Recurrent flow patterns as a basis for two-dimensional turbulence: Predicting statistics from structures

Significance A long-standing challenge in turbulence has been to connect individual coherent structures to the more well-known statistical properties of the flow. Here, we establish such a connection by representing two-dimensional turbulence as a Markov chain between exact unstable periodic orbits, which are realized transiently in the flow. To find the dynamically relevant solutions, we develop a method based on optimization of a scalar loss function, which overcomes the restrictions of previous algorithms and is effective at a high Reynolds number. The Markovian representation is then achieved by using a neural network to label turbulent snapshots according to the nearest unstable solution, and its invariant measure reproduces full PDFs of the chaotic flow.


Latent distance to UPOs
As described in the main text, the distance to periodic orbit j is determined according to d j ψ (ω) := minm,q ψ(ω) − ψ(S m R q f t (ωj)) T 2, [1] with the closest UPO following from j * = arg min j d j ψ (ω).In figure S1 we report the distance (1) to all UPOs found at Re = 40 for a short turbulent trajectory (the same one considered in figure 5 in the main text), along with the time evolution of the dissipation rate.We highlight two particular solutions which are frequently visited in this turbulent episode, a high dissipation solution (blue) which plays a dominant role in the burst, and a common low dissipation state (orange) which is shadowed for an extensive period around t ∼ 80.

Further results at Re = 100
The higher Re = 100 transition matrix is included here in figure S2 (compare to the Re = 40 results in figure 6 of the main paper), along with the invariant measure used to compute the statistics in figure 7 of the manuscript.There is no clear distinction between low/high dissipation states and transitions can apparently occur between widely separated UPOs (in terms of dissipation).However, these results are clearly likely to be impacted significantly by the large number of missing states -e.g.see the I − D plot in figure 3 of the main paper, and a clearer picture will likely emerge as we continue to converge new solutions in future calculations.The relationship between the weights defined by the invariant measure of the Re = 100 transition matrix and the unstable growth rates of the associated UPOs are also reported in figure S2.Unlike the Re = 40 results, it is challenging to identify a clear relationship, which again may be resolved after computation of more solutions.
In addition to the transition matrix, we also report snapshots of spanwise vorticity for five further UPOs at Re = 100 in figure S3 to demonstrate the wealth of vorticity dynamics contained in the UPO library.These include localised co-rotating three-vortex states, as well as large quiescent vortex patches.

Periodic orbit details
Here we report details of the UPOs we have found via automatic differentiation at both Re = 40 and Re = 100 in tables S1 and S2 respectively, including their leading Floquet exponent and the dimension of the unstable manifold.
In addition, we also include a visualisation of the UPOs at both Re = 40 and Re = 100 in the energy-dissipation plane (figure S4) to better illustrate the difficulty in reconstruction of the PDF of E (particularly at Re = 40 -see figure 7a in the main paper) compared to the dissipation rate.It is clear from the figure that at Re = 40 we are missing low-energy, low-dissipation states in our solution library, which can be reasonably attributed to either (i) our focus on searching for short orbits or (ii) our lack of a search over discrete symmetries for pre-periodic orbits, which we hope to address in future work.
Fig. S1.(Top) Distance d ψ (f t (ω)) to all UPOs along a short turbulent trajectory at Re = 40 (grey lines).The values shown have been normalised by ψ(f t (ω)) , where • indicates and average over the long 2.5 × 10 4 trajectory described in the main text.Two frequently visited UPOs are highlighted in orange (T = 2.829, low dissipation D/D l T = 0.095) and blue (T = 5.062, high dissipation D/D l T = 0.253).(Bottom) Dissipation normalised by the laminar value along the same trajectory.Coloured portions of the curve highlight where the two UPOs are being shadowed according to the criteria discussed in the text.Note the trajectory used in this figure is the same one used in figure 5 of the main manuscript.

Fig. S2 .
Fig. S2.(a) Invariant measure π and transition matrix P at Re = 100 (log of transition probabilities is shown, spacing between snapshots is δt = 0.25).States are ordered from lowest to highest average dissipation rate (lowest at top/leftmost).(b) The weights in the expansion (7) -which are also the invariant measure of the Markov chain wj = πj -plotted against the (real part of the) sum of growing Floquet exponents j σj , σj > 0, for each UPO.

Known solutions as listed in (1) are listed in the 'UPO' column; T is the period, α the shift. N indicates the number of unstable directions and σr is the growth rate in the leading Floquet exponent. We also report the average dissipation normalized by the laminar value
, D/D l .

Known solutions as listed in (1) are listed in the 'UPO' column; T is the period, α the shift. N indicates the number of unstable directions and σr is the growth rate in the leading Floquet exponent. We also report the average dissipation normalised by the laminar value
, D/D l .