3.1. Large-scale streams

Figures 5(a) and 5(b) are heat maps for the fit of the vortex and dipole templates to a typical flow snapshot. The intensity in a heat map is proportional to the goodness of fit of the template centred at that point. The darker regions in figure 5 represent better fits, measured by $\varPhi _q$. Figure 5(a) is drawn for vortex templates, and figure 5(b) is drawn for dipoles. In both cases, the size of the template is chosen to be optimum, which in this particular case is different for the two templates (see the cyan boxes in the figure). Inspection of the figures shows that the features extracted by the two templates have much in common, although they differ in detail. This is not surprising because a dipole is formed by two vortices, and it was to be expected that at least some of the vortices identified by a vortex template are part of a dipole.

Figure 5.

(a,b) Heat maps for the fit error of a typical flow field. Case T768 at $\omega '_0 t=22$. Colour represents the approximation error, $\varPhi _q$, for a template centred at each point. Line contours are the vorticity magnitude. The cyan box is the size of the template, chosen in each case to optimise $P_q$. (a) Vortex template. (b) Dipole template. (c) Velocity field for the flow in (a,b). The scale bars above the figure are as in figure 2. The background colour is the velocity magnitude, lighter for faster velocities.

However, the main use of heat maps is not to identify individual features, but to highlight the organisation of the features themselves. Thus, although we have mentioned that figures 5(a) and 5(b) differ in detail, it is visually clear that they cluster around a common large-scale structure, present in both figures. Its nature is clearer in figure 5(c) which shows that it is a meandering stream that spans the full box. This is interesting for two reasons. The first one is that it suggests that the feature detected by the dipole template is not the pair of vortices, but the jet between them. The second one is that those jet segments are part of a larger stream, too large and too irregular to be represented by any local template, but which can be recovered by the concatenation of several of them.

Figure 6 shows how this large-scale flow organisation can be defined by thresholding the heat map below a given error level. The resulting points are collected into individual objects, defined by contiguity along the four directions of the coordinate axes. Figure 6(a) is the percolation diagram (Moisy & Jiménez Reference Moisy and Jiménez2004). The solid lines represent the fraction of the total thresholded area contained in the largest contiguous thresholded object. It is unity for very high thresholds, where a single object fills the whole field, and also for a lowest limit in which a single point represents both the whole thresholded region and its largest object. Neither limit gives information about the structure of the flow, and both are left outside figure 6(a), which centres on the intermediate range in which several individual low-error objects first appear and then merge into larger ones as the threshold is raised. The dashed lines are the number of individual objects, normalised to unit maximum. The percolation diagram is averaged over all the times of each simulation, and varies little among simulations. After some experimentation, the reference threshold is chosen to be $\varPhi _{q,th}=0.85$, which is used in figure 6(b) to threshold the map in figure 5(b). There is a dark largest connected object, and several smaller ones in a lighter colour, which are also below the error threshold but which are not connected to the largest one. They will not be used when compiling the statistics of the large flow scales. The percolation transition is narrow, and changing the threshold by any large amount moves the result into either an empty or a completely full map, but thresholds in the range 0.8–0.9 yield approximately the same results as those presented below.

Figure 6.

Definition of the largest thresholded structure in each snapshot. (a) Percolation diagram for $\varPhi _q\le \varPhi _{q,th}$, as a function of the threshold. Dipole templates optimised for $P_{q}$. Solid lines are the area of the largest thresholded object divided by the total thresholded area. Dashed lines are the number of objects, normalised to unit maximum. Individual lines are for each simulation, with symbols as in table 1. The vertical dashed line is the standard threshold used below. (b) Thresholded heat map from figure 5(b). The darker object is the largest connected structure, used in the following to represent the large-scale flow organisation, and the purple box is the template size.

It should be understood that heat maps and their thresholded versions are at most ‘skeletons’ of flow properties. Each point of the map is an element of the $N_t\times N_t$ ‘test’ grid used to test the template approximation properties. It marks the centre of a template box, but the optimum template size is generally wider than the spacing of the test grid, as shown in figures 5 and 6. Any geometric property of the skeletons should be interpreted with this in mind. For example, the solid lines in figure 7(a) show the inner ‘width,’ $\rho _1$, of the largest thresholded object in each frame, defined as the side of the largest square that completely fits within the object (Catrakis & Dimotakis Reference Catrakis and Dimotakis1996; Moisy & Jiménez Reference Moisy and Jiménez2004). The figure is compiled over a test grid with $N_t=20$, so that the minimum possible value is $\rho _1/L=0.05$. This is smaller than the widths in figure 7(a), which are of the order of $\lambda _{\omega 0}\approx 0.15$ (table 1), but close enough to it to recommend testing whether $\rho _1$ is influenced by the test grid. Limited testing with $N_t=30$ shows that the results in figure 7(a,b) could change by approximately 15–20 % on a much finer test grid.

Figure 7.

Properties of the largest thresholded structure of low $P_q$ in each snapshot. (a) Temporal evolution of the geometry of the largest structure. , Inner scale normalised with the initial enstrophy scale, $\rho _1/\lambda _{\omega 0}$; , area of the largest structure normalised with its initial value, $S/S_0$; symbols without lines are the integral length $L_{int}/\lambda _{\omega 0}$ for the correlations of $\varPhi _q$, defined in (3.2). Symbols as in table 1. Open symbols are vortex templates; closed symbols are dipoles. (b) Flow properties within the largest structures. , Kinetic-energy density; , enstrophy density. (c) As in (a), but unnormalised, versus the kinetic-energy wavelength.

The symbols without lines in figure 7(a) are the integral length,

(3.2)\begin{equation} L_{int}= \int_0^\infty C_{\varPhi\varPhi}(r) \,\textrm{d} r,\end{equation}

derived from the radial autocorrelation function of the approximation error, $\varPhi _q$, which typically measures the narrowest dimension of the structures of that variable. It is interesting that, even if we have seen that the approximation error is a marker for the largest, energy-containing scales in the flow, the integral length is narrower than the enstrophy peak in figure 1. In fact, its typical value, $0.3\lambda _{\omega 0}\approx 0.05 L$, is of the order of the spacing, $L/N_t$, of the test grid over which it is computed, and could actually be shorter in a finer grid. If we take $L_{int}$ to represent the average thickness of the structures of $\varPhi _q$, it would imply that the snapshot in figure 6(b), where the thickness of the structures is of the order of the test cell, is indeed typical.

Note that, since the length of the structures can be estimated by $S/\rho _1$, the fact that $S$ and $\rho _1$ grow at a similar rate in figure 7(a) implies that the length of the structures changes little as the flow evolves $(S/\rho _1L=1.3\to 1.5)$, but that their aspect ratio grows ‘fatter’ $(S/\rho _1^2=11\to 8)$. We will see below that the longitudinal scale of the structures is typically of the order of the box size, and, therefore, presumably limited by it.

Figure 7(b) shows kinetic energy and enstrophy averaged over dipole templates centred on points within the largest thresholded object in each snapshot. As suggested by figure 5(c), and by the definition of the dipole template in figure 3(b), dipoles contain locally high kinetic energy, but also moderately high enstrophy. Interestingly, enstrophy is less concentrated than the energy. It could be expected that, since the edge of a jet is necessarily the seat of high vorticity, the ‘fringe’ of grid points surrounding the large-scale thresholded streams would be regions of especially high enstrophy, but this is not true. When energy and enstrophy are conditioned to that fringe, the kinetic-energy density is lower than the average, and the enstrophy density is indistinguishable from the mean (not shown). A similar result holds for other flow regions outside the streams. The only high-energy regions are apparently those detected by the dipole template, including the lighter-grey regions in figure 6(b), and the only moderately high enstrophy regions are also, on average, associated with the streams.

Perhaps the strongest indication of the connection of the streams with the structure of the kinetic energy is figure 7(c). The quantities in figure 7(a) are normalised with their initial value because they do not otherwise collapse. The initial conditions, which are chosen to provide a variety of scale combinations, are too different to allow it. For example, the average value of the area of the largest stream at $t=0$ varies by 60 % among the different simulations. On the other hand, figure 7(c) shows the unnormalised area and thickness of the large streams versus the energy wavelength, $\lambda _q$. They collapse well, strongly suggesting that the streams are the support of the kinetic energy of the flow, and that the process of stream formation is tantamount to the flux of the energy to larger scales.

Figure 8 tests whether the structures detected by the concatenation of dipoles are elongated and aligned with the flow velocity, as suggested by figure 5(c). Consider the sketch in figure 8(a). For a given flow snapshot and template size, each cell in the $N_t\times N_t$ test grid has an associated flow velocity, ${\boldsymbol u}$, and an approximation error, $\varPhi$, defined by averaging over the domain of the optimal template centred on it. Choosing a displacement vector ${\boldsymbol r}$ with respect to this point, figure 8(b,c) shows the average of the approximation error corresponding to the cell whose centre is closest to the end of ${\boldsymbol r}$. The figure shows the mean error conditioned to the magnitude of ${\boldsymbol r}$ and to its angle, $\angle _{\boldsymbol{\textit{ur}}}$, with respect to ${\boldsymbol u}$. When this conditional mean is averaged over all the points of the test grid, the result is essentially independent of the angle, and similar to the unconditional mean of the error. But, when the conditional centre is chosen within the largest low-error structure in each flow field, figure 8(b) shows that points aligned with the velocity, $\angle _{\boldsymbol{\textit{ur}}}=0$ or $\angle _{\boldsymbol{\textit{ur}}}={\rm \pi}$, preferentially contain low approximation errors, while those perpendicular to it have high ones. This effect weakens with the length of $r$, but figure 8(c) shows that it persists for a distance of the order of $0.25 L$, which is four or five times longer than the width $\rho _1$ in figure 7(a). It is interesting that the distance at which the minimum error in figure 8(c) reverts to its unconditional value is approximately the same for the four cases included in the figure. In fact, it changes little among all the cases tested, showing that the longitudinal scale of the high-velocity streams is always of the order of the box size. The growth of the energy wavelength, $\lambda _q$, in figure 1(b) and of the area of the high-energy region in figure 7(a) is presumably due to more convoluted streams, rather than to longer-range ones.

Figure 8.

Mean approximation error conditioned to the orientation with respect to the velocity. (a) Definition sketch. (b) Mean error as a function of the orientation angle. T1024 at $\omega '_0 t=39$. From bottom to top, distance to the reference point: $r/L=0.1\ (0.05)\ 0.25$. (c) Streamwise and transverse conditional errors as functions of the distance to the reference point. Symbols as in table 1, with each case at its final simulation time. Closed symbols are measured aligned with the velocity, and open ones are measured at right angles to it. The dashed horizontal lines in (b,c) are unconditional errors.

An intriguing feature of figure 8(b) is the asymmetry between the conditional error at $\angle _{\boldsymbol{\textit{ur}}}=0$ and $\angle _{\boldsymbol{\textit{ur}}}={\rm \pi}$. Since each point in the low-error structures is both the origin and the end of some conditioning vector, both directions could be expected to be equivalent. But they are not, and the error is lower in front of the conditioning point than behind it. The effect is small but statistically significant; the estimated standard deviation of the lines in figure 8(b) is only slightly larger than the width of the lines. It is also consistently found in all the cases examined, not only in the one chosen for the figure. This result is difficult to interpret, but the implication is that the direction of the velocity is a better predictor of the downstream location of the low-error structures than of their upstream location. We will come back to this point in § 4.2.

3.2. Vortices

Even if this paper is mostly concerned with the large energy-containing structures of the flow, there is no doubt that two-dimensional turbulence can also be described as a collection of coherent vortices (McWilliams Reference McWilliams1984, Reference McWilliams1990a; Benzi et al. Reference Benzi, Colella, Briscolini and Santangelo1992). The question that interests us here is whether the large-scale structure discussed above for the kinetic energy can be described in terms of the organisation of these vortices. In this section we first address the properties and evolution of the vortices themselves.

Figure 9(a) shows a segmentation of a typical flow field into individual vortices, defined as connected regions in which $|\omega |\ge H\omega '$. As in the case of figure 7, the vorticity of the flow separates into a few large connected objects for $H\ll 1$, and breaks into more numerous smaller objects as $H$ increases. Beyond a certain threshold, the number of vortices decreases again, and eventually vanishes when no vorticity satisfies the thresholding condition. The value $H=0.9$ used in figure 9 is chosen to maximise the number of individual vortices (Moisy & Jiménez Reference Moisy and Jiménez2004). To gain some sense of the importance of vortex interactions, the vortices in figure 9(a) are grouped into co- and counter-rotating pairs. Two vortices are considered a potential pair if their area, $s$, differs by less than a factor of $m^2$, which is an adjustable parameter. The underlying rationale for this restriction is that very dissimilar vortices are unlikely to form long-lived pairs, because the larger one would tear the smaller one apart (Meunier, Le Dizès & Leweke Reference Meunier, Le Dizès and Leweke2005). The figure uses $m=2$, but statistics compiled with $m=1.5$ and $m=3$ show no substantial differences (see Jiménez Reference Jiménez2020b). Vortices are paired to the closest unpaired neighbour within their area class, and no vortex can have more than one partner. Some vortices find no suitable partner, and are left unpaired.

Figure 9.

Properties of the thresholded vorticity structures. (a) Typical segmented image. Case T1024, $\omega '_0t=39$. In all the panels in this figure, unless otherwise noted, red are dipoles, blue are co-rotating pairs and green are isolated vortices. Other symbols as in table 1. (b) The open symbols are the mean diameter of the vortices, and the closed coloured ones are the distance between vortices in a pair. Lengths are normalised with the vorticity wavelength at $t=0$, and the trend lines are the polynomial fits from figure 1(b) to: , $0.3\lambda _\omega$; , $0.22 \lambda _q$. (c) Area fraction covered by all the vortices of a given class. Open symbols include all the classes. (d) Number fraction of vortices in different associations. (e) Probability density function of the distances between vortices in a pair. Colours as in (a), but the chain-dotted black line is the distribution for a Poisson point set. (f) Mean vorticity of the vortex cores. Symbols in (f–i) are as in (c). (g) Average circulation of the vortex cores. (h) Enstrophy fraction contained in the different vortex associations. (i) As in (h), for the kinetic energy.

Figure 9(b) displays mean values of the diameter, $s^{1/2}$, and of the distance $d$ between the centres of gravity of the component vortices of the pairs, compiled at several evolution times for each set of simulations. The diameter of the vortices depends very little on how they are paired, and, in agreement with the estimations from the vorticity and velocity statistics in figure 1, is described well by a fraction of the vorticity wavelength, $0.3 \lambda _\omega$. The distance between the vortices of a pair is initially somewhat larger, of the order of $0.5 \lambda _\omega$, but it grows with $\lambda _q$, faster than $\lambda _\omega$. The approximate similarity between the inter-vortex distance and their diameter implies that vortex pairs remain tightly packed at this stage of their evolution, although the faster growth of the average distance means that the area fraction covered by the vortices slowly decays (figure 9c). It should be noted at this point that the absolute value of the intra-pair distance in figure 9(b) offers a possible explanation of the observation in § 3.1 that the edges of the high-speed streams are not a concentration of high enstrophy. It follows from comparing figures 7(a) and 9(b) that $d$ is 2–3 times narrower than the width $\rho _1$ of the jets, so that, if these jets are defined as in figure 7, the dipoles are contained within them, rather than at their border. This is consistent with the visual inspection of the snapshot in figure 5(b).

Figure 9(d) shows the number of vortices involved in different kinds of pairings, giving a rough measure of the importance of the different interactions. Most vortices are in the form of pairs. Of the approximately $2\times 10^6$ vortices represented in figure 9, 43 % form dipoles, 29 % are in co-rotating pairs and 28 % are isolated, with a tendency of the number of dipoles and co-rotating pairs to converge as the Reynolds number increases. Similar values were found by Jiménez (Reference Jiménez2020b) at the lowest of the four Reynolds numbers used in the present paper. The difference between the number of co-rotating and counter-rotating pairs is also encountered in the covering fraction in figure 9(c), and is a property of the flow that disappears if the vortex position is randomised. On the other hand, the scarcity of unpaired vortices is a geometric property that persists when the pairing algorithm is applied to a set of Poisson-distributed points.

In fact, a random distribution of the vortex position is a reasonable lowest-order model for their local organisation. Even if the vortex diameter and the intra-pair distance grow by a factor of approximately two during the simulation time, the form of their p.d.f. stays remarkably constant. For example, figure 9(e) shows the p.d.f. of the distance among the components of vortex pairs, normalised by their mean for each individual experiment. The figure also includes the p.d.f of a set of Poisson-distributed points with the same mean. The distribution of dipoles and co-rotating pairs are very similar, with a weak tendency of dipoles to be farther apart. The Poisson distribution is wider, but most of the discrepancy can be explained by the exclusion of pairs whose separation is smaller than the vortex diameter (see appendix A). This explanation is confirmed by the distribution of the distance $d$ between all possible couples of vortices (not shown). This distribution should be proportional to $d$ for any homogeneous flow, but this is only true for $d \gtrsim 2\langle s^{1/2}\rangle$. Shorter distances are essentially missing.

In the same way, there is relatively little difference between the properties of vortices in dipoles and those in co-rotating pairs. We saw in figure 9(b) that their diameters are similar, and figure 9(f) shows that so are their vorticities, which decay at the same rate as the r.m.s. vorticity of the flow. On the other hand, some explanation is needed for figure 9(c,d), which shows that both the number of dipoles and the total area covered by them is larger than those of co-rotating pairs. A similar difference appears in figure 9(g), which shows that the average vortex circulation magnitude grows linearly for all classes. The circulation of a vortex core can only grow by merging with other cores or by entraining background vorticity, since there is no vorticity source in two dimensions, but the average over a class is also influenced by the transfer among classes. Figure 9(g) shows that the circulation of dipoles is typical of the overall average (they are the largest contributors to it), but that co-rotating pairs grow more slowly, while unpaired vortices grow slightly faster than the average. The simplest explanation is that co-rotating pairs tend to merge into single (initially unpaired) cores (Meunier et al. Reference Meunier, Le Dizès and Leweke2005), thus depleting their number, while dipoles are longer lasting (Flierl et al. Reference Flierl, Larichev, McWilliams and Reznik1980; McWilliams Reference McWilliams1980).

Figure 9(h) shows the fraction of overall enstrophy contained in the thresholded vortices. Over 85 % of the total enstrophy is contained in them, and this fraction is remarkably constant among Reynolds numbers and time, no doubt, in part, because the vorticity threshold used to identify them is a constant fraction of the r.m.s. vorticity. The contribution to the enstrophy of the different vortex classes is in line with the area fraction in figure 9(c), as could be expected from the similarity of vortex properties discussed above.

The fraction of the kinetic energy due to the vortices is harder to define. The simplest definition is the fraction of $q'^2$ retained by a flow reconstructed from the vorticity contained within a given class of thresholded vortices. Figure 9(i) shows that keeping all the thresholded vortices retains 65–70 % of $q'^2$, while keeping only the dipoles or the co-rotating pairs retains 25–30 % of the energy, and keeping only the unpaired vortices retains 30–40 %. The change in the relative contribution of dipoles to the enstrophy in figure 9(h) and to the energy in figure 9(i) is interesting. While dipoles predominate in number, area and enstrophy, their contribution to the energy is of the same order as the co-rotating pairs, and substantially less than the unpaired vortices. In fact, even if dipoles contain a local jet of high velocity, their overall kinetic energy is lower than for co-rotating pairs, because the total circulation of a dipole is zero, and its induced velocity falls with distance much faster than for a co-rotating pair (Batchelor Reference Batchelor1967). Even if decaying turbulence is very far from an equilibrium system, Benzi et al. (Reference Benzi, Colella, Briscolini and Santangelo1992) and Dritschel et al. (Reference Dritschel, Scott, Macaskill, Gottwald and Tran2008) have shown that an approximately Hamiltonian system of point vortices, punctuated by the occasional merger of like-signed vortices, is a good approximation to the late stages of two-dimensional enstrophy decay. The difference in the interaction energy of dipoles and co-rotating pairs is probably also part of the reason for their different behaviour, and for the slower decay of the former through amalgamation.

In summary, the vortex evolution discussed in this section is consistent with the description of decaying two-dimensional turbulence as a system of discrete vorticity structures (McWilliams Reference McWilliams1984; Benzi et al. Reference Benzi, Patarnello and Santangelo1987; Brachet et al. Reference Brachet, Meneguzzi, Politano and Sulem1988), although, at the relatively early stage of the decay studied here, the structures cannot be described as either equilibrium or isolated. Their self-similar growth through amalgamation accounts for the gradual increase in the vorticity scale, $\lambda _\omega$, but the sizes involved are always much smaller than the energy scale, $\lambda _q$ (see the scale bars in figure 2).

It is interesting that the Reynolds number of the individual vortex cores stays relatively low during the simulations, $Re_\gamma =\langle \gamma \rangle /4{\rm \pi} \nu \approx 1.5 \to 7$. Since the maximum vorticity of a viscously decaying circular vortex satisfies $\omega t =Re_\gamma$ (Batchelor Reference Batchelor1967), where the left-hand side is time measured in turnovers, it follows that the size of the vortices in the simulations, even as they grow by accretion, is typically controlled by viscosity.

4.1. Vortex organisation

In fact, the question of how vortices organise themselves to create the streams discussed in § 3.1 remains open. One possibility, already mentioned, is that the streams are concatenations of dipoles that are responsible for short segments of the stream. We saw in the discussion of figure 9 that this model is compatible with the observed vortex distances and dimensions, but the open question is how the individual dipoles align themselves into longer units. Another model is that the streams are contact interfaces between large-scale vortices. We saw in figure 9(b) that the mean vortex diameter increases only weakly during even the longest flow evolution, and that it is always much smaller than the kinetic-energy wavelength, but it is possible that the small cores organise themselves into large-scale vortex ‘bags’ that fill the space between the streams, acting as coherent structures from the point of view of the kinetic energy (Paret & Tabeling Reference Paret and Tabeling1998; Tabeling Reference Tabeling2002). However, we saw in § 3.1 that the only concentration of vorticity is within the coherent jets, not between them.

The crucial uncertainty is the intensity of the interaction among vortex cores, and whether, for example, the advection velocity of the cores is mainly due to their closest neighbour, or to a background of ‘field’ vortices. The former would support the first of the two models above, while the latter would support the second. Consider vortex pairs. The result of the mutual induction among two vortices of the same sign and similar circulation is a rotation around each other. If the vortices are denoted by $A$ and $B$, and we estimate the (vector) ‘mobility’ of a vortex by averaging the flow velocity over its core,

(4.1)\begin{equation} {\boldsymbol u}_{vor}=s_{vor}^{-1}\int_{vor} {\boldsymbol u} \,\textrm{d} s, \end{equation}

the induced mobilities of the components of a co-rotating pair would be ${\boldsymbol u}_A= -{\boldsymbol u}_B$, and the velocity of the centre of the pair would vanish, ${\boldsymbol u}_{cg}=({\boldsymbol u}_A+{\boldsymbol u}_B)/2=0$. On the other hand, dipoles self-induce a common translation velocity, and ${\boldsymbol u}_A={\boldsymbol u}_B={\boldsymbol u}_{cg}$. If we define $q_{vor}= \|{\boldsymbol u}_{vor}\|$, a self-inducing co-rotating pair would be characterised by $q_{cg}\ll q_{pair}\equiv (q_A+q_B)/2$, while a dipole would satisfy $q_{cg}\approx q_{pair}$. This is tested in figure 10(a), and it is only satisfied by slow-moving vortices. The behaviour of fast pairs for which $q\gtrsim q'$ is independent of whether they are co- or counter-rotating, and they can therefore be assumed to be mostly advected by a background velocity field.

Figure 10.

Properties of the vortex pairs. (a) Joint p.d.f. of the velocity of the centre of gravity of a vortex pair versus the averaged velocity magnitude of its two component vortices. In all the panels in this figure: , T256; , T512; , T768; , T1024. Red lines are dipoles, and black ones are co-rotating pairs. The two probability contours in each case enclose 50 % and 95 % of the probability mass. (b) Mean circulation magnitude of the vortex components of the pair, versus the inter-component distance. (c) Vortex circulation versus vortex mobility.

Figure 10(b) shows that vortex pairs can be classified into two groups. The ‘nose’ extending to the lower right in the figure represents a family of strong cores with large circulations, whose intra-pair distance is relatively small. This family exists for co-rotating pairs and for dipoles, although it is most marked for the latter. The vertical band to the left of the figure contains relatively weak vortices with no clear preference for a particular coupling distance. Most vortex pairs are in this latter family, but they are relatively unimportant for the flow. Approximately 66 % of the cores have $|\gamma |<\langle |\gamma |\rangle$, but they only contain 15–25 % of the total circulation magnitude. A similar distinction can be based on vortex area (not shown), since the mean vorticity is relatively uniform among the vortices. The classification into vortex types can be based on either property.

One could hypothesise that the large, tightly coupled vortices in the ‘nose’ family would be the ones with the fastest mobilities, which they would induce on each other, but figure 10(c) shows that the opposite is true. The vortices with the largest circulations move relatively slowly, and the high mobilities tend to be associated with the weak circulations to the left of the figure. This somewhat surprising observation leads to a model in which a set of organising large vortices in an ‘approximate equilibrium’ configuration are responsible for organising the flow into streams where weaker vortices are advected at relatively high speed.

In fact, the large and small vortices have very different properties. Figure 11(a,b) shows the one-dimensional p.d.f. of the core circulation, which collapses reasonably well for all the simulations and evolution times. Figure 11(a) shows that the weak vortices, $|\gamma |< \langle |\gamma |\rangle$, follow a power-law distribution $P(|\gamma |)\sim |\gamma |^{-3/4}$, while figure 11(b) shows that the vortices above that limit follow an exponential one. The separation of decaying two-dimensional turbulence into coherent vortices evolving under mutual induction, and a ‘chaotic’ background has been discussed often (Benzi et al. Reference Benzi, Patarnello and Santangelo1987; Benzi, Patarnello & Santangelo Reference Benzi, Patarnello and Santangelo1988; McWilliams Reference McWilliams1990b), but the background is usually not characterised in term of vortices, and we are not aware of any previous characterisation of the larger vortices as slowly moving. For example, Benzi et al. (Reference Benzi, Patarnello and Santangelo1987) discuss the large vortices as the only coherent structures in the flow and report that their areas follow a power-law distribution, although with a different exponent than the one above, $|\gamma | ^{-3/2}$. The reason for this difference is not clear, but we mentioned in § 2 that Benzi et al. (Reference Benzi, Patarnello and Santangelo1987) study a later stage of the decay, and that they use a higher-order viscous dissipation, which presumably creates different vortex cores (McWilliams Reference McWilliams1990b; Jiménez Reference Jiménez1994). It is also unclear why the exponential range of the distribution in figure 11(b) is not discussed by Benzi et al. (Reference Benzi, Patarnello and Santangelo1987), although this may be partly due to their different sample size. The analysis in their paper is based on 17 vortices, while each of the distributions in figure 11 represents $10^4$–$10^5$ objects.

Figure 11.

(a) Probability density function of the vortex circulation magnitude. In all cases, time increases from blue to red. The dashed line is proportional to $|\gamma |^{-0.75}$. (b) As in (a). The dashed line is proportional to $\exp (-0.666 |\gamma |/ \langle |\gamma |\rangle )$. (c) Kinetic energy contained in different vortex classes versus mean vortex mobility. Open symbols are all the thresholded vortices; red symbols are large vortices, $s_{vor} \ge \langle s_{vor}\rangle$; blue symbols are $s_{vor} < \langle s_{vor}\rangle$. The red arrow is the direction of time advance for the red symbols.

Figure 11(c) shows that the kinematics of the weak and strong cores is very different. The vertical axis in this figure is the fraction of the kinetic energy carried by the thresholded vortices, defined as in figure 9(i). The open symbols are the contribution from all the vortices, as in that figure, and the red symbols are the contribution from vortices whose area is larger than the average. They contain most of the kinetic energy. The blue symbols in the lower-right corner are the contribution from vortices smaller than the average, which is much smaller. The horizontal axis is the average vortex mobility defined as the modulus of (4.1). The large and small vortices lie in very different parts of the plot, as already suggested by figure 10(c). Large vortices are responsible for most of the kinetic energy of the flow, but are themselves relatively immobile, while small ones move fast, but are only responsible for a small fraction of the kinetic energy.

The power-law and exponential probability distributions suggest that the cores grow by two different aggregation mechanisms of smaller units. While a power law implies self-similar scale-free growth, in which cores merge with other cores of similar size (Benzi et al. Reference Benzi, Colella, Briscolini and Santangelo1992), an exponential has a definite scale, which is proportional to the mean of the distribution, and to the size of the elements being accreted (Jiménez & Kawahara Reference Jiménez, Kawahara, Davidson, Kaneda and Sreenivasan2013). In figure 11 the lower limit of the exponential is indeed of the order of $\langle |\gamma |\rangle$, and it is interesting to speculate about an aggregation model in which cores merge self-similarly with each other until they grow to be large enough to ‘freeze’ in a quasi-equilibrium slowly evolving pattern. The motion of these large vortices is not chaotic, at least over short times, and these cores stop merging among themselves. But they keep absorbing the remaining fast-moving vortices of the background, and the largest of these field vortices determine the scale of the exponential distribution.

Over a much longer time scale, it is to be expected that even the ‘frozen’ vortices would merge among themselves, in an amalgamation process similar to the one described by Carnevale et al. (Reference Carnevale, McWilliams, Pomeau, Weiss and Young1991, Reference Carnevale, McWilliams, Pomeau, Weiss and Young1992), but the appearance of a collective equilibrium signals that the energy scale has become of the order of the box size. As mentioned in § 2, this is the limit of our simulations, and of the analysis in this paper.

Vortex arrangements that remain stationary in some frame of reference have been studied for over a century (see the review in Aref et al. Reference Aref, Newton, Stremler, Tokieda and Vainchtein2002). Some of them are stable, and form spontaneously in experiments. In particular, forced two-dimensional turbulence is known to settle to stationary vortex ‘crystals’ which are partly determined by the forcing method and by the boundary conditions (Fine et al. Reference Fine, Cass, Flynn and Driscoll1995; Jin & Dubin Reference Jin and Dubin2000; Jiménez & Guegan Reference Jiménez and Guegan2007), and beautiful examples of equilibrium vortex polygons have been observed in the polar regions of planetary atmospheres (Tabataba-Vakilia et al. Reference Tabataba-Vakilia, Rogers, Eichstädt, Orton, Hansen, Momary, Sinclair, Giles, Caplinger and Ravine2020). Most known equilibrium systems are regular arrangements of vortices of a single sign in a background of opposite-sign vorticity, but mixed-sign stable systems are also known. The von Kármán vortex street is probably the best known example of the latter, and it is known that two-dimensional turbulence in a square box converges to a quasi-equilibrium single dipole, arranged diagonally in the box, which only decays slowly by viscosity (Smith & Yakhot Reference Smith and Yakhot1993).

The circulation of the slow-moving vortices discussed here is essentially in balance (with a residue of 1 %–3 % of $\omega '$), but they are still far from equilibrium, and may perhaps be considered an intermediate stage to a final steady state. Attempts to extract a regular arrangement for them failed, beyond the four-way symmetry induced by the computational box, but the drop in mobility can be considered diagnostic of incipient ‘crystallisation.’

4.2. The up-scale energy flux

The discussion in the previous section raises the question of how the dipoles get organised into long streams. Three examples are given in figure 12. These examples were chosen for didactic purposes to demonstrate the aggregation process, and are not truly random. But they are fairly representative. Of the several thousand simulations available, approximately 30 % were found visually to display an initial evolution similar to those in the figure. Three of those were chosen at random from about 50 ‘good’ cases. Each row in the figure is a simulation, displayed at four approximately equidistant times, which are the same for the three cases. Vortices are represented in colour, according to their sign, and the arrows are the velocities. The intensity of the grey background is the velocity magnitude. To best display the evolution, only the largest vortices $(s>\langle s\rangle )$ are included in each plot, and the Reynolds number is purposely chosen low.

Figure 12.

Three examples of the organisation of vortices into streams. Time runs in each row from left to right, $\omega _0't=0$, 2, 4.6, 6.5. Case T256. The three rows are independent realisations, and only vortices with $s>\langle s \rangle$ are included in the figure. Line contours are positive (red) and negative (blue) vorticity, the arrows are velocity, and the grey background is the velocity magnitude.

Each simulation starts with a relatively disorganised arrangement of vortices but, at the end of each sequence, positive (red) vortices have sorted themselves to one side of the flow, and negative (blue) ones to the other, supporting a jet between them. Some merging of like-signed vortices takes place in all cases.

The question is how this happens, because, while continuity probably implies that the velocity of any elongated velocity structure should be aligned with its axis, the opposite is not true. Compact jet-like vortex dipoles with aspect ratios of order one (modons) are well-known stable solutions of the Euler and Navier–Stokes equations (Flierl et al. Reference Flierl, Larichev, McWilliams and Reznik1980; McWilliams Reference McWilliams1980).

Simulations of point vortex systems (not shown) spontaneously form tight dipoles and co-rotating pairs, but the dipoles do not organise into trains or jets, as is the case in figure 12(a,b). Neither do they form stronger dipoles of vortex ‘clouds,’ as in figure 12(c). When a point dipole collides with a solitary point vortex or with a co-rotating pair, it often loses one of its vortices, possibly breaking the colliding couple and forming a new association. Very seldom the resulting arrangement involves more than two vortices.

Of course, sets of point vortices are Hamiltonian systems whose interactions conserve energy (Batchelor Reference Batchelor1967). Both the merging of like-signed vortices and the breaking of an existing dipole involve energy exchanges, which are much simpler if viscosity or filamentation can be used as an energy dump. This would statistically favour the formation of the lower-energy dipoles, but any such selection criterion requires a local mechanism to implement it. In particular, it is unclear why a positive vortex being overtaken by a dipole would tend to reinforce the positive component of the dipole, strengthening it, rather than merging with the negative one, weakening it.

A possible mechanism is explained in figure 13. Consider the point vortex dipole in figure 13(a). In the comoving frame of reference, it forms a recirculation bubble separated from infinity by an approximately elliptical dividing streamline (see appendix B). Any sufficiently weak point vortex being overtaken by the dipole follows the streamlines around the bubble, independently of its sign, and it is eventually left behind. There is no preference for which side of the dipole its path takes, and it is therefore unlikely to statistically strengthen or weaken it.

Figure 13.

(a) Velocity field of a dipole of equal point vortices of circulation $\pm \gamma _0$ at distance $2H$. The dipole is moving to the right, but is shown in the frame of reference linked to the vortices. The blue lines are streamlines, as well as the trajectories of an advected point vortex. The black lines are the dividing streamline in this frame of reference. (b) As in (a), but the blue lines are the trajectories of the centre of gravity of a vortex patch of positive circulation $\gamma$ and area $s$, such that $\gamma _0 s^2/\gamma H^4=2$. A patch with negative circulation would be entrained to the negative vortex in the dipole. Only trajectories coming from large positive $x$ are included. See text and appendix B for details.

The situation is different for the entrainment of an extended vortex, as detailed in appendix B for the case of a uniform vortex patch. Such patches drift with respect to the advecting streamlines, as shown by the trajectories in figure 13(b). Positive patches drift towards the positive component of the dipole, and negative ones towards the negative component. The result is an average strengthening of the dipole, and the formation of organised jets.

Note that this accretion model could explain the asymmetry observed in figure 8, where it was shown that the direction of the velocity of a stream is a better predictor of the direction taken by the stream ahead than behind the position at which it is measured. In the model presented here dipoles propagate forward, and take some time to incorporate new vorticity. The jet of a growing dipole is strongest at its trailing edge, and, therefore, predicts better the position of the leading dipole ahead of it. Examples of this asymmetry between the front and back of dipoles can be seen in figures 12(a) and 12(c), where a growing head dipole is followed by a trailing jet of elongated filaments that have been created during their interaction with the dipole.

On the other hand, although accretion explains why existing dipoles become stronger, and thus how they segregate into classes, it does not explain how they organise into longer streams. Several examples of both processes can be seen in figure 12. For example, the flow in figure 12(c) organises into a single strong dipole but not into a long stream. Figure 12(b) forms a horizontal leftwards jet in the middle of the frame, but none of its dipoles is especially strong, and the same is true of the oblique jet forming in the lower-right corner of figure 12(a). The growth of the longitudinal scale in a set of two-dimensional vortices first came under consideration when modelling the spreading of free shear layers in Brown & Roshko (Reference Brown and Roshko1974). Two mechanisms were proposed at the time. In ‘vortex pairing’ two vortices of the same sign merge into a larger one (Winant & Browand Reference Winant and Browand1974), while in ‘vortex tearing’ a weaker vortex between two stronger ones is strained into a filament that is eventually entrained or dissipated (Moore & Saffman Reference Moore and Saffman1975). The accretion model developed above can be considered a case of pairing, and a clear tearing can be seen between the second and third frames of figure 12(b). Both processes are relevant in experimental shear layers (Hernán & Jiménez Reference Hernán and Jiménez1982), but that flow involves vorticity of a single sign, and the scale growth mostly takes place along a single axis. Repeating a similar analysis in the present mixed-sign isotropic situation would involve a more thorough processing than is possible here, both theoretically and observationally.

The interaction of fast-moving dipoles with smaller vortices, akin to tearing, has been invoked by Dritschel et al. (Reference Dritschel, Scott, Macaskill, Gottwald and Tran2008) as an important step of the forward enstrophy cascade. It takes place when a third vortex is shredded by a dipole, but, to our knowledge, such interactions have not been connected with the up-scale cascade. In a related observation, random forcing of two-dimensional turbulence induces a self-similar enstrophy cascade, but the cascade disappears (to form vortex crystals) when the forcing is partially deterministic, and the character of the flow again changes when the forcing is fully deterministic, in which case the flow becomes a ‘dilute gas of dipoles’ that cleans most of the background low-level vorticity (Jiménez & Guegan Reference Jiménez and Guegan2007).