2. Analogy with electrodynamics

We return to the two-dimensional form of our equations, (1.7) and (1.8). The key step is to realize that (1.8) is analogous to the homogeneous subset of Maxwell’s equations, and may be automatically satisfied by the proper choice of potentials. That is, (1.8) is equivalent to the statement

(2.1) $$\begin{eqnarray}(\partial _{t},\partial _{x},\partial _{y})\boldsymbol{\cdot }({\it\alpha},u,v)=0\end{eqnarray}$$

that the three-dimensional space–time divergence of the vector $({\it\alpha},u,v)$ must vanish. This condition is automatically satisfied by the potential representation

(2.2) $$\begin{eqnarray}({\it\alpha},u,v)=(\partial _{t},\partial _{x},\partial _{y})\times ({\it\phi},A,B)=(B_{x}-A_{y},{\it\phi}_{y}-B_{t},A_{t}-{\it\phi}_{x}),\end{eqnarray}$$

where ${\it\phi}(\boldsymbol{x},t)$ and $\boldsymbol{A}(\boldsymbol{x},t)=(A,B)$ and are analogues of the scalar and vector potentials in electrodynamics. The representation (2.2) is analogous to (1.12). Since the curl of a gradient vanishes, the potential representation (2.2) is not unique; the substitutions

(2.3a,b ) $$\begin{eqnarray}{\it\phi}\rightarrow {\it\phi}+{\it\lambda}_{t},\quad \boldsymbol{A}\rightarrow \boldsymbol{A}+\boldsymbol{{\rm\nabla}}{\it\lambda},\end{eqnarray}$$

where ${\it\lambda}$ is an arbitrary function, cause no change in the physical variables $({\it\alpha},u,v)$ . With (1.8) automatically satisfied, (1.7) takes the form

(2.4) $$\begin{eqnarray}(\partial _{tt}-c^{2}{\rm\nabla}^{2})\boldsymbol{A}-\boldsymbol{{\rm\nabla}}({\it\phi}_{t}-c^{2}\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{A})=-q\boldsymbol{u}.\end{eqnarray}$$

We use the gauge arbitrariness (2.3) to require that

(2.5) $$\begin{eqnarray}{\it\phi}_{t}-c^{2}\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{A}=0.\end{eqnarray}$$

Then (2.4) takes the form

(2.6) $$\begin{eqnarray}(\partial _{tt}-c^{2}{\rm\nabla}^{2})\boldsymbol{A}=-q\boldsymbol{u}.\end{eqnarray}$$

By the definition of vorticity

(2.7) $$\begin{eqnarray}{\it\zeta}=v_{x}-u_{y}=(A_{t}-{\it\phi}_{x})_{x}-({\it\phi}_{y}-B_{t})_{y}=c^{-2}(\partial _{tt}-c^{2}{\rm\nabla}^{2}){\it\phi},\end{eqnarray}$$

after use of the gauge condition (2.5).

To make further progress in the direction of electrodynamics we must introduce the analogue of point charges. We assume that the vorticity takes the form

(2.8) $$\begin{eqnarray}{\it\zeta}=\mathop{\sum }_{i}{\it\Gamma}_{i}{\it\delta}(\boldsymbol{x}-\boldsymbol{x}_{i}(t)),\end{eqnarray}$$

where ${\it\delta}$ denotes the Dirac delta function. This assumption is, as it must be, consistent with the conservation of $\iint \text{d}\boldsymbol{x}{\it\zeta}$ , which is one of our conserved Casimirs. Since the point vortices represent singularities in the potential vorticity $q$ , they must, according to (1.9), move at the velocity $\boldsymbol{u}/{\it\alpha}$ . Thus

(2.9) $$\begin{eqnarray}\dot{\boldsymbol{x}}_{i}(t)=\frac{\boldsymbol{u}(\boldsymbol{x}_{i}(t),t)}{{\it\alpha}(\boldsymbol{x}_{i}(t),t)},\end{eqnarray}$$

where the overdot denotes time differentiation. From (2.8) and (2.9) it follows that

(2.10) $$\begin{eqnarray}q\boldsymbol{u}={\it\zeta}\frac{\boldsymbol{u}}{{\it\alpha}}=\mathop{\sum }_{i}{\it\Gamma}_{i}{\it\delta}(\boldsymbol{x}-\boldsymbol{x}_{i}(t))\dot{\boldsymbol{x}}_{i}(t)\end{eqnarray}$$

depends only on the trajectories of the point vortices. Thus (2.6) and (2.7) may be written in the forms

(2.11) $$\begin{eqnarray}\displaystyle & \displaystyle (\partial _{tt}-c^{2}{\rm\nabla}^{2}){\it\phi}=c^{2}\mathop{\sum }_{i}{\it\Gamma}_{i}{\it\delta}(\boldsymbol{x}-\boldsymbol{x}_{i}(t)), & \displaystyle\end{eqnarray}$$

(2.12) $$\begin{eqnarray}\displaystyle & \displaystyle (\partial _{tt}-c^{2}{\rm\nabla}^{2})\boldsymbol{A}=-\mathop{\sum }_{i}{\it\Gamma}_{i}{\it\delta}(\boldsymbol{x}-\boldsymbol{x}_{i}(t))\dot{\boldsymbol{x}}_{i}(t), & \displaystyle\end{eqnarray}$$

(2.13) $$\begin{eqnarray}\displaystyle & (B_{x}-A_{y}){\dot{x}}_{i}(t)={\it\phi}_{y}-B_{t}, & \displaystyle\end{eqnarray}$$

(2.14) $$\begin{eqnarray}\displaystyle & (B_{x}-A_{y}){\dot{y}}_{i}(t)=A_{t}-{\it\phi}_{x}. & \displaystyle\end{eqnarray}$$

To complete the analogy with classical electrodynamics, we seek a variational principle equivalent to (2.4), (2.7) and (2.13), (2.14). (The variational principle should not refer to a particular gauge.) It is not difficult to guess that the Lagrangian takes the form

(2.15) $$\begin{eqnarray}L[{\it\phi}(\boldsymbol{x},t),\boldsymbol{A}(\boldsymbol{x},t),\boldsymbol{x}_{i}(t)]=L_{1}[{\it\phi},\boldsymbol{A}]+L_{2}[{\it\phi},\boldsymbol{A},\boldsymbol{x}_{i}],\end{eqnarray}$$

where

(2.16) $$\begin{eqnarray}L_{1}=\frac{1}{2}\iiint \text{d}t\text{d}\boldsymbol{x}\left(c^{2}(B_{x}-A_{y})^{2}-(A_{t}-{\it\phi}_{x})^{2}-(B_{t}-{\it\phi}_{y})^{2}\right)\end{eqnarray}$$

depends only on the fields, and

(2.17) $$\begin{eqnarray}L_{2}=\iiint \text{d}t\text{d}\boldsymbol{x}\mathop{\sum }_{i}{\it\Gamma}_{i}{\it\delta}(\boldsymbol{x}-\boldsymbol{x}_{i}(t))\left({\it\phi}(\boldsymbol{x},t)+\boldsymbol{A}(\boldsymbol{x},t)\boldsymbol{\cdot }\dot{\boldsymbol{x}}_{i}(t)\right)\end{eqnarray}$$

represents the coupling between the vortices and the fields. We obtain the field equations by requiring that ${\it\delta}L=0$ for arbitrary variations ${\it\delta}{\it\phi}(\boldsymbol{x},t)$ and ${\it\delta}\boldsymbol{A}(\boldsymbol{x},t)$ in the fields, and we obtain the equations (2.13) and (2.14) for the evolution of the vortices by requiring that ${\it\delta}L=0$ for arbitrary variations ${\it\delta}\boldsymbol{x}_{i}(t)$ in the locations of the vortices. To see the latter, we rewrite

(2.18) $$\begin{eqnarray}L_{2}=\mathop{\sum }_{i}{\it\Gamma}_{i}\int \text{d}t\left({\it\phi}(\boldsymbol{x}_{i}(t),t)+A(\boldsymbol{x}_{i}(t),t){\dot{x}}_{i}(t)+B(\boldsymbol{x}_{i}(t),t){\dot{y}}_{i}(t)\right)\end{eqnarray}$$

and consider variations ${\it\delta}x_{i}(t)$ in the $x$ -component of the location of the $i$ th vortex. We find that

(2.19) $$\begin{eqnarray}\displaystyle {\it\delta}L_{2}\! & = & \displaystyle \!\int \text{d}t\left[\left({\it\phi}_{x}+A_{x}{\dot{x}}_{i}(t)+B_{x}{\dot{y}}_{i}(t)\right){\it\delta}x_{i}(t)+A\frac{\text{d}}{\text{d}t}{\it\delta}x_{i}(t)\right]\nonumber\\ \displaystyle \! & = & \displaystyle \!\int \text{d}t\left[{\it\phi}_{x}+A_{x}{\dot{x}}_{i}(t)+B_{x}{\dot{y}}_{i}(t)-\frac{\text{d}}{\text{d}t}A\right]{\it\delta}x_{i}(t)\nonumber\\ \displaystyle \! & = & \displaystyle \!\int \text{d}t\left[{\it\phi}_{x}+B_{x}{\dot{y}}_{i}(t)-A_{t}-A_{y}{\dot{y}}_{i}(t)\right]{\it\delta}x_{i}(t).\end{eqnarray}$$

Since ${\it\delta}L_{2}=0$ for arbitrary ${\it\delta}x_{i}(t)$ , (2.14) follows. Similarly, ${\it\delta}L_{2}/{\it\delta}y_{i}(t)=0$ implies (2.13).

Our variational principle bears a striking resemblance to the corresponding variational principle for classical electrodynamics (e.g. Landau & Lifshitz, Reference Landau and Lifshitz1975, p. 69). Our $L_{1}$ corresponds to the terms conventionally written as $F_{{\it\mu}{\it\nu}}F^{{\it\mu}{\it\nu}}$ , and our $L_{2}$ corresponds to the terms conventionally written as $A_{{\it\mu}}j^{{\it\mu}}$ . Moreover, there is a gauge symmetry corresponding to (2.3). Specifically, variations of the form ${\it\delta}{\it\phi}=({\it\delta}{\it\lambda})_{t}$ and ${\it\delta}\boldsymbol{A}=\boldsymbol{{\rm\nabla}}({\it\delta}{\it\lambda})$ do not affect $L_{1}$ , but ${\it\delta}L_{2}/{\it\delta}{\it\lambda}=0$ implies a conservation law for the vorticity (2.8). Thus the gauge symmetry of the WV Lagrangian corresponds to vorticity conservation in the same way that the gauge symmetry of classical electrodynamics corresponds to charge conservation.

The main difference between the variational principles for WV dynamics and electrodynamics is that the WV Lagrangian lacks a term that depends only on the vortex locations. That is, terms analogous to those that represent the inertia of charged particles are missing from our formulation. This corresponds to the fact that the point vortices move at a velocity that is determined by the fields and not, as in electrodynamics, with an acceleration determined by the fields.

We note that the solutions of (2.11) and (2.12) are singular at the locations of the point vortices. Thus, in applying (2.9), we must omit the contribution of the $i$ th point vortex to the fields that appear on the right-hand side. This restriction, which also occurs in conventional point vortex dynamics, is analogous to neglecting the force of charged particles on themselves. It has (I believe) no fundamental justification.

If we neglect the $\partial _{tt}$ -terms in (2.11) and (2.12), then the resulting equations are elliptic. Taking the solutions of these elliptic equations as the leading order terms in an ‘action-at-a-distance’ approximation, we obtain ${\it\phi}(\boldsymbol{x},t)={\it\phi}_{0}(\boldsymbol{x},t)+O(c^{-2})$ , where

(2.20) $$\begin{eqnarray}{\it\phi}_{0}(\boldsymbol{x},t)=-\mathop{\sum }_{i}\frac{{\it\Gamma}_{i}}{2{\rm\pi}}\ln (|\boldsymbol{x}-\boldsymbol{x}_{i}(t)|),\end{eqnarray}$$

and $\boldsymbol{A}(\boldsymbol{x},t)=(0,x)+O(c^{-2})$ , where we have been careful to add the homogeneous solution $B=x$ required to satisfy the boundary condition ${\it\alpha}=B_{x}-A_{y}\rightarrow 1$ as $r\rightarrow \infty$ . Substituting these approximations back into (2.16) and (2.17), and neglecting irrelevant constants and terms of order $c^{-2}$ , we obtain

(2.21) $$\begin{eqnarray}L_{1}=\frac{1}{2}\iiint \text{d}t\text{d}\boldsymbol{x}\left(-\boldsymbol{{\rm\nabla}}{\it\phi}_{0}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}{\it\phi}_{0}\right)=\frac{1}{2}\int \text{d}t\mathop{\sum }_{i}\mathop{\sum }_{j}\frac{{\it\Gamma}_{i}{\it\Gamma}_{j}}{2{\rm\pi}}\ln r_{ij}(t)\end{eqnarray}$$

and

(2.22) $$\begin{eqnarray}\displaystyle L_{2}\! & = & \displaystyle \!\int \text{d}t\mathop{\sum }_{i}{\it\Gamma}_{i}\left({\it\phi}_{0}(\boldsymbol{x}_{i}(t))+x_{i}(t){\dot{y}}_{i}(t)\right)\nonumber\\ \displaystyle \! & = & \displaystyle \!\int \text{d}t\mathop{\sum }_{i}{\it\Gamma}_{i}x_{i}(t){\dot{y}}_{i}(t)-\int \text{d}t\mathop{\sum }_{i}\mathop{\sum }_{j}\frac{{\it\Gamma}_{i}{\it\Gamma}_{j}}{2{\rm\pi}}\ln r_{ij}(t),\end{eqnarray}$$

where $r_{ij}(t)=|\boldsymbol{x}_{i}(t)-\boldsymbol{x}_{j}(t)|$ . The resulting

(2.23) $$\begin{eqnarray}L=L_{1}+L_{2}=\int \text{d}t\mathop{\sum }_{i}{\it\Gamma}_{i}x_{i}(t){\dot{y}}_{i}(t)-\mathop{\sum }_{i}\mathop{\sum }_{j>i}\frac{{\it\Gamma}_{i}{\it\Gamma}_{j}}{2{\rm\pi}}\int \text{d}t\ln r_{ij}(t)\end{eqnarray}$$

is the Lagrangian for conventional point vortex dynamics, which we henceforth call Kirchhoff dynamics, after one of its originators. For a review of Kirchhoff dynamics, see Aref (Reference Aref2007). In writing out the sums in (2.23), we have thrown away the self-interaction corresponding to $i=j$ .

3. Fields generated by a point vortex with a prescribed trajectory

Now we consider a single point vortex with prescribed trajectory $\boldsymbol{x}_{1}(t)$ on the infinite plane. We adopt the gauge condition (2.5) so that the fields are determined by (2.11) and (2.12). These equations take the form of two-dimensional wave equations with a prescribed source term. However, it is convenient to consider them as special cases of the three-dimensional wave equation

(3.1) $$\begin{eqnarray}(\partial _{tt}-c^{2}(\partial _{xx}+\partial _{yy}+\partial _{zz}))f(x,y,z,t)=Q(x,y,z,t),\end{eqnarray}$$

in which $Q$ , and hence $f$ , is independent of $z$ . The point vortices are lines parallel to the $z$ -axis. The causal solution of (3.1) is

(3.2) $$\begin{eqnarray}f(\boldsymbol{X},t)=\frac{1}{4{\rm\pi}c^{2}}\iiiint \text{d}\boldsymbol{X}_{0}\text{d}t_{0}\frac{Q(\boldsymbol{X}_{0},t_{0})}{|\boldsymbol{X}-\boldsymbol{X}_{0}|}{\it\delta}(t_{0}-t_{r}),\end{eqnarray}$$

where

(3.3) $$\begin{eqnarray}t_{r}=t-|\boldsymbol{X}-\boldsymbol{X}_{0}|/c\end{eqnarray}$$

is the ‘retarded time’. Our notation is $\boldsymbol{X}=(\boldsymbol{x},z)=(x,y,z)$ . With no loss in generality we may set $z=0$ and write (3.2) as

(3.4) $$\begin{eqnarray}f(\boldsymbol{x},t)=\frac{1}{4{\rm\pi}c^{2}}\iiiint \text{d}\boldsymbol{x}_{0}\text{d}Z_{0}\text{d}t_{0}\frac{Q(\boldsymbol{x}_{0},t_{0})}{|\boldsymbol{X}-\boldsymbol{X}_{0}|}{\it\delta}(t_{0}-t_{r}).\end{eqnarray}$$

Then by (2.11) and (2.12) we have

(3.5) $$\begin{eqnarray}({\it\phi}(\boldsymbol{x},t),A(\boldsymbol{x},t),B(\boldsymbol{x},t))=\frac{1}{4{\rm\pi}c^{2}}\iiint \text{d}\boldsymbol{x}_{0}\text{d}z\frac{{\it\Gamma}_{1}(c^{2},-{\dot{x}}_{1}(t_{r}),-{\dot{y}}_{1}(t_{r}))}{\sqrt{(x-x_{0})^{2}+(y-y_{0})^{2}+z^{2}}}{\it\delta}(\boldsymbol{x}_{0}-\boldsymbol{x}_{1}(t_{r})),\end{eqnarray}$$

where now

(3.6) $$\begin{eqnarray}t_{r}=t-\sqrt{(x-x_{0})^{2}+(y-y_{0})^{2}+z^{2}}/c,\end{eqnarray}$$

and we have replaced the dummy integration variable $Z_{0}$ by $z$ . Next we use the delta functions to eliminate the integration over $x_{0}$ and $y_{0}$ . This requires a transformation of the integration variables from $\boldsymbol{x}_{0}$ to $\boldsymbol{x}_{0}-\boldsymbol{x}_{1}(t_{r})$ . The Jacobian of transformation is

(3.7) $$\begin{eqnarray}\frac{\partial (\boldsymbol{x}_{0}-\boldsymbol{x}_{1}(t_{r}))}{\partial (\boldsymbol{x}_{0})}=1-\frac{\dot{\boldsymbol{x}}_{1}(t_{r})\boldsymbol{\cdot }(\boldsymbol{x}-\boldsymbol{x}_{0})}{c\sqrt{(x-x_{0})^{2}+(y-y_{0})^{2}+z^{2}}}.\end{eqnarray}$$

Thus (3.5) becomes

(3.8) $$\begin{eqnarray}({\it\phi}(\boldsymbol{x},t),A(\boldsymbol{x},t),B(\boldsymbol{x},t))=\frac{{\it\Gamma}_{1}}{4{\rm\pi}c^{2}}\int _{-\infty }^{+\infty }\text{d}z\frac{(c^{2},-{\dot{x}}_{1}(t_{r}),-{\dot{y}}_{1}(t_{r}))}{D_{1}},\end{eqnarray}$$

where

(3.9) $$\begin{eqnarray}\displaystyle & D_{1}=\sqrt{r_{1}^{2}+z^{2}}-\dot{\boldsymbol{x}}_{1}(t_{1})\boldsymbol{\cdot }\boldsymbol{r}_{1}/c, & \displaystyle\end{eqnarray}$$

(3.10) $$\begin{eqnarray}\displaystyle & t_{1}=t-\sqrt{r_{1}^{2}+z^{2}}/c, & \displaystyle\end{eqnarray}$$

(3.11) $$\begin{eqnarray}\displaystyle & \boldsymbol{r}_{1}=\boldsymbol{x}-\boldsymbol{x}_{1}(t_{1}), & \displaystyle\end{eqnarray}$$

$r_{1}=|\boldsymbol{r}_{1}|$

$\boldsymbol{r}_{1}(x,y,z,t)$

$t_{1}(x,y,z,t)$

$D_{1}(x,y,z,t)$

Our interest is in the ‘physical fields’ $({\it\alpha},u,v)$ , and these require derivatives of $({\it\phi},A,B)$ with respect to $x,y$ and $t$ . For example, ${\it\alpha}=B_{x}-A_{y}$ , where

(3.12) $$\begin{eqnarray}\boldsymbol{A}(\boldsymbol{x},t)=(0,x)-\frac{{\it\Gamma}_{1}}{4{\rm\pi}c^{2}}\int _{-\infty }^{+\infty }\text{d}z\frac{\dot{\boldsymbol{x}}_{1}(t_{r})}{D_{1}},\end{eqnarray}$$

and we have added a homogeneous solution needed to satisfy the boundary condition that ${\it\alpha}\rightarrow 1$ as $|\boldsymbol{x}|\rightarrow \infty$ . To evaluate

(3.13) $$\begin{eqnarray}{\it\alpha}(\boldsymbol{x},t)=1+\frac{{\it\Gamma}_{1}}{4{\rm\pi}c^{2}}\int _{-\infty }^{+\infty }\text{d}z\left(\frac{\partial }{\partial y}\frac{{\dot{x}}_{1}(t_{1})}{D_{1}}-\frac{\partial }{\partial x}\frac{{\dot{y}}_{1}(t_{1})}{D_{1}}\right)\end{eqnarray}$$

we must compute $\boldsymbol{{\rm\nabla}}D_{1}$ and $\boldsymbol{{\rm\nabla}}t_{1}$ . From the definitions (3.10) and (3.11) we obtain

(3.14) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{{\rm\nabla}}r_{1}=-\frac{\sqrt{r_{1}^{2}+z^{2}}}{r_{1}}c\boldsymbol{{\rm\nabla}}t_{1}, & \displaystyle\end{eqnarray}$$

(3.15) $$\begin{eqnarray}\displaystyle & c\boldsymbol{{\rm\nabla}}t_{1}=-\boldsymbol{r}_{1}/D_{1}. & \displaystyle\end{eqnarray}$$

$D_{1}$

(3.16) $$\begin{eqnarray}\boldsymbol{{\rm\nabla}}D_{1}=-\frac{\boldsymbol{v}_{1}}{c}+\frac{1}{D_{1}}\left(1+\frac{\boldsymbol{a}_{1}\boldsymbol{\cdot }\boldsymbol{r}_{1}}{c^{2}}-\frac{\boldsymbol{v}_{1}\boldsymbol{\cdot }\boldsymbol{v}_{1}}{c^{2}}\right)\boldsymbol{r}_{1},\end{eqnarray}$$

where we have adopted the shorthand notation $\boldsymbol{v}_{1}\equiv \dot{\boldsymbol{x}}_{1}(t_{1})$ and $\boldsymbol{a}_{1}\equiv \ddot{\boldsymbol{x}}_{1}(t_{1})$ . Then (3.13) becomes

(3.17) $$\begin{eqnarray}{\it\alpha}(\boldsymbol{x},t)=1+\frac{{\it\Gamma}_{1}}{4{\rm\pi}c^{2}}\int _{-\infty }^{+\infty }\text{d}z\left(\frac{1}{c{D_{1}}^{2}}(\boldsymbol{r}_{1}\times \boldsymbol{a}_{1})+\frac{1}{{D_{1}}^{3}}\left(1-\frac{\boldsymbol{v}_{1}\boldsymbol{\cdot }\boldsymbol{v}_{1}}{c^{2}}+\frac{\boldsymbol{a}_{1}\boldsymbol{\cdot }\boldsymbol{r}_{1}}{c^{2}}\right)(\boldsymbol{r}_{1}\times \boldsymbol{v}_{1})\right).\end{eqnarray}$$

Since the equations (2.11) and (2.12) determining the fields are linear equations, we obtain the pressure field corresponding to $N$ point vortices by superposition: change all the 1-subscripts above to $i$ -subscripts and sum the integral term in (3.17) from $i=1$ to $N$ . The $z$ -integration in (3.17) corresponds to an integration back in time, as the signal from larger $|z|$ requires a greater time to arrive at $z=0$ . The acceleration terms (i.e. the $\boldsymbol{a}_{1}$ -terms) in (3.17) radiate energy away to infinity; the remaining terms represent the pressure field carried along by the point vortex. Thus, as in electrodynamics, only accelerating charges (vortices) radiate. The $\boldsymbol{r}_{1}\times \boldsymbol{a}_{1}$ term is responsible for dipole radiation. The $\boldsymbol{a}_{1}\boldsymbol{\cdot }\boldsymbol{r}_{1}$ term is responsible for quadrupole radiation. We give several examples to illustrate these points.

As a first example, let

(3.18) $$\begin{eqnarray}\boldsymbol{x}_{1}(t)=(V_{0}t,0)\end{eqnarray}$$

be the location of a single vortex moving along the $x$ -axis at the constant prescribed speed $V_{0}$ . Then $\boldsymbol{v}_{1}=(V_{0},0)$ , $\boldsymbol{a}_{1}=0$ , and (3.17) reduces to

(3.19) $$\begin{eqnarray}{\it\alpha}(\boldsymbol{x},t)=1-\frac{yV_{0}{\it\Gamma}_{1}}{4{\rm\pi}c^{2}}(1-{\it\epsilon}^{2})\int _{-\infty }^{+\infty }\text{d}z\frac{1}{{D_{1}}^{3}},\end{eqnarray}$$

where ${\it\epsilon}=V_{0}/c$ is the Froude number. Solving (3.9)–(3.11) for

(3.20) $$\begin{eqnarray}{D_{1}}^{2}=(x-V_{0}t)^{2}+(1-{\it\epsilon}^{2})(y^{2}+z^{2}),\end{eqnarray}$$

putting (3.20) into (3.19), and carrying out the integration we finally obtain

(3.21) $$\begin{eqnarray}{\it\alpha}(\boldsymbol{x},t)=1-\frac{V_{0}{\it\Gamma}_{1}}{2{\rm\pi}c^{2}}\frac{\sqrt{1-{\it\epsilon}^{2}}y}{(x-V_{0}t)^{2}+(1-{\it\epsilon}^{2})y^{2}}\end{eqnarray}$$

and

(3.22) $$\begin{eqnarray}\boldsymbol{u}(\boldsymbol{x},t)=-c^{2}\int _{-\infty }^{t}\text{d}t^{\prime }\boldsymbol{{\rm\nabla}}{\it\alpha}(\boldsymbol{x},t^{\prime })=\frac{{\it\Gamma}_{1}\sqrt{1-{\it\epsilon}^{2}}}{2{\rm\pi}}\frac{(-y,x-V_{0}t)}{(x-V_{0}t)^{2}+(1-{\it\epsilon}^{2})y^{2}}.\end{eqnarray}$$

The results (3.21) and (3.22) are exact within the context of WV dynamics. As ${\it\epsilon}\rightarrow 0$ , ${\it\alpha}\rightarrow 1$ and the velocity (3.22) approaches that of Kirchhoff dynamics. For ${\it\epsilon}\neq 0$ the velocity field (3.22) is divergent. We find that

(3.23) $$\begin{eqnarray}\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{u}=\frac{{\it\epsilon}^{2}{\it\Gamma}_{1}}{{\rm\pi}}\frac{(x-V_{0}t)\sqrt{1-{\it\epsilon}^{2}}y}{((x-V_{0}t)^{2}+(1-{\it\epsilon}^{2})y^{2})^{2}}.\end{eqnarray}$$

Of course, a single vortex cannot move by itself; for the solution (3.21) and (3.22) to have meaning we must imagine that the vortex is being pushed along by an outside force. However, we obtain a consistent solution by admitting a second vortex with ${\it\Gamma}_{2}=-{\it\Gamma}_{1}$ located on $x=x_{1}$ at the point where $u_{1}/{\it\alpha}_{1}=V_{0}$ and remembering that neither vortex experiences its own field. This two-vortex solution is a generalization of the corresponding solution for counter-rotating vortices in Kirchhoff dynamics.

As a second example, we consider the pressure field generated by a point vortex oscillating at frequency ${\it\omega}$ along the $x$ -axis,

(3.24) $$\begin{eqnarray}\boldsymbol{x}_{1}(t)=\left(\frac{V_{0}}{{\it\omega}}\sin {\it\omega}t,0\right)\!.\end{eqnarray}$$

In this case it is impossible to evaluate (3.17) exactly. We obtain the solution for $r=|\boldsymbol{x}|\rightarrow \infty$ by the method of stationary phase. The details of this calculation are standard, and we present only the result. To the leading order in $r^{-1}$ and $c^{-1}$ , only the $\boldsymbol{r}_{1}\times \boldsymbol{a}_{1}$ term in (3.17) contributes, and we find that

(3.25) $$\begin{eqnarray}{\it\alpha}(\boldsymbol{x},t)\sim 1+\frac{{\it\epsilon}{\it\omega}{\it\Gamma}_{1}}{4{\rm\pi}c^{2}}\sqrt{\frac{2{\rm\pi}}{Kr}}\sin {\it\theta}\sin ({\it\omega}t+{\rm\pi}/4),\end{eqnarray}$$

where $K={\it\omega}/c$ is the wavenumber associated with the oscillation and ${\it\theta}=\tan ^{-1}y/x$ . The solution (3.25) corresponds to dipole radiation at right angles to the trajectory of the vortex. Again, it is impossible for a vortex to oscillate in such a manner by itself, but we may consider (3.25) as the Born approximation for the response of a vortex to an incoming wave.

As a final example, we consider a pair of co-rotating vortices with $\boldsymbol{x}_{1}(t)=-\boldsymbol{x}_{2}(t)$ and equal strength ${\it\Gamma}$ . We assume that the vortices rotate in a circle of radius $r_{0}$ at velocity $V_{0}={\it\Gamma}/4{\rm\pi}r_{0}$ and angular speed ${\it\Omega}={\it\Gamma}/4{\rm\pi}{r_{0}}^{2}$ . This is the Kirchhoff approximation to the near field; by making it we escape the difficult task of solving (2.11)–(2.14) as a coupled set. This calculation is very similar to that of Powell (Reference Powell1964); see Howe (Reference Howe2003, p. 120). Again we are interested in the solution as $r\rightarrow \infty$ , and we present only the final result. To leading order in $r^{-1}$ and $c^{-1}$ the contributions of the vortices to the dipole term in (3.17) cancel, and only the $\boldsymbol{a}_{i}\boldsymbol{\cdot }\boldsymbol{r}_{i}$ terms contribute. We find that

(3.26) $$\begin{eqnarray}{\it\alpha}(r,{\it\theta})\sim 1+\frac{{\it\Gamma}V_{0}^{2}{\it\Omega}}{4{\rm\pi}c^{4}}\sqrt{\frac{{\rm\pi}}{Kr}}\cos (2Kr-2{\it\Omega}t-2{\it\theta}),\end{eqnarray}$$

where $K={\it\Omega}/c$ , corresponding to a weak quadrupole radiation. In the exact solution of the problem, the two co-rotating vortices would gradually move apart at a rate determined by the radiative loss of energy.

4. Remarks

WV dynamics certainly omit phenomena of physical importance. Wave breaking and shock formation are only the most obvious of these. Howe (Reference Howe1999) considers the scattering of an acoustic wave by a point vortex in the Born approximation, using dynamics analogous to (1.1) and (1.2), in which wave refraction can occur in regions of vanishing vorticity. He finds that the scattered wave includes a dipole radiation term identical to (3.25). However, there is an additional, equally important, scattering contribution from the flow circulating around the point vortex. Thus the property of WV dynamics that the waves interact only with vorticity can lead to inaccurate results.

On the other hand, one could hardly expect to produce a close analogy between fluid mechanics and classical electrodynamics without making a strong approximation. Electromagnetic waves do not ‘break’; they interact only with charged particles. Hence electrodynamics and fluid mechanics can never be made equivalent. However, there are situations in which the unrealistic properties of WV dynamics may be desirable. In a study of the interaction between waves and mean flows using the shallow-water equations, Bühler (Reference Bühler1998) found it convenient to remove shock formation from the dynamics by an artificial modification of the potential energy in (1.5). WV dynamics achieves the same end by a modifying the kinetic energy to that in (1.6).

WV dynamics may be a useful new starting point for applying renormalized perturbation theories such as Kraichnan’s (Reference Kraichnan1959) direct interaction approximation to the study of two-dimensional turbulence. In their usual formulation, these techniques apply to any field theory with quadratic nonlinearity, and thus ignore the special status of vorticity in fluid mechanics. Similar methods applied to electrodynamics pay great attention to electric charge.