2 The process of pyramid reconnection

Figure 2.

Sketch indicating the nature of the pyramid-reconnection process: the sense of the surviving vortex filaments (solid blue, with associated flux $\unicode[STIX]{x1D6E4}_{s}(\unicode[STIX]{x1D70F})$ and direction as indicated by the arrows) is such that they propagate upwards under their mutual interaction towards the apex of the prism, where they reconnect; the reconnected vortex filaments (dashed red, with associated flux $\unicode[STIX]{x1D6E4}_{r}(\unicode[STIX]{x1D70F})$ ) propagate downwards on the complementary faces of the prism away from the apex.

The ‘pyramid-reconnection’ process, as anticipated in MK19, is represented schematically by the sketch of figure 2. As $\unicode[STIX]{x1D6FF}/s$ increases to $O(1)$ it is necessary to take account of the process of viscous vortex reconnection on the symmetry plane $x=0$ . This leads to a stripping away of some of the $y$ -component of vorticity at the tipping points in each half-space $x<0$ and $x>0$ . Thus, the initial circulation $-\unicode[STIX]{x1D6E4}$ in the vortex ${\mathcal{V}}_{1}$ centred on the curve $C_{1}$ in the half-space $x<0$ branches into two circulations: ‘surviving circulation’ $-\unicode[STIX]{x1D6E4}_{s}(\unicode[STIX]{x1D70F})$ (blue, solid) and ‘reconnected circulation’ $-\unicode[STIX]{x1D6E4}_{r}(\unicode[STIX]{x1D70F})$ (red, dashed), with $\unicode[STIX]{x1D6E4}_{s}(\unicode[STIX]{x1D70F})+\unicode[STIX]{x1D6E4}_{r}(\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D6E4}$ (and similarly of course for the vortex ${\mathcal{V}}_{2}$ in $x>0$ ). The arrows in figure 2 indicate the compatible directions of vorticity $\unicode[STIX]{x1D74E}$ in the surviving and reconnected domains. The reconnected circulation is oriented so that it falls away from the interaction region; this presumably exerts a mild braking effect on the continuing progress of the surviving circulation towards the incipient singularity. Here, in exploratory vein, we shall neglect this braking effect, which could, however, be included in an improved model.

These branching and falling-away processes are essentially the same as the ‘bridging’ and ‘stripping’ processes envisaged by Melander & Hussain (Reference Melander and Hussain1989) and Hussain & Duraisami (Reference Hussain and Duraisami2011). The processes have been recently described in similar terms by Kerr (Reference Kerr2018), through DNS computations on vortex reconnection for a trefoil vortex and for the more conventional situation of perturbed anti-parallel vortices. For the latter case, Kerr describes an ‘exchange of circulations’, similar to the viscous destruction of surviving circulation and its replacement by reconnected circulation, a process that we now analyse. We note that Hussain & Duraisami (Reference Hussain and Duraisami2011) describe vortex reconnection at vortex Reynolds numbers up to 7000; the principal vortices remain coherent both before and after reconnection, and they observe that ‘viscous reconnection is never complete, leaving behind a part of the initial tubes as threads, which then undergo successive reconnections’. Flattening of the primary viscous core cross-sections is not apparent in this study, but flattening does occur in the interaction of the bridging threads after reconnection. We cannot capture these ‘detritus effects’ in our analytical model, which is primarily concerned with the build-up to reconnection and to the time when the stretched vorticity is maximal.

The rate of decrease of the surviving circulation $\unicode[STIX]{x1D6E4}_{s}(\unicode[STIX]{x1D70F})$ results from viscous diffusion across the plane $x=0$ , and is given in dimensionless form by

(2.1) $$\begin{eqnarray}\frac{\text{d}\unicode[STIX]{x1D6E4}_{s}}{\text{d}\unicode[STIX]{x1D70F}}=\frac{\text{d}}{\text{d}\unicode[STIX]{x1D70F}}\oint _{C_{-}}\boldsymbol{v}\boldsymbol{\cdot }\text{d}\boldsymbol{x}=\unicode[STIX]{x1D716}\oint _{C_{-}}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{v}\boldsymbol{\cdot }\text{d}\boldsymbol{x},\end{eqnarray}$$

where $C_{-}$ is the closed circuit consisting of the $z$ -axis ( $x=y=0$ ) and the semicircle at infinity in the half-plane $y=0$ , $x<0$ . (Note that when $\unicode[STIX]{x1D716}=0$ this equation is an expression of Kelvin’s circulation theorem for the stationary circuit $C_{-}$ ; when $\unicode[STIX]{x1D716}>0$ , we need merely include the term $\unicode[STIX]{x1D716}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{v}$ , from the Navier–Stokes equation.) It is evidently only the contribution from the $z$ -axis here that is non-zero, and this gives

(2.2) $$\begin{eqnarray}\frac{\text{d}\unicode[STIX]{x1D6E4}_{s}}{\text{d}\unicode[STIX]{x1D70F}}=\unicode[STIX]{x1D716}\int _{-\infty }^{\infty }\left.\left(\frac{\unicode[STIX]{x2202}^{2}v_{z}}{\unicode[STIX]{x2202}x^{2}}+\frac{\unicode[STIX]{x2202}^{2}v_{z}}{\unicode[STIX]{x2202}z^{2}}\right)\right|_{x=0}\,\text{d}z=\unicode[STIX]{x1D716}\int _{-\infty }^{\infty }\left.\frac{\unicode[STIX]{x2202}^{2}v_{z}}{\unicode[STIX]{x2202}x^{2}}\right|_{x=0}\text{d}z,\end{eqnarray}$$

(the second term integrating to zero because $\unicode[STIX]{x2202}v_{z}/\unicode[STIX]{x2202}z=0$ at $z=\pm \infty$ ).

Now neglecting the core deformation, the vorticity field on the plane $y=0$ at time  $\unicode[STIX]{x1D70F}$ is that of two vortices of Gaussian structure, with the single component

(2.3) $$\begin{eqnarray}\unicode[STIX]{x1D714}_{y}(x,z,\unicode[STIX]{x1D70F})=\frac{\unicode[STIX]{x1D6E4}_{s}(\unicode[STIX]{x1D70F})}{4\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}^{2}}\left(\exp \left[-\frac{(x+s)^{2}+z^{2}}{4\unicode[STIX]{x1D6FF}^{2}}\right]-\exp \left[-\frac{(x-s)^{2}+z^{2}}{4\unicode[STIX]{x1D6FF}^{2}}\right]\right),\end{eqnarray}$$

as sketched in figure 3(a) for $s=1$ , $\unicode[STIX]{x1D6FF}=0.9456$ . The corresponding $z$ -component of velocity is

(2.4) $$\begin{eqnarray}\displaystyle v_{z}(x,z,\unicode[STIX]{x1D70F}) & = & \displaystyle \frac{\unicode[STIX]{x1D6E4}_{s}(\unicode[STIX]{x1D70F})}{2\unicode[STIX]{x03C0}}\left\{\frac{x+s}{(x+s)^{2}+z^{2}}\left(1-\exp \left[-\frac{(x+s)^{2}+z^{2}}{4\unicode[STIX]{x1D6FF}^{2}}\right]\right)\right.\nonumber\\ \displaystyle & & \displaystyle -\left.\frac{x-s}{(x-s)^{2}+z^{2}}\left(1-\exp \left[-\frac{(x-s)^{2}+z^{2}}{4\unicode[STIX]{x1D6FF}^{2}}\right]\right)\right\},\end{eqnarray}$$

(figure 3(b), for $s=1,2,3,4$ and fixed $\unicode[STIX]{x1D6FF}=0.9456$ ), from which we may calculate

(2.5) $$\begin{eqnarray}\displaystyle \!\!\left.\frac{\unicode[STIX]{x2202}^{2}v_{z}}{\unicode[STIX]{x2202}x^{2}}\right|_{x=0} & = & \displaystyle -\frac{s^{5}\unicode[STIX]{x1D6E4}_{s}(\unicode[STIX]{x1D70F})[(s^{6\!}+2s^{4}z^{2\!}-6z^{4}\unicode[STIX]{x1D6FF}^{2\!}-24z^{2}\unicode[STIX]{x1D6FF}^{2\!}+s^{2}z^{4\!}-4s^{2}z^{2}\unicode[STIX]{x1D6FF}^{2\!}+8s^{2}\unicode[STIX]{x1D6FF}^{4})\exp [-(s^{2\!}+z^{2})/4\unicode[STIX]{x1D6FF}^{2}]]}{2\unicode[STIX]{x03C0}(s^{2}+z^{2})^{3}\unicode[STIX]{x1D6FF}^{2}}\nonumber\\ \displaystyle & & \displaystyle -\,\frac{2s\unicode[STIX]{x1D6E4}_{s}(\unicode[STIX]{x1D70F})(3z^{2}-s^{2})}{\unicode[STIX]{x03C0}(s^{2}+z^{2})^{3}}.\end{eqnarray}$$

Here, the second term is just what survives in the ‘point-vortex’ limit $\unicode[STIX]{x1D6FF}\rightarrow 0$ , and it integrates to zero over the range $z\in (-\infty ,\infty )$ . On this integration, the first term contributes the result

(2.6) $$\begin{eqnarray}\int _{-\infty }^{\infty }\left.\frac{\unicode[STIX]{x2202}^{2}v_{z}}{\unicode[STIX]{x2202}x^{2}}\right|_{x=0}\text{d}z=-\frac{s\unicode[STIX]{x1D6E4}_{s}(\unicode[STIX]{x1D70F})}{2\surd \unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}^{3}}\exp [-s^{2}/4\unicode[STIX]{x1D6FF}^{2}],\end{eqnarray}$$

and it follows from (2.2) that

(2.7) $$\begin{eqnarray}\frac{\text{d}\unicode[STIX]{x1D6E4}_{s}}{\text{d}\unicode[STIX]{x1D70F}}=-\unicode[STIX]{x1D716}\frac{s\unicode[STIX]{x1D6E4}_{s}(\unicode[STIX]{x1D70F})}{2\surd \unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}^{3}}\exp [-s^{2}/4\unicode[STIX]{x1D6FF}^{2}].\end{eqnarray}$$

Figure 3.

(a) The vorticity $\unicode[STIX]{x1D714}_{y}(x,0,\unicode[STIX]{x1D70F})/\unicode[STIX]{x1D6E4}_{s}(\unicode[STIX]{x1D70F})$ given by (2.3), at a time $\unicode[STIX]{x1D70F}$ at which $s=1$ , $\unicode[STIX]{x1D6FF}/s=0.9456$ (the limiting value stated above); the position of the vortex centres is indicated by the two red dots; (b) the velocity $v_{z}(x,0,\unicode[STIX]{x1D70F})/\unicode[STIX]{x1D6E4}_{s}(\unicode[STIX]{x1D70F})$ given by (2.4) for $s=1$ (solid), 2 (green, dashed, 3 (blue, dotted) and 4 (red, dot-dashed).

The above neglect of core deformation calls for comment. There is evidence from DNS, such as in the recent investigations of Kerr (Reference Kerr2018) and McKeown et al. (Reference McKeown, Ostilla-Monico, Pumir, Brenner and Rubinstein2018) and in earlier studies, that there is a flattening of the vortex cores during the interaction process when $\unicode[STIX]{x1D6FF}/s$ increases to $O(1)$ . We have argued in MK19 that this flattening should decrease with increasing Reynolds number, on the grounds that the vortices are then spinning so rapidly that they experience an effectively axisymmetric strain near the tipping points. Nevertheless, the curvature of the vortices at the tipping points may affect this conclusion, a possibility that calls for further analysis. A modest degree of core flattening could easily be taken into account through simple modification of the vorticity structure (2.3) assumed above; this would change the details of the above calculation, but is unlikely to modify the result (2.7) in any fundamental way. We therefore adopt this result as it stands. It is important to note that this result is firmly based on (2.1), which itself is a direct consequence of the Navier–Stokes equation.

During the very short reconnection phase, we thus need to replace $\unicode[STIX]{x1D6E4}$ by $\unicode[STIX]{x1D6E4}_{s}(\unicode[STIX]{x1D70F})$ in calculating the instantaneous velocity, rate of stretching, and rate of change of curvature at the tipping points. This simply means that, defining $\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D70F})\equiv \unicode[STIX]{x1D6E4}_{s}(\unicode[STIX]{x1D70F})/\unicode[STIX]{x1D6E4}$ , we have to replace (1.2) by

(2.8a-c ) $$\begin{eqnarray}\frac{\text{d}s}{\text{d}\unicode[STIX]{x1D70F}}=-\frac{\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D705}\cos \unicode[STIX]{x1D6FC}}{4\unicode[STIX]{x03C0}}\left[\log \left(\frac{s}{\unicode[STIX]{x1D6FF}}\right)+\unicode[STIX]{x1D6FD}_{1}\right],\quad \frac{\text{d}\unicode[STIX]{x1D705}}{\text{d}\unicode[STIX]{x1D70F}}=\frac{\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D705}\cos \unicode[STIX]{x1D6FC}\sin \unicode[STIX]{x1D6FC}}{4\unicode[STIX]{x03C0}s^{2}},\quad \frac{\text{d}\unicode[STIX]{x1D6FF}^{2}}{\text{d}\unicode[STIX]{x1D70F}}=\unicode[STIX]{x1D716}-\frac{\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D705}\cos \unicode[STIX]{x1D6FC}}{4\unicode[STIX]{x03C0}s}\unicode[STIX]{x1D6FF}^{2},\end{eqnarray}$$

which, from (2.7), must now be coupled with

(2.9) $$\begin{eqnarray}\frac{\text{d}\unicode[STIX]{x1D6FE}}{\text{d}\unicode[STIX]{x1D70F}}=-\unicode[STIX]{x1D716}\frac{s\unicode[STIX]{x1D6FE}}{2\surd \unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}^{3}}\exp [-s^{2}/4\unicode[STIX]{x1D6FF}^{2}].\end{eqnarray}$$

We may now integrate this 4th-order dynamical system with the initial conditions (1.4), together with $\unicode[STIX]{x1D6FE}(0)=1$ .

3 Integration of the system (2.8), (2.9)

We have integrated this system, using Mathematica with 56-point precision, with a view to determining the variation of $\unicode[STIX]{x1D714}(\unicode[STIX]{x1D70F})\equiv \unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D70F})/\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D70F})^{2}$ ; this is a measure of the maximum vorticity at the centre of the surviving vortex core. The function $\unicode[STIX]{x1D714}(\unicode[STIX]{x1D70F})/\unicode[STIX]{x1D714}(0)$ then represents the amplification of vorticity. There are two competing effects here: $\unicode[STIX]{x1D6FF}^{2}$ decreases, at least initially, towards zero, but $\unicode[STIX]{x1D6FE}$ also decreases due to the reconnection process.

Figure 4(a) shows results for $\unicode[STIX]{x1D716}=1/2000$ , i.e. $R_{\unicode[STIX]{x1D6E4}}=2000$ . This shows that $\unicode[STIX]{x1D6FF}^{2}$ increases initially due to viscous diffusion, but then vortex stretching dominates, causing a decrease to very near (but not quite) zero; in fact, $\unicode[STIX]{x1D6FF}^{2}$ reaches a minimum of $2.67412\times 10^{-8}$ at $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}_{c}\approx 9.12455$ , and then for $\unicode[STIX]{x1D70F}\gtrsim \unicode[STIX]{x1D70F}_{c}$ , increases almost linearly due to viscous diffusion. The variation of $\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D70F})/100$ is shown, the factor $1/100$ being introduced to make the variation more visible; $\unicode[STIX]{x1D6FE}$ decreases from 1 to about 0.02 as $\unicode[STIX]{x1D70F}$ approaches $\unicode[STIX]{x1D70F}_{c}$ , and then drops very rapidly to zero, indicating rapid completion of the reconnection process. Figure 4(b) shows the corresponding variation of $\unicode[STIX]{x1D714}(\unicode[STIX]{x1D70F})/\unicode[STIX]{x1D714}(0)$ , which first increases due to the decrease of  $\unicode[STIX]{x1D6FF}^{2}$ , but reaches a modest maximum of 3.711. Thus, there is certainly no singularity for this value of  $R_{\unicode[STIX]{x1D6E4}}$ .

Figure 4.

Evolution governed by (2.8) with $\unicode[STIX]{x1D716}=1/2000$ : (a)  $\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D70F})^{2}$ (black) which appears to fall to zero at $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}_{c}\approx 9.12455$ , but does not quite reach zero, as shown by the expanded curve of $10^{3}\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D70F})^{2}$ (blue); $\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D70F})/100$ (green) decreases to about 0.0002 at $\unicode[STIX]{x1D70F}\approx \unicode[STIX]{x1D70F}_{c}$ then falls rapidly to very near zero; (b) the corresponding variation of vorticity amplification $\unicode[STIX]{x1D714}(\unicode[STIX]{x1D70F})/\unicode[STIX]{x1D714}(0)$ in the neighbourhood of  $\unicode[STIX]{x1D70F}_{c}$ , where it reaches a maximum of approximately 3.7.

Figure 5 shows corresponding results with $\unicode[STIX]{x1D716}=1/3000$ . Again, $\unicode[STIX]{x1D6FF}^{2}$ increases initially but then decreases to near zero at $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}_{c}\approx 1.9391619710016794917768$ , but still just fails to reach zero; in fact, $\unicode[STIX]{x1D6FF}^{2}$ evaluates to $2.7278\times 10^{-24}$ at $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}_{c}$ ! $\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D70F})$ has decreased to ${\sim}0.05$ at $\unicode[STIX]{x1D70F}=1.93$ and drops to 0.000451 at $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}_{c}$ . Now, $\unicode[STIX]{x1D714}(\unicode[STIX]{x1D70F})/\unicode[STIX]{x1D714}(0)$ (figure 5 b) shows a very sharp peak just before $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}_{c}$ where, in fact, $\unicode[STIX]{x1D714}(\unicode[STIX]{x1D70F}_{c})/\unicode[STIX]{x1D714}(0)$ evaluates to $2.425\times 10^{16}$ , a huge increase caused by vortex stretching, but still not a singularity.

Figure 5.

Same as figure 4, but with $\unicode[STIX]{x1D716}=1/3000$ .

Figure 6.

Evolution for $\unicode[STIX]{x1D716}=1/4000$ , and for $0<\unicode[STIX]{x1D70F}<\unicode[STIX]{x1D70F}_{c}\approx 0.917576564746480\ldots$ (marked by red dotted line); (a)  $\unicode[STIX]{x1D6FF}^{2}(\unicode[STIX]{x1D70F})$ and $\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D70F})$ both appear to fall to zero at $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}_{c}$ ; (b) zoom of $10^{12}\unicode[STIX]{x1D6FF}^{2}(\unicode[STIX]{x1D70F})$ very near  $\unicode[STIX]{x1D70F}_{c}$ ; (c,d)  $\unicode[STIX]{x1D714}(\unicode[STIX]{x1D70F})/\unicode[STIX]{x1D714}(0)$ and $\unicode[STIX]{x1D714}(0)/\unicode[STIX]{x1D714}(\unicode[STIX]{x1D70F})$ near $\unicode[STIX]{x1D70F}_{c}$ , as well as can be resolved; (e,f) other relevant variables near  $\unicode[STIX]{x1D70F}_{c}$ , as well as can be resolved.

Consider now the situation when $\unicode[STIX]{x1D716}=1/4000$ (i.e. $R_{\unicode[STIX]{x1D6E4}}=4000$ ). Here, Mathematica stalls at $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}_{c}\approx 0.91757656474648$ . The reason is that now $\unicode[STIX]{x1D6FF}^{2}$ falls so close to zero that (2.9) runs into an apparent singularity. Figure 6(a) shows $\unicode[STIX]{x1D6FF}^{2}(\unicode[STIX]{x1D70F})$ and $\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D70F})/5000$ (again the factor $1/5000$ is included to make the comparison clear), showing that, while $\unicode[STIX]{x1D6FF}^{2}$ seems to go to zero at the apparent singularity time $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}_{c}$ (marked by the dotted line), $\unicode[STIX]{x1D6FE}$ is still finite; figure 6(b) shows $\unicode[STIX]{x1D6FF}^{2}$ expanded by a factor $10^{12}$ in the immediate neighbourhood of $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}_{c}$ ( $0.91757656474639<\unicode[STIX]{x1D70F}<0.91757656474655$ ); $\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D70F}_{c})^{2}$ is certainly extremely small if not exactly zero. Figure 6(c) shows (over the same exceedingly small range of  $\unicode[STIX]{x1D70F}$ ) that $\unicode[STIX]{x1D714}(\unicode[STIX]{x1D70F})/\unicode[STIX]{x1D714}(0)$ increases to a very large value near  $\unicode[STIX]{x1D70F}_{c}$ ; but is this really infinite? Figure 6(d) shows the inverse function $\unicode[STIX]{x1D714}(0)/\unicode[STIX]{x1D714}(\unicode[STIX]{x1D70F})$ , which does indeed appear to be zero at this point as far as this computation can determine.

4 Asymptotic behaviour for small $\unicode[STIX]{x1D716}$

A necessary condition for a finite-time singularity of vorticity $\unicode[STIX]{x1D714}(\unicode[STIX]{x1D70F})\equiv \unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D70F})/\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D70F})^{2}$ is obviously that $\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D70F})^{2}$ should decrease to exactly zero at some finite time. In order to further investigate this possibility, we consider the $\unicode[STIX]{x1D716}$ -dependence of $\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D70F})_{min}^{2}$ as $\unicode[STIX]{x1D716}$ decreases towards zero from $1/2000$ . For each  $\unicode[STIX]{x1D716}$ , we first locate the time $\unicode[STIX]{x1D70F}_{c}$ at which $\text{d}\unicode[STIX]{x1D6FF}^{2}/\text{d}\unicode[STIX]{x1D70F}$ ( $=\unicode[STIX]{x1D716}-(\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D705}\cos \unicode[STIX]{x1D6FC}/4\unicode[STIX]{x03C0}s)\unicode[STIX]{x1D6FF}^{2}$ ) is zero, and then evaluate the corresponding minimum value of $\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D70F})^{2}$ ; this requires a very sharp refinement process to accurately identify the time $\unicode[STIX]{x1D70F}_{c}$ at which this minimum actually occurs.

Table 1.

Dependence of the minimum attained by $\unicode[STIX]{x1D6FF}^{2}(\unicode[STIX]{x1D70F})$ ; as $\unicode[STIX]{x1D716}$ decreases the time $\unicode[STIX]{x1D70F}_{c}$ at which this minimum is attained has to be determined with ever increasing accuracy as indicated by the number of decimal points.

The results are shown in table 1 and in the corresponding plot of $\log \log \unicode[STIX]{x1D6FF}^{-2}|_{max}$ as a function of $\log \unicode[STIX]{x1D716}^{-1}$ (figure 8 a). The straight line fit provides the relationship

(4.1) $$\begin{eqnarray}\log \log \unicode[STIX]{x1D6FF}^{-2}|_{max}\sim 3\log \unicode[STIX]{x1D716}^{-1}-20.03,\end{eqnarray}$$

or equivalently

(4.2) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}^{2}|_{min}\sim \exp [-53(R_{\unicode[STIX]{x1D6E4}}/3000)^{3}].\end{eqnarray}$$

This does indeed indicate an extremely rapid decrease of $\unicode[STIX]{x1D6FF}^{2}|_{min}$ as $R_{\unicode[STIX]{x1D6E4}}=\unicode[STIX]{x1D716}^{-1}$ increases beyond about 3000, as shown in figure 8(c); for example, when $R_{\unicode[STIX]{x1D6E4}}=4000$ , it gives $\unicode[STIX]{x1D6FF}^{2}|_{min}=2.75\times 10^{-55}$ , which appears to be zero in figure 6; but in fact the asymptotic relation (4.2), if valid as $\unicode[STIX]{x1D716}\rightarrow 0$ , indicates that for any $\unicode[STIX]{x1D716}>0$ , $\unicode[STIX]{x1D6FF}^{2}|_{min}$ does not vanish, and we do not therefore have a strict mathematical singularity.

Figure 7.

Behaviour of $\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D70F})$ , $s(\unicode[STIX]{x1D70F})$ and the ratio $\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D70F})/s(\unicode[STIX]{x1D70F})$ during the very small time interval when $\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D70F})$ (shown in green) decreases rapidly to near zero; (a,c)  $\unicode[STIX]{x1D716}=1/2500$ ; the abscissa runs from 3.71518149 to 3.7151818 and $\unicode[STIX]{x1D6FE}(3.72)=3.21945\times 10^{-8}$ ; (b,d)  $\unicode[STIX]{x1D716}=1/3000$ ; the abscissa runs from 1.93916197100149 to 1.93916197100182 and $\unicode[STIX]{x1D6FE}(1.194)=6.05673\times 10^{-13}$ . In both cases, $\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D70F})$ falls to an extremely low value, but not exactly zero; the remnant weak vorticity expands solely under the action of viscous diffusion.

The behaviour of $\unicode[STIX]{x1D6FF}/s$ near $\unicode[STIX]{x1D70F}_{c}$ is not well resolved in figure 6(f), despite the extreme expansion of scale in this region. Figure 7 shows the situation for two larger values of $\unicode[STIX]{x1D716}(1/2500~\text{and}~1/3000)$ , which are more easily resolvable; these show that $\unicode[STIX]{x1D6FF}/s$ actually increases during the very short time interval when $\unicode[STIX]{x1D6FE}$ decreases rapidly to very near zero. This increase is, however, bounded: it rises to near 2 in both cases, and remains at this level in the subsequent purely viscous diffusion of the remnant vorticity; $\unicode[STIX]{x1D6FE}$ is not exactly zero for $\unicode[STIX]{x1D70F}>\unicode[STIX]{x1D70F}_{c}$ , as indicated in the figure caption.

Figure 8.

(a) Plot of the data of table 1, fitted by the straight line (4.1); (b) same for the data of table 2, fitted by the quadratic curve (4.3); (c,d) inferred dependence of $\unicode[STIX]{x1D6FF}_{min}^{2}$ and $\unicode[STIX]{x1D714}_{max}/\unicode[STIX]{x1D714}_{0}$ on vortex Reynolds number  $R_{\unicode[STIX]{x1D6E4}}$ .

Table 2.

Dependence of the maximum attained by $\unicode[STIX]{x1D714}(\unicode[STIX]{x1D70F})/\unicode[STIX]{x1D714}(0)$ ; the time $\unicode[STIX]{x1D70F}_{m}$ at which this maximum is attained is slightly less than  $\unicode[STIX]{x1D70F}_{c}$ , and $\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D70F}_{m})$ is correspondingly slightly greater than $\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D70F}_{c})$ .

Although we do not have a strict mathematical singularity, we may nevertheless seek to determine the maximum vorticity attained, as a function of  $\unicode[STIX]{x1D716}$ . Computed values are shown in table 2, and plotted in figure 8(b) together with the closely fitting curve

(4.3) $$\begin{eqnarray}\log [\unicode[STIX]{x1D714}_{max}/\unicode[STIX]{x1D714}_{0}]\sim 1+220(\log [1/\unicode[STIX]{x1D716}]-7.6)^{2},\end{eqnarray}$$

or equivalently

(4.4) $$\begin{eqnarray}\unicode[STIX]{x1D714}_{max}/\unicode[STIX]{x1D714}_{0}\sim \exp [1+220(\log [R_{\unicode[STIX]{x1D6E4}}/2000])^{2}].\end{eqnarray}$$

Actually this formula underestimates the values computed for $R_{\unicode[STIX]{x1D6E4}}\gtrsim 3200$ , and should be regarded as a lower bound on $\unicode[STIX]{x1D714}_{max}/\unicode[STIX]{x1D714}_{0}$ for larger values of  $R_{\unicode[STIX]{x1D6E4}}$ . The dependence (4.4), shown in figure 8, shows a rapid increase for $R_{\unicode[STIX]{x1D6E4}}\gtrsim 3200$ ; and if we assume that this trend continues for larger $R_{\unicode[STIX]{x1D6E4}}$ , then for $R_{\unicode[STIX]{x1D6E4}}=4000$ it gives $\unicode[STIX]{x1D714}_{max}/\unicode[STIX]{x1D714}_{0}\sim 2\times 10^{46}$ , and for $R_{\unicode[STIX]{x1D6E4}}=5000$ it gives $\unicode[STIX]{x1D714}_{max}/\unicode[STIX]{x1D714}_{0}\sim 4.5\times 10^{80}$ ! We should note, however, that this peak of vorticity occurs when a large fraction of the original circulation has already reconnected (for example, $\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D70F}_{m})=0.008$ when $R_{\unicode[STIX]{x1D6E4}}=3400$ (table 2)); it is just the residual surviving vorticity that is apparently so intensely stretched.