2 Proper labelling

The result of applying the above strategy to a given flow field is its segmentation into subsets labelled as more or less significant for some particular application, and we will refer to labellings that are optimal in this sense as ‘proper’. We consider the following algorithm.

1. (PL1) Construct a reference initial flow field, $g_{ref}(\boldsymbol{x},t=t_{ref})$ , and choose a target time, $t=t_{ref}+T$ .

2. (PL2) Create a test field $\widetilde{g}$ by substituting $g_{ref}$ by something else over a subset $A$ covering a fixed fraction of the domain volume, e.g. $\widetilde{g}(A,t=t_{ref})=0$ .

3. (PL3) Evolve both $g_{ref}$ and $\widetilde{g}$ to $t=t_{ref}+T$ , using the equations of motion.

4. (PL4) Iterate over $A$ , and label as most significant the subset maximising (or minimising) the magnitude of the resulting perturbation, $\unicode[STIX]{x1D700}(T)=\Vert \widetilde{g}(t_{ref}+T)-g_{ref}(t_{ref}+T)\Vert$ .

This definition is ambiguous in several respects, the most obvious being the freedom of how to modify the test field in step PL2. This freedom is not absolute, because the resulting flow field has to satisfy, for example, continuity, but the way in which such constraints are enforced only adds choices to the procedure. Another choice is the perturbation norm in PL4. We use in this paper the $L_{2}$ norm of the velocity field, but enstrophy or some measure of scalar mixing could be equally valid, and would probably give different results.

Another question is whether to consider as significant the subsets resulting in the maximum or the minimum deviation. Both cases are interesting, because one of the questions to be answered is whether there really exist parts of the flow that are more significant than others. The ratio between the two extreme deviations, $R_{\unicode[STIX]{x1D700}}=\unicode[STIX]{x1D700}_{max}/\unicode[STIX]{x1D700}_{min}$ , is a figure of merit that characterises the existence of significant structures, and therefore the coherence properties of the flow field as a whole. Finally, the above algorithm labels a single reference initial condition. The most interesting questions depend on applying it to many such cases, and abstracting from the results the statistical characteristics that define significant subsets.

The above procedure is related to the Lyapunov analysis of dynamical systems, whose aim is to identify the maximum growth rate of infinitesimal perturbations. However, it differs from it in three important aspects. In the first place, our perturbations are not infinitesimal, since they are intended to model the nonlinear evolution of turbulence. The mathematical theory for finite-amplitude perturbations is not as developed as for infinitesimal ones (Eckmann & Ruelle Reference Eckmann and Ruelle1985), but they have been used to characterise predictability (Cencini & Vulpiani Reference Cencini and Vulpiani2013). Second, Lyapunov analysis searches for properties of the dynamical system in the long-time limit, while we are interested in labelling significant regions of individual realisations after a finite evolution time. Any information is thus linked to a particular flow field and a given time horizon, and generic properties only arise after ensemble averaging over many realisations (Ding & Li Reference Ding and Li2007). Lastly, and most importantly, Lyapunov analysis seeks to identify optimal perturbations, while we restrict ourselves to perturbations within the prescribed support of the subset $A$ . This is required to represent individual structures, of which several may exist in any given flow realisation, but the perturbation growth is necessarily suboptimal.

Figure 1.

(a) Premultiplied vorticity spectra of the simulations at the two times used as references. The wavelength, $\ell =2\unicode[STIX]{x03C0}/k$ , is defined from the wavenumber magnitude $k$ ; ——, $t=0$ ; – – –, $\unicode[STIX]{x1D714}_{0}^{\prime }t=11$ ; – ⋅ – ⋅ –, $\unicode[STIX]{x1D714}_{0}^{\prime }t=26$ . The line with symbols is the premultiplied energy spectrum at $\unicode[STIX]{x1D714}_{0}^{\prime }t=11$ . (b) Velocity and vorticity magnitude of the reference field at $\unicode[STIX]{x1D714}_{0}^{\prime }t=11$ . The overlaid grid is used to select the test subsets. (c) As in (b), for $\unicode[STIX]{x1D714}_{0}^{\prime }t=26$ .

3 Application to two-dimensional turbulence

To illustrate the procedure, we apply it to the relatively simple case of identifying the most and least significant features of decaying isotropic two-dimensional turbulence. Simulations are performed in a doubly periodic square box of side $L=2\unicode[STIX]{x03C0}$ using a standard spectral Fourier code dealiased by the $3/2$ rule. Time advance is third-order Runge–Kutta. The flow is defined by its velocity field $\boldsymbol{u}=(u,v)$ as a function of $\boldsymbol{x}=(x,y)$ . The one-component vorticity is $\unicode[STIX]{x1D714}=\unicode[STIX]{x2202}_{x}v-\unicode[STIX]{x2202}_{y}u$ and the rate of strain tensor is $\unicode[STIX]{x1D634}_{ij}=(\unicode[STIX]{x2202}_{i}u_{j}+\unicode[STIX]{x2202}_{j}u_{i})/2$ , where the subindices of the partial derivatives range over $(x,y)$ and those of the velocity components over $(u,v)$ . Its magnitude is defined by $S^{2}=\unicode[STIX]{x1D634}_{ij}\unicode[STIX]{x1D634}_{ij}$ , where repeated indices imply summation. The natural time and velocity scales are respectively the root-mean-square vorticity magnitude $\unicode[STIX]{x1D714}^{\prime }=\langle \unicode[STIX]{x1D714}^{2}\rangle ^{1/2}$ , where the time-dependent average $\langle \cdot \rangle$ is taken over the full computational box, and $q^{\prime }=(u^{\prime 2}+v^{\prime 2})^{1/2}$ . A Taylor length scale can be defined as $\unicode[STIX]{x1D706}=q^{\prime }/\unicode[STIX]{x1D714}^{\prime }$ .

The flow is initialised with a fixed spectrum and random Fourier phases, and allowed to evolve until structures have established themselves (figure 1). To investigate the scaling of the different quantities, two initial reference times are used, $\unicode[STIX]{x1D714}_{0}^{\prime }t_{11}=11$ and $\unicode[STIX]{x1D714}_{0}^{\prime }t_{26}=26$ , where the subindex denotes the moment at which a variable is measured. As the simulation proceeds, the vorticity decays to $\unicode[STIX]{x1D714}_{11}^{\prime }/\unicode[STIX]{x1D714}_{0}^{\prime }=0.67$ and $\unicode[STIX]{x1D714}_{26}^{\prime }/\unicode[STIX]{x1D714}_{0}^{\prime }=0.47$ . The kinetic energy decreases more slowly, and the Taylor scale grows from $\unicode[STIX]{x1D706}_{0}/L=0.03$ to $\unicode[STIX]{x1D706}_{11}/L=0.04$ and $\unicode[STIX]{x1D706}_{26}/L=0.05$ . Comparing these values with the vorticity fields in figure 1(b,c) and the spectra in figure 1(a), $\unicode[STIX]{x1D706}$ corresponds to the thickness of the thinnest vorticity filaments, and is a few times smaller than the structures containing most of the enstrophy. The kinematic viscosity $\unicode[STIX]{x1D708}$ is set to $q_{0}^{\prime }L/\unicode[STIX]{x1D708}=2500\,(q_{11}^{\prime }\unicode[STIX]{x1D706}_{11}/\unicode[STIX]{x1D708}\approx 90)$ and the simulation uses $85^{2}$ complex Fourier modes. The numerical resolution expressed in terms of Fourier modes improves from $\unicode[STIX]{x0394}x/\unicode[STIX]{x1D706}=0.14$ at $t_{11}$ to 0.11 at $t_{26}$ , and there are approximately eight grid points across the smallest vorticity structures during the time of the experiment. The energy spectrum included in figure 1(a) shows that the scale disparity between energy and enstrophy is approximately half a decade.

As shown in figure 1(b,c), the field is divided into a regular grid for the modification step PL2, and subsets of a given number of cells of this grid are tested by zeroing their vorticity at time $T=0$ . This is convenient in two-dimensional flow because vorticity is not subject to continuity constraints, but spatial periodicity requires that $\langle \unicode[STIX]{x1D714}\rangle =0$ . This is enforced by adding an appropriate constant, $\unicode[STIX]{x1D714}_{00}$ , to the whole flow field,

(3.1) $$\begin{eqnarray}\displaystyle \widetilde{\unicode[STIX]{x1D714}}=\left\{\begin{array}{@{}ll@{}}\unicode[STIX]{x1D714}_{00} & \text{in}~A,\\ \unicode[STIX]{x1D714}_{ref}+\unicode[STIX]{x1D714}_{00} & \text{otherwise}.\end{array}\right. & & \displaystyle\end{eqnarray}$$

To test the effect of cell size, we use grids with $N=16~(4\times 4)$ , $N=36~(6\times 6)$ and $N=100~(10\times 10)$ . The effect of changing the modification strategy is discussed in § 3.1.

Figure 2.

(a) Evolution of the largest (solid) and smallest (dashed) deviation norms after deleting subsets of ○, $n=1$ ; ▫, 3; ▵, 7 cells, out of 36. The bars show the standard deviation of the variability over 48 initial conditions (192 for $n=1$ ). (b) As in (a), for the significance ratio $R_{\unicode[STIX]{x1D700}}=\unicode[STIX]{x1D700}_{max}/\unicode[STIX]{x1D700}_{min}$ . The dashed line with circles is $n=1$ , with cells deleted by zeroing the velocity instead of the vorticity, as in (3.3). (c) Outlined in white is the most significant subset of $n=3$ cells out of $N=36$ , defined at $\unicode[STIX]{x1D714}_{11}^{\prime }(t-t_{11})=3.4$ and overlaid on the vorticity magnitude. The least significant subset is outlined in red. (d) As in (c). Darkened cells belong to at least one of the 10 most significant subsets. The two cells heavily outlined in white belong to all of them.

Figure 2(a) shows the evolution of the perturbation magnitude for the most and least significant subsets of $n$ cells in an $N=36$ test grid. It suggests that significance is largely a matter of representation, and that the deviation after some time is roughly proportional to the representation difference at $T=0$ , but figure 2(b) tells a slightly different story. It shows that the ratio $R_{\unicode[STIX]{x1D700}}$ between the largest and smallest deviation norms from subsets of a given size is relatively large, and that it increases with the evolution time at which it is measured, at least up to $\unicode[STIX]{x1D714}^{\prime }T\approx 4$ . The former proves that there are flow regions that are substantially more significant than others, while the latter shows that significance to the flow evolution involves more than initial representativeness. There are flow regions that have a stronger effect on the evolution of the flow than what could be expected from the energy that they contain. Figure 2(b) also shows that the subsets that are most effective in determining significance are the smallest ones, $n=1$ . This is reasonable, at least on average, because there would be in general a most and a least significant grid point in each flow field, and modifying more grid points dilutes their significance.

Figure 2(c) shows an example of the most and least significant subsets with $n=3$ and $N=36$ , measured at $\unicode[STIX]{x1D714}_{11}T=3.4$ , and illustrates the tendency of even these small subsets to be formed by collections of smaller ones. It turns out that there are often several subsets with very similar significance. For the case in figure 2(c,d), $\unicode[STIX]{x1D700}_{max}/\unicode[STIX]{x1D700}_{min}=5.45$ , but the ratio between the lowest and the 10th lowest deviation (out of approximately 7000 possible combinations) is approximately 1.1, and that between the highest and the 10th highest deviation is 1.07. This is a common feature of optimisation over many degrees of freedom (LeCun et al. Reference LeCun, Bengio and Hinton2015), but, at least in this particular case, it appears to be connected with the sharing of significant substructures among subsets. Figure 2(d) shows the union of the 10 most significant subsets. Their component cells are scattered across the flow, but the two cells heavily outlined in white are present in all of the subsets, suggesting that the search algorithm identifies individual significant cells and forms with them larger connected composite substructures.

Connected substructures of more than one cell form by chance even in Poisson-distributed sets of cells, and how different significant sets are from random ones can be characterised by comparing the probability distribution (p.d.f.) of the number, $n_{s}$ , of cells in their connected substructures. Defining connectivity by the eight nearest neighbours, this is done in figure 3(a,b) for the most and least significant subsets respectively. They behave differently. Figure 3(a) shows that the connected substructures of the most significant sets tend to be larger than random ones. This difference scales best with the ratio of the length $n_{s}\unicode[STIX]{x1D6E5}_{c}$ to the Taylor microscale, where $\unicode[STIX]{x1D6E5}_{c}=L/N^{1/2}$ is the cell size, and is maximum for intermediate sizes of the order of the diameter of individual vortices. Not shown in the figure is another interesting difference. If the inertial tensor of a connected substructure is computed by substituting cells by unit point masses, an elongation can be defined by the ratio of its two principal moments of inertia. Random sets tend to be approximately circular, with elongations close to unity, but the most significant subsets are more elongated. This is especially true for the grid that is finest with respect to the Taylor scale, which is also the one in which the distribution of component size differs most from the random case (triangles in figure 3 a). The simplest interpretation is that finer grids are able to adapt the shape and size of significant sets to the shape of the strained vortices seen in figure 1. The tendency to group cells into linear filaments is also reflected in the empirically best scaling length for the abscissae in figure 3(a,b), which is the linear length of $n_{s}$ elements of size $\unicode[STIX]{x1D6E5}_{c}$ , rather than the diameter, $n_{s}^{1/2}\unicode[STIX]{x1D6E5}_{c}$ , of a blob with the same number of elements.

Figure 3(b) shows that most of these observations do not apply to the least significant subsets, whose distribution of connected substructures is essentially the same as for Poisson sets. It is also true that their elongations are much closer to unity than for the most significant subsets. The comparison suggests that, while the most significant sets reflect the structure of the flow, with specific geometries and sizes, the least significant ones tend to be random collections of background cells.

The effect of deleting cells is not simply additive. Different cells tend to cancel one another, and the effect of a small number of cells is weaker than the sum of its individual components. Additivity improves for longer test times, presumably because the effect of different components decorrelates. For example, in the cases in figure 2, the effect at $T=0$ of deleting the most significant subset of three cells is approximately 1.6 times weaker than the sum of the effects of the three most significant single cells, but that factor is 1.3 for $\unicode[STIX]{x1D714}_{11}T=3.4$ and almost unity for $\unicode[STIX]{x1D714}_{11}T=6.7$ .

Figure 3.

(a) Size distribution of the connected components in the 10 most significant subsets with $n=5$ (solid), compared with the equivalent distribution for random sets of cells (dashed). (b) As in (a), for the 10 least significant subsets. (c) Significance ratio measured at $T=t-t_{ref}$ , normalised with its value at $T=0$ . Here, $n=1$ . Time is scaled with the eddy turnover at the cell size, $\unicode[STIX]{x1D6E5}_{c}$ . In all panels: ▵, $(N,t_{ref},\unicode[STIX]{x1D6E5}_{c}/\unicode[STIX]{x1D706})=(100,t_{11},2.4)$ ; ▫, $(36,t_{26},2.8)$ ; ○, $(36,t_{11},4)$ ; ▿, $(16,t_{11},6)$ .

Figure 3(c) shows the temporal evolution of the significance ratio with the test time $T$ for several test cases and $n=1$ . It follows from figure 2(b) that the evolution of this ratio is relatively independent of $n$ , although its absolute value depends strongly on the size of the subset and on other details of the initial conditions. To compensate for this dependence, the ratio is normalised in figure 3(c) with its value at $T=0$ . The time in this figure scales best with the turnover time associated with the cell size, $\unicode[STIX]{x1D6E5}_{c}$ , rather than with the blob diameter, $n_{s}^{1/2}\unicode[STIX]{x1D6E5}_{c}$ , especially for times of the order of a turnover. This suggests that the labelling procedure isolates features of the order of $\unicode[STIX]{x1D6E5}_{c}$ , implying again thin filaments rather than blobs.

3.1 The effect of different modification strategies

Up to now we have discussed tests in which the whole flow field is modified in step PL2 of the labelling algorithm, following (3.1). A different option would be

(3.2) $$\begin{eqnarray}\displaystyle \widetilde{\unicode[STIX]{x1D714}}=\left\{\begin{array}{@{}ll@{}}\unicode[STIX]{x1D714}_{00} & \text{in}~A,\\ \unicode[STIX]{x1D714}_{ref} & \text{otherwise},\end{array}\right. & & \displaystyle\end{eqnarray}$$

where only the test subset is changed. This was found to make no appreciable difference with respect to (3.1), most probably because $\unicode[STIX]{x1D714}_{00}$ is typically much smaller than $\unicode[STIX]{x1D714}^{\prime }$ .

A more interesting possibility is to zero the velocity field within the chosen cells, in which case continuity has to be taken into account because streamlines are truncated at the edge of the cell. This option was also tested, projecting the modified field onto the solenoidal subspace by means of a pressure-like gradient,

(3.3) $$\begin{eqnarray}\displaystyle \widetilde{\boldsymbol{u}}=\left\{\begin{array}{@{}ll@{}}\unicode[STIX]{x1D735}p & \text{in}~A,\\ \boldsymbol{u}_{ref}+\unicode[STIX]{x1D735}p & \text{otherwise},\end{array}\right. & & \displaystyle\end{eqnarray}$$

where $p$ is adjusted so that $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\widetilde{\boldsymbol{u}}=0$ . It turns out that this procedure is less effective than (3.1). The resulting significance ratio for $n=1$ is included as a dashed line with circles in figure 2(b), and is approximately two times weaker than the comparable solid line for (3.1). The reason is that zeroing the velocity within a cell does not modify the surrounding cells, and, in particular, it does not change the circulation around the cell being modified. The vorticity originally contained in the cell is merely displaced to a sheet along its edge, which the irrotational gradient added in (3.3) cannot modify. Therefore, zeroing the velocity is equivalent to the less effective process of rearranging the vorticity, instead of deleting it.

This option is not explored further in this paper, although the results are used in the next section, but an equivalent choice has to be faced in generalising the present procedure to three-dimensional flows. Not only are the length scales of velocity and vorticity typically farther apart in three dimensions than in two, but vorticity itself is subject to a solenoidality constraint.

3.2 Properties of the significant regions

Figure 4.

(a) P.d.f.s of the ratio of mean vorticity to mean rate of strain over individual significant cells, for 192 initial conditions with $n=1,N=36,\unicode[STIX]{x1D714}_{11}^{\prime }(t-t_{11})=3.4$ . The vertical dashed line is the global average that separates vortex- from strain-dominated regions, $\langle \unicode[STIX]{x1D714}^{2}\rangle /\langle S^{2}\rangle =2$ ; ——, the 10 most significant cells for each initial condition; – – –, the 10 least significant cells; – - – - –, random cells. Lines without symbols are initialised using (3.1). Those with symbols use (3.3). (b) As in (a), for the kinetic energy, $q^{2}=u^{2}+v^{2}$ .

Although the main result of the previous analysis is the demonstration that there are subsets of the flow with very different dynamical significance, it also allows us to determine the properties of the most and least significant ones. Figure 4(a) displays the p.d.f. of the ratio of the enstrophy to the magnitude of the strain rate, averaged over the individual cells that have been labelled as most and least significant, compared with the p.d.f. compiled over randomly chosen cells. When this ratio is taken over the whole box, it is kinematically true that $\langle \unicode[STIX]{x1D714}^{2}\rangle /\langle S^{2}\rangle =2$ , which separates vorticity-dominated regions from those dominated by the rate of strain. The figure includes subsets modified by zeroing the vorticity, as in (3.1), together with others obtained by zeroing the velocity, as in (3.3). In both cases, the most significant sets are vortices and the least significant ones are the strain-dominated gaps between them, although the distinction is clearest for the cases that initially manipulate the vorticity. This agrees with the qualitative impression from figure 2(c), and with previous intuition (McWilliams Reference McWilliams1990), and applies to all combinations of cell and subset size. Figure 4(b) shows similar results for the kinetic energy. The tendency is also for highly significant cells to be energetic, although, in this case, it is clearest for the cases that initially manipulate the velocity. The tendency for significance to follow representativeness is to be expected, but it is reassuring that similar conclusions are reached from cross-manipulation, in which the behaviour of a variable is obtained from the modification of a different one.

4 Discussion and conclusions

We have shown that fully automatic comparison of simultaneous undisturbed and modified simulations can be used to segment a flow into regions properly labelled as more or less significant from the point of view of flow dynamics. Moreover, significance can be quantified as the growth of the perturbation after some time, and because regions are labelled individually, their properties can be statistically identified. In our application to two-dimensional turbulence, we find that the significant structures are vortices or vortex filaments, while the interstitial strain-dominated regions are less significant. We also show that significance is different from representativeness at a fixed time. Some flow features are more dynamically significant than what could be expected from their instantaneous energy content, and it is reassuring that tests based on the modification of different variables give similar results. Moreover, the results of the segmentation can be used to characterise the time scales and geometries of significant features. It follows from figure 3 that the key quantity controlling the scales being observed is the size, $\unicode[STIX]{x1D6E5}_{c}$ , of the test cells.

The problem is posed as a computer ‘game’ in which the machine tests the effect of different modifications of the initial conditions, and selects those that maximise or minimise it. The main difficulty of the procedure is cost, due to the large number of initial conditions and modifications that have to be tested, but it can benefit from the body of recent research on similar problems in data analytics and artificial intelligence. Even simple approximations help. While the purpose of the present paper is to test the strategy rather than to optimise processing time, considerable savings are achieved by substituting exhaustive searches by reasonable approximations, and the results presented above only represent a few thousand hours of computer time. In addition, the tests of different initial conditions are essentially independent, and can be trivially parallelised.

In retrospect, it should be clear that the problem of structure identification has only been partially automatised. In true machine learning, the result of the game is fed back into the initial conditions to optimise the result. In our experiment, this ‘back propagation’ step is left to the researcher, who evaluates the result and proposes modifications. The choice of test grids, initial flow conditions, observation times and deletion procedures discussed in § 3 was motivated in this way. The automatic part of the procedure described here can thus be seen as a fast and relatively exhaustive ‘forward’ search over experiments, resulting in the automatic labelling of flow subsets in terms of their significance.

Its main utility is probably its relative independence from preconceived ideas, which may mislead the researcher into confusing dynamical significance with representation of a particular quantity. While it would have been surprising if two-dimensional turbulence had been found to be non-intermittent, or if its significant eddies had been found to be other than vortices, the situation in more complex flows is, at least to the author, less clear. For example, while it is probable that we have identified by now the most significant types of structures of wall turbulence, especially near the wall, it is not impossible that we have missed some, and the situation is even less certain in homogeneous three-dimensional turbulence. Generalisation of the present results to these cases would require additional work, especially in the choice of initial perturbations, but, at least for moderate Reynolds numbers, both problems should be accessible to the current generation of computers.