat{\unicode[STIX]{x1D73D}}\!\cdot \hat{\boldsymbol{y}}+O(\unicode[STIX]{x1D716}^{2}r_{0}^{2}r^{-2})\}=\mathsf{p}_{1}k_{0}^{-1}\{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F7}/\unicode[STIX]{x2202}y+O(\unicode[STIX]{x1D716}^{2}r_{0}^{2}r^{-2})\}$ with $\unicode[STIX]{x1D6F7}$ as in (1.2), and where $\mathsf{p}_{1}$ can now be read as the incident pseudomomentum, neglecting refraction, that is, $\mathsf{p}_{1}=$ const. as in §§ 4–5. Correct to $O(a^{2}\unicode[STIX]{x1D716}^{1})$ , therefore, the first integral on the right of (6.2) reduces to $c_{0}\mathsf{p}_{1}k_{0}^{-1}\lim \int _{-\unicode[STIX]{x1D6FC}\tilde{\unicode[STIX]{x1D703}}}^{+\unicode[STIX]{x1D6FC}\tilde{\unicode[STIX]{x1D703}}}\text{d}\unicode[STIX]{x1D6F7}=c_{0}\mathsf{p}_{1}k_{0}^{-1}\lim (2\unicode[STIX]{x1D6FC}\tilde{\unicode[STIX]{x1D703}})=(\unicode[STIX]{x1D6E4}\mathsf{p}_{1}/\unicode[STIX]{x1D70B})\lim \tilde{\unicode[STIX]{x1D703}}$ , where $\tilde{\unicode[STIX]{x1D703}}=\arctan (W/L)>0$ . The non-interchangeability of the limits $L\rightarrow \infty$ and $W\rightarrow \infty$ is now evident.

At each fixed $y$ , and correct to $O(a^{2}\unicode[STIX]{x1D716}^{1})$ , $\mathsf{p}_{2}$ is an odd function of $x$ , and $\hat{\boldsymbol{y}}\!\cdot \unicode[STIX]{x1D5D5}_{\text{o}}\!\cdot \hat{\boldsymbol{x}}$ also. Therefore the second and third integrals in (6.2) add up to a contribution equal to that from the first integral, so that altogether

(6.3) $$\begin{eqnarray}R_{\text{o}}\;=\;\frac{2\unicode[STIX]{x1D6E4}\mathsf{p}_{1}}{\unicode[STIX]{x1D70B}}\lim \arctan (W/L)\end{eqnarray}$$

correct to $O(a^{2}\unicode[STIX]{x1D716}^{1})$ , in agreement with (5.5). Notice incidentally that problem (i) now appears as a trivial variant of the above, obtained by selecting appropriate subsets of rays in cases where ray theory is valid all along the wavetrain.

7 Wave refraction in problem (ii): the far field within the wake

To complete the work on problem (ii) we need to evaluate the remaining contribution to the $y$ component of (6.1),

(7.1) $$\begin{eqnarray}R_{\text{w}}\;=\;-\lim \int _{\text{wake}}\hat{\boldsymbol{y}}\!\cdot \unicode[STIX]{x1D5D5}_{\text{w}}\!\cdot \hat{\boldsymbol{x}}\text{d}y\biggm\vert_{x=\frac{1}{2}L},\end{eqnarray}$$

and to verify that it agrees with (5.2). Here $\unicode[STIX]{x1D5D5}_{\text{w}}$ stands for $\unicode[STIX]{x1D5D5}$ within the wake.

We evaluate (7.1) in two cases that are analytically tractable, $k_{0}r_{0}\ll 1$ and $k_{0}r_{0}\gg 1$ . In the second case we use ray tracing across the vortex core, and in the first we draw on the work of Ford & Llewellyn Smith (Reference Ford and Llewellyn Smith1999), hereafter FLS), who carried out a careful and thorough asymptotic analysis of weak refraction and scattering in that case, building on earlier contributions including that of Sakov (Reference Sakov1993); see also Belyaev & Kopiev (Reference Belyaev and Kopiev2008).

7.1 The long-wave case $k_{0}r_{0}\ll 1$

FLS’s results are subject to a severe restriction on the range of $\unicode[STIX]{x1D6FC}$ values for which they are valid. From (1.3) we see that $\unicode[STIX]{x1D6FC}$ is now the product of two small quantities $k_{0}r_{0}$ and $\unicode[STIX]{x1D716}$ . The results are nevertheless attractive for our purposes because, when valid, they show that the wake has a simple Fresnel-diffractive structure with width-scale $w\sim k_{0}^{-1/2}L^{1/2}\gg k_{0}^{-1}$ , and angular scale asymptotically zero, as $L\rightarrow \infty$ . Across the wake there is a smooth phase transition such that the ray-theoretic formulae used in § 6 still hold, including $\unicode[STIX]{x1D5FD}\propto \boldsymbol{k}$ and $\mathsf{p}_{2}=\mathsf{p}_{1}k_{0}^{-1}\unicode[STIX]{x2202}\unicode[STIX]{x1D6F7}/\unicode[STIX]{x2202}y$ , where $\unicode[STIX]{x1D6F7}$ now denotes the phase within the wake, as distinct from the phase given by (1.2). The wave field within the wake still has the form $A\exp (\text{i}\unicode[STIX]{x1D6F7})$ , where the amplitude $A$ is still real and constant to sufficient accuracy across the wake, with relative error $O(\unicode[STIX]{x1D716})$ , and where the phase $\unicode[STIX]{x1D6F7}$ increases by $2\unicode[STIX]{x1D70B}\unicode[STIX]{x1D6FC}$ going anticlockwise across the wake. Therefore we can evaluate $\int \hat{\boldsymbol{y}}\!\cdot \unicode[STIX]{x1D5D5}_{\text{w}}\!\cdot \hat{\boldsymbol{x}}\text{d}y$ across the wake as $\int c_{0}\mathsf{p}_{2}\text{d}y=c_{0}\mathsf{p}_{1}k_{0}^{-1}\int \!\text{d}\unicode[STIX]{x1D6F7}=2\unicode[STIX]{x1D70B}\unicode[STIX]{x1D6FC}c_{0}\mathsf{p}_{1}k_{0}^{-1}=\unicode[STIX]{x1D6E4}\mathsf{p}_{1}$ , verifying that (7.1) with its minus sign does agree with (5.2).

FLS’s solution also contains a Born-scattering term with amplitude $O(r^{-1/2})$ , which however contributes nothing. The magnitude $O(r^{-1})$ of its pseudomomentum flux makes it potentially able to contribute to $-\oint \unicode[STIX]{x1D5D5}\!\cdot \hat{\boldsymbol{n}}\text{d}s$ . But for our rectangular contour of integration the contribution to $\hat{\boldsymbol{y}}\!\cdot \unicode[STIX]{x1D5D5}\!\cdot \hat{\boldsymbol{n}}$ is an odd function of $y$ , which integrates to zero.

The reader who wishes to check the foregoing against FLS in more detail may find the following notes useful. Outside the wake, the leading far-field term in FLS’s solution agrees with the foregoing for any $\unicode[STIX]{x1D6FC}\ll 1$ because we then have, from (1.2),

(7.2) $$\begin{eqnarray}\displaystyle & & \displaystyle \exp (\text{i}\unicode[STIX]{x1D6F7})~=~\exp (-\text{i}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D703})\exp \{\text{i}(k_{0}(x-c_{0}t))+\text{const.}\}\nonumber\\ \displaystyle & & \displaystyle ~=~(1-\text{i}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D703})\exp \{\text{i}(k_{0}(x-c_{0}t))+\text{const.}\}+O(\unicode[STIX]{x1D6FC}^{2}).\end{eqnarray}$$

This agrees with FLS’s (2.14), (2.20) and the leading term in their (5.7), after allowing for their different definition of $\unicode[STIX]{x1D703}$ and remembering that our $\unicode[STIX]{x1D703}$ jumps from $+\unicode[STIX]{x1D70B}$ to $-\unicode[STIX]{x1D70B}$ , going anticlockwise across the positive $x$  axis. In their dimensionless notation, our $\unicode[STIX]{x1D6FC}$ is written as $M^{2}\unicode[STIX]{x1D6E4}\unicode[STIX]{x1D714}/2\unicode[STIX]{x1D70B}$ , where their $M$ is our $\unicode[STIX]{x1D716}$ and their $\unicode[STIX]{x1D6E4}\unicode[STIX]{x1D714}/2\unicode[STIX]{x1D70B}$ is our $k_{0}r_{0}/\unicode[STIX]{x1D716}$ , all taken positive. The Born scattering term is the next, $O(r^{-1/2})$ term in their (5.7), with outgoing waves $\propto r^{-1/2}\exp \{\text{i}k_{0}(r-c_{0}t)\}$ . The Fresnel wake is described by their (5.12). The phase transition at fixed $x$ is given by the sum $C+S$ of two real-valued Fresnel integrals, suitably scaled, and is therefore an odd function of $y/w$ , with $\unicode[STIX]{x1D6F7}$ asymptoting toward $\pm \unicode[STIX]{x1D70B}\unicode[STIX]{x1D6FC}+\text{const}.$ with gentle oscillations on the scale $w$ . The factor $\exp (-\text{i}\unicode[STIX]{x1D702}^{2})$ in FSL’s (5.11), where $\unicode[STIX]{x1D702}^{2}={\textstyle \frac{1}{2}}(\,y/w)^{2}$ , converts a Born-like factor $\exp \{\text{i}k_{0}(r-c_{0}t)\}$ into a plane-wave factor  $\exp \{\text{i}k_{0}(x-c_{0}t)\}$ , to sufficient accuracy within the wake, matching up with our (7.2).

Belyaev & Kopiev (Reference Belyaev and Kopiev2008) reconsider FLS using a different solution technique, that of Aharonov & Bohm (Reference Aharonov and Bohm1959) and Berry et al. (Reference Berry, Chambers, Large, Upstill and Walmsley1980). They also discuss the conceptual issue of whether (1.2) above can usefully be regarded as a plane wave outside the Fresnel wake, in the limit $r\rightarrow \infty$ . However, the wave field properties required by the foregoing analysis of the pseudomomentum budget are unaffected. Those properties are, most crucially, the validity of (1.2) outside the wake, the phase continuity across the wake, and the $y$ -antisymmetry of the Born contribution to $\hat{\boldsymbol{y}}\!\cdot \unicode[STIX]{x1D5D5}\!\cdot \hat{\boldsymbol{n}}$ . And we note again that the analysis is independently confirmed by the end-to-end cross-check from §§ 3–5.

7.2 The short-wave case $k_{0}r_{0}\gg 1$

We now evaluate (7.1) in the case $kr_{0}\gg 1$ , using ray theory. Attention is restricted to the simplest case, the Rankine vortex model in which the core is in solid rotation, with constant angular velocity $\unicode[STIX]{x1D6FA}={\textstyle \frac{1}{2}}|\unicode[STIX]{x1D74E}_{0}|=|\boldsymbol{u}_{0}(r)|/r=\unicode[STIX]{x1D6E4}/2\unicode[STIX]{x1D70B}r_{0}^{2}$ , taken positive, i.e. anticlockwise, for definiteness, as in the figures.

Correct to $O(\unicode[STIX]{x1D716})$  the rays outside the core and its lee are straight, as already remarked, with absolute group velocities $\boldsymbol{C}^{\text{abs}}$ parallel to $\hat{\boldsymbol{x}}$ , but slightly rotated wavecrests, as seen in figure 2. Inside the core, the ray-tracing equations, e.g. (2.14) of Reference Bühler and McIntyreBM03, verify what is obvious from rotating the reference frame while keeping the same intrinsic phase and group velocity $c=c_{0}\{1+O(\unicode[STIX]{x1D716}^{2})\}$ , namely that the wavenumber vector $\boldsymbol{k}$ rotates with angular velocity $\unicode[STIX]{x1D6FA}$ as a ray point crosses the core at velocity $c_{0}\hat{\boldsymbol{x}}+O(\unicode[STIX]{x1D716})$ . Thus the rays bend slightly to the left as they cross the core into its lee, where they are straight again but no longer quite parallel to the $x$  axis. We note that the group velocity $\boldsymbol{C}^{\text{abs}}$ rotates twice as fast as $\boldsymbol{k}$ , with angular velocity $2\unicode[STIX]{x1D6FA}$ , because of the changing $O(\unicode[STIX]{x1D716})$  contribution from the $y$ component of $\boldsymbol{u}_{0}=\unicode[STIX]{x1D6FA}(-y,x)$ in (2.8) as the ray point crosses the core. This and the straightness of the rays outside the core are special cases of the curl-curvature formula mentioned below (2.8). The formula implies generally that, correct to $O(\unicode[STIX]{x1D716})$ , the group velocity vector rotates with angular velocity $\unicode[STIX]{x1D74E}_{0}=\unicode[STIX]{x1D735}\!\times \boldsymbol{u}_{0}$ .

Figure 5.

Rays emerging from the vortex core and forming a wake with a caustic, computed correct to $O(\unicode[STIX]{x1D716})$  from the ray-tracing equations. The $O(\unicode[STIX]{x1D716})$  deflections are exaggerated in the plot. The rays entering the core, not shown, are initially parallel to the $x$  axis at $y/r_{0}=0$ , $\pm 0.1$ , $\pm 0.2$ , $\pm 0.3$ , $\pm 0.4$ , $\pm 0.5$ , $\pm 0.6$ , $\pm 0.7$ , $\pm 0.8$ , $\pm 0.9$ , $\pm 0.95$ , $\pm 0.98$ , $\pm 0.995$ , $\pm 1$ .

For weak refraction the ray undergoing the greatest deflection is that crossing the widest part of the core, at $y=0$ . The rays from $-r_{0}<y<0$ therefore splay out slightly, while those from $0<y<r_{0}$ cross one another and form a caustic, extending slightly outside the tangent line $y=+r_{0}$ , as shown in figure 5 with the deflections exaggerated. A full analysis is beyond our scope here; to evaluate (7.1) we will simply add up the leading-order pseudomomentum fluxes $\hat{\boldsymbol{y}}\!\cdot \unicode[STIX]{x1D5D5}_{\text{w}}\!\cdot \hat{\boldsymbol{x}}=c_{0}\mathsf{p}_{2}$ as if carried by each ray independently. This assumes that the refraction term in the $O(a^{2}\unicode[STIX]{x1D716}^{1})$  pseudomomentum law (3.4) works in the same way, to leading order at least, whether or not the rays go through a caustic.

The treatment of the ray deflections as small of order $\unicode[STIX]{x1D716}$ , and the neglect of diffractive effects, becomes delicate and very restrictive when combined with the formal limit $L\rightarrow \infty$ . It will nevertheless yield the correct result, in agreement with (5.2), as will now be shown. The agreement will also lend support to our assumption about (3.4) and caustics.

The following is a shortcut to the results from the ray-tracing equations shown in figure 5, using the rotation-rate of $\boldsymbol{k}$ already mentioned. A ray point entering the core at $y=-r_{0}\sin \unicode[STIX]{x1D703}$ , say, for some fixed $\unicode[STIX]{x1D703}$ with $|\unicode[STIX]{x1D703}|<\unicode[STIX]{x1D70B}/2$ , already has a wavenumber with non-vanishing $y$ component $\boldsymbol{k}\!\cdot \hat{\boldsymbol{y}}=-\unicode[STIX]{x1D6FC}r_{0}^{-1}\hat{\unicode[STIX]{x1D73D}}\!\cdot \hat{\boldsymbol{y}}=+\unicode[STIX]{x1D6FC}r_{0}^{-1}\cos \unicode[STIX]{x1D703}$ , from (1.2) or (2.4) giving, at that location, $\hat{\boldsymbol{y}}\!\cdot \unicode[STIX]{x1D5D5}_{\text{w}}\!\cdot \hat{\boldsymbol{x}}=c_{0}\mathsf{p}_{2}=c_{0}\mathsf{p}_{1}k_{0}^{-1}\unicode[STIX]{x1D6FC}r_{0}^{-1}\cos \unicode[STIX]{x1D703}=\mathsf{p}_{1}\unicode[STIX]{x1D6E4}(2\unicode[STIX]{x1D70B}r_{0})^{-1}\!\cos \unicode[STIX]{x1D703}=\mathsf{p}_{1}\unicode[STIX]{x1D6FA}r_{0}\cos \unicode[STIX]{x1D703}$ . As the ray point crosses the core, over a distance $2r_{0}\cos \unicode[STIX]{x1D703}$ and taking a time $2c_{0}^{-1}r_{0}\cos \unicode[STIX]{x1D703}$ , the vectors $\boldsymbol{k}$ and $\unicode[STIX]{x1D5FD}$ rotate with angular velocity $\unicode[STIX]{x1D6FA}$ , so that $c_{0}\mathsf{p}_{2}$ increases by a further small amount $2\mathsf{p}_{1}\unicode[STIX]{x1D6FA}r_{0}\cos \unicode[STIX]{x1D703}$ (equal to $c_{0}\mathsf{p}_{1}$ times the net rotation angle $2c_{0}^{-1}\unicode[STIX]{x1D6FA}r_{0}\cos \unicode[STIX]{x1D703}$ ), and again by a further small amount $\mathsf{p}_{1}\unicode[STIX]{x1D6FA}r_{0}\cos \unicode[STIX]{x1D703}$ after exiting the core and reaching sufficiently large $x>0$ . This last increment is the same increment as in (1.2) between the far edge of the core and $x\rightarrow \infty$ . However, it is to be added to the new far-core-edge value $3\mathsf{p}_{1}\unicode[STIX]{x1D6FA}r_{0}\cos \unicode[STIX]{x1D703}$ rather than to the original value $-\mathsf{p}_{1}\unicode[STIX]{x1D6FA}r_{0}\cos \unicode[STIX]{x1D703}$ implied by (1.2) and (2.4). With our ray still nearly parallel to the $x$  axis after exiting the core, we are using the fact that the $O(\unicode[STIX]{x1D716})$  rates of change of $\boldsymbol{k}\!\cdot \hat{\boldsymbol{y}}$ outside the core are the same as those implied by (1.2), (2.4) and (2.8).

So, adding all the contributions just noted, we have that the ray has a total end-to-end change $\mathsf{p}_{1}\unicode[STIX]{x1D6FA}r_{0}\cos \unicode[STIX]{x1D703}+2\mathsf{p}_{1}\unicode[STIX]{x1D6FA}r_{0}\cos \unicode[STIX]{x1D703}+\mathsf{p}_{1}\unicode[STIX]{x1D6FA}r_{0}\cos \unicode[STIX]{x1D703}=4\mathsf{p}_{1}\unicode[STIX]{x1D6FA}r_{0}\cos \unicode[STIX]{x1D703}$ , in $c_{0}\mathsf{p}_{2}$ , corresponding to an end-to-end deflection angle $\unicode[STIX]{x1D6FD}$ , say, $=\mathsf{p}_{2}/\mathsf{p}_{1}|_{x\rightarrow \infty }=4c_{0}^{-1}\unicode[STIX]{x1D6FA}r_{0}\cos \unicode[STIX]{x1D703}=4\unicode[STIX]{x1D716}\cos \unicode[STIX]{x1D703}$ , with maximum value $4\unicode[STIX]{x1D716}$ , in an anticlockwise sense. Integrating the change in $c_{0}\mathsf{p}_{2}$ over all the rays that cross the core, from $y=-r_{0}$ to $y=r_{0}$ , noting that $\text{d}y=-r_{0}\cos \unicode[STIX]{x1D703}\text{d}\unicode[STIX]{x1D703}$ and that $\int _{-\unicode[STIX]{x1D70B}/2}^{\unicode[STIX]{x1D70B}/2}\cos ^{2}\unicode[STIX]{x1D703}\text{d}\unicode[STIX]{x1D703}=\unicode[STIX]{x1D70B}/2$ , we find from (7.1) with due care over signs that $R_{\text{w}}=-2\unicode[STIX]{x1D70B}\mathsf{p}_{1}\unicode[STIX]{x1D6FA}r_{0}^{2}=-\unicode[STIX]{x1D6E4}\mathsf{p}_{1}$ , agreeing with (5.2) as expected.

To summarize, (6.1) gives us, both for $k_{0}r_{0}\ll 1$ and for $k_{0}r_{0}\gg 1$ ,

(7.3) $$\begin{eqnarray}R\;=\;R_{\text{w}}+\,R_{\text{o}}\;=\;-\left\{1-\frac{2}{\unicode[STIX]{x1D70B}}\lim \arctan \left(\frac{W}{L}\right)\right\}\unicode[STIX]{x1D6E4}\mathsf{p}_{1}\end{eqnarray}$$

for arbitrary limiting values of $W/L$ , agreeing with the independent derivations in §§ 3–5. Furthermore, recalling that those independent derivations are valid for arbitrary $k_{0}r_{0}$ , we see also that (7.3) must be a result far more robust than is suggested by the delicacy of the flux computations for $k_{0}r_{0}\ll 1$ and $k_{0}r_{0}\gg 1$ . On the other hand, everything still depends on the smallness of $\unicode[STIX]{x1D716}$ . Numerical solutions for cases of stronger refraction show wakes oriented at substantial angles away from the $x$ axis; see for instance figures 2–3 of Coste, Lund & Umeki (Reference Coste, Lund and Umeki1999). And it is still an open question as to what might or might not replace the impulse–pseudomomentum theorem for arbitrary $\unicode[STIX]{x1D716}$ .

8 Problem (iii)

The main reason for being interested in this rapidly-rotating version of problem (i), in which the waves are deep-water gravity waves, is the existence of the Ursell anti-Stokes flow. This is an Eulerian-mean flow $\overline{\boldsymbol{u}}$ that largely cancels the strongly $z$ -dependent Stokes drift $\overline{\boldsymbol{u}}^{\text{S}}$ of the waves. Indeed the cancellation is exact, for finite-amplitude waves, when the wave field is exactly steady and exactly homogeneous across an infinite $xy$ domain (Ursell Reference Ursell1950; Pollard Reference Pollard1970). (In a thought-experiment starting with irrotational waves, in such a domain, the mean flow undergoes a free inertial oscillation about the anti-Stokes state. It is sometimes forgotten that this thought-experiment was clearly analysed and understood in Ursell’s pioneering work.) In problem (iii), however, contrary to what might at first be thought, the anti-Stokes flow fails to suppress remote recoil.

We consider an unstratified rapidly-rotating system of finite depth $H$ under gravity $-g\hat{\boldsymbol{z}}$ , so that the $z$ direction is vertically upward. The vector Coriolis parameter $\boldsymbol{f}$ is parallel to $\hat{\boldsymbol{z}}$ . The waves have a wavenumber $k=|\boldsymbol{k}|$ that is large enough to make $\exp (-kH)$ negligible. The intrinsic wave frequency $kc=(gk)^{1/2}\gg f=|\,\boldsymbol{f}|$ , so that rotation affects the wave dynamics only weakly.

In addition to $a$ and $\unicode[STIX]{x1D716}$ we now have another small parameter, the mean-flow Rossby number

(8.1) $$\begin{eqnarray}\text{Ro}~=~U/fr_{0}~\ll ~1,\end{eqnarray}$$

whose smallness will bring in the Taylor–Proudman effect and give us quasigeostrophic mean-flow dynamics. As before, the velocity scale $U$ will be taken as the velocity of the vortex flow at the edge of the core, $r=r_{0}$ . The core will be defined by non-vanishing quasigeostrophic potential vorticity, $(\unicode[STIX]{x1D6FB}_{\!\text{H}}^{2}-L_{\text{D}}^{-2})\tilde{\unicode[STIX]{x1D713}}_{0}$ in the notation of § 2.

The anti-Stokes flow can be regarded as a consequence of the Taylor–Proudman effect together with the exact advection property expressed by the mean vorticity equations (2.10) and (2.11). This is no more than a rephrasing of Ursell’s original argument, putting it within the GLM framework. Focusing on the present case $\text{Ro}\ll 1$ , we can regard inertia waves as fast waves with a strong Coriolis restoring force. The Taylor–Proudman effect arises from the corresponding stiffness of the lines of absolute vorticity $\boldsymbol{f}+\widetilde{\unicode[STIX]{x1D74E}}$ , which must tend to stay vertical, on average at least. In particular, they cannot be continually sheared over by the mean flow; and for this purpose the mean flow is the Lagrangian-mean flow $\overline{\boldsymbol{u}}^{\text{L}}$ , as equations (2.10) and (2.11) make clear. A Lagrangian-mean flow without vertical shear is a Stokes drift plus an Eulerian-mean anti-Stokes flow, plus an additional contribution that is independent of $z$ – in this case the vortex flow plus the Bretherton flow that mediates remote recoil.

To tackle problem (iii) we must first derive (2.18). The starting point is the vertical component of (2.11). Writing $f+\widetilde{\unicode[STIX]{x1D714}}$ for the vertical component of $\boldsymbol{f}+\widetilde{\unicode[STIX]{x1D74E}}$ , and $\overline{w}^{\text{L}}$ for the vertical component of $\overline{\boldsymbol{u}}^{\text{L}}$ , we have

(8.2) $$\begin{eqnarray}\frac{\overline{\text{D}}^{\text{L}}\widetilde{\unicode[STIX]{x1D714}}}{\text{D}t}~+~(f+\widetilde{\unicode[STIX]{x1D714}})\unicode[STIX]{x1D735}\!\cdot \overline{\boldsymbol{u}}^{\text{L}}~=~(\,\boldsymbol{f}+\widetilde{\unicode[STIX]{x1D74E}})\!\cdot \unicode[STIX]{x1D735}\overline{w}^{\text{L}}\end{eqnarray}$$

exactly. Upon cancelling a pair of terms in $\unicode[STIX]{x2202}\overline{w}^{\text{L}}/\unicode[STIX]{x2202}z$ , this reduces to

(8.3) $$\begin{eqnarray}\frac{\overline{\text{D}}^{\text{L}}\widetilde{\unicode[STIX]{x1D714}}}{\text{D}t}~+~(f+\widetilde{\unicode[STIX]{x1D714}})\unicode[STIX]{x1D735}_{\!\text{H}}^{\phantom{L}}\!\cdot \overline{\boldsymbol{u}}_{\text{H}}^{\text{L}}~=~\widetilde{\unicode[STIX]{x1D74E}}_{\text{H}}\!\cdot \unicode[STIX]{x1D735}_{\!\text{H}}^{\phantom{L}}\overline{w}^{\text{L}}.\end{eqnarray}$$

As before, suffix H denotes horizontal projection. Writing $\widetilde{\unicode[STIX]{x1D714}}=\unicode[STIX]{x1D714}_{0}+\widetilde{\unicode[STIX]{x1D714}}_{\text{B}}$ and $\overline{\boldsymbol{u}}^{\text{L}}=\boldsymbol{u}_{0}+\overline{\boldsymbol{u}}_{\text{B}}^{\text{L}}$ where the vortex-only contributions $\unicode[STIX]{x1D714}_{0}$ and $\boldsymbol{u}_{0}$ are $z$ -independent, with $\boldsymbol{u}_{0}$ horizontal, and the wave-induced contributions $\widetilde{\unicode[STIX]{x1D714}}_{\text{B}}$ and $\overline{\boldsymbol{u}}_{\text{B}}^{\text{L}}$ are $O(a^{2})$ , we note that in the first term on the left the contribution $\overline{\boldsymbol{u}}_{\text{B}}^{\text{L}}\!\cdot \unicode[STIX]{x1D735}\widetilde{\unicode[STIX]{x1D714}}_{\text{B}}\,=\,\overline{\boldsymbol{u}}_{\text{B}}^{\text{L}}\!\cdot \unicode[STIX]{x1D735}_{\!\text{H}}^{\phantom{L}}\widetilde{\unicode[STIX]{x1D714}}_{\text{B}}\,+\,\overline{w}^{\text{L}}\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D714}}_{\text{B}}/\unicode[STIX]{x2202}z\,=\,$ $O(a^{4})$  and is therefore negligible. (We need not restrict $\unicode[STIX]{x1D716}$ at this stage.) There are two further such $O(a^{4})$  contributions in (8.3), namely the right-hand side, and on the left $\widetilde{\unicode[STIX]{x1D714}}_{\text{B}}\unicode[STIX]{x1D735}_{\!\text{H}}^{\phantom{L}}\!\cdot \overline{\boldsymbol{u}}_{\text{H}}^{\text{L}}=\widetilde{\unicode[STIX]{x1D714}}_{\text{B}}\unicode[STIX]{x1D735}_{\!\text{H}}^{\phantom{L}}\!\cdot \overline{\boldsymbol{u}}_{\text{B}}^{\text{L}}$ since $\unicode[STIX]{x1D735}_{\!\text{H}}^{\phantom{L}}\!\cdot \boldsymbol{u}_{0}=0$ . The $O(a^{2})$  contribution $\unicode[STIX]{x1D714}_{0}\unicode[STIX]{x1D735}_{\!\text{H}}^{\phantom{L}}\!\cdot \overline{\boldsymbol{u}}_{\text{H}}^{\text{L}}=\unicode[STIX]{x1D714}_{0}\unicode[STIX]{x1D735}_{\!\text{H}}^{\phantom{L}}\!\cdot \overline{\boldsymbol{u}}_{\text{B}}^{\text{L}}$ is also negligible, against $f\unicode[STIX]{x1D735}_{\!\text{H}}^{\phantom{L}}\!\cdot \overline{\boldsymbol{u}}_{\text{H}}^{\text{L}}$ , because of the smallness of $\text{Ro}$ . After neglecting all these contributions we can take the vertical average of (8.3), using the $z$ -independence of $\boldsymbol{u}_{0}$ and $\unicode[STIX]{x1D74E}_{0}$ . Denoting vertical averages by angle brackets as before and noting that $\boldsymbol{u}_{0}\!\cdot \unicode[STIX]{x1D735}_{\!\text{H}}^{\phantom{L}}\unicode[STIX]{x1D714}_{0}=0$ we get

(8.4) $$\begin{eqnarray}\left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+\boldsymbol{u}_{0}\!\cdot \unicode[STIX]{x1D735}_{\!\text{H}}^{\phantom{L}}\right)\langle \widetilde{\unicode[STIX]{x1D714}}_{\text{B}}\rangle \;+\;\langle \overline{\boldsymbol{u}}_{\text{B}}^{\text{L}}\rangle \!\cdot \unicode[STIX]{x1D735}_{\!\text{H}}^{\phantom{L}}\unicode[STIX]{x1D714}_{0}\;+\;f\unicode[STIX]{x1D735}_{\!\text{H}}^{\phantom{L}}\!\cdot \langle \overline{\boldsymbol{u}}_{\text{H}}^{\text{L}}\rangle ~=~0\end{eqnarray}$$

or, written more compactly, again with negligible error $O(a^{4})$ ,

(8.5) $$\begin{eqnarray}\frac{\overline{\text{D}}_{\text{H}}^{\text{L}}\langle \widetilde{\unicode[STIX]{x1D714}}\rangle }{\text{D}t}~+~f\unicode[STIX]{x1D735}_{\!\text{H}}^{\phantom{L}}\!\cdot \langle \overline{\boldsymbol{u}}_{\text{H}}^{\text{L}}\rangle ~=~0,\end{eqnarray}$$

where we have defined $\overline{\text{D}}_{\text{H}}^{\text{L}}/\text{D}t=\unicode[STIX]{x2202}/\unicode[STIX]{x2202}t+\langle \overline{\boldsymbol{u}}_{\text{H}}^{\text{L}}\rangle \!\cdot \unicode[STIX]{x1D735}_{\!\text{H}}^{\phantom{L}}$ . In a closely similar way, the vertical average of the three-dimensional mass-conservation equation, Reference BühlerB14 equation (10.47), simplifies to a vertically-averaged version of (2.13),

(8.6) $$\begin{eqnarray}\frac{\overline{\text{D}}_{\text{H}}^{\text{L}}\tilde{h}}{\text{D}t}~+~\tilde{h}\unicode[STIX]{x1D735}_{\!\text{H}}^{\phantom{L}}\!\cdot \langle \overline{\boldsymbol{u}}_{\text{H}}^{\text{L}}\rangle ~=~0,\end{eqnarray}$$

again with negligible error $O(a^{4})$ . As before, the mean layer depth $\tilde{h}=\tilde{h}(x,y,t)$ is defined such that $\unicode[STIX]{x1D70C}\tilde{h}\text{d}x\,\text{d}y$ is the areal mass element, where $\unicode[STIX]{x1D70C}$ is the constant mass density. Elimination of $\unicode[STIX]{x1D735}_{\!\text{H}}^{\phantom{L}}\!\cdot \langle \overline{\boldsymbol{u}}_{\text{H}}^{\text{L}}\rangle$ between (8.5) and (8.6) gives us that $\langle \widetilde{\unicode[STIX]{x1D714}}\rangle -f\ln \tilde{h}$ , plus an arbitrary additive constant, is a material invariant under advection by $\langle \overline{\boldsymbol{u}}_{\text{H}}^{\text{L}}\rangle$ . Fractional changes in $\tilde{h}$ are small, $(\tilde{h}-H)/H=O(\text{Ro})$ , and so taking the additive constant to be $f\ln H$ and using $\ln \tilde{h}-\ln H=\ln (\tilde{h}/H)=\ln \{(\tilde{h}-H+H)/H\}=(\tilde{h}-H)/H+O(\text{Ro}^{2})$ , we get

(8.7) $$\begin{eqnarray}\frac{\overline{\text{D}}_{\text{H}}^{\text{L}}\tilde{q}}{\text{D}t}~=~0,\end{eqnarray}$$

where

(8.8) $$\begin{eqnarray}\tilde{q}~=~\tilde{q}(x,y,t)~=~\langle \widetilde{\unicode[STIX]{x1D714}}\rangle -\frac{f}{H}(\tilde{h}-H),\end{eqnarray}$$

which is the appropriate form of the quasigeostrophic potential vorticity of the mean flow. In any thought-experiment in which the waves are switched on after the vortex is established, (8.7) implies that the $\tilde{q}$ field is unchanged by the presence of the waves, apart from the advection of the vortex core by the Bretherton flow. See also appendix B. So in problem (iii) we have $\tilde{q}=q_{0}$ where $q_{0}$ is the potential vorticity of the vortex alone.

The final step in deriving (2.18) is to make explicit use of hydrostatic and geostrophic balance. Some delicate scale analysis is involved at this stage. The full details are given in appendix C, in which the key points are as follows. Hydrostatic balance, meaning the overall balance for a complete fluid column, implies that horizontal pressure gradients on the bottom, underneath the wavetrain, are given by $\unicode[STIX]{x1D70C}g\unicode[STIX]{x1D735}_{\!\text{H}}^{\phantom{L}}\tilde{h}$ , again because $\unicode[STIX]{x1D70C}\tilde{h}\text{d}x\,\text{d}y$ is the areal mass element. Geostrophic balance then gives (2.15) with $\tilde{\unicode[STIX]{x1D713}}=g(\tilde{h}-H)/f$ . Then $\tilde{q}=q_{0}$ together with (2.9) and (2.16) gives (2.18).  The Taylor–Proudman effect extends the geostrophic relation upward into the wavetrain, $\langle \overline{\boldsymbol{u}}_{\text{H}}^{\text{L}}\rangle =\overline{\boldsymbol{u}}_{\text{H}}^{\text{L}}$ .  Radiation stresses within the wavetrain cannot break the overall hydrostatic balance because such stresses have no foothold on the bottom boundary, in virtue of our assumption that $\exp (-kH)$ is negligibly small. That allows us to neglect the net vertical, radiation-stress-induced external force on the fluid column – in contrast, it should be noted, with the situation of figure 3. For further comments see appendix C. In appendix C we also note that the exact wave solution of Pollard (Reference Pollard1970) provides some useful cross-checks.

With (2.18) in place, we can now invoke the impulse-pseudomomentum theorem to assert that recoil forces can be computed either from Bretherton flows correct to $O(a^{2}\unicode[STIX]{x1D716}^{0})$  or from net pseudomomentum fluxes correct to $O(a^{2}\unicode[STIX]{x1D716}^{1})$ . In the remainder of this section we carry out both computations, in the case of a small vortex core with $r_{0}\ll L_{\text{D}}$ , providing mechanistic insight as well as an end-to-end cross-check on our derivation of (2.18).

First consider the Bretherton flow. Because it satisfies (2.18), it decays sideways like $\exp (-|y|/L_{\text{D}})$ , on the fixed length-scale $L_{\text{D}}$ . Therefore there is no dilution effect like that in problem (i). With $L\rightarrow \infty$ , and with a narrow wavetrain for which $W\ll L_{\text{D}}$ and $W\ll Y$ , in the notation of § 4, we have, for $|y-Y|>W$ , outside the unrefracted wavetrain, with $Y$ the distance to the vortex core,

(8.9) $$\begin{eqnarray}\overline{\boldsymbol{u}}_{\text{B}}^{\text{L}}(x,y)~=~(S/2L_{\text{D}})\exp (-|y-Y|/L_{\text{D}})(-\hat{\boldsymbol{x}}),\end{eqnarray}$$

where $S$ is still defined by (4.1) but with vertical averaging understood.  So, for our small vortex core with $r_{0}\ll L_{\text{D}}$ , carried bodily by the $z$ -independent Bretherton flow, the recoil force is

(8.10) $$\begin{eqnarray}\boldsymbol{R}=(\unicode[STIX]{x1D6E4}S/2L_{\text{D}})\exp (-|Y|/L_{\text{D}})(+\hat{\boldsymbol{y}}),\end{eqnarray}$$

with $\unicode[STIX]{x1D6E4}$ evaluated at the edge of the core. The signs are the same as those in problem (i).

Second, we compute $\boldsymbol{R}$ from the $O(a^{2}\unicode[STIX]{x1D716}^{1})$  pseudomomentum flux $\mathsf{B}_{21}=\hat{\boldsymbol{y}}\!\cdot \unicode[STIX]{x1D5D5}\!\cdot \hat{\boldsymbol{x}}$ , using ray theory. As in § 7.2, the rays start exactly parallel to the $x$  axis, with $\mathsf{p}_{2}\rightarrow 0$ as $x\rightarrow -\infty$ , and finish after bending slightly, through an $O(\unicode[STIX]{x1D716})$  end-to-end deflection angle $\unicode[STIX]{x1D6FD}=\mathsf{p}_{2}/\mathsf{p}_{1}|_{x\rightarrow \infty }$ . The vortex flow has velocity $\boldsymbol{u}_{0}(r)=\hat{\unicode[STIX]{x1D73D}}\unicode[STIX]{x2202}\unicode[STIX]{x1D713}_{0}/\unicode[STIX]{x2202}r$ , say, where the quasigeostrophic streamfunction $\unicode[STIX]{x1D713}_{0}$ satisfies $(\unicode[STIX]{x1D6FB}^{2}-L_{\text{D}}^{-2})\unicode[STIX]{x1D713}_{0}=0$ outside the core. Defining $r^{\prime }=r/L_{\text{D}}$ , we have

(8.11) $$\begin{eqnarray}\unicode[STIX]{x1D713}_{0}~=\;-\frac{\unicode[STIX]{x1D6E4}}{2\unicode[STIX]{x1D70B}}K_{0}(r^{\prime })\quad \text{outside the core},\end{eqnarray}$$

where $K_{0}(r^{\prime })$ is the modified Bessel function asymptoting to $(\unicode[STIX]{x1D70B}/2r^{\prime })^{1/2}\exp (-r^{\prime })$ for $r^{\prime }\gg 1$ and to $-\ln (r^{\prime })$ for $r^{\prime }\ll 1$ , near the core. The Kelvin circulation $\unicode[STIX]{x1D6E4}$ is again defined to be the circulation at the core edge $r=r_{0}$ , namely $\pm 2\unicode[STIX]{x1D70B}r_{0}|\boldsymbol{u}_{0}(r_{0})|=\pm 2\unicode[STIX]{x1D70B}r_{0}U$ , with positive sign when the vortex is cyclonic as in the figures. For $r>r_{0}$ the circulation is not constant, but decays exponentially like $r^{\prime 1/2}\exp (-r^{\prime })$ .

To verify agreement with (8.10) we need only calculate $\unicode[STIX]{x1D6FD}$ . The curl-curvature formula tells us that $\unicode[STIX]{x1D6FD}$ is non-zero at $O(\unicode[STIX]{x1D716})$ , because the relative vorticity $\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D713}_{0}=L_{\text{D}}^{-2}\unicode[STIX]{x1D713}_{0}$ is non-zero outside the core. A cyclonic vortex core is surrounded by anticyclonic vorticity and the rays therefore bend to the right, rather than to the left as in § 7, so that $\text{sgn}\unicode[STIX]{x1D6FD}=-\text{sgn}\unicode[STIX]{x1D6E4}$ . Notice incidentally that there will no longer be any far-field subtleties, or issues with non-interchangeable limits, thanks to the exponential decay of $\unicode[STIX]{x1D713}_{0}$ . Another effect of that decay is that the right-bending rays must splay out slightly when they pass to the left of the vortex, but cross one another and form a caustic when to the right. The presence or absence of a caustic makes no difference to the results.

The deep water waves in problem (iii) have intrinsic frequency $kc=(gk)^{1/2}$ and intrinsic group velocity $\boldsymbol{C}=C\boldsymbol{k}/k$ where $C={\textstyle \frac{1}{2}}c={\textstyle \frac{1}{2}}(g/k)^{1/2}$ . The absolute group velocity $\boldsymbol{C}^{\text{abs}}={\textstyle \frac{1}{2}}c_{0}\hat{\boldsymbol{x}}+$ $O(\unicode[STIX]{x1D716})$ , with $O(\unicode[STIX]{x1D716})$  contributions coming both from $\boldsymbol{u}_{0}$ and from refractive changes in wavenumber $\boldsymbol{k}$ . Following a ray point moving at speed ${\textstyle \frac{1}{2}}c_{0}+$ $O(\unicode[STIX]{x1D716})$ , the curl-curvature formula says that the direction of $\boldsymbol{C}^{\text{abs}}$ rotates clockwise away from the $x$ direction at an angular velocity equal to the (negative) relative vorticity $\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D713}_{0}=L_{\text{D}}^{-2}\unicode[STIX]{x1D713}_{0}$ . So for weak refraction we have

(8.12) $$\begin{eqnarray}\unicode[STIX]{x1D6FD}~=~({\textstyle \frac{1}{2}}c_{0})^{-1}\int _{-\infty }^{\infty }L_{\text{D}}^{-2}\unicode[STIX]{x1D713}_{0}(x,Y)\text{d}x~=\;-\frac{\unicode[STIX]{x1D6E4}\;}{\unicode[STIX]{x1D70B}c_{0}L_{\text{D}}^{2}}\int _{-\infty }^{\infty }K_{0}\{(x^{2}+Y^{2})^{1/2}/L_{\text{D}}\}\text{d}x.\end{eqnarray}$$

The integral on the right is exactly equal to $L_{\text{D}}\unicode[STIX]{x1D70B}\exp (-|Y|/L_{\text{D}})$ , as will be shown shortly. Hence $\unicode[STIX]{x1D6FD}=-(\unicode[STIX]{x1D6E4}/c_{0}L_{\text{D}})\exp (-|Y|/L_{\text{D}})$ . Remembering that $C={\textstyle \frac{1}{2}}c_{0}$ , we see that there is an end-to-end difference in pseudomomentum fluxes representing a rate of import $-{\textstyle \frac{1}{2}}c_{0}\mathsf{p}_{2}|_{x\rightarrow \infty }=-{\textstyle \frac{1}{2}}c_{0}\unicode[STIX]{x1D6FD}\mathsf{p}_{1}=+(\unicode[STIX]{x1D6E4}\mathsf{p}_{1}/2L_{\text{D}})\exp (-|Y|/L_{\text{D}})$ of $y$ -pseudomomentum per unit $y$ -distance, correct to $O(a^{2}\unicode[STIX]{x1D716}^{1})$ . Recalling the definition of $S$ in (4.1), with vertical averaging understood, we sum over all the rays to find the total recoil force in the $y$  direction as

(8.13) $$\begin{eqnarray}\boldsymbol{R}~=~(\unicode[STIX]{x1D6E4}S/2L_{\text{D}})\exp (-|Y|/L_{\text{D}})(+\hat{\boldsymbol{y}}),\end{eqnarray}$$

in agreement with (8.10).

The integral on the right of (8.12) is equal to $L_{\text{D}}$ times the value at $y^{\prime }=Y/L_{\text{D}}$ of the function $J(\,y^{\prime })$ defined by $J(\,y^{\prime })=\int _{-\infty }^{\infty }K_{0}(r^{\prime })\text{d}x^{\prime }$ where $x^{\prime }=x/L_{\text{D}}$ and $y^{\prime }=y/L_{\text{D}}$ so that $r^{\prime 2}=x^{\prime 2}+y^{\prime 2}$ . Now $K_{0}(r^{\prime })$ is equal to its Laplacian in the $x^{\prime },y^{\prime }$ plane, except at the origin where the Laplacian has a delta function $-2\unicode[STIX]{x1D70B}\unicode[STIX]{x1D6FF}(x^{\prime })\unicode[STIX]{x1D6FF}(\,y^{\prime })$ in place of the integrable logarithmic singularity in $K_{0}$ itself. For any $y^{\prime }\neq 0$ we therefore have

(8.14) $$\begin{eqnarray}J(\,y^{\prime })~=~\int _{-\infty }^{\infty }\left(\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}x^{\prime 2}}+\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}y^{\prime 2}}\right)K_{0}(r^{\prime })\text{d}x^{\prime }~=~\frac{\text{d}^{2}}{\text{d}y^{\prime 2}}\int _{-\infty }^{\infty }K_{0}(r^{\prime })\text{d}x^{\prime }~=~\frac{\text{d}^{2}}{\text{d}y^{\prime 2}}J(\,y^{\prime })\end{eqnarray}$$

and, taking the delta function into account, we have for all $y^{\prime }$ from $-\infty$ to $+\infty$

(8.15) $$\begin{eqnarray}\frac{\text{d}^{2}}{\text{d}y^{\prime 2}}J(\,y^{\prime })-J(\,y^{\prime })~=\;-2\unicode[STIX]{x1D70B}\unicode[STIX]{x1D6FF}(\,y^{\prime }),\end{eqnarray}$$

whose solution evanescent at infinity is $J(\,y^{\prime })=\unicode[STIX]{x1D70B}\exp (-|y^{\prime }|)$ , corresponding to the result asserted.

9 Concluding remarks

Despite their restricted parameter range, the problems studied here are enough to remind us that remote recoil, as such, is generic and ubiquitous. Remote recoil will occur whenever wave-induced mean flows extend outside wavetrains or wave packets and advect coherent vortices. Remote recoil is excluded, or made subdominant, by asymptotic theories of wave–current interactions that assume slowly-varying mean currents with a single large length-scale and correspondingly weak vorticity or PV anomalies.

The main question left open by this work concerns the scope of the pseudomomentum rule. As remarked at the end of § 3, the rule is known to be valid in a wider range of cases than those considered here, even though in a still wider context there are known exceptions including the case of one-dimensional sound waves in a rigid tube, as noted long ago in Brillouin’s classic works on radiation stress. Further exceptions include the internal-gravity-wave problem studied in McIntyre (Reference McIntyre1973) and the rotating problems studied in Thomas et al. (Reference Thomas, Bühler and Smith2018, & refs.), in some ways similar to our problem (iii). The failure of the rule in these latter cases, and in Brillouin’s, is related to $O(a^{2})$  mean pressure reactions from confining boundaries (more detail in appendix C below). We may similarly expect failure of the rule in laboratory experiments such as those of Humbert, Aumaître & Gallet (Reference Humbert, Aumaître and Gallet2017), conducted in tanks or channels with confining walls that can support $O(a^{2})$  mean pressures. Section 3 reminds us that the impulse–pseudomomentum theorem depends on having a sufficiently large fluid domain enclosing the regions occupied by waves and vortices.

For the reasons indicated at the end of § 3, even in a large domain the scope of the pseudomomentum rule in wave–vortex interactions is very much a non-trivial question calling for further research, probably involving numerical experimentation along the lines of the strong-refraction experiments of Coste et al. (Reference Coste, Lund and Umeki1999). Even though the Kelvin impulse concept depends on banishing large-scale pressure-field adjustments to infinity, the basic thought-experiment associated with the Magnus relation (3.2), that of applying a force to move a vortex core, works, by contrast, in a relatively local way. This poses not only a technical but also a non-trivial conceptual challenge.

Regarding quantum vortices, it would be interesting to see how the present analysis of problem (ii) extends to the Gross–Pitaevskii superfluid model, a context in which problem (i) was studied in Guo & Bühler (Reference Guo and Bühler2014). In the corresponding version of problem (ii) we can expect to find the same non-interchangeability of limits and the same caveats regarding the Aharonov–Bohm effect, pointing to a remote-recoil contribution in addition to the Iordanskii force. The Gross–Pitaevskii model provides a simple representation of quantum vortex cores (Berloff Reference Berloff2004), whose supersonic flow velocities might vitiate any attempt at a weak-refraction theory, even though the small core size might, on the other hand, imply that bodily advection of the core – back and forth by a larger-scale wavemotion as well as persistently by the mean flow – could still be a useful simplifying feature.