2.1.1 Travelling wave solutions

We consider the case where the vortex pair are separated by a constant distance; that is, $D(t)=d\gg 1$ , and study travelling wave solutions to (2.6). We do this by introducing the independent variable $\unicode[STIX]{x1D709}=s-vt$ , where $v$ is the velocity of the travelling wave (which is assumed to be constant and $O(1)$ ), and find that the transformed equation becomes

We make use of the fact that we are studying large separation distances and expand $\unicode[STIX]{x1D707}(\unicode[STIX]{x1D709})$ as a series in $d^{-1}$ . In particular, $\unicode[STIX]{x1D707}=\sum _{m=0}^{\infty }(d)^{-m}\unicode[STIX]{x1D707}^{(m)}$ , so that the second-order asymptotic solution is (see § A.1.1 for details)

where $c_{n}\in \mathbb{C}$ are integration constants and $(~)^{\ast }$ is the complex conjugate of the variable.

Equation (2.8) suggests that if the vortices are held at infinite separation, such that only self-induction effects are included, the resulting shape of the individual filaments is that of a helix with wavenumber $v$ that oscillates at an angular frequency of $v^{2}$ . As the pair is brought closer together, the vortex interaction is manifested in two ways. First, there is repulsion in the perpendicular direction due to the $-4\text{i}(s-vt)/v$ term, which is independent of any imposed conditions on $\unicode[STIX]{x1D707}(\unicode[STIX]{x1D709})$ ; an expected result in light of point vortices. In a 2D flow, each co-rotating point vortex will feel a velocity field that is opposite in direction to the field felt by the other vortex, both of which are perpendicular to the direction of separation, thus driving the pair apart in the same manner as equation (2.8) predicts. Noting that $\unicode[STIX]{x1D6E4}_{1}=\unicode[STIX]{x1D6E4}_{2}>0$ , the direction of repulsion is such that the filament described by $\unicode[STIX]{x1D713}_{1}(t,s)$ will drift in the positive $y$ direction while the filament described by $\unicode[STIX]{x1D713}_{2}(t,s)$ will drift in the negative $y$ direction. If we consider the case where $\unicode[STIX]{x1D6E4}_{1}<0$ , such that $t\rightarrow -t$ , the vortices are repelled in the opposite direction along the $y$ -axis.

Second, an additional wave is generated in the configuration as a response to the zeroth-order solution; one that has the form $-2c_{1}^{\ast }\text{exp}(-\text{i}v\unicode[STIX]{x1D709})/v$ . If we suppose that $\text{Im}(c_{1})=0$ , then for $\text{Re}(c_{1})>0$ , the secondary wave suppresses oscillations in the $y$ direction, while vibrations in the $x$ direction are similarly subdued when $\text{Re}(c_{1})<0$ . Physically, the helical shapes of the filaments flatten into ellipses due to the pairwise vortex interaction.

Furthermore, there is additional attraction or repulsion at $O(d^{-2})$ that depends on the initial offset of the helical vortices; that is, it depends on $4\text{i}c_{2}^{\ast }\unicode[STIX]{x1D709}/v$ . An offset that is in the $x$ direction results in repulsion in the $y$ direction, as seen in the first-order correction. However, if the constant offset is in the $y$ direction, then the opposite holds true; namely, the vortex pair is translated in the $x$ direction in either repulsion or attraction. For $\unicode[STIX]{x1D6E4}_{1}>0$ , if $\text{Im}(c_{2})>0$ , then $4\text{i}c_{2}^{\ast }\unicode[STIX]{x1D709}/v>0$ and the pair attract each other, while if $\text{Im}(c_{2})<0$ , then $4\text{i}c_{2}^{\ast }\unicode[STIX]{x1D709}/v<0$ and the vortices repel each other. The opposite applies when $\unicode[STIX]{x1D6E4}_{1}<0$ . This secondary repulsion or attraction is not predicted by the 2D theory and is a consequence of the vortex filaments existing in a 3D flow.

We numerically solve (2.7) for the parameters $v=4$ and $d=10$ , with initial conditions $\unicode[STIX]{x1D707}(\unicode[STIX]{x1D709}=0)=-\text{i}$ and $\unicode[STIX]{x1D707}^{\prime }(\unicode[STIX]{x1D709}=0)=v$ , and plot the vortex filament shapes that result when we set $\unicode[STIX]{x1D708}(t,s)=0$ (shown in figure 2(a–d) for times $t=\{0,40,80,120\}$ respectively). We observe that (2.8) breaks down for times past the initial repulsion of the vortex pair and does not capture the slowly oscillatory behaviour of the numerical solution. This periodic solution on the slow wave scale physically corresponds to the binary orbit of the vortex pair, whereby the filaments rotate around each other (see figure 2). This result is predicted if one considers the same problem from the perspective of 2D vortices. As the vortices separate vertically from their initial repulsion (figure 2 a), the velocity field felt by each vortex becomes increasingly horizontal and decreasingly vertical, resulting in the pair becoming vertically displaced from each other (figure 2 b). The velocity field for each vortex is now horizontal, with the repulsion now leading to increasing vertical contributions and decreasing horizontal contributions in the velocity. The pair become horizontally separated again (figure 2 c) and the cycle repeats.

Given that the pair are orbiting on a slow wave scale and are separated by a constant $d$ through all orientations, we instead look for a slowly varying solution of the form

where $\unicode[STIX]{x1D6FD}$ is a positive constant and $\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D703}\in \mathbb{R}$ . Ansatz (2.9) replaces the helical shape of each vortex in the pair, which individually rotate on a fast time scale, with straight filaments that no longer exhibit fast oscillations.

Substituting (2.9) into (2.7), we find the general solution (see § A.1.2 for details)

where $c_{7},c_{8},c_{9}\in \mathbb{R}$ are integration constants determining the initial orientation of the vortex pair. The odd powers of $d^{-n}$ in (2.10) are corrections so that (2.9) satisfies the initial conditions imposed on $\unicode[STIX]{x1D707}_{\mathit{slow}}(\unicode[STIX]{x1D709})$ , while even powers give corrections to the frequency of the vortex pair orbit.

We compare $\unicode[STIX]{x1D707}_{\mathit{slow}}(\unicode[STIX]{x1D709})$ in (2.9), with $\unicode[STIX]{x1D703}(\unicode[STIX]{x1D709})$ calculated from (2.10), with the numerical solution of (2.7) for the same initial conditions and value for $v$ as seen previously. To impose $\unicode[STIX]{x1D707}_{\mathit{slow}}(\unicode[STIX]{x1D709}=0)=-\text{i}$ , we expand the exponential in (2.9) as a series and match powers of $d^{-1}$ on both sides, giving $c_{7}=-1$ , $c_{8}=0$ and $c_{9}=0$ , which incurs an error of $O(d^{-2})$ at $\unicode[STIX]{x1D709}=0$ . We show these comparisons for $d=10$ in figure 3(a,b) and $d=5$ in figure 3(c,d) for the real and imaginary parts of $\unicode[STIX]{x1D707}(\unicode[STIX]{x1D709})$ . We observe that the slowly varying approximation agrees very well with the numerical solutions showing the vortex pair binary orbit, even for $d=5$ . However, we note that (2.9) and (2.10) begin to break down as $\unicode[STIX]{x1D709}\approx 2000$ , suggesting that higher-order frequency corrections are necessary to keep this solution valid for large $\unicode[STIX]{x1D709}$ .

Figure 2.

The time evolution of travelling wave solutions in co-rotating vortices held at a constant distance, with $\unicode[STIX]{x1D713}_{1}(t,s)$ and $\unicode[STIX]{x1D713}_{2}(t,s)$ being denoted by blue and red respectively. In particular, $\unicode[STIX]{x1D708}(t,s)=0$ and the numerical solution to (2.7) have been plotted with initial conditions $\unicode[STIX]{x1D707}(\unicode[STIX]{x1D709}=0)=-\text{i}$ and $\unicode[STIX]{x1D707}^{\prime }(\unicode[STIX]{x1D709}=0)=v$ , for $v=4$ and $d=10$ , over $s\in [30,40]$ . The times shown are (a) $t=0$ , (b) $t=40$ , (c) $t=80$ and (d) $t=120$ .

Figure 3.

Real and imaginary components of $\unicode[STIX]{x1D707}(\unicode[STIX]{x1D709})$ for the slowly varying solution (2.9) and (2.10) (denoted by black) and numerical solution of the full problem (2.7) (denoted by red). The initial conditions imposed are $\unicode[STIX]{x1D707}(\unicode[STIX]{x1D709}=0)=-\text{i}$ and $\unicode[STIX]{x1D707}^{\prime }(\unicode[STIX]{x1D709}=0)=v$ for parameters $v=4$ and (a,b) $d=10$ and (c,d) $d=5$ .

2.1.2 Self-similar solutions

We consider the case where the separation of the vortex pair grows in time; that is, $D(t)=d\sqrt{t}$ . As in § 2.1.1, $d\in \mathbb{R}$ is a constant that is very large such that $1/d=\unicode[STIX]{x1D716}\ll 1$ . For a vortex separation of this form, equation (2.6) has self-similar solutions given by $\unicode[STIX]{x1D707}(t,s)=\sqrt{t}\unicode[STIX]{x1D712}(\unicode[STIX]{x1D702})$ , where $\unicode[STIX]{x1D702}=s/\sqrt{t}$ . Transforming so that $\unicode[STIX]{x1D702}$ is the independent variable, (2.6) becomes

Taking $d^{-1}\ll 1$ , we expand $\unicode[STIX]{x1D712}(\unicode[STIX]{x1D702})$ as a power series in $d^{-1}$ , so that $\unicode[STIX]{x1D712}(\unicode[STIX]{x1D702})=\sum _{n=0}^{\infty }d^{-n}\unicode[STIX]{x1D712}^{(n)}$ , to find the first-order asymptotic solution (see § A.2)

where $a_{n}\in \mathbb{C}$ are integration constants and $\text{erfi}(z)=\text{i}\text{erf}(\text{i}z)$ is the complex error function.

We plot the vortex filament shapes obtained by numerically solving (2.11) for $\unicode[STIX]{x1D712}(\unicode[STIX]{x1D702}=100)=1+\text{i}$ , $\unicode[STIX]{x1D712}^{\prime }(\unicode[STIX]{x1D702}=100)=1$ and $d=10$ with $\unicode[STIX]{x1D708}(t,s)=0$ for times $t=\{0.1,0.5,1,2\}$ in figure 4. These conditions correspond to $\unicode[STIX]{x1D707}(t,s)$ having an initial shape that satisfies the relations

evaluated at $100=s_{0}/\sqrt{t_{0}}$ .

Figure 4.

The time evolution of self-similar solutions in co-rotating vortices held apart by a separation function $D(t)=\sqrt{t}$ , with $\unicode[STIX]{x1D713}_{1}(t,s)$ and $\unicode[STIX]{x1D713}_{2}(t,s)$ being denoted by blue and red respectively. In particular, $\unicode[STIX]{x1D708}(t,s)=0$ and (2.11), with conditions satisfying (2.13) and (2.14), have been plotted for $d=10$ over $s\in [10,20]$ . The times shown are (a) $t=0.1$ , (b) $t=0.5$ , (c) $t=1$ and (d) $t=2$ .

Figure 5.

Solutions of $\unicode[STIX]{x1D712}(\unicode[STIX]{x1D702})$ calculated from the first-order asymptotic solution (2.12) (shown in black) and the numerical solution of (2.11) (shown in red). In particular, the initial conditions $\unicode[STIX]{x1D712}(\unicode[STIX]{x1D702}=100)=1+\text{i}$ and $\unicode[STIX]{x1D712}^{\prime }(\unicode[STIX]{x1D702}=100)=1$ were used with $d=10$ .

We further compare the numerical solution with the asymptotic solution in (2.12) for the same boundary conditions in figure 5 and describe the form of solutions predicted for $t>0$ and $s>0$ as follows. At zeroth order, the initial shape of the pair is that of individual helical cones along an arbitrary orientation whose amplitude, wavelength and separation grow with time. The conical nature of the helix arises from the complex error function, while the increasing helix amplitude is due to the complex exponential in (2.12). Such conical-type solutions were previously seen for isolated vortex filaments (Van Gorder Reference Van Gorder2016). The inclusion of vortex interaction at first order gives the same general shape as the zeroth-order solution, with the exception of the monotonically increasing term in $\sqrt{t}$ . As in § 2.1.1, the vortex pair exhibit repulsion in opposite directions along the $y$ -axis; however, in this instance, the asymptotic solution does not show significant disagreement with the numerical solution for $t\rightarrow \infty$ . This is due to the separation function $D(t)$ used, which makes binary orbits, as seen with travelling wave solutions, impossible due to the horizontal distance between the vortex pair increasing with time rather than staying constant. Finally, we note that the rotation of the vortex helices is such that they rotate in the same direction, with the orientation of one helix being a reflection of the other in both the $x$ - and the $y$ -axes. In figure 4, both vortices have circulation $\unicode[STIX]{x1D6E4}>0$ ; however, the helices rotate in a clockwise direction, suggesting that the direction of a helix rotation is opposite to an individual filament circulation.

2.2.1 Travelling wave solutions

As in § 2.1.1, we consider vortices that are separated by a constant distance $D(t)=d$ such that we have the small parameter $d^{-1}\ll 1$ . Transforming (2.17) into wave coordinates $\unicode[STIX]{x1D709}=s-vt$ , with $v=O(1)$ , we find

where there is only one non-zero integration constant due to the relationship between $\unicode[STIX]{x1D707}(t,s)$ and $\unicode[STIX]{x1D708}(t,s)$ , which imposes the constraints

with the latter equation used to determine $\unicode[STIX]{x1D6FF}$ from conditions on $\unicode[STIX]{x1D707}(\unicode[STIX]{x1D709})$ and $\unicode[STIX]{x1D708}^{\prime }(\unicode[STIX]{x1D709})$ .

We note that (2.18) has solutions up to $O(d^{-2})$ that remain finite for all time and along the entire arclength of the filament pair, provided that the velocity of the travelling wave $v$ is such that $v^{2}>4d^{-2}$ . As a result, we are able to find solutions of the oscillatory Poincaré–Lindstedt form, which, to second order, are

and

where $X,Y\in \mathbb{R}$ and are related to $\unicode[STIX]{x1D707}$ by $\unicode[STIX]{x1D707}=X+\text{i}Y$ , $\unicode[STIX]{x1D6EF}_{1}=(1+2/v^{2}d^{2})(s-vt)$ , $\unicode[STIX]{x1D6EF}_{2}=(1-2/v^{2}d^{2})(s-vt)$ , $\unicode[STIX]{x1D6FF}=\unicode[STIX]{x1D6FF}_{1}+\text{i}\unicode[STIX]{x1D6FF}_{2}$ , and all $c_{n}\in \mathbb{R}$ (see § A.3.1 for further details).

We comment on some important aspects of (2.21) and (2.22). For counter-rotating vortices, there is no secular vortex repulsion or attraction, unlike what was previously seen with the co-rotating pair in (2.8). Instead, the deviations of the vortex pair have a constant offset, which is determined by the speed of the travelling wave solution, as well as $\unicode[STIX]{x1D6FF}$ , which is dependent on the conditions imposed on $\unicode[STIX]{x1D707}(\unicode[STIX]{x1D709})$ and $\unicode[STIX]{x1D708}^{\prime }(\unicode[STIX]{x1D709})$ . However, we note that by integrating (2.20), the constant offsets seen in $\unicode[STIX]{x1D707}(\unicode[STIX]{x1D709})$ become secular in $\unicode[STIX]{x1D709}$ for $\unicode[STIX]{x1D708}(\unicode[STIX]{x1D709})$ . Although the deviations of the vortex pair may not grow for all time by means of attraction or repulsion, the centre of the vortex pair will move from its initial position; that is, the filaments will move together at a constant velocity through the 3D flow. However, the deviations in the $y$ direction will tend to $-\infty$ and $\infty$ if $v^{2}<4\unicode[STIX]{x1D716}^{2}$ , which suggests that the velocity of the travelling wave serves to stabilise the vortex pair so that they remain together. It should be noted that (2.21) and (2.22) will not approximate such a regime accurately because of the assumption made at the beginning of § 2.2.1, namely that $v=O(1)$ .

Finally, the oscillation of the filaments is subject to dispersion that arises from the vortex interaction at $O(1/d^{2})$ . Vibrations in the $x$ direction will increase in frequency by a factor of $2/v^{2}d^{2}$ while oscillations in $y$ will decrease by $2/v^{2}d^{2}$ . This orientation-dependent oscillation causes a vortical helix to shear to a planar vibration and back to a helix again.

To illustrate some of the phenomena outlined, we solve (2.18) and (2.20) numerically for $d=10$ and $v=1$ with the initial conditions $\unicode[STIX]{x1D707}(\unicode[STIX]{x1D709}=0)=0$ , $\unicode[STIX]{x1D707}^{\prime }(\unicode[STIX]{x1D709}=0)=v+\text{i}v$ , $\unicode[STIX]{x1D708}(\unicode[STIX]{x1D709}=0)=0$ and $\unicode[STIX]{x1D708}^{\prime }(\unicode[STIX]{x1D709}=0)=0$ . The corresponding vortex filament shapes are shown in figure 6(a–d) for times $t=\{25,80,135,190\}$ respectively. In this case, the inclusion of $\unicode[STIX]{x1D708}(t,s)$ causes the vortex pair to travel together in the $y$ direction as $t>0$ . This result is again predicted by considering the simpler problem of point vortices in a 2D flow. A pair that has opposite circulation will generate a velocity field that is the same direction for both vortices, thus pushing them together. For the case shown in figure 6, $\unicode[STIX]{x1D713}_{1}(t,s)$ has positive circulation while $\unicode[STIX]{x1D713}_{2}$ has negative circulation, leading to the respective vortices feeling a velocity field that is in the positive $y$ direction.

Figure 6.

The time evolution of travelling wave solutions in counter-rotating vortices held apart by a constant distance $d$ , with $\unicode[STIX]{x1D713}_{1}(t,s)$ and $\unicode[STIX]{x1D713}_{2}(t,s)$ being denoted by blue and red respectively. In particular, numerical solutions to (2.18) and (2.20) are plotted for $v=2$ , $d=10$ and initial conditions $\unicode[STIX]{x1D707}(\unicode[STIX]{x1D709}=0)=0$ , $\unicode[STIX]{x1D707}^{\prime }(\unicode[STIX]{x1D709}=0)=v+\text{i}v$ , $\unicode[STIX]{x1D708}(\unicode[STIX]{x1D709}=0)=0$ and $\unicode[STIX]{x1D708}^{\prime }(\unicode[STIX]{x1D709}=0)=0$ over $s\in [20,40]$ . The times shown are (a) $t=25$ , (b) $t=80$ , (c) $t=135$ and (d) $t=190$ .

Figure 7.

Real and imaginary components of $\unicode[STIX]{x1D707}(\unicode[STIX]{x1D709})$ and $\unicode[STIX]{x1D708}(\unicode[STIX]{x1D709})$ predicted by the second-order asymptotic correction (2.21) and (2.22) (shown in black) and numerical solutions of (2.18) and (2.20) (shown in red), with the parameters $v=2$ and $d=10$ and initial conditions $\unicode[STIX]{x1D707}(\unicode[STIX]{x1D709}=0)=0$ , $\unicode[STIX]{x1D707}^{\prime }(\unicode[STIX]{x1D709}=0)=v+\text{i}v$ , $\unicode[STIX]{x1D708}(\unicode[STIX]{x1D709}=0)=0$ and $\unicode[STIX]{x1D708}^{\prime }(\unicode[STIX]{x1D709}=0)=0$ .

However, there is an additional effect that is not predicted by the 2D theory; that is, the oscillations of $\unicode[STIX]{x1D708}(t,s)$ . These vibrations cause the amplitude of the helices to periodically vary in a manner such that the individual helices are out of phase with each other. We observe that as time evolves, $\unicode[STIX]{x1D713}_{1}(t,s)$ , which initially has a helical shape (figure 6 a), has an amplitude that shrinks (figure 6 b), before returning to its original shape (figure 6 c). The vortex described by $\unicode[STIX]{x1D713}_{2}(t,s)$ is out of phase in that its amplitude shrinks when $\unicode[STIX]{x1D713}_{1}(t,s)$ grows (figure 6 d).

We compare the asymptotic solutions given by (2.21) and (2.22) with the numerical solution of (2.18) for the same parameters and initial conditions on $\unicode[STIX]{x1D707}(\unicode[STIX]{x1D709})$ and $\unicode[STIX]{x1D708}(\unicode[STIX]{x1D709})$ (shown in figure 7). We note that the asymptotic solutions provide a very good approximation to the full vortex interaction, with a notable exception: the solution with the untruncated vortex interaction predicts that $\unicode[STIX]{x1D708}(\unicode[STIX]{x1D709})$ exhibits slow amplitude modulation when compared with the second-order asymptotic correction in figure 7(c). This result suggests that, in a similar manner to that for the co-rotating vortices, we require a slow-scale analysis of the problem to capture this effect.

We look for a slow solution of the form

where we have included the previously mentioned constant translation of the vortex pair centre from the $4/dv^{2}$ term in (2.21).

We substitute (2.23) into (2.20) to obtain $\unicode[STIX]{x1D707}_{\mathit{slow}}(\unicode[STIX]{x1D709})$ , the use of which gives the following asymptotic solution upon using the result in (2.18) (see § A.3.2 for mathematical details):

for $c_{n}\in \mathbb{R}$ .

We compare (2.24) and (2.23) with the numerical solution of (2.18) and (2.20) for the same parameters and initial conditions on $\unicode[STIX]{x1D707}(\unicode[STIX]{x1D709})$ and $\unicode[STIX]{x1D708}(\unicode[STIX]{x1D709})$ as previously but with $d=\{10,7.5\}$ in figures 8(a) and 8(b) respectively. In particular, the constants are $c_{13}=-\unicode[STIX]{x03C0}/2$ and all other $c_{n}=0$ . It should be noted that as $\unicode[STIX]{x1D709}\rightarrow \infty$ , the monotonically increasing term in $\unicode[STIX]{x1D709}$ dominates in (2.24). Furthermore, we have vertically offset the slowly varying solution to show its fit to the slowly varying envelope. We observe very good quantitative agreement for $d=10$ ; however, breakdown is apparent when $d=7.5$ for $\unicode[STIX]{x1D709}\approx 200$ .

Figure 8.

Real components of $\unicode[STIX]{x1D708}(\unicode[STIX]{x1D709})$ predicted by the slowly varying solution (2.24) and (2.23) (shown in black) and the numerical solution of (2.18) and (2.20) (shown in red) for $v=2$ and (a) $d=10$ and (b) $d=7.5$ .

2.2.2 Self-similar solutions

We proceed in the same manner as § 2.1.2 by considering a distance function $D(t)=d\sqrt{t}$ and self-similar solutions of the form $\unicode[STIX]{x1D707}=\sqrt{t}\unicode[STIX]{x1D712}(\unicode[STIX]{x1D702})$ , with the addition of $\unicode[STIX]{x1D708}=\sqrt{t}\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D702})$ , for $\unicode[STIX]{x1D702}=s/\sqrt{t}$ . Transformation of (2.15) and (2.16) gives

which can be combined by differentiating (2.25) once with respect to $\unicode[STIX]{x1D702}$ and substituting (2.26) into this result to obtain

where the additional initial condition on $\unicode[STIX]{x1D712}^{\prime \prime }(\unicode[STIX]{x1D702})$ is calculated by the conditions imposed on $\unicode[STIX]{x1D712}(\unicode[STIX]{x1D702})$ , $\unicode[STIX]{x1D712}^{\prime }(\unicode[STIX]{x1D702})$ and $\unicode[STIX]{x1D70C}^{\prime }(\unicode[STIX]{x1D702})$ in (2.25).

We expand the nonlinear interaction term in (2.27) and use the standard perturbation expansion for $\unicode[STIX]{x1D712}(\unicode[STIX]{x1D702})=\sum _{q=0}^{\infty }d^{-q}\unicode[STIX]{x1D712}^{(q)}$ to find

which gives the first-order solution

The absence of the $O(d^{-1})$ vortex interaction term results in no repulsion of the vortex pair, as seen for the co-rotating pair. The interaction between the pair is $O(d^{-2})$ , so that the shape of the filaments in (2.29) is wholly determined by self-induction. The corresponding shape is similar to the co-rotating case in (2.12); that is, the vortices form conical spirals of increasing amplitude and wavelength. We show the vortex filament shapes obtained by numerically solving (2.25) and (2.26) for boundary conditions on $\unicode[STIX]{x1D707}(t,s)$ satisfying (2.13) and (2.14) with $\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D702}=100)=\unicode[STIX]{x1D70C}^{\prime }(\unicode[STIX]{x1D702}=100)=0$ and $d=20$ in figure 9 for times $t=\{0.05,0.5,1,2\}$ . We observe that, as with the co-rotating case, the helices rotate in the opposite direction to the respective circulation of the vortex, with $\unicode[STIX]{x1D713}_{1}(t,s)$ and $\unicode[STIX]{x1D713}_{2}(t,s)$ having circulations $\unicode[STIX]{x1D6E4}_{1}>0$ and $\unicode[STIX]{x1D6E4}_{2}<0$ . The helices are now reflections of each other across the $y$ -axis.

We compare the real and imaginary components of the asymptotic solution (2.29) with the numerical solutions for the same conditions in figure 10. In this case, the boundary condition necessary to determine the third condition on $\unicode[STIX]{x1D712}(\unicode[STIX]{x1D702})$ is $\unicode[STIX]{x1D712}^{\prime \prime }(\unicode[STIX]{x1D702}=100)=4d^{-1}(1+d^{-1}(1-\text{i}))^{-1}$ by (2.25). Expansion of the condition for $d^{-1}\ll 1$ gives $\unicode[STIX]{x1D712}^{\prime \prime }(\unicode[STIX]{x1D702}=100)=O(d^{-2})$ , which we impose to find the zeroth-order solution $\unicode[STIX]{x1D712}^{(0)}(\unicode[STIX]{x1D702})$ and subsequently find the corresponding $\unicode[STIX]{x1D70C}^{(0)}(\unicode[STIX]{x1D702})$ from (2.26). We note that the asymptotic solution shows good agreement with the numerical solutions, but breaks down as longer time scales are considered.

Figure 9.

The time evolution of self-similarity solutions in counter-rotating vortices held apart by a distance function $D(t)=d\sqrt{t}$ , with $\unicode[STIX]{x1D713}_{1}(t,s)$ and $\unicode[STIX]{x1D713}_{2}(t,s)$ being denoted by blue and red respectively. In particular, numerical solutions of (2.25) and (2.26), with conditions satisfying (2.13), (2.14), and $\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D702}=100)=\unicode[STIX]{x1D70C}^{\prime }(\unicode[STIX]{x1D702}=100)=0$ , have been plotted for $d=20$ over $s\in [10,20]$ . The times shown are (a) $t=0.05$ , (b) $t=0.5$ , (c) $t=1$ and (d) $t=2$ .

3.1.1 Plane-wave solutions

We consider rotating helix solutions to (3.4) of the form

where $A_{0}$ is the constant amplitude of separation, $k$ is the wavenumber related to the twist of the helix, $\unicode[STIX]{x1D714}$ is the rotational velocity and $A_{0},k,\unicode[STIX]{x1D714}\in \mathbb{R}$ .

Substitution of (3.9) into (3.4) gives the dispersion relation

a result that has previously been found by Klein et al. (Reference Klein, Majda and Damodaran1995). However, (3.10) implies that the vortex pair can rotate clockwise, anticlockwise or not at all depending on the twist of the helical solution and its amplitude, regardless of the individual circulation of the vortices.

3.1.2 Travelling wave solutions

We introduce the same wave coordinate $\unicode[STIX]{x1D709}=s-vt$ seen in the aforementioned analysis, for $v$ a constant. Transforming relations (3.6) and (3.8), we find

Solutions to these equations will effectively generalise the plane-wave or helix solutions of Klein et al. (Reference Klein, Majda and Damodaran1995) mentioned above, as those solutions correspond to the case where $A$ is constant.

We note that (3.11) and (3.12) allow the exact implicit solution

where $\unicode[STIX]{x1D709}_{0},E\in \mathbb{R}$ are integration constants and $V(A)$ is a Hamiltonian potential given by

where $\unicode[STIX]{x1D6FE}\in \mathbb{R}$ is another integration constant that is a measure of coupling between the vortex pair separation and changes in its orientation (see § B.1).

We comment on some important aspects of (3.14). The potential function implies that all travelling wave solutions will be attracted to $A\rightarrow 0^{+}$ as $\unicode[STIX]{x1D709}\rightarrow \pm \infty$ if $\unicode[STIX]{x1D6FE}=0$ , resulting in collapse of the vortex pair as $t\rightarrow \pm \infty$ . Coupling between the amplitude of separation and the angular velocity of the pair is necessary to sustain travelling wave solutions for co-rotating pairs. When $\unicode[STIX]{x1D6FE}\neq 0$ , (3.8) predicts that $A$ will have periodic trajectories such that $A>0$ for all time. However, for $\unicode[STIX]{x1D6FE}\ll 1$ , there are three different asymptotic regimes in (3.11) and (3.12) (for full details of the other regimes, see § B.1), the most noteworthy of which is $A(\unicode[STIX]{x1D709})=O(\unicode[STIX]{x1D6FE}^{n})$ for $n>1$ . In this case, there exists an initial layer of width $\unicode[STIX]{x1D6FE}^{2n-1}$ so that $A^{\prime \prime }\sim 1/A^{3}$ and the leading-order behaviour is

for $c_{1},c_{2},\unicode[STIX]{x1D703}_{0}\in \mathbb{R}$ .

We illustrate the vortex pair by numerically solving (3.11) and (3.12) and compare these results with those from the leading-order behaviour and the implicit solution (3.13). In particular, we consider the conditions $A(\unicode[STIX]{x1D709}=0)=1$ , $A^{\prime }(\unicode[STIX]{x1D709}=0)=0$ and $\unicode[STIX]{x1D703}(\unicode[STIX]{x1D709}=0)=0$ , with $\unicode[STIX]{x1D6FE}=0.001$ and $v=1$ . For these parameters, the trajectories of $A$ are such that they never exceed the potential given by $v^{2}/8+\unicode[STIX]{x1D6FE}^{2}/2$ , according to (3.14), constraining the values of $A$ to the range $A\in [1.17\times 10^{-5},1]$ , and, furthermore, the period of oscillation in $A(\unicode[STIX]{x1D709})$ is $T\approx 1.226$ by solving (3.13). We plot the potential (3.14), corresponding $A(\unicode[STIX]{x1D709})$ , and the real and imaginary components of $\text{exp}(\text{i}\unicode[STIX]{x1D703}(\unicode[STIX]{x1D709}))$ in figures 12(a), 12(b) and 12(c) respectively as well as the shape of the vortex pair at $t=\{0,2,4,6\}$ with $\unicode[STIX]{x1D708}(t,s)=0$ (shown in figure 13(a–d)).

From figure 12(b), we observe that the period of oscillation is indeed $T\approx 1.226$ , and furthermore note that the numerical solutions show similarities to (3.15) for small $A(\unicode[STIX]{x1D709})$ ; in particular, the sharp cusp that was predicted in the regime $A=O(\unicode[STIX]{x1D6FE}^{n})$ for $n>1$ . As the vortices approach each other, their attraction occurs on a very fast time scale, for $\unicode[STIX]{x1D6FE}\ll 1$ , until the pair reaches their minimum separation whereby they repel. The orientation of the vortices at this moment changes almost instantaneously, as $\unicode[STIX]{x1D6FE}\tan ^{-1}(c_{1}c_{2}+c_{1}\unicode[STIX]{x1D709}/\unicode[STIX]{x1D6FE}^{2n-1})$ dominates in (3.16). As the vortices repel and $A\rightarrow O(1)$ , the time scale slows down and the orientation of the pair evolves according to the $v\unicode[STIX]{x1D709}/2$ term, which now plays a role in (B 6). The vortices reach their maximum separation before attracting each other again and repeating the cycle.

Figure 12.

(a) Potential of the separation equation as defined in (3.14), and numerical solutions of (b) $A(\unicode[STIX]{x1D709})$ and (c) the real and imaginary parts of the complex exponential $\text{exp}(\text{i}\unicode[STIX]{x1D703}(\unicode[STIX]{x1D709}))$ (shown in black and red respectively). The numerical solutions were obtained by solving (B 1) and (B 2) for $v=1$ and $\unicode[STIX]{x1D6FE}=0.001$ , with initial conditions $A(\unicode[STIX]{x1D709}=0)=1$ , $A^{\prime }(\unicode[STIX]{x1D709}=0)=0$ and $\unicode[STIX]{x1D703}(\unicode[STIX]{x1D709}=0)=0$ . The trajectories of $A(\unicode[STIX]{x1D709})$ are restricted in the potential well such that they never exceed the potential $v^{2}/8+\unicode[STIX]{x1D6FE}^{2}/2$ , shown by the dashed line, and oscillate with period $T\approx 1.226$ , by (3.13).

Figure 13.

Time evolution of travelling wave solutions in intertwining co-rotating vortices, with $\unicode[STIX]{x1D713}_{1}(t,s)$ and $\unicode[STIX]{x1D713}_{2}(t,s)$ shown in blue and red respectively. Numerical solutions of (B 1) and (B 2), with $\unicode[STIX]{x1D708}(\unicode[STIX]{x1D709})=0$ , are plotted over $s\in [0,5]$ . In particular, the initial conditions are $A(\unicode[STIX]{x1D709}=0)=1$ , $A^{\prime }(\unicode[STIX]{x1D709}=0)=0$ and $\unicode[STIX]{x1D703}(\unicode[STIX]{x1D709}=0)=0$ , with parameters $v=1$ and $\unicode[STIX]{x1D6FE}=0.001$ . The times shown are (a) $t=0$ , (b) $t=2$ , (c) $t=4$ and (d) $t=8$ .

3.1.3 Purely rotating-wave solutions

We study solutions of the rotating-wave ansatz

where $\unicode[STIX]{x1D714}$ is the constant angular velocity of the rotating wave. The resulting solutions are each effectively planar filaments, as they maintain their spatial structure while the only motion is pure rotation; see Van Gorder (Reference Van Gorder2015b ) and references therein.

Substitution of this ansatz into (3.6) gives

which, following the treatment of § B.1, has the exact implicit solution

where $s_{0},E\in \mathbb{R}$ are integration constants and the Hamiltonian potential is given by

We note that (3.21) is not bounded from below, suggesting that, for all rotating-wave solutions with $\unicode[STIX]{x1D714}>0$ , $A\rightarrow 0^{+}$ as $s\rightarrow \pm \infty$ ; the vortex pair will collapse if their respective circulations are in the same direction as the pair rotates in. If $\unicode[STIX]{x1D714}<0$ , then another possibility arises, namely $A\rightarrow \infty$ as $s\rightarrow \infty$ , as the potential is not bound from above. If we consider a vortex pair with circulations in the opposite direction, then rotating pairs with $\unicode[STIX]{x1D714}<0$ will collapse or, if $\unicode[STIX]{x1D714}>0$ , may also repel without bound.

To illustrate the aforementioned description of the vortex shapes, we plot the numerical solution of (3.19) with $\unicode[STIX]{x1D708}(t,s)=0$ for the case of $\unicode[STIX]{x1D714}>0$ and $\unicode[STIX]{x1D714}<0$ . In particular, we consider $\unicode[STIX]{x1D714}=1$ for the initial conditions $A(s=0)=1$ and $A(s=0)=5$ (shown in figure 14 a,b), and $\unicode[STIX]{x1D714}=-1$ for conditions $A(s=0)=1$ and $A(s=0)=1.8$ , both at times $t=\{0,2\}$ . We note that, as predicted, the solutions for $\unicode[STIX]{x1D714}>0$ exhibit eventual collapse of the pair for increasing $s$ , while solutions for $\unicode[STIX]{x1D714}<0$ show unbounded growth of the pair for increasing $s$ .

Figure 14.

Time evolution of planar rotating filament solutions in intertwining co-rotating vortices, with $\unicode[STIX]{x1D713}_{1}(t,s)$ and $\unicode[STIX]{x1D713}_{2}(t,s)$ shown in blue and red respectively. Numerical solutions of (3.19), with $\unicode[STIX]{x1D708}(t,s)=0$ , are plotted over $s\in [0,9]$ . In particular, the initial conditions and parameters are $A(\unicode[STIX]{x1D709}=0)=1$ , $A^{\prime }(\unicode[STIX]{x1D709}=0)=5$ and $\unicode[STIX]{x1D714}=1$ (a,b), and $A(\unicode[STIX]{x1D709}=0)=1$ , $A^{\prime }(\unicode[STIX]{x1D709}=0)=1.8$ and $\unicode[STIX]{x1D714}=-1$ (c,d), both of which are plotted at times $t=\{0,2\}$ .

3.1.4 Self-similar solutions

We transform (3.4) into the variables $\unicode[STIX]{x1D707}(t,s)=s\unicode[STIX]{x1D712}(\unicode[STIX]{x1D702})$ , with $\unicode[STIX]{x1D702}=s/\sqrt{t}$ ,

and use the amplitude–phase ansatz for $\unicode[STIX]{x1D712}(\unicode[STIX]{x1D702})$ to simplify the vortex interaction term,

which, substituting into (3.22) and separating real and imaginary components, gives

By multiplying (3.25) through by $A(\unicode[STIX]{x1D702})$ and dividing by $\unicode[STIX]{x1D702}^{2}$ , we obtain

which is a linear equation in $A^{2}\unicode[STIX]{x1D703}^{\prime }(\unicode[STIX]{x1D702})$ that can be integrated to give

where $\unicode[STIX]{x1D6FE}\in \mathbb{R}$ is an integration constant. It should be noted that, compared with the travelling wave solutions in § 3.1.2, $\unicode[STIX]{x1D6FE}$ does not quantify the degree of coupling between the changing orientation of the vortex separation and the amplitude of separation, because if $\unicode[STIX]{x1D6FE}=0$ , there is still dependence of $\unicode[STIX]{x1D703}^{\prime }(\unicode[STIX]{x1D702})$ on $A(\unicode[STIX]{x1D702})$ .

We numerically solve (3.24) and (3.27) for the boundary conditions $A(\unicode[STIX]{x1D702}=0.001)=5$ , $A^{\prime }(\unicode[STIX]{x1D702}=0.001)=1$ , $\unicode[STIX]{x1D703}(\unicode[STIX]{x1D702}=0.001)=0$ , $\unicode[STIX]{x1D703}^{\prime }(\unicode[STIX]{x1D702}=0.001)=0$ , and plot $A(\unicode[STIX]{x1D702})$ and $\text{exp}(\text{i}\unicode[STIX]{x1D703}(\unicode[STIX]{x1D702}))$ in figures 15(a) and 15(b) respectively, and the corresponding vortex filament shapes in figure 16 for $\unicode[STIX]{x1D708}(t,s)=0$ at times $t=\{10,50,100,500,1000,5000\}$ .

We observe a number of interesting aspects. First, given the conditions on $A(\unicode[STIX]{x1D702})$ and $\unicode[STIX]{x1D703}(\unicode[STIX]{x1D702})$ , we note that $\unicode[STIX]{x1D6FE}\neq 0$ , which implies that when $A(\unicode[STIX]{x1D702})\rightarrow 0^{+}$ , then $\unicode[STIX]{x1D703}^{\prime }(\unicode[STIX]{x1D702})\rightarrow \pm \infty$ . In a similar manner to that in § 3.1.2 for the case of $\unicode[STIX]{x1D6FE}\neq 0$ , when the vortex pair come into close proximity, the orientation of separation changes rapidly, as we note in figure 15(a,b). Furthermore, as $\unicode[STIX]{x1D702}\rightarrow 0^{+}$ , corresponding to time $t\rightarrow \infty$ , the amplitude and the wavelength of the vortex separation become larger and the changes in orientation become slower. We note that these results follow directly from the self-similar nature of the vortex pair, which exhibit an initial shape (figure 16 a) that becomes increasingly zoomed in (figure 16 b–f) for increasing times. As $t\rightarrow \infty$ , the separation of the vortices themselves will grow to infinity, and their orientation will be planar along a single direction.

Figure 15.

Numerical solutions of (a) $A(\unicode[STIX]{x1D702})$ and (b) real and imaginary parts of the complex exponential $\text{exp}(\text{i}\unicode[STIX]{x1D703}(\unicode[STIX]{x1D702}))$ (shown in black and red respectively), obtained by solving (3.24) and (3.27) with initial conditions $A(\unicode[STIX]{x1D702}=0.001)=5$ , $A^{\prime }(\unicode[STIX]{x1D702}=0.001)=1$ , $\unicode[STIX]{x1D703}(\unicode[STIX]{x1D702}=0.001)=0$ and $\unicode[STIX]{x1D703}^{\prime }(\unicode[STIX]{x1D702}=0.001)=0$ .

Figure 16.

Time evolution of self-similar solutions in intertwining co-rotating vortices, with $\unicode[STIX]{x1D713}_{1}(t,s)$ and $\unicode[STIX]{x1D713}_{2}(t,s)$ shown in blue and red respectively. Numerical solutions of (3.24) and (3.27), with $\unicode[STIX]{x1D708}(t,s)=0$ , are plotted over $s\in [0.5,20]$ . In particular, the initial conditions are $A(\unicode[STIX]{x1D702}=0.001)=5$ , $A^{\prime }(\unicode[STIX]{x1D702}=0.001)=1$ , $\unicode[STIX]{x1D703}(\unicode[STIX]{x1D702}=0.001)=0$ and $\unicode[STIX]{x1D703}^{\prime }(\unicode[STIX]{x1D702}=0.001)=0$ . The times shown are (a) $t=10$ , (b) $t=50$ , (c) $t=100$ , (d) $t=500$ , (e) $t=1000$ and (f) $t=5000$ .

3.2.1 Plane-wave solutions

We consider solutions to (3.30) of the form given in (3.9). The vortex centre $\unicode[STIX]{x1D708}(t,s)$ is not arbitrary in this case, and is explicitly given by solving (3.29),

for $c_{1},c_{2}\in \mathbb{C}$ and where $A_{0}$ , $k$ and $\unicode[STIX]{x1D714}$ have the same meaning as in § 3.1.1. The centre of the pair traces out a helix with the same wavenumber and rotates at the same angular velocity as the separation between the vortex pair, up to an arbitrary orientation determined by $c_{1}$ and translation defined by $c_{2}$ .

Substitution of (3.9) into (3.30) gives the dispersion relation

The angular velocity is again determined by the wavenumber $k$ and the amplitude of vortex separation $A_{0}$ . Similarly to § 3.1.1, $\unicode[STIX]{x1D714}$ can be positive, negative or zero; however, for all non-zero values, the angular velocity can be either clockwise or anticlockwise rotating, regardless of the parameters imposed on the system. To illustrate, we plot the filament shapes of the vortex pair in figure 17 for $A_{0}=1$ with (a,b) $k=1$ with $\unicode[STIX]{x1D714}=-\sqrt{5}$ , and (c,d) $k=1$ with $\unicode[STIX]{x1D714}=\sqrt{5}$ . The positive root of (3.32) was taken in both cases.

Figure 17.

Time evolution of plane-wave solutions in intertwining counter-rotating vortices, with $\unicode[STIX]{x1D713}_{1}(t,s)$ and $\unicode[STIX]{x1D713}_{2}(t,s)$ shown in blue and red respectively. In particular, (3.9) and (3.31), with the dispersion relation determined by (3.32), are plotted over $s\in [-5,5]$ . The parameters are $A_{0}=1$ and (a,b) $k=-1$ with $\unicode[STIX]{x1D714}=-\sqrt{5}$ and (c,d) $k=1$ with $\unicode[STIX]{x1D714}=\sqrt{5}$ . The times shown are (a,c) $t=0$ and (b,d) $t=0.5$ .

3.2.2 Travelling wave solutions

We transform (3.30) into wave coordinates $\unicode[STIX]{x1D709}=s-vt$ and integrate twice with respect to $\unicode[STIX]{x1D709}$ to obtain

for $\unicode[STIX]{x1D6FF}\in \mathbb{C}$ , which has constraints

As before, we determine $\unicode[STIX]{x1D6FF}$ by imposing conditions on $\unicode[STIX]{x1D707}(\unicode[STIX]{x1D709})$ and $\unicode[STIX]{x1D708}^{\prime }(\unicode[STIX]{x1D709})$ . However, for the forthcoming analysis, we only consider cases where $\unicode[STIX]{x1D6FF}=0$ .

We study solutions to (3.33) of the amplitude–phase ansatz (3.5) and find, upon separating real and imaginary components,

Solutions to these equations naturally generalise the plane-wave helix solutions of the previous subsection, as those solutions correspond to constant $A$ .

We note that (3.36) and (3.37) have an exact solution given by (3.13) with a potential function given by

where $\unicode[STIX]{x1D6FE}\in \mathbb{R}$ is an integration constant that has the same physical meaning as in § 3.1.2, measuring the coupling between the vortex pair separation and its change in orientation.

We note some important differences compared with the co-rotating case. First, if $\unicode[STIX]{x1D6FE}=0$ , then the orientation of the counter-rotating pair does not change and the oscillation remains completely planar for all time. For example, vortices that only vibrate in the $x$ -axis will do so for all time. Furthermore, the pair no longer collapses as $t\rightarrow \infty$ , according to (3.38), as the lower bound of the potential is determined by $-4\text{ln}(A)$ in (3.38). For any value of $v$ or $\unicode[STIX]{x1D6FE}$ , the counter-rotating vortices will not meet; however, if the velocity of the travelling wave solution $v=0$ , the pair will repel as $A\rightarrow \infty$ for $\unicode[STIX]{x1D709}\rightarrow \pm \infty$ . If $v\neq 0$ , then periodic solutions in $A$ are guaranteed for all values of $\unicode[STIX]{x1D6FE}$ . This aspect leads to the counter-rotating vortices having a different asymptotic structure that no longer depends on $\unicode[STIX]{x1D6FE}$ , unlike what was previously seen for the co-rotating vortices (see § B.3 for further details).

We illustrate the potential (3.38) and numerically solve (3.36) and (3.37) for $A(\unicode[STIX]{x1D709})$ and $\unicode[STIX]{x1D703}(\unicode[STIX]{x1D709})$ with the conditions $A(\unicode[STIX]{x1D709}=0)=2$ , $A^{\prime }(\unicode[STIX]{x1D709}=0)=0$ and $\unicode[STIX]{x1D703}(\unicode[STIX]{x1D709}=0)=0$ , for parameters $v=2$ and $\unicode[STIX]{x1D6FE}=1$ (shown in figures 18 a, 18 b and 18 c respectively). In this case, the trajectories of $A$ are restricted such that they always remain within the range $A\in [0.494376,2]$ for all $t$ and $s$ , with period given by $T\approx 1.925$ . We note that both figures 18(b) and 18(c) show that the separation of the vortex pair is periodic, which, by (3.35), suggests that the position of the pair centre will periodically oscillate too.

We numerically solve (3.35) to obtain $\unicode[STIX]{x1D708}(\unicode[STIX]{x1D709})$ , imposing the condition $\unicode[STIX]{x1D708}(\unicode[STIX]{x1D709}=0)=0$ , and plot the associated filament shapes at times $t=\{1,1.5,2,2.5\}$ to obtain figures 19(a)–19(d) respectively. Furthermore, we show the shape the vortex pair traces on the $x$ – $y$ plane, for $s=0$ during the time $t\in [0,25]$ , in figure 20. We observe that $\unicode[STIX]{x1D713}_{2}(t,s)$ oscillates in a trefoil knot-like shape while $\unicode[STIX]{x1D713}_{1}(t,s)$ traces out a helical shape that is deformed such that its curvature minima coincide with the maximal tips of the trefoil shape, a result that arises due to the coupling of $\unicode[STIX]{x1D707}(t,s)$ and $\unicode[STIX]{x1D708}(t,s)$ , which causes the centre of the vortex pair to form the periodically deformed helical shape for all time and filament arclength.

Figure 18.

(a) Potential of the separation equation as defined in (3.38), and numerical solutions of (b) $A(\unicode[STIX]{x1D709})$ and (c) the real and imaginary parts of the complex exponential $\text{exp}(\text{i}\unicode[STIX]{x1D703}(\unicode[STIX]{x1D709}))$ (shown in black and red respectively) obtained by solving (3.37) and (3.36) for $v=2$ , $\unicode[STIX]{x1D6FE}=1$ and $\unicode[STIX]{x1D6FF}=0$ with initial conditions $A(\unicode[STIX]{x1D709}=0)=2$ , $A^{\prime }(\unicode[STIX]{x1D709}=0)=0$ and $\unicode[STIX]{x1D703}(\unicode[STIX]{x1D709}=0)=0$ . The trajectories of $A$ are restricted in the potential well such that they never exceed the potential $2v^{2}/8-4\ln (2)+\unicode[STIX]{x1D6FE}^{2}/8$ , shown by the dashed line, and oscillate with period $T\approx 1.925$ according to (3.13).

Figure 19.

Time evolution of travelling wave solutions in intertwining counter-rotating vortices, with $\unicode[STIX]{x1D713}_{1}(t,s)$ and $\unicode[STIX]{x1D713}_{2}(t,s)$ shown in blue and red respectively. Numerical solutions of (3.35), (3.37) and (3.36) are plotted over $s\in [0,10]$ . In particular, the initial conditions are $A(\unicode[STIX]{x1D709}=0)=2$ , $A^{\prime }(\unicode[STIX]{x1D709}=0)=0$ , $\unicode[STIX]{x1D703}(\unicode[STIX]{x1D709}=0)=0$ and $\unicode[STIX]{x1D708}(\unicode[STIX]{x1D709}=0)=0$ with parameters $v=2$ , $\unicode[STIX]{x1D6FE}=1$ and $\unicode[STIX]{x1D6FF}=0$ . The times shown are (a) $t=0$ , (b) $t=2$ , (c) $t=4$ and (d) $t=8$ .

Figure 20.

The shape that the filament pair in figure 19 traces out in the $x$ – $y$ plane during the time $t\in [0,25]$ at $s=0$ , with $\unicode[STIX]{x1D713}_{1}(t,s)$ and $\unicode[STIX]{x1D713}_{2}(t,s)$ corresponding to black and red lines respectively.

3.2.3 Purely rotating-wave solutions

We study solutions of the rotating-wave ansatz of (3.5), (3.17) and (3.18) and apply it to (3.30) to obtain

where the general solution to $\unicode[STIX]{x1D708}(t,s)$ is given by a combination (3.28) and (3.29) as

for $c_{1},c_{2}\in \mathbb{C}$ . Again, the solutions we obtain will correspond to planar vortex filaments that maintain their spatial form as the only motion is purely rotational.

We observe that, with the characteristic coupling between the deviations and the centre of the counter-rotating vortices, the pair not only rotate around each other, but also around a centre point in the fluid, determined by the conditions on $\unicode[STIX]{x1D708}(t,s)$ .

We numerically solve (3.39) and (3.40) with boundary conditions given by $A(s=0)=2$ , $A(s=10)=2$ , $A^{\prime }(s=0)=0$ , $A^{\prime }(s=10)=0$ and $\unicode[STIX]{x1D708}(t,0)=\unicode[STIX]{x2202}\unicode[STIX]{x1D708}(t,0)/\unicode[STIX]{x2202}s=0$ , with $\unicode[STIX]{x1D714}=0.5$ and $c_{1}=c_{2}=0$ . We plot $\unicode[STIX]{x1D707}(t,s)$ and $\unicode[STIX]{x1D708}(t,s)$ at $t=0$ in figure 21(a,b), and the corresponding filament shapes over $s\in [0,10]$ at times $t=\{2.5,7.5\}$ in figure 21(c,d). We observe that, for the boundary conditions chosen, the separation of the vortex pair is oscillatory along the arclength (figure 21 a), a solution not possible for the co-rotating case seen in § 3.1.3. By (3.40), $\unicode[STIX]{x1D708}(t,s)$ is also bounded and non-zero (figure 21 b). The filament shapes are therefore asymmetrical when compared with the symmetrical co-rotating vortices (figure 21), a direct result of the coupling of $\unicode[STIX]{x1D707}(t,s)$ and $\unicode[STIX]{x1D708}(t,s)$ in counter-rotating vortices.

Figure 21.

Numerical solutions of (3.39) and (3.40) at $t=0$ (shown in (a,b)) for the conditions $A(s=0)=2$ , $A(s=10)=2$ , $A^{\prime }(s=0)=0$ , $A^{\prime }(s=10)=0$ and $\unicode[STIX]{x1D708}(t,0)=\unicode[STIX]{x2202}\unicode[STIX]{x1D708}(t,0)/\unicode[STIX]{x2202}s=0$ , with $\unicode[STIX]{x1D714}=0.5$ and $c_{1}=c_{2}=0$ . The corresponding filament shapes of $\unicode[STIX]{x1D713}_{1}(t,s)$ (shown in blue) and $\unicode[STIX]{x1D713}_{2}(t,s)$ (shown in red) are plotted over $s\in [0,10]$ for times (c) $t=2.5$ and (d) $t=7.5$ . The solutions clearly take on a planar filament structure.

3.2.4 Self-similar solutions

We transform (3.28) and (3.29) into the variables $\unicode[STIX]{x1D707}(t,s)=s\unicode[STIX]{x1D712}(\unicode[STIX]{x1D702})$ and $\unicode[STIX]{x1D708}(t,s)=s\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D702})$ , for $\unicode[STIX]{x1D702}=s/\sqrt{t}$ , to find

and a single equation for $\unicode[STIX]{x1D712}(\unicode[STIX]{x1D702})$ upon substituting (3.41) into (3.42),

with the additional initial condition $\unicode[STIX]{x1D712}^{\prime \prime }(\unicode[STIX]{x1D702})$ obtained from (3.41) using conditions on $\unicode[STIX]{x1D712}(\unicode[STIX]{x1D702})$ , $\unicode[STIX]{x1D712}^{\prime }(\unicode[STIX]{x1D702})$ and $\unicode[STIX]{x1D70C}^{\prime }(\unicode[STIX]{x1D702})$ .

We study solutions to $\unicode[STIX]{x1D712}(\unicode[STIX]{x1D702})$ of the amplitude–phase ansatz in (3.23) and find, upon substitution into (3.43) and separating real and imaginary parts,

To obtain physical intuition into the types of solutions predicted by (3.43), we numerically solve (3.44) and (3.45) with the conditions $A(\unicode[STIX]{x1D702}=3)=1$ , $A^{\prime }(\unicode[STIX]{x1D702}=3)=0$ , $A^{\prime \prime }(\unicode[STIX]{x1D702}=3)=0$ , $\unicode[STIX]{x1D703}(\unicode[STIX]{x1D702}=3)=0$ , $\unicode[STIX]{x1D703}^{\prime }(\unicode[STIX]{x1D702}=3)=1$ and $\unicode[STIX]{x1D703}^{\prime \prime }(\unicode[STIX]{x1D702}=1)=-5$ , and plot the real and imaginary parts of $\unicode[STIX]{x1D712}(\unicode[STIX]{x1D702})$ and $\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D702})$ , using the condition $\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D702}=3)=0$ , in figures 22(a), 22(b) and 22(c), 22(d) respectively. Furthermore, we illustrate the corresponding filament shapes over the arclength $s\in [5,15]$ at times $t=\{0.1,1,2,2.5\}$ in figure 23. We observe some parallels to self-similar solutions in other vortex configurations (i.e. § 2.1.2, 2.2.2, and 3.1.4), namely the progression of the vortex pair shape from a straight filament in some orientation to oscillatory solutions with increasing amplitude and wavelength as $t\rightarrow \infty$ . However, in contrast to the self-similar solutions in § 3.1.4, the vortices are never in very close proximity, so that the orientation of the pair does not rapidly change according to the dependence of $1/A$ in (3.45). The vortex pair remains separated due to the constant offset in $\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D702})$ in figure 22(c,d), but as $t\rightarrow \infty$ , the centre of the vortex pair moves such that the filaments begin to intertwine and rotate around each other (figure 23 d).

Figure 22.

Real and imaginary components of $\unicode[STIX]{x1D712}(\unicode[STIX]{x1D702})$ and $\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D702})$ calculated by the numerical solution of (3.44), (3.45) and (3.42). The initial conditions are $A(\unicode[STIX]{x1D702}=3)=1$ , $A^{\prime }(\unicode[STIX]{x1D702}=3)=0$ , $A^{\prime \prime }(\unicode[STIX]{x1D702}=3)=0$ , $\unicode[STIX]{x1D703}(\unicode[STIX]{x1D702}=3)=0$ , $\unicode[STIX]{x1D703}^{\prime }(\unicode[STIX]{x1D702}=3)=1$ , $\unicode[STIX]{x1D703}^{\prime \prime }(\unicode[STIX]{x1D702}=3)=-5$ and $\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D702}=3)=0$ .

Figure 23.

Time evolution of self-similar solutions in intertwining counter-rotating vortices, with $\unicode[STIX]{x1D713}_{1}(t,s)$ and $\unicode[STIX]{x1D713}_{2}(t,s)$ shown in blue and red respectively. Numerical solutions of (3.44), (3.45) and (3.42) are plotted over $s\in [5,15]$ with the initial conditions $A(\unicode[STIX]{x1D702}=3)=1$ , $A^{\prime }(\unicode[STIX]{x1D702}=3)=0$ , $A^{\prime \prime }(\unicode[STIX]{x1D702}=3)=0$ , $\unicode[STIX]{x1D703}(\unicode[STIX]{x1D702}=3)=0$ , $\unicode[STIX]{x1D703}^{\prime }(\unicode[STIX]{x1D702}=3)=1$ , $\unicode[STIX]{x1D703}^{\prime \prime }(\unicode[STIX]{x1D702}=3)=-5$ and $\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D702}=3)=0$ . The times shown are (a) $t=0.1$ , (b) $t=1$ , (c) $t=2$ and (d) $t=2.5$ .