Mesoscale and submesoscale features with Rossby and Richardson numbers near unity indicate a breakdown of geostrophic balance. This gives rise to ageostrophic flows that drive circulation across density gradients and produce vertical motions, transporting heat and biogeochemical tracers below the mixed layer. During winter 2022, high resolution multiplatform in situ observations and realistic numerical simulations captured the evolution of mesoscale and submesoscale features in the northwestern Mediterranean Sea. A mesoscale front in the Balearic Sea was observed progressing from intensification to decay, culminating in the formation of two submesoscale cyclonic frontal eddies (SCEs). These formed as the front elongated and interacted with a mesoscale ridge, illustrating the dynamic interplay between mesoscale and submesoscale processes. The front intensified due to strain-induced frontogenesis. A strong down-front wind event triggered submesoscale instabilities and the nonlinear Ekman effect, enhancing vertical motions through an ageostrophic secondary circulation and contributing to restratification. As the front weakened, isopycnal slopes flattened, and energy cascaded toward smaller scales, forming the SCEs. This energy transfer was primarily driven by submesoscale instabilities, with additional contributions from centrifugal and gravitational instabilities. A Lagrangian analysis revealed that horizontal parcel transport was dominated by mesoscale circulation, while vertical displacements were controlled by submesoscale processes. The evolving SCEs exhibited a three-dimensional helical-spiral recirculation pattern, promoting vertical transport. Submesoscale eddy-induced frontogenesis drove subduction into the mixed layer, intensified by submesoscale instabilities and guided by downward-sloping isopycnal surfaces at the eddy periphery.

Typically, numerical simulations of Earth systems are coarse, and Earth observations are sparse and gappy. We apply four generative diffusion modeling approaches to super-resolution and inference of forced two-dimensional quasi-geostrophic turbulence on the beta-plane from coarse, sparse, and gappy observations. Two guided approaches minimally adapt a pre-trained unconditional model: SDEdit modifies the initial condition, and Diffusion Posterior Sampling (DPS) modifies the reverse diffusion process score. Two conditional approaches, a vanilla variant and classifier-free guidance, require training with paired high-resolution and observation data. We consider multiple test cases spanning: two regimes, eddy and anisotropic-jet turbulence; two Reynolds numbers, 10³ and 10⁴; and two observation types, 4x coarse-resolution fields and coarse, sparse and gappy observations. Our comprehensive skill metrics include norms of the reconstructed vorticity fields, turbulence statistical quantities, and quantifications of the super-resolved probabilistic ensembles and their errors. We also study the sensitivity to tuning parameters such as guidance strength. Results show that the generated super-resolution fields of SDEdit are unphysical, while those of DPS are reasonable but with smoothed fine-scale features; however, neither of these lower-cost models propagates observational information effectively to unobserved regions. The two conditional models require re-training, but reconstruct missing fine-scale features, are cycle-consistent with observations, and predict correct turbulence statistics, including the tails. Further, their mean errors are highly correlated with and predictable from their ensemble standard deviations. Results highlight the tradeoffs between ease of implementation, fidelity (sharpness), and cycle-consistency of the diffusion models, and offer practical guidance for deployment.

In this work, we analyze Lagrangian material coherence in dynamic open domains. We derive and apply new theory and efficient schemes to extract material sets in dynamic flow fields that remain the most or the least coherent throughout the time interval of interest, with special attention to realistic ocean domains that have multiple time-dependent inlets and/or outlets. First, the partial differential equation (PDE)-based method of composition is extended to efficiently compute flow maps in open domains, evolving a dynamic mask field without compounding numerical errors. This permits the use of existing grid based PDE solvers to compute flow maps on their dynamic non-regular domain. Inherent parallelization capabilities with accuracy as trajectory-based schemes but importantly with also an optimal grid-based resolution make this method very attractive. Second, we derive a novel approach to compute material sets in dynamic fluid flows that undergo minimal stress throughout the considered time interval. The level sets of the proposed metric, called the ‘extended polar distance’, yield material subdomains that remain rigid (i.e. only undergo translation and rotation) throughout the time interval of interest up to a certain tolerance. This metric and the corresponding persistently coherent sets and incoherent sets are computed using the PDE-based flow map computation. We further relate the extended polar distance and the diffusion barrier strength metric and show that the extended polar distance rigorously cumulates the tendency of a material subdomain to be prone to diffusion and the average strain it undergoes. We utilize the new theory and numerical methods to analyze Lagrangian coherence in analytical and realistic scenarios – an analytical unsteady double gyre flow and a realistic simulation in the Southern Pacific Ocean. The former helps us better understand the proposed theory in practice, and highlights the evolution of coherent, persistently coherent, and incoherent sets. In the latter Southern Pacific Ocean application, we find that the surface regions around Palau island are highly incoherent due to the steep topography and complex interactive dynamics. However, we also find a rigid set advected by the larger-scale currents around the Island, retrieving its shape at the end, as well as a persistently rigid set that approximately maintains its shape throughout the time interval, maximally resisting advective stretching and diffusive transport.

A theoretical synthesis of forward reachability for minimum–time control of anisotropic vehicles operating in strong and dynamic flows is provided. The synthesis relies on the computation of the forward reachable set of states. Using ideas rooted in the theory of non–smooth calculus, we prove that this set is governed by the viscosity solution of an unsteady Hamilton–Jacobi (HJ) equation. We show that the minimum arrival time satisfies a static HJ equation, when a special local controllability condition holds. Results are exemplified by applications to a sailboat moving in a uniform wind–field and autonomous underwater gliders operating in the Sulu Archipelago.

This paper focuses on the extractions of Lagrangian Coherent Sets from realistic velocity fields obtained from ocean data and simulations, each of which can be highly resolved and non volume-preserving. We introduce two novel methods for computing two formulations of such sets. First, we propose a new “diffeomorphism-based” criterion to extract “rigid sets”, defined as sets over which the flow map acts approximately as a rigid transformation. Second, we develop a matrix-free methodology that provides a simple and efficient framework to compute “coherent sets” with operator methods. Both new methods and their resulting rigid sets and coherent sets are illustrated and compared using three numerically simulated flow examples, including a high-resolution realistic, submesoscale to large-scale dynamic ocean current field in the Palau Island region of the western Pacific Ocean.