2.1 The axisymmetric geometry

In AFWOS, we may study the growth of vorticity by expansive stretching in its simplest setting. It is known that no finite-time singularity can then be formed (Majda & Bertozzi Reference Majda and Bertozzi2001), but one may still pose an initial value problem in $\mathbb{R}^{3}$ and ask how fast vorticity can grow at large times. The problem was taken up in Childress (Reference Childress2008), and we now summarize the results. The ideas outlined here will be developed further in § 3.

Since the vorticity consists of rings with a common axis, maximal growth of vorticity as $t\rightarrow \infty$ can be determined by considering a symmetric anti-parallel bundle of vortex rings. Optimization under the condition of conservation of vorticity volume then bounds (for large $t$ ) the maximum of vorticity as a multiple of $t^{2}$ . This estimate can be understood as follows. Imagine a torus with a centreline $\mathscr{C}$ of radius $R$ and a circular cross-section of radius $a$ . Let the angular vorticity component (the only component to be considered here) be $\pm \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D703}}$ in the two half-discs of the cross-section, the signs such as to produce expansive stretching. By conservation of volume, $a^{2}R\sim 1$ in order of magnitude. Also, vortex dynamics ensures $\text{d}R/\text{d}t\sim \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D703}}a$ . Finally, conservation of vorticity flux requires $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D703}}a^{2}\sim 1$ . Thus $\text{d}R/\text{d}t\sim \sqrt{R}$ leading to the $t^{2}$ estimate. We now set $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D703}}=\unicode[STIX]{x1D714}$ for the axisymmetric case.

The growth as $t^{2}$ cannot be obtained by solutions of Euler’s equations since the volume-conserving optimizer does not conserve total kinetic energy $E$ . Indeed, $E\sim Ra^{2}(\unicode[STIX]{x1D714}a)^{2}\sim R$ . If conservation of energy is also imposed, and a cross-sectional scale determined again by a single length $a$ , one must take $a\sim R^{-3/4}$ so that vorticity volume is lost. In fact we can only maintain kinetic energy approximately, with loss of volume and energy occurring through the shedding of a ‘tail’ of vorticity-laden fluid from the vortex pair, of thickness $H\sim R^{-5/2}$ .

To see this, suppose then that we seek a structure with $a\sim R^{-p},p>0$ , which extrudes a tail in the form of a sheet of thickness $H\sim R^{-q}$ . The rate of change of dipole volume is of order $\text{d}(a^{2}R)/\text{d}t\sim R^{-2p}{\dot{R}}$ and the flux of volume into the tail is (since vorticity is proportional to $R$ ) equal to $HR{\dot{R}}\sim R^{1-q}{\dot{R}}$ . Conservation of volume requires that

so that $q=2p+1$ . For the kinetic energy, considered relative to the fluid at infinity, we first compute the flux of energy into the tail. The velocity in the tail is of order $\unicode[STIX]{x1D714}H\sim R^{1-q}$ and therefore, as the circular band of height $H$ and radius $R$ expands at a rate ${\dot{R}}$ , energy is created in the tail at a rate

If this must equal the energy decrease in the ‘head’ of the structure, estimated as

and also $p\neq 3/4$ , then $R^{2p+2}\sim 1$ , which is impossible. The only recourse is to set $3-4p=0$ to make $E_{head}\sim 1$ . Thus $(p,q)=(3/4,5/2)$ and $\text{d}R/\text{d}t\sim \unicode[STIX]{x1D714}R^{-3/4}\sim R^{1/4}$ , yielding a maximum growth for large $t$ as $R\sim t^{4/3}$ . Kinetic energy is lost to the tail at a rate $R^{1-7p}=R^{-17/4}$ from (2.2) and so is extremely small at large $R$ , consistent with $E_{head}\sim 1$ in (2.1). We have thus established that the condition of a negligible loss of kinetic energy uniquely determines the exponents $p,q$ .

In Childress (Reference Childress2008) we described the solution to the variational problem for conserved energy and volume. Details, and related work without energy conservation, are given in Childress (Reference Childress2009). The form of the optimizing dipole is shown in figure 1. The extruded ‘tail’ conserves volume while negligibly reducing kinetic energy.

Figure 1. Vorticity distribution for the dipole which maximizes the velocity of the centreline ‘target’ (the centre dot), subject to constraints on volume, $\unicode[STIX]{x1D714}/r$ , and total kinetic energy. Here $C$ denotes the maximum initial value of $|\unicode[STIX]{x1D714}/r|$ .

Despite their origin from a problem with axial symmetry, these last estimates provide a crucial piece of information concerning the structure of fast-growing vortical structures of this kind. For an anti-parallel, symmetric pair of adjacent eddies, conservation of energy forces a contraction of eddy cross-section over and above that imposed by conservation of volume. Relative to a co-moving frame with coordinates suitably normalized (here by a factor $R^{3/4}$ ), the apparent flow now contains a small non-solenoidal component, effectively feeding volume into the tail as the true cross-section contracts. The key point is that this component breaks the constraint of closed streamlines that prevails without it. In effect conservation of energy turns a structurally unstable topology into a structurally stable, spiral topology. We sketch the proposed flow lines of the upper eddy, a structure we shall refer to as the ‘snail’, in figure 2. (Note that the direction of motion of the vortices is opposite to the direction of crawling of the ‘snail’.)

Since we shall not consider here the calculation of the ‘optimizer’ shown in figure 1, it is perhaps useful to explain how it relates to the construction to follow. To obtain the fastest growth of vorticity under the constraint of constant kinetic energy we maximize the velocity at a ‘target’ vortex ring by arranging the available vorticity in an optimal way. Since we are interested only in the maximal final rate of growth at large times, we examine the maximum absolute value $k$ of $\unicode[STIX]{x1D714}$ divided by distance $R$ from the axis of symmetry. The $\pm k$ is used in eddies of uniform vorticity, and the shape of the dipole chosen to maximize the velocity of the target ring. Since at large $R$ we know that vorticity on every ring is proportional to $R$ , we may bound the growth of vorticity as $t\rightarrow \infty$ . It is shown in Childress (Reference Childress2009) that for $R$ large compared to the diameter of the dipole cross-section, the optimizer satisfies $|\unicode[STIX]{x1D714}|\leqslant Ck^{5/3}E^{1/3}t^{4/3}+O(t^{1/3})$ , where $C$ is a positive number and $E$ is the total kinetic energy of the toroidal dipole structure (with unit density). For the Sadovskii snail $R$ would be chosen so large that the asymptotic Sadovskii state had been reached, and so a similar estimate, applied for the uniform vorticity eddies, would apply. But the number $C$ would presumably be smaller. We have not computed this number since the exponent $4/3$ is the only property of interest here.

Figure 2. The upper half of the snail, showing the instantaneous flow lines relative to a co-moving frame, in local coordinates $(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702})=R^{3/4}(x,y)$ . The spiral shown is the unique interior flow line terminating at a stagnation point on $\unicode[STIX]{x1D702}=0$ . Note that the streamfunction is multivalued on this spiral, jumping across it by an amount equal to the flux into the tail, as a result of the apparent distribution of sources across the dipole. In the asymptotics to be developed the spiral is much ‘tighter’ than depicted here.

2.2 The change of topology in AFWOS

We now summarize results to be derived in the next section. An asymptotic analysis of the maximal growth of vorticity in time results in the leading-order flow and vorticity field $\unicode[STIX]{x1D714}$ taking the form of an arbitrary 2-D eddy with closed streamlines. At the next order, however, contour averaging introduces a compatibility constraint on the leading vorticity term, needed to ensure the existence of the solution at second order.

We are dealing, therefore, with a singular perturbation in the topology of the flow. Realizing that the actual structure at second order is that of the snail, we see that the compatibility constraints on individual closed streamlines are removed, provided that the perturbed velocity field is taken as the ‘leading flow field’, establishing the spiral flow lines. Thus we have an example of bringing forward a second-order effect, in order to completely reorder a calculation, here for the purpose of correctly identifying the spiral topology. Once this is done, contour integration along spiral flow lines terminating in the tail determines the tail, that is, determines the vorticity shed to the wake, irrespective of the particular form of the forcing at second order.

Analysis of the snail configuration will lead naturally to the hypothesis that the preferred ultimate vorticity distribution is one which is piecewise constant, corresponding to zeroth-order eddies which are of the form of the Sadovskii vortex. We use ‘the’ here to refer to the special case where velocity (and the total pressure) is continuous at the dipole boundary. This dipole is one of a family, allowing a discontinuity of velocity at the boundary, considered by Sadovskii (Reference Sadovskii1971). The solution of interest here was independently studied by Saffman & Tanveer (Reference Saffman and Tanveer1982); for a review of these problems see Moore, Saffman & Tanveer (Reference Moore, Saffman and Tanveer1988). The constant vorticity regions emerge from any other dipolar configuration by the stripping away of vorticity into the tail, leaving tubular neighbourhoods of two symmetric anti-parallel vortex lines, thus giving the Sadovskii structure.

3.1 Local analysis of eroding dipoles

In cylindrical coordinates $(r,z,\unicode[STIX]{x1D703})$ (this order being chosen as we shall be working largely in the $(r,z)$ -plane), the vorticity equation is

The conservation of volume is expressed, for an incompressible fluid of unit density, by

We now pass to local coordinates $(x,y)$ though the transformation $r=R(t)+x$ , $z=y$ , $u_{r}={\dot{R}}+u$ , $u_{z}=v$ , with ${\dot{R}}=\text{d}R/\text{d}t$ . We then have

The time derivative is now understood to be for $x$ fixed.

Writing

where $R_{0}$ is a reference length and $\unicode[STIX]{x1D714}_{0}$ a reference vorticity, for example associated with initial conditions, we have

Next, let $a(t)=R_{0}(R(t)/R_{0})^{-p}$ be a lateral scale for the dipole, and set $(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702})=a^{-1}(x,y)$ . Here $p$ is an exponent we expect to be $3/4$ from the previous section, but we leave it unspecified for the moment as we wish to emphasize the independence of the constraint of conservation of energy from the form of the dipole topology. We may assume for eroding vortices that $p>1/2$ (so volume $a^{2}R$ decreases). Lastly, we set

and define a dimensionless time $\unicode[STIX]{x1D70F}$ by

In these variables we set $h=1+x/R$ and have

Here we have chosen $\unicode[STIX]{x1D714}_{0},R_{0}$ so that

and therefore

One useful point to note is that the tail thickness here is $R^{-(1+2p)}$ and so the vorticity in the tail contributes a velocity of order $R\times R^{-(1+2p)}\sim R^{-2p}$ , whereas relative to the head the fluid flow exits the tail with velocity of order $R^{1-p}$ so as to match with the free stream velocity, consistent with the above estimates. Thus the vorticity in the tail contributes a velocity which is negligible compared to the free stream.

3.2 Formal expansion

We now return to (3.9) and carry out a formal expansion in $\unicode[STIX]{x1D700}$ (or $\unicode[STIX]{x1D70F}^{-1}$ ) for the flow in the upper half of the dipole. We introduce the new time variable $\unicode[STIX]{x1D70F}^{\ast }$ by

Thus $\unicode[STIX]{x1D70F}^{\ast }=(1+p)^{-1}\log [1+(1+p)\unicode[STIX]{x1D70F}]\sim (1+p)^{-1}\log \unicode[STIX]{x1D70F}$ . Then (3.9) becomes

The reasoning here is that the scaling of the coordinates and velocity components has already absorbed the dominant time dependence, and we are left with the slower dependence on $\unicode[STIX]{x1D70F}^{\ast }$ . Of course now $\unicode[STIX]{x1D700}$ may be regarded as a function of $\unicode[STIX]{x1D70F}^{\ast }$ , with

We now let $\boldsymbol{Q}\equiv (U,V)=\boldsymbol{Q}_{0}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702},\unicode[STIX]{x1D70F}^{\ast })+\unicode[STIX]{x1D700}\boldsymbol{Q}_{1}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702},\unicode[STIX]{x1D70F}^{\ast })+\cdots \,$ and $\unicode[STIX]{x1D6FA}=\unicode[STIX]{x1D6FA}_{0}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702},\unicode[STIX]{x1D70F}^{\ast })+\unicode[STIX]{x1D700}\unicode[STIX]{x1D6FA}_{1}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702},\unicode[STIX]{x1D70F}^{\ast })+\cdots \,$ . We then obtain the equations

Let us first solve (3.15) simply by setting $\unicode[STIX]{x1D6FA}_{0}=$ constant in a lobe of the vortex. This is a special case which, however, will be shown in the following subsection to be the only allowed solution for the eroding dipole. At next order a particular solution is seen to satisfy

For example, we can take

where the streamfunction $\unicode[STIX]{x1D6F9}_{0}$ is specified by

Any potential flow can be added to this solution and the result matched with an exterior potential flow to make the velocity continuous on the bounding streamline of the vortex.

The point is then that we have a way of extending the zeroth-order solution. In fact we know that there exists a $\boldsymbol{Q}_{0}$ of the desired form, namely the Sadovskii vortex with continuous total pressure.

3.3 The general case

We shall say that the dipole vortex is compatible if an asymptotic solution exists inclusive of the terms of order $\unicode[STIX]{x1D700}$ . We now establish a simple but somewhat surprising result:

To prove this we note from (3.15) that the general solution has the form $\unicode[STIX]{x1D6FA}_{0}=F(\unicode[STIX]{x1D6F9}_{0})$ where $F$ is an arbitrary function, Then, from (3.16) we see that

We now restrict attention to one of the two regions of closed streamlines of the zeroth-order dipole, and introduce the contour average

taken along the direction of flow around a streamline of the flow $\boldsymbol{Q}_{0}$ in the upper eddy. Then it is easy to see from (3.20) that

However, from the divergence theorem and (3.16),

( $U_{0}$ makes no contribution), where $A$ is area within a contour of constant $\unicode[STIX]{x1D6F9}_{0}$ in the $(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702})$ plane. Also

It then follows from (3.22) and $p>1/2$ that $F^{\prime }(\unicode[STIX]{x1D6F9}_{0})=0$ and the lemma is proved.

This lemma brings to mind the classical Prandtl–Batchelor result concerning the constancy of vorticity in steady flow in a region of closed streamlines at large Reynolds number; see Batchelor (Reference Batchelor1956). Indeed, that work inspired investigation of the associated Euler flows (Childress Reference Childress1966). However, the proof of the Prandtl–Batchelor theorem uses the Navier–Stokes equation, since the result depends upon the small but persistent diffusion of vorticity, whereas here the drift to the constant state follows from erosion of vorticity in an inviscid flow and thus convergence to the vorticity value at the vortex centre.

Now we allow $\unicode[STIX]{x1D6FA}_{0}$ to depend upon $\unicode[STIX]{x1D70F}^{\ast }$ . The contour average then gives

But

Since (see e.g. Childress (Reference Childress1987))

we have

Consequently

for some function $G$ , But $\text{e}^{-(2p-1)\unicode[STIX]{x1D70F}^{\ast }}\sim \unicode[STIX]{x1D70F}^{-(2p-1)/(1+p)}\sim R^{1-2p}$ and $R^{1-2p}A$ is equal to $R$ times the dimensional area. Thus we obtain a steady flow-preserving constant total volume, with area shrinking as $R^{-1}$ . This is a compatible dipole for arbitrary $F(\unicode[STIX]{x1D6F9}_{0})$ since it corresponds to (3.22), (3.23) with $p=1/2$ . The dependence on $\unicode[STIX]{x1D70F}^{\ast }$ when $p>1/2$ results from observing a steady volume-preserving structure within coordinates shrinking faster than is required by conservation of volume.

Using the term ‘steady’ in the above sense, meaning that $\unicode[STIX]{x1D6FA}_{0}$ is independent of $\unicode[STIX]{x1D70F}^{\ast }$ , we thus have the following result:

We emphasize that compatibility is a fairly weak measure of dynamic consistency, leaving the requirement of constant kinetic energy as an added and independent constraint. The exponent $p$ needs to be fixed by a full asymptotic solution for large $R$ involving matching an eroding vortex to an external potential flow, as well as proper treatment of the vorticity ‘tail’, and this requires a numerical solution for the perturbed Sadovskii vortex, a problem we take up in the next section. We know of course that the unique compatible structure preserving total kinetic energy is the steady eroding vortex with $p=3/4$ .

In spite of the limited implications of compatibility, we do gain a basic constraint of the zeroth-order structure. We know that the vorticity squared of the dipole, times the area of one vortex, divided by the speed of propagation squared, must equal 37.11 (Saffman & Tanveer Reference Saffman and Tanveer1982). Let the dipole be at position $R\gg R_{0}$ , moving with speed ${\dot{R}}=\unicode[STIX]{x1D714}_{0}R_{0}(R/R_{0})^{1/4}$ , and having vorticity $\unicode[STIX]{x1D714}_{0}R/R_{0}$ and area $2A(R_{0}/R)^{3/2}$ . It then follows that

is our reference length.

We remark that the structure of our preferred dipole with $p=3/4$ can be studied directly in the stable topology. The idea is simply to take the ‘zeroth-order’ term of the snail velocity field, $(U_{s},V_{s})$ say, to include the apparent fluid source to order $\unicode[STIX]{x1D700}$ :

where again $(U_{0},V_{0})$ is the unperturbed Sadovskii velocity field. Our ‘zeroth-order’ problem then becomes;

Our result is now immediate. All flow lines of each vortex are spirals out of a common centre. Since $\unicode[STIX]{x1D6FA}_{s}$ is constant on these flow lines, $\unicode[STIX]{x1D6FA}_{s}$ must be equal everywhere to the value at this centre. We thus may understand the perturbed Sadovskii structure as a result of eroding away the outer layers of the initial structure.

3.4 Summary of the composite solution at leading order

We have seen that the snail emerges as the only compatible vortex structure conserving kinetic energy. We now shall describe the ‘leading-order’ structure in its entirety, including the external potential flow. By ‘leading order’ we here mean that the nominally higher-order effect needed to capture the vortex shrinkage is to be included as a leading-order effect. We thus will describe a perturbed Sadovskii vortex. We are here neglecting entirely the dynamics of erosion. We assume a contracting Sadovskii dipole, at a rate determined by the assumption of energy conservation, and match this with an external flow. To justify this leading-order solution one must derive the erosion from the equations of motion, and this problem we will take up in the following section.

4.1 Inner expansion about a steady Sadovskii vortex

We seek a solution for the vortex pair evolution at large radii $R(t)$ for which it is helpful to set $r\unicode[STIX]{x1D71B}=\unicode[STIX]{x1D714}=\unicode[STIX]{x2202}u_{z}/\unicode[STIX]{x2202}r-\unicode[STIX]{x2202}u_{r}/\unicode[STIX]{x2202}z$ and to solve (3.2) using a Stokes streamfunction $\unicode[STIX]{x1D713}$ , $u_{r}=-r^{-1}\unicode[STIX]{x2202}\unicode[STIX]{x1D713}/\unicode[STIX]{x2202}z$ , $u_{z}=r^{-1}\unicode[STIX]{x2202}\unicode[STIX]{x1D713}/\unicode[STIX]{x2202}r$ . Introducing these into (3.2), we seek to solve the vorticity equation

This simplifies a little if we replace the radial coordinate $r$ by $(1/2)r^{2}$ and this motivates the change of variables from $(r,z,t)$ to $(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702},\unicode[STIX]{x1D70F})$ given by

with $R_{0}$ and $\unicode[STIX]{x1D714}_{0}$ dimensional reference quantities as before. We have introduced dimensionless radii given by $a(t)=R_{0}f(\unicode[STIX]{x1D70F})$ and $R(t)=R_{0}g(\unicode[STIX]{x1D70F})$ . The transformation differs only in minor ways from that introduced earlier in § 3.1. For the fields we set

Dropping any tildes leaves the vorticity equation and vorticity–streamfunction link as

where we use $\mathscr{J}$ for a Jacobian with respect to the $(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702})$ coordinates and a dot (in this section only) for a $\unicode[STIX]{x1D70F}$ -derivative of $f$ or $g$ . This formulation is exact but we have in mind $g=R/R_{0}\gg 1$ and $f=a/R_{0}\ll 1$ for large times and that these are slowly varying, that is,

these may be verified a posteriori. We remark that the assumption $p=3/4$ in § 2 leads to $f\sim t^{-1}$ , $g\sim t^{4/3}$ , so that, as before, an expansion for large $t$ is implied. Our aim now is to obtain a value for $p$ asymptotically by analysis of the shedding of vorticity into the tail.

Thus for an inner solution, that is valid for $(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702})=O(1)$ , we drop the $2fg^{-1}\unicode[STIX]{x1D709}$ term in (4.5) and the frame contraction terms involving ${\dot{f}}/f$ and ${\dot{g}}/g$ in (4.4) at leading order. We use an inner expansion

in which we will find that the $\unicode[STIX]{x1D71B}_{1}$ and $\unicode[STIX]{x1D713}_{1}$ are of order unity. This gives, at leading order, equations for purely 2-D Euler flow,

with $c_{0}$ defined by

Now all we have done so far is valid for any $f_{0}(\unicode[STIX]{x1D70F})$ and $g_{0}(\unicode[STIX]{x1D70F})$ and so without further information $c_{0}$ could also depend on $\unicode[STIX]{x1D70F}$ . However we are seeking a leading-order approximation as the steady Sadovskii vortex with continuous velocity. We thus set the vorticity, streamfunction and speed, that is $(\unicode[STIX]{x1D71B}_{0},\unicode[STIX]{x1D713}_{0},c_{0})$ , to be one of the family of such vortices, with $c_{0}$ taken as constant. We will later choose one with $\unicode[STIX]{x1D71B}_{0}=\pm 1$ in the two lobes, and $c_{0}=1$ , but for the moment the choice is arbitrary. Thus, (4.9) provides a single ordinary differential equation (ODE) linking $f_{0}$ and $g_{0}$ ; we need a further ODE to close the system, and this will emerge at the next order.

Although the choice of our leading-order steady solution is arbitrary, the fact that it is one of a family has important implications. It means that an infinitesimal translation,

satisfies the linear equations

For a solution $(\unicode[STIX]{x1D71B}_{0},\unicode[STIX]{x1D713}_{0},c_{0})$ , a rescaled solution is $(\unicode[STIX]{x1D71B}_{0}(\unicode[STIX]{x1D706}\unicode[STIX]{x1D709},\unicode[STIX]{x1D706}\unicode[STIX]{x1D702}),\unicode[STIX]{x1D706}^{-2}\unicode[STIX]{x1D713}_{0}(\unicode[STIX]{x1D706}\unicode[STIX]{x1D709},\unicode[STIX]{x1D706}\unicode[STIX]{x1D702}),\unicode[STIX]{x1D706}^{-1}c_{0})$ for any $\unicode[STIX]{x1D706}$ . Thus, taking the derivative with respect to $\unicode[STIX]{x1D706}$ at $\unicode[STIX]{x1D706}=1$ , we obtain a solution giving an infinitesimal change of scale

which obeys

Here we have introduced $\mathscr{G}$ in (4.11), (4.13) as the operator inverting the Laplacian in (4.8), that is integration against the kernel

in infinite $(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702})$ space.

Having dealt with the leading-order problem we now write down the first-order equation, in which the neglected terms involving ${\dot{f}}/f$ , ${\dot{g}}/g$ in (4.4) and $2fg^{-2}\unicode[STIX]{x1D709}$ in (4.5) are reintroduced to drive corrections $(\unicode[STIX]{x1D71B}_{1},\unicode[STIX]{x1D713}_{1})$ to the fields:

Here we have made use of (4.6) and (4.9), and defined

To deal with this first-order problem, first invert (4.16) as

and then use (4.10), (4.12) to rearrange (4.15) as

On the left-hand side we have advection of vorticity $\unicode[STIX]{x1D71B}_{1}$ in the basic flow field of the Sadovskii vortex; on the right-hand side are the remaining terms. This equation is ‘driven’ by the terms $c_{0}\unicode[STIX]{x1D709}\unicode[STIX]{x2202}\unicode[STIX]{x1D71B}_{0}/\unicode[STIX]{x2202}\unicode[STIX]{x1D709}$ and $\mathscr{J}(\unicode[STIX]{x1D713}_{1}^{\mathit{sq}},\unicode[STIX]{x1D71B}_{0})$ , in that if these terms were absent a solution would be $\unicode[STIX]{x1D71B}_{1}=0$ , $c_{1}=p_{1}=0$ . Although the driving terms are constant (independent of $\unicode[STIX]{x1D70F}$ ), the solution $\unicode[STIX]{x1D71B}_{1}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702},\unicode[STIX]{x1D70F})$ will generally not be steady, as it will acquire pieces of $\unicode[STIX]{x1D71B}_{0}^{trans}$ corresponding to drift in the $\unicode[STIX]{x1D709}$ -direction and $\unicode[STIX]{x1D71B}_{0}^{scale}$ corresponding to a change in scale; see (4.10)–(4.13).

However we can eliminate these terms by suitable choice of $c_{1}$ and $p_{1}$ – we will check this numerically in due course – and with this choice we expect to be able to obtain a solution $\unicode[STIX]{x1D71B}_{1}$ independent of $\unicode[STIX]{x1D70F}$ . Note that this choice is available to us as the functions $f$ and $g$ are arbitrary rescalings and so we can choose to fix them order by order. This imposition of a solvability condition gives a solution representing the modified Sadovskii vortex, travelling outwards according to $g_{0}(\unicode[STIX]{x1D70F})$ and shrinking through shedding vorticity according to $f_{0}(\unicode[STIX]{x1D70F})$ .

So, we suppose we have converged to a steady solution $\unicode[STIX]{x1D71B}_{1}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702})$ with constants $c_{1}$ (which is not of use to us as it involves $f_{1}$ as a new unknown) and $p_{1}$ which gives a second ODE linking $f_{0}$ and $g_{0}$ in (4.17). Together with (4.2), (4.9) we obtain

and we anticipate $p_{1}>0$ so that $g_{0}$ increases with $t$ and the approximations are all self-consistent. Here we may identify $p_{1}=p$ , the exponent introduced in § 2.1.

We comment that the term in $\unicode[STIX]{x1D713}^{\mathit{sq}}$ corresponds to the leading-order effect of curved vortex lines creating a flow that drives the two lobes of the Sadovskii vortex together, a weak but controlling effect in our expansion. We should also note that when we invert minus the Laplacian and write down $\unicode[STIX]{x1D713}_{0}=\mathscr{G}\unicode[STIX]{x1D71B}_{0}$ and $\mathscr{G}\unicode[STIX]{x1D71B}_{1}$ , for example in (4.18) we could add on a component which is harmonic in the $(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702})$ plane. In fact fundamentally this is how the distant structure of the vortex would feed into the inner solution, modifying vortex shape and motion. It is clear that the terms that would be incorporated would take the form of a multipole expansion: the first would appear at the level of $\mathscr{G}\unicode[STIX]{x1D71B}_{1}$ and would correspond to uniform flow. This could be absorbed into $c_{1}$ (a Galilean transformation) but would not affect the vortex structure or $p_{1}$ .

Figure 4. Schematic of the perturbed Sadovskii vortex indicating the local Cartesian coordinates $(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702})$ and the curvilinear coordinates $(\unicode[STIX]{x1D712},\unicode[STIX]{x1D70E})$ (4.25) adapted to follow the bounding contour $C_{0}$ of the unperturbed vortex. The oncoming velocity at infinity is $-c_{0}\boldsymbol{i}$ .

4.2 Formulation in terms of contours and numerical solution

Now the above is written as if the vorticity fields are smooth, but in fact we are working about the Sadovskii vortex in which the vorticity field is piecewise constant, and so to actually solve the above problem we need to work, not with the fields $\unicode[STIX]{x1D71B}_{0}$ , $\unicode[STIX]{x1D71B}_{1}$ , but instead using contour dynamics. We have in mind here the asymptotic state of the dipole pair for large radius $R$ , where erosion has led to the vorticity being effectively constant in each lobe and the eroded edge is taken as a discontinuity. We thus need to further manipulate the equations, working in the $(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702})$ plane with the use of polar coordinates $(\unicode[STIX]{x1D70C},\unicode[STIX]{x1D703})$ in this plane when needed. Note that (4.19) takes the form

(we will express $\boldsymbol{U}_{0}$ and $\boldsymbol{U}_{1}$ explicitly below), which is the linear piece of the full equation

The leading piece of this is $\boldsymbol{U}_{0}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D71B}_{0}=0$ and gives the steady Sadovskii vortex with the velocity $\boldsymbol{U}_{0}=(U_{0},V_{0})$ and vorticity linked to the total streamfunction $\unicode[STIX]{x1D6F9}_{0}=\unicode[STIX]{x1D713}_{0}+c_{0}\unicode[STIX]{x1D702}$ (including the flow past the vortex) via

The vortex has vorticity $\unicode[STIX]{x1D71B}_{0}=1$ in a region bounded by the $\unicode[STIX]{x1D709}$ -axis and a contour $C_{0}$ in the half-plane $\unicode[STIX]{x1D702}>0$ and $\unicode[STIX]{x1D71B}_{0}=-1$ in the mirror image region; see figure 4. Using the divergence theorem the corresponding streamfunction $\unicode[STIX]{x1D713}_{0}=\mathscr{G}\unicode[STIX]{x1D71B}_{0}$ can be obtained by integrating over the boundaries and gives

with the latter term giving the contribution from the integral along the base, that is the piece $-\unicode[STIX]{x1D709}_{0}\leqslant \unicode[STIX]{x1D709}\leqslant \unicode[STIX]{x1D709}_{0}$ of the $\unicode[STIX]{x1D709}$ axis. For this to represent a steady vortex dipole embedded in a flow $(-c_{0},0)$ at infinity we need the total streamfunction $\unicode[STIX]{x1D6F9}_{0}=\unicode[STIX]{x1D713}_{0}+c_{0}\unicode[STIX]{x1D702}$ to be zero on the contour $C_{0}$ . This condition enables $C_{0}$ to be found for a given $c_{0}$ , for example using a collocation method as described in Saffman & Tanveer (Reference Saffman and Tanveer1982). (In equation (4.24) we correct a misprint in Saffman & Tanveer (Reference Saffman and Tanveer1982), noting that their streamfunction is taken with the opposite sign to ours.)

With $C_{0}$ known at least numerically, we can use a coordinate system based on arc-length $\unicode[STIX]{x1D70E}$ along the contour and a coordinate $\unicode[STIX]{x1D712}$ that measures distance perpendicular to the contour $C_{0}$ ; see figure 4. The corresponding metric is then

where $\unicode[STIX]{x1D705}$ is the curvature of the curve $C_{0}$ at the point given by $\unicode[STIX]{x1D70E}$ . We need to recast the first equation of (4.22) in a contour dynamics setting. We take the non-zero constant vorticity to be $\unicode[STIX]{x1D71B}=1$ in the upper half-plane, confined by a time-dependent contour which we call $C$ and suppose (with mild abuse of notation) given by a function $\unicode[STIX]{x1D712}=C(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F})$ (see figure 4). The situation in the lower half-plane is mirror symmetric. Now $C$ is a material curve and so

or

with $\boldsymbol{t}=h\unicode[STIX]{x1D735}\unicode[STIX]{x1D70E}$ and $\boldsymbol{n}=\unicode[STIX]{x1D735}\unicode[STIX]{x1D712}$ being tangential and normal unit vectors. The unperturbed problem has $C(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F})=C_{0}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F})\equiv 0$ and $\boldsymbol{U}_{0}\boldsymbol{\cdot }\boldsymbol{n}=0$ . At the first order we set $C(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F})=C_{1}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F})+\cdots \,$ , $\boldsymbol{U}=\boldsymbol{U}_{0}+\boldsymbol{U}_{1}+\cdots \,$ to obtain in the linear approximation,

evaluated on the curve $C_{0}$ given by $\unicode[STIX]{x1D712}=0$ . With the use of the streamfunction we write

and after a short calculation obtain

again evaluated on $C_{0}$ .

Setting

we can write the equation in perhaps the most intuitive form

This represents advection of vorticity flux $\unicode[STIX]{x1D6F7}_{1}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F})$ between curves $C_{0}$ and $C_{1}$ along the unperturbed curve $C_{0}$ , with a source term that involves the perpendicular component of the perturbation velocity $\boldsymbol{U}_{1}$ . Note that as we approach the trailing stagnation point, where vorticity will peel off into the flow, $\boldsymbol{U}_{0}\boldsymbol{\cdot }\boldsymbol{t}\rightarrow 0$ and so the source term on the right-hand side is suppressed and the quantity $\unicode[STIX]{x1D6F7}_{1}$ will be seen to converge, even though $C_{1}$ must diverge there.

Figure 5. In (a) the flow field for the Sadovskii vortex is depicted, given by curves of constant $\unicode[STIX]{x1D6F9}_{0}=\unicode[STIX]{x1D713}_{0}+c_{0}\unicode[STIX]{x1D702}$ (4.24) in the $(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702})$ plane, and in (b) the flow $\unicode[STIX]{x1D713}_{1}^{\mathit{sq}}$ (4.18) driven by vortex curvature, which is roughly of stagnation point form.

With the key machinery in place, we indicate the numerical solution that aims to fix $p_{1}$ , through time stepping the partial differential equation (PDE) (4.32) until it can be made to converge to a steady state. Before we time step we evaluate the boundary of the Sadovskii vortex from (4.24) following Saffman & Tanveer and express this as a curve $\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D70C}_{0}(\unicode[STIX]{x1D703})$ in polar coordinates in the $(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702})$ -plane; the resulting flow field is depicted in figure 5(a). From this we may evaluate $\boldsymbol{t}$ and $\boldsymbol{n}$ along $C_{0}$ relative to polar coordinates. Then, for the left-hand side of (4.32) we need $\boldsymbol{U}_{0}\boldsymbol{\cdot }\boldsymbol{t}$ which is obtained from the Sadovskii streamfunction in (4.24) with $\unicode[STIX]{x1D6F9}_{0}=\unicode[STIX]{x1D713}_{0}+c_{0}\unicode[STIX]{x1D702}$ by finite differencing of $\unicode[STIX]{x1D713}_{0}$ as obtained numerically. Turning to the right-hand side of (4.32), $\boldsymbol{U}_{1}$ contains several components from (4.19), in order,

The most straightforward of these are expressed in polar coordinates as

The term arising from vortex line curvature is $\unicode[STIX]{x1D713}_{1}^{\mathit{sq}}$ which is a fixed flow field that can be evaluated once at the start of the computation. This is done rather crudely by evaluating $\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D713}_{0}/\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{2}$ using finite differences, then applying $\mathscr{G}$ by approximating the integral as a finite sum over grid points and finally finite differencing again. Streamlines of the resulting flow field are shown in figure 5(b); this has an approximate stagnation point form, pressing the two lobes of the vortex together.

Finally as we time step the PDE (4.32) the only term that cannot be pre-calculated is the feedback $\boldsymbol{U}_{1}^{\mathscr{G}}$ which is the flow arising from $\mathscr{G}\unicode[STIX]{x1D71B}_{1}$ , from the perturbed contour and a functional of $C_{1}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F})$ . Now the unperturbed contour is $\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D70C}_{0}(\unicode[STIX]{x1D703})$ in polar coordinates, and the gap between this and the perturbed contour, $\unicode[STIX]{x1D712}=C_{1}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F})$ , gives essentially a vortex sheet which has to be integrated as in (4.14) to obtain the corresponding flow. At a point $(\unicode[STIX]{x1D70C}_{0}(\unicode[STIX]{x1D703}),\unicode[STIX]{x1D703})$ on the contour the normal component that we need may be written as an integral over the contour, in terms of the dummy variable $\unicode[STIX]{x1D703}^{\prime }$ ,

where $\unicode[STIX]{x2202}_{\unicode[STIX]{x1D703}}\unicode[STIX]{x1D70C}_{0}=\text{d}\unicode[STIX]{x1D70C}_{0}/\text{d}\unicode[STIX]{x1D703}$ , a prime denotes evaluation with respect to the dummy variable $\unicode[STIX]{x1D703}^{\prime }$ and the Jacobian is given by

With this in place, we time step the PDE (4.32) in terms of $\unicode[STIX]{x1D6F7}_{1}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F})$ by evaluating $\boldsymbol{U}_{1}^{\mathscr{G}}\boldsymbol{\cdot }\boldsymbol{n}$ at each time $\unicode[STIX]{x1D70F}$ and looking up all the other components of the flow field. We need to allow $c_{1}$ and $p_{1}$ to converge so that $\unicode[STIX]{x1D6F7}_{1}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F})$ becomes steady as $\unicode[STIX]{x1D70F}\rightarrow \infty$ , thus avoiding secular behaviour. We have freedom about how this is done: any two conditions that fix the scale and the $\unicode[STIX]{x1D709}$ -location of the vortex will suffice. We choose

and so once all the components of $\boldsymbol{U}_{1}\boldsymbol{\cdot }\boldsymbol{n}$ are found the calculation of $c_{1}$ and $p_{1}$ is straightforward. The result of time stepping is that $\unicode[STIX]{x1D6F7}_{1}$ converges to a $\unicode[STIX]{x1D70F}$ -independent profile, depicted in figure 6(a) with $p_{1}\simeq 0.74$ . This is in line with the theoretical value $p_{1}=3/4$ needed for energy conservation. Note that of the two driving terms, with only $\boldsymbol{U}_{1}^{\mathit{sq}}$ we obtain $p_{1}=0.37$ , and with only $\boldsymbol{U}_{1}^{frame}$ , $p_{1}=0.35$ , so these each account for approximately half the effect.

Figure 6. The steady correction to the Sadovskii vortex is shown by plots of (a) $\unicode[STIX]{x1D6F7}_{1}$ and (b) $C_{1}$ (see (4.31)) as functions of the polar angle $\unicode[STIX]{x1D703}$ in the $(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702})$ plane, valid in the asymptotic limit $\unicode[STIX]{x1D70F}\rightarrow \infty$ .

Finally, we remark on the formula (4.37): the feedback on the flow because the contour $C$ differs a little from $C_{0}$ amounts to calculating the flow from a vortex sheet of strength $C_{1}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F})$ along the curve $C_{0}$ . Such an integral has to be taken as a principal value, and here we have done this by removing explicitly the singular components from the integrands in (4.37), which are then placed in the final $\log (\tan (1/2)\unicode[STIX]{x1D703})$ term. Taking the principal value is appropriate as at a point $\unicode[STIX]{x1D70E}$ on the vortex sheet/thin layer the transverse flows generated by the vorticity for $\unicode[STIX]{x1D70E}^{\prime }>\unicode[STIX]{x1D70E}$ and $\unicode[STIX]{x1D70E}^{\prime }<\unicode[STIX]{x1D70E}$ locally cancel out. However at $\unicode[STIX]{x1D703}=0$ and $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}$ , this argument fails: the vortex sheet comes to an abrupt end and in fact changes sign. (The curvature singularities and singular flow field here in the underlying Sadovskii vortex are explained in depth in Saffman & Tanveer (Reference Saffman and Tanveer1982)). This explains the presence of the logarithmic singularity at $\unicode[STIX]{x1D703}=0$ , $\unicode[STIX]{x03C0}$ in the term $\log (\tan (1/2)\unicode[STIX]{x1D703})$ . In our calculations we have taken $\boldsymbol{U}_{1}\boldsymbol{\cdot }\boldsymbol{n}=0$ at $\unicode[STIX]{x1D703}=0$ in (4.39) which keeps $C_{1}=0$ there (see (4.30)) and removes immediate difficulties with this term. For $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}$ the singular term is present, and is part of the flow field that leads to the ejection of vorticity from the rear of the vortex pair.

5.1 Set-up

We simulate AFWOS subject to the following conditions: vorticity is anti-symmetric about the $z=0$ plane and vorticity is non-zero only in a small region (which possibly moves over time, and may lie far from the $r=0$ axis). The configuration of the simulation is represented at each time step by the values of $\unicode[STIX]{x1D703}$ -vorticity within a small square region of the plane. Time stepping is implemented via the fourth-order Runge–Kutta scheme, with the components of the Euler equation recovered from the vorticity via the Biot–Savart law. Calculations are undertaken in a local Fourier basis to preserve as much spatial accuracy as possible, both in the simulation, and in the computation of quantities of interest afterwards. While this Fourier basis has many convenient features favouring the speed and accuracy of the simulation, there are a number of complications introduced by the mismatch between the periodic nature of the basis and the infinite domain of the cylindrical coordinate system.

Every initial configuration we simulated evolved into a ‘snail’; we here examine the trajectory of one configuration in depth. We describe the initial conditions here and explain why they could be expected to evolve into a snail in a particularly direct and smooth way. Specifically, the initial condition consists of two anti-parallel vortex rings, each of which has vorticity which is the product of the radial coordinate $r$ , with a smooth transition function which is close to 1 inside a torus and close to 0 outside it: an appropriately shifted and scaled error function applied to the distance from circular centreline of each torus. Specifically, given cylindrical coordinates $(r,z,\unicode[STIX]{x1D703})$ , the initial condition for the vorticity in the $\unicode[STIX]{x1D703}$ direction is defined by

Thus our initial conditions have vortex rings at $(r,z)=(0.7,\pm 0.32)$ , each of radius $3\times 0.06=0.18$ , and with the transition from the interior to the exterior of each ring occurring over roughly 0.06 distance. The vorticity is chosen to be nearly homogeneous (before the $r$ scaling) inside each vortex ring so that when the snail sheds the outer layers of each ring, the vorticity will become increasingly constant inside the evolving snail.

Starting with vortex rings at radius $0.7$ , the simulation was run until the radius at the centre of the snail was 9.6. The diameter of the vortex tube was initially $0.36$ and was finally 0.12 in the radial direction and 0.039 in the $z$ direction, having shed 55 % of its circulation. Because our simulation repeatedly increases resolution so that the snail remains several hundred grid points across, the grid size decreases from $0.002$ down to $0.00035$ over the course of a run, and would continue to decrease as $r$ increases. The grid used is $800\times 800$ pixels and this is centred on the vortex rings; the tail is passive, and allowed to trail behind the vortex dipole, and out of the box in the simulation. As we do many operations in a local ‘periodic’ box, i.e. using a Fourier basis, in many operations we use masking around the edge to avoid the numerical periodicity interfering with the actual cylindrical geometry. Stability concerns dictate that the evolving vortex rings can move at most a fraction of a grid space in each simulation time step, which employs a fourth-order Runge–Kutta scheme, meaning that the cost of continuing for larger $r$ would continue to increase rapidly, and running over a much wider range of radii is infeasible. Nonetheless, we present results in § 5.2 showing that already within the scope of this simulation, the configuration converges rapidly to the expected behaviour of the snail.

The most time consuming part of the code is, given the vorticity distribution in the box in the local $(r,z)$ -plane, to compute the corresponding flow field. This involves a convolution in the $z$ -direction which may be done in Fourier space, the problem being separable in the axial direction. (A box of twice the vertical extent and masking are used to avoid any spurious effects of the numerical periodicity interfering with the convolution.) However in the radial direction it is necessary to undertake the convolution explicitly and to compute the appropriate elliptic integrals in this Green’s function (with the use of both Matlab’s built-in function and an expansion valid for points close to the source). Because of the expense in evaluating the elliptic integrals, as the window following the vortex moves and also zooms in (through six different resolutions in the run presented here), much of this Green’s function data are reused by resampling on the new grid.