arXiv subjects

Andrew Cleary

Publications and source records attributed to Andrew Cleary.

4 recordsLinked to original sources

arXiv2025

Latent-space variational data assimilation in two-dimensional turbulence

Starting from limited measurements of a turbulent flow, data assimilation (DA) attempts to estimate all the spatio-temporal scales of motion. Success is dependent on whether the system is observable from the measurements, or how much of the initial turbulent field is encoded in the available measurements. Adjoint-variational DA minimises the discrepancy between the true and estimated measurements by optimising the initial velocity or vorticity field (the `state space'). Here we propose to instead optimise in a lower-dimensional latent space which is learned by implicit rank minimising autoencoders. Assimilating in latent space, rather than state space, redefines the observability of the measurements and identifies the physically meaningful perturbation directions which matter most for accurate prediction of the flow evolution. When observing coarse-grained measurements of two-dimensional Kolmogorov flow at moderate Reynolds numbers, the proposed latent-space DA approach estimates the full turbulent state with a relative error improvement of two orders of magnitude over the standard state-space DA approach. The small scales of the estimated turbulent field are predicted more faithfully with latent-space DA, greatly reducing erroneous small-scale velocities typically introduced by state-space DA. Furthermore, latent-space DA is demonstrated to be robust to noisy measurements at the range of Reynolds numbers considered. These findings demonstrate that the observability of the system from available data can be greatly improved when turbulent measurements are assimilated in the right space, or coordinates.

physics.flu-dyn↗

arXiv2025

Characterizing the Reynolds number dependence of the chaotic attractor in two-dimensional turbulence with dimension-minimizing autoencoders

Deep autoencoder neural networks can generate highly accurate, low-order representations of turbulence. We design a new family of autoencoders which are a combination of a 'dense-block' encoder-decoder structure (Page et al, J. Fluid Mech. 991, 2024), an 'implicit rank minimization' series of linear layers acting on the embeddings (Zeng et al, Mach. Learn. Sci. Tech. 5, 2024) and a full discrete+continuous symmetry reduction. These models are applied to two-dimensional turbulence in Kolmogorov flow for a range of Reynolds numbers $25 \leq Re \leq 400$, and used to estimate the dimension of the chaotic attractor, $d_{\mathcal A}(Re)$. We find that the dimension scales like $\sim Re^{1/3}$ -- much weaker than known bounds on the global attractor which grow like $Re^{4/3}$. In addition, two-dimensional maps of the latent space in our models reveal a rich structure not seen in previous studies, including multiple classes of high-dissipation events at lower $Re$ which guide bursting trajectories. We visualize the embeddings of large numbers of "turbulent" unstable periodic orbits, which the model indicates are distinct (in terms of features) from any flow snapshot in a large turbulent dataset, suggesting their dynamical irrelevance. This is in sharp contrast to their appearance in more traditional low-dimensional projections, in which they appear to lie within the turbulent attractor.

physics.flu-dyn↗

arXiv2025

Dynamical relevance of periodic orbits under increasing Reynolds number and connections to inviscid dynamics

Large numbers of relative periodic orbits (RPOs) have been found recently in doubly-periodic, two-dimensional Kolmogorov flow at moderate Reynolds numbers $Re \in \{40, 100\}$. While these solutions lead to robust statistical reconstructions at the $Re$-values where they were obtained, it is unclear how their dynamical importance evolves with increasing $Re$. We perform arclength continuation on this library of solutions to show that large numbers of RPOs quickly become dynamically irrelevant, reaching dissipation values either well above or below those associated with the turbulent attractor at high $Re$. The scaling of the high dissipation RPOs is shown to be consistent with a direct connection to solutions of the unforced Euler equation, and is observed for a wide variety of states beyond the 'unimodal' solutions considered in previous work (Kim & Okamoto, Nonlinearity 28, 2015). On the other hand, the weakly dissipative states have properties indicating a connection to exact solutions of a forced Euler equation. The apparent dynamical irrelevance is associated with poor statistical reconstructions away from the $Re$ values where the RPOs were originally converged. Motivated by the connection to solutions of the Euler equation, we show that many of these states can be well described by exact relative periodic solutions in a system of point vortices. The point vortex RPOs are converged via gradient-based optimisation of a scalar loss function which (1) matches the dynamics of the point vortices to the turbulent vortex cores and (2) insists the point vortex evolution is itself time-periodic.

physics.flu-dyn↗

arXiv2023

Exploring the free-energy landscape of a rotating superfluid

The equilibrium state of a superfluid in a rotating cylindrical vessel is a vortex crystal -- an array of vortex lines which is stationary in the rotating frame. Experimental realisations of this behaviour typically show a sequence of transient states before the free-energy minimising configuration is reached. Motivated by these observations, we construct a new method for a systematic exploration of the free-energy landscape via gradient-based optimisation of a scalar loss function. Our approach is inspired by the pioneering numerical work of Campbell & Ziff (Phys. Rev. B 20, 1979), and makes use of automatic differentiation (AD) which crucially allows us to include entire solution trajectories in the loss. We first use the method to converge thousands of low-free-energy relative equilibria for vortex numbers in the range $10 \leq N \leq 30$, which reveals an extremely dense set of mostly saddle-like solutions. As part of this search, we discover new continuous families of relative equilibria (in the unbounded domain) which are often global minimisers of the free energy. These continuous families all consist of crystals arranged in a double-ring configuration, and we assess which state from the family is most likely to be observed experimentally by computing energy-minimising pathways from nearby local minima -- identifying a common entry point into the family. Finally, we develop an approach to compute homoclinic orbits and use it to examine the dynamics in the vicinity of the minimising state by converging connections for low-energy saddles.

physics.flu-dyn↗