??–??

# Towards a finite-time singularity of the Navier-
Stokes equations. Part 2. Vortex reconnection
and singularity evasion

H. K. Moffatt1 and Yoshifumi Kimura2 1Department of Applied Mathematics and
Theoretical Physics,
Wilberforce Road, Cambridge CB3 0WA, UK
2Graduate School of Mathematics, Nagoya University,
Furo-cho, Chikusa-ku, Nagoya 464-8602 Japan

(??)

###### Abstract

In Part 1 of this work, we have derived a dynamical system describing the
approach to a finite-time singularity of the Navier-Stokes equations. We now
supplement this system with an equation describing the process of vortex
reconnection at the apex of a pyramid, neglecting core deformation during the
reconnection process. On this basis, we compute the maximum vorticity
$\omega_{max}$ as a function of vortex Reynolds number $R_{\Gamma}$ in the
range $2000\leq R_{\Gamma}\leq 3400$, and deduce a compatible behaviour
$\omega_{max}\sim\omega_{0}\exp{\left[1+220\left(\log\left[R_{\Gamma}/2000\right]\right)^{2}\right]}$
as $R_{\Gamma}\rightarrow\infty$. This may be described as a physical
(although not strictly mathematical) singularity, for all $R_{\Gamma}\gtrsim
4000$.

††volume: ??

##  1 Introduction

In Part 1 of this work (Moffatt & Kimura 2019, hereafter MK19), we have
derived a dynamical system governing the evolution of two initially circular
vortices of radius $R$ and circulations $\pm\Gamma$. The vortices are
symmetrically located on planes $x=\pm z\tan\alpha$, and are assumed to have
Gaussian cores of initial radial scale $\delta_{0}$. The Navier-Stokes
equations are non-dimensionalised in terms of length-scale $R$ and time-scale
$R^{2}/\Gamma$. The ‘tipping points’ are the points of closest approach of the
vortices, and it turns out that the key variables are the separation
$2s(\tau)$ of the tipping points, and the curvature $\kappa(\tau)$ and
evolving radial scale $\delta(\tau)$ at either tipping point, where
$\tau=(\Gamma/R^{2})\,t$ is dimensionless time. It is assumed that

$\delta(0)\ll s(0)\ll 1,$ (1)

and that the vortex Reynolds number $R_{\Gamma}=\Gamma/\nu$ is large (where
$\nu$ is the usual kinematic viscosity of the fluid). The dynamical system
derived in MK19 (eqns. 6.9a-c) is then

$\frac{\textnormal{d}s}{\textnormal{d}{\tau}}=-\frac{\kappa\cos\alpha}{4\pi}\left[\log\left(\frac{s}{\delta}\right)+\beta_{1}\right]\,,\quad\frac{\textnormal{d}\kappa}{\textnormal{d}{\tau}}=\frac{\kappa\cos\alpha\sin\alpha}{4\pi{s}^{2}}\,,\quad\frac{\textnormal{d}\,\delta^{2}}{\textnormal{d}{\tau}}=\epsilon-\frac{\kappa\cos\alpha}{4\pi{s}}\,\delta^{2}\,,$
(2)

where $\epsilon\equiv R_{\Gamma}^{-1}\ll 1$, and the parameter $\beta_{1}$
takes the value $\beta_{1}=0.4417$ for the assumed Gaussian vortex core
structures. It was argued that the rate of strain in the neighbourhood of
either tipping point is such as to turn the planes always towards angle
$\alpha=\pi/4$, and that this constant value may therefore be adopted.
Equations (2) may then be integrated with initial conditions

$s(0)=s_{0},\quad\kappa(0)=\kappa_{0}=1,\quad\delta(0)=\delta_{0}\,.$ (3)

In the present paper, we shall, by way of illustration, adopt the initial
values

$s_{0}=0.1,\quad\kappa_{0}=1,\quad\delta_{0}=0.01.$ (4)

This choice allows reasonable resolution of the behaviour near the critical
time $\tau=\tau_{c}$ when consideration of vortex reconnection becomes
essential. (Other values compatible with (1) may be similarly treated, as done
in MK19, where the case
$s(0)=0.05,\delta(0)=10^{-5},\kappa(0)=1,\epsilon=10^{-20}$ is described. In
all cases, it is found that, when $\epsilon\ll 1$, solutions of (1.2) become
singular at a finite time $\tau_{c}$ that depends on $s(0)$, and that
$\delta/s$ and $\kappa s$ approach asymptotic values as
$\tau\rightarrow\tau_{c}$ near to the values $\surd{2}$ and $0.943367$
predicted by the asymptotic similarity solution obtained in §10 of MK19).
Figure 1 shows the evolution of these variables from the initial conditions
(4) with the choice $\epsilon=1/3000$, up to the critical time
$\tau=\tau_{c}\approx 0.25452$. Figure 1b shows the corresponding variation of
$\kappa(\tau)s(\tau)$ and $\delta(\tau)/s(\tau)$, which rise in tandem to
limiting values 1.4208 and 0.9456 respectively at $\tau=\tau_{c}$. It is clear
that when $\delta(\tau)/s(\tau)$ rises to about 0.2 and greater, the Biot-
Savart law on which the first two equations of (2) are based requires
modification to take account of inter-diffusion of the two vortices, and
associated vortex reconnection. It is the purpose of this follow-up paper to
take due account of this effect, and to determine the resulting behaviour as
the Reynolds number $R_{\Gamma}$ increases.

(a) (b) (c) (d)

Figure 1: Evolution governed by (2) with initial conditions (4) and with
$\epsilon=1/3000$; (a) the three curves as labelled fall to zero at
$\tau=\tau_{c}\approx 0.25452$; (b) the curves of $\kappa(\tau)s(\tau)$
(dashed) and $\delta(\tau)/s(\tau)$ (solid) rise to limiting values 1.4208 and
0.9456 respectively (blue, dashed) at $\tau=\tau_{c}$ (red dotted); (c)
$s(\tau)$ and $\delta(\tau)$, showing how both fall to zero at
$\tau=\tau_{c}$; (d) very near $\tau_{c}$, where $s(\tau)^{2}$ and
$\delta(\tau)^{2}$ decrease linearly at constant ratio in the limit
$\tau\rightarrow\tau_{c}$.

##  2 The process of pyramid reconnection

The ‘pyramid-reconnection’ process, as anticipated in MK19, is represented
schematically by the sketch of figure 2. As $\delta/s$ 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 $-\Gamma$ in the vortex ${\cal V}_{1}$
centred on the curve $C_{1}$ in the half-space $x<0$ branches into two
circulations: ‘surviving circulation’ $-\Gamma\\!_{s}(\tau)$ (blue, solid) and
‘reconnected circulation’ $-\Gamma\\!_{r}(\tau)$ (red, dashed), with
$\Gamma\\!_{s}(\tau)+\Gamma\\!_{r}(\tau)=\Gamma$ (and similarly of course for
the vortex ${\cal V}_{2}$ in $x>0$). The arrows in Figure 2 indicate the
compatible directions of vorticity $\omega$ 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 (1989)
and Hussain & Duraisami (2011). The processes have been recently described in
similar terms by Kerr (2018), 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 (2011) 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.

Figure 2: Sketch indicating the nature of the pyramid-reconnection process:
the sense of the surviving vortex filaments (solid blue, with associated flux
$\Gamma\\!_{s}(\tau)$ 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 $\Gamma\\!_{r}(\tau)$) propagate downwards on the
complementary faces of the prism away from the apex.

The rate of decrease of the surviving circulation $\Gamma\\!_{s}(\tau)$
results from viscous diffusion across the plane $x=0$, and is given in
dimensionless form by

$\frac{\textnormal{d}\Gamma\\!_{s}}{\textnormal{d}\tau}=\frac{\textnormal{d}}{\textnormal{d}\tau}\oint_{C_{-}}{\bf{v}}\,\cdot\textnormal{d}{\bf{x}}=\epsilon\oint_{C_{-}}\nabla^{2}{\bf{v}}\,\cdot\textnormal{d}{\bf{x}}\,,$
(5)

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

$\frac{\textnormal{d}\Gamma\\!_{s}}{\textnormal{d}\tau}=\epsilon\int_{-\infty}^{\infty}\left.\left(\frac{\partial^{2}\varv_{z}}{\partial
x^{2}}+\frac{\partial^{2}\varv_{z}}{\partial
z^{2}}\right)\right|_{x=0}\textnormal{d}z=\epsilon\int_{-\infty}^{\infty}\left.\frac{\partial^{2}\varv_{z}}{\partial
x^{2}}\right|_{x=0}\textnormal{d}z,$ (6)

(the second term integrating to zero because $\partial\varv_{z}/\partial z=0$
at $z=\pm\infty$.)

(a) (b)

Figure 3: (a) The vorticity $\omega_{y}(x,\,0,\tau)/\Gamma\\!_{s}(\tau)$ given
by (7), at a time $\tau$ at which $s=1,\delta/s=0.9456$ (the limiting value
stated above); the position of the vortex centres is indicated by the two red
dots; (b) the velocity $\varv_{z}(x,\,0,\tau])/\Gamma\\!_{s}(\tau)$ given by
(8) for $s=1$ (solid), 2 (green, dashed, 3 (blue, dotted) and 4 (red, dot-
dashed).

Now neglecting the core deformation, the vorticity field on the plane $y=0$ at
time $\tau$ is that of two vortices of Gaussian structure, with the single
component

$\omega_{y}(x,\,z,\,\tau)=\frac{\Gamma\\!_{s}(\tau)}{4\pi\delta^{2}}\left(\exp{\left[-\frac{(x+s)^{2}+z^{2}}{4\delta^{2}}\right]}-\exp{\left[-\frac{(x-s)^{2}+z^{2}}{4\delta^{2}}\right]}\,\right)\,,$
(7)

as sketched in figure 3a for $s=1,\delta=0.9456$. The corresponding
$z$-component of velocity is

$\varv_{z}(x,z,\tau)\\!=\\!\frac{\Gamma\\!_{s}(\tau)}{2\pi}\\!\left\\{\\!\frac{x+s}{{(x\\!+\\!s)^{2}\\!+\\!z^{2}}}\left(\\!1\\!-\\!\exp{\left[-\frac{(x\\!+\\!s)^{2}\\!+\\!z^{2}}{4\delta^{2}}\right]}\right)-\frac{x-s}{{(x\\!-\\!s)^{2}\\!+\\!z^{2}}}\left(\\!1\\!-\\!\exp{\left[-\frac{(x\\!-\\!s)^{2}\\!+\\!z^{2}}{4\delta^{2}}\right]}\right)\right\\},$
(8)

(figure 3b, for $s=1,2,3,4$ and fixed $\delta=0.9456$), from which we may
calculate

$\displaystyle\left.\frac{\partial^{2}\varv_{z}}{\partial
x^{2}}\right|_{x=0}\\!\\!=\\!$ $\displaystyle-$
$\displaystyle\frac{s^{5}\,\Gamma\\!_{s}(\tau)\left[\left(s^{6}\\!+\\!2s^{4}z^{2}\\!-\\!6z^{4}\delta^{2}\\!-\\!24z^{2}\delta^{2}\\!+\\!s^{2}z^{4}\\!-\\!4s^{2}z^{2}\delta^{2}\\!+\\!8s^{2}\delta^{4}\right)\exp{\left[-(s^{2}\\!+\\!z^{2})/4\delta^{2}\right]}\right]}{2\pi(s^{2}+z^{2})^{3}\delta^{2}}$
(9) $\displaystyle-$
$\displaystyle\frac{2s\,\Gamma\\!_{s}(\tau)(3z^{2}-s^{2})}{\pi(s^{2}+z^{2})^{3}}\,.$

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

$\int_{-\infty}^{\infty}\left.\frac{\partial^{2}\varv_{z}}{\partial
x^{2}}\right|_{x=0}\textnormal{d}z=-\frac{s\,\Gamma\\!_{s}(\tau)}{2\surd{\pi}\,\delta^{3}}\exp[-s^{2}/4\delta^{2}]\,,$
(10)

and it follows from (6) that

$\frac{\textnormal{d}\Gamma\\!_{s}}{\textnormal{d}\tau}=-\epsilon\,\frac{s\,\Gamma\\!_{s}(\tau)}{2\surd{\pi}\,\delta^{3}}\exp[-s^{2}/4\delta^{2}]\,.$
(11)

The above neglect of core deformation calls for comment. There is evidence
from DNS, such as in the recent investigations of McKeown et al. (2018) and
Kerr (2018) and in earlier studies, that there is a flattening of the vortex
cores during the interaction process when $\delta/s$ 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 (7) assumed above; this would
change the details of the above calculation, but is unlikely to modify the
result (11) in any fundamental way. We therefore adopt this result as it
stands. It is important to note that this result is firmly based on (5), which
itself is a direct consequence of the Navier-Stokes equation.

During the very short reconnection phase, we thus need to replace $\Gamma$ by
$\Gamma\\!_{s}(\tau)$ in calculating the instantaneous velocity, rate of
stretching, and rate-of-change of curvature at the tipping points. This simply
means that, defining $\gamma(\tau)\equiv\Gamma\\!_{s}(\tau)/\Gamma$, we have
to replace (2) by

$\frac{\textnormal{d}s}{\textnormal{d}{\tau}}=-\frac{\gamma\,\kappa\cos\alpha}{4\pi}\left[\log\left(\frac{s}{\delta}\right)+\beta_{1}\right]\,,\quad\frac{\textnormal{d}\kappa}{\textnormal{d}{\tau}}=\frac{\gamma\,\kappa\cos\alpha\sin\alpha}{4\pi{s}^{2}}\,,\quad\frac{\textnormal{d}\,\delta^{2}}{\textnormal{d}{\tau}}=\epsilon-\frac{\gamma\,\kappa\cos\alpha}{4\pi{s}}\,\delta^{2}\,,$
(12)

which, from (11), must now be coupled with

$\frac{\textnormal{d}\gamma}{\textnormal{d}\tau}=-\epsilon\,\frac{s\,\gamma}{2\surd{\pi}\,\delta^{3}}\exp[-s^{2}/4\delta^{2}]\,.$
(13)

We may now integrate this 4th-order dynamical system with the initial
conditions (4), together with $\gamma(0)=1$.

##  3 Integration of the system (12, 13)

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

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

(a) (b)

Figure 4: Evolution governed by (12) with $\epsilon=1/2000$: (a)
$\delta(\tau)^{2}$ (black) which appears to fall to zero at
$\tau=\tau_{c}\approx 9.12455$, but does not quite reach zero as shown by the
expanded curve of $10^{3}\delta(\tau)^{2}$ (blue); $\gamma(\tau)/100$ (green)
decreases to about 0.0002 at $\tau\approx\tau_{c}$ then falls rapidly to very
near zero; (b) the corresponding variation of vorticity amplification
$\omega(\tau)/\omega(0)$ in the neigbourhood of $\tau_{c}$, where it reaches a
maximum of approximately 3.7.

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

(a) (b)

Figure 5: Same as figure 4, but with $\epsilon=1/3000$.

Consider now the situation when $\epsilon=1/4000$ (i.e. $R_{\Gamma}=4000$).
Here, Mathematica stalls at $\tau=\tau_{c}\approx 0.917\,576\,564\,746\,48$.
The reason is that now $\delta^{2}$ falls so close to zero that equation (13)
runs into an apparent singularity. Figure 6a shows $\delta^{2}(\tau)$ and
$\gamma(\tau)/5000$ (again the factor $1/5000$ is included to make the
comparison clear), showing that, while $\delta^{2}$ seems to go to zero at the
apparent singularity time $\tau=\tau_{c}$ (marked by the dotted line),
$\gamma$ is still finite; figure 6b shows $\delta^{2}$ expanded by a factor
$10^{12}$ in the immediate neighbourhood of $\tau=\tau_{c}$
($0.917\,576\,564\,746\,39<\tau$ $<0.917\,576\,564\,746\,55$);
$\delta(\tau_{c})^{2}$ is certainly extremely small if not exactly zero.
Figure 6c shows (over the same exceedingly small range of $\tau$) that
$\omega(\tau)/\omega(0)$ increases to a very large value near $\tau_{c}$; but
is this really infinite? Figure 6d shows the inverse function
$\omega(0)/\omega(\tau)$, which does indeed appear to be zero at this point as
far as this computation can determine.

(a) (b)

(c) (d)

(e) (f)

Figure 6:  Evolution for $\epsilon=1/4000$, and for $0<\tau<\tau_{c}\approx
0.917\,576\,564\,746\,480\dots$ (marked by red dotted line; (a)
$\delta^{2}(\tau)$ and $\gamma(\tau)$ both appear to fall to zero at
$\tau=\tau_{c}$; zoom of $10^{12}\delta^{2}(\tau)$ very near $\tau_{c}$; (c)
and (d) $\omega(\tau)/\omega(0)$ and $\omega(0)/\omega(\tau)$ near $\tau_{c}$,
as well as can be resolved; (e) and (f) other relevant variables near
$\tau_{c}$, as well as can be resolved.

(a)  (b)

(c)  (d)

Figure 7: Behaviour of $\delta(\tau),\,s(\tau)$ and the ratio
$\delta(\tau)/s(\tau)$ during the very small time-interval when $\gamma(\tau)$
(shown in green) decreases rapidly to near zero; (a) and (c)
$\epsilon=1/2500$; the abscissa runs from 3.71518149 to 3.7151818 and
$\gamma(3.72)=3.21945\times 10^{-8}$; (b) and (d) $\epsilon=1/3000$; the
abscissa runs from 1.93916197100149 to 1.93916197100182 and
$\gamma(1.194)=6.05673\times 10^{-13}$. In both cases, $\gamma(\tau)$ falls to
an extremely low value, but not exactly zero; the remnant weak vorticity
expands solely under the action of viscous diffusion.

##  4 Asymptotic behaviour for small $\epsilon$

A necessary condition for a finite-time singularity of vorticity
$\omega(\tau)\equiv\gamma(\tau)/\delta(\tau)^{2}$ is obviously that
$\delta(\tau)^{2}$ should decrease to exactly zero at some finite time. In
order to further investigate this possibility, we consider the
$\epsilon$-dependence of $\delta(\tau)^{2}_{min}$ as $\epsilon$ decreases
towards zero from $1/2000$. For each $\epsilon$, we first locate the time
$\tau_{c}$ at which
$\textnormal{d}\,\delta^{2}/\textnormal{d}{\tau}\,\,(=\epsilon-(\gamma\,\kappa\cos\alpha/4\pi{s})\,\delta^{2})$
is zero, and then evaluate the corresponding minimum value of
$\delta(\tau)^{2}$; this requires a very sharp refinement process to
accurately identify the time $\tau_{c}$ at which this minimum actually occurs.

$\epsilon$ | $\,\,\,\,\,\,\,\,\,\,\,\,\,\tau_{c}$ | $\delta^{2}_{min}$ |  $\gamma(\tau_{c})$
---|---|---|---
1/2000 | 9.12478705 | $2.67412\times 10^{-8}$ |  0.000677
1/2250 | 5.64196432008 | $8.34641\times 10^{-11}$ |  0.000602
1/2500 | 3.71518168478948 | $3.69041\times 10^{-14}$ |  0.000542
1/2750 | 2.60587589889538 | $1.37078\times 10^{-18}$ |  0.000501
1/3000 | 1.9391619710016794917768 | $2.72780\times 10^{-24}$ |  0.000451
1/3050 | 1.83975957787485971515631271 | $1.34992\times 10^{-25}$ |  0.000444
1/3100 | 1.748954708554952698937886817 | $5.85955\times 10^{-27}$ |  0.000437
1/3200 | 1.5896543686535978867264670724 | $7.41570\times 10^{-30}$ |  0.000423
1/3300 | 1.4552611229433966364089007674807 | $5.49450\times 10^{-33}$ |  0.000410
1/3400 | 1.341068010282669618710465713447975557 | $2.37870\times 10^{-36}$ |  0.000399
Table 1: Dependence of the minimum attained by $\delta^{2}(\tau)$; as
$\epsilon$ decreases the time $\tau_{c}$ at which this minimum is attained has
to be determined with ever increasing accuracy as indicated by the number of
decimal points.

(a) (b)

(c) (d)

Figure 8: (a) Plot of the data of table 1, fitted by the straight line (14); (b) same for the data of table 2,fitted by the quadratic curve (16); (c) and (d) inferred dependence of $\delta^{2}_{min}$ and $\omega_{max}/\omega_{0}$ on vortex Reynolds number $R_{\Gamma}$. $\epsilon$ | $\,\,\,\,\,\,\,\,\,\,\,\,\,\tau_{m}$ | $\omega_{max}/\omega_{0}$ |  $\gamma(\tau_{m})$
---|---|---|---
1/2000 | 9.12456266 | $3.711$ |  0.00139
1/2250 | 5.64196352000 | $1057$ |  0.00125
1/2500 | 3.71518168440194 | $2.151\times 10^{6}$ |  0.00112
1/2750 | 2.605875898895364555 | $5.266\times 10^{10}$ |  0.00101
1/3000 | 1.9391619710016794917425 | $2.425\times 10^{16}$ |  0.000930
1/3050 | 1.8397595778748597151545 | $4.820\times 10^{17}$ |  0.000910
1/3100 | 1.748954708554952698937812055 | $1.092\times 10^{19}$ |  0.000889
1/3200 | 1.589654368653597886726466973 | $8.363\times 10^{21}$ |  0.000871
1/3300 | 1.455261122943396636408900767405 | $1.095\times 10^{25}$ |  0.000843
1/3400 | 1.341068010282669618710465713447942 | $2.454\times 10^{28}$ |  0.000800
Table 2: Dependence of the maximum attained by $\omega(\tau)/\omega(0)$; the
time $\tau_{m}$ at which this maximum is attained is slightly less than
$\tau_{c}$, and $\gamma(\tau_{m})$ is correspondingly slightly greater than
$\gamma(\tau_{c})$.

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

$\log{\log{\delta^{-2}|_{max}}}\sim 3\log{\epsilon^{-1}}-20.03,$ (14)

or equivalently

${\delta^{2}|_{min}}\sim\exp{\left[-53\left(R_{\Gamma}/3000\right)^{3}\right]}\,.$
(15)

This does indeed indicate an extremely rapid decrease of $\delta^{2}|_{min}$
as $R_{\Gamma}=\epsilon^{-1}$ increases beyond about $3000$, as shown in
figure 8c ; for example, when $R_{\Gamma}=4000$, it gives
$\delta^{2}|_{min}=2.75\times 10^{-55}$, which appears to be zero in figure 6;
but in fact the asymptotic relation (15), if valid as $\epsilon\rightarrow 0$,
indicates that for any $\epsilon>0$, $\delta^{2}|_{min}$ does not vanish, and
we do not therefore have a strict mathematical singularity.

The behaviour of $\delta/s$ near $\tau_{c}$ 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 $\epsilon\,(1/2500$ and $1/3000)$,
which are more easily resolvable; these show that $\delta/s$ actually
increases during the very short time-interval when $\gamma$ 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; $\gamma$ is not exactly zero for
$\tau>\tau_{c}$, as indicated in the figure caption.

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

$\log{[\omega_{max}/\omega_{0}]}\sim 1+220(\log{[1/\epsilon]}-7.6)^{2},$ (16)

or equivalently

$\omega_{max}/\omega_{0}\sim\exp{\left[1+220\left(\log\left[R_{\Gamma}/2000\right]\right)^{2}\right]}\,.$
(17)

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

##  5 Conclusion

We initially thought that the failure of Mathematica to resolve the behaviour
of the system (12, 13) for $\epsilon=1/4000$ and smaller was an indication of
a singularity at the time $\tau_{c}$ at which the programme stalled. However,
computation of the dependence of $\delta^{2}_{min}$ on decreasing $\epsilon$
has revealed a smooth, although extremely sharp, cutoff in the range of
$\epsilon$ in the interval $(1/3000,1/4000)$, such that, inevitably, there is
a point beyond which Mathematica interprets $\delta^{2}_{min}$ as zero. Our
inference however is that $\delta^{2}_{min}$ is nonzero for any $\epsilon>0$,
although so small for $\epsilon\lesssim 1/4000$ that the core cross-sectional
scale $\delta$ is less than the inter-molecular scale below which the
continuum hypothesis (on which the Navier-Stokes equation is based) is no
longer justifiable. Moreover, the maximum vorticity amplification, attained
just before this minimum of $\delta^{2}$ is reached, is so large that we may
reasonably describe this as a physical, if not a mathematical, singularity.
The situation is reminiscent of the free-surface cusp singularity described by
Jeong & Moffatt (1992), again physical rather that mathematical, in which the
radius of curvature at the ‘cusp’ is O$(10^{-42})$ times the input scale of
the problem. The input scale of the present problem is the radius $R$ of the
two vortices at time $\tau=0$, so that if for example $R=10^{-1}$m, then even
at $R_{\Gamma}=2750$, the figures of table 1 give $\delta\sim 10^{-10}$m, and
we are down at the scale of the hydrogen atom. This is therefore quite
evidently a physical singularity. In the light of this behaviour, the question
“can there be a finite-time singularity of the Navier-Stokes equations?” is to
some extent academic; but it continues to provide a legitimate academic
challenge nonetheless!

Two physical processes have been ignored in the foregoing analysis: (i)
deformation of the vortex cores during the reconnection process; and (ii) the
possible braking effect exerted by the strands of reconnected vorticity on the
continuing reconnection process. These processes may limit the growth of
vorticity (Hussain & Duraisami 2011); the fact that a singularity is averted
even when they are ignored provides compelling evidence that, for the pyramid
geometry considered here, a strict finite-time mathematical singularity does
not occur, no matter how large the vortex Reynolds number may be. YK
acknowledges support from JSPS KAKENHI Grant Numbers JP18H04443, JP24247014,
JP16H01175. We acknowledge the comments of three referees, which have led to
improvements in the presentation.

## References

  * Hussain (2011) Hussain, F. & Duraisami 2011 Mechanics of viscous vortex reconnection. Physics of Fluids 23, 021701.
  * Jeong & Moffatt (1992) Jeong, J. T. & Moffatt, H. K. 1992 Free-surface cusps associated with flow at low Reynolds-number. J. Fluid Mech. 241, 1–22.
  * Kerr (2018) Kerr, R. M. 2018 Enstrophy and circulation scaling for Navier-Stokes reconnection. J. Fluid Mech. 839, R2.
  * McKeown et al. (2018) McKeown, R., Ostilla-Monico, R., Pumir, A., Brenner, M.P. & Rubinstein, S.M. 2018 A cascade leading to the emergence of small structures in vortex ring collisions. Phys. Rev. Fluids 3, 124702.
  * Melander (1989) Melander, M. V. & Hussain, F. 1989 Cross-linking of two antiparallel vortex tubes. Phys. Fluids A: Fluid Dynamics 1, 633–636.
  * Moffatt (2019) Moffatt, H. K. & Kimura, Y. 2019 Towards a finite-time singularity of the Navier-Stokes equations. Part 1. J. Fluid Mech. 861, 930-967.