arXiv subjects

Theo Diamantakis

Publications and source records attributed to Theo Diamantakis.

4 recordsLinked to original sources

arXiv2025

Lagrangian averaging of singular stochastic actions for fluid dynamics

We construct sub-grid scale models of incompressible fluids by considering expectations of semi-martingale Lagrangian particle trajectories. Our construction is based on the Lagrangian decomposition of flow maps into mean and fluctuation parts, and it is separated into the following steps. First, through Magnus expansion, the fluid velocity field is expressed in terms of fluctuation vector fields whose dynamics are assumed to be stochastic. Second, we use Malliavin calculus to give a regularised interpretation of the product of white noise when inserting the stochastic velocity field into the Lagrangian for Euler's fluid. Lastly, we consider closures of the mean velocity by making stochastic analogues of Talyor's frozen-in turbulence hypothesis to derive a version of the anisotropic Lagrangian averaged Euler equation.

math-ph↗

arXiv2024

Variational closures for composite homogenised fluid flows

Homogenisation theory has seen recent applications in deriving stochastic transport models for fluid dynamics. In this work, we first derive the stochastic Lagrange-to-Euler map that underpins stochastic transport noise in fluid dynamics as the homogenisation limit of a parameterised flow map decomposing into rapidly fluctuating and slow components. Specifically, we prove convergence of this parameterised flow map to a scale-separated limit under the assumptions of a weak invariance principle for the rapidly fluctuating component and path continuity for the slow component. In this limit, the rapidly fluctuating component converges to a stochastic flow of diffeomorphisms that transforms the full flow dynamics into an SDE-governed stochastic flow through composition, while the slow component requires closure. Our second contribution formulates two distinct variational closures for the slow component of the homogenised flow that exploit the composite structure of the stochastic flow. For the first closure, the critical points of a new variational principle satisfy a system of random-coefficient PDEs, which can be transformed into a system of stochastic PDEs via the coadjoint action of the stochastic flow map obtained from homogenising the fluctuating component. We show that these equations coincide with the stochastic Euler-Poincar\'e equations previously derived in Holm, Proc. Royal Soc. (2015). For the second closure, we modify the assumptions on the slow component and the associated variational principle to derive averaged models inspired by previous work on mean flow dynamics such as the Generalised Lagrangian Mean.

math-ph↗

arXiv2024

L\'evy areas, Wong Zakai anomalies in diffusive limits of Deterministic Lagrangian Multi-Time Dynamics

Stochastic modelling necessitates an interpretation of noise. In this paper, we describe the loss of deterministically stable behaviour in a fundamental fluid mechanics problem, conditional to whether noise is introduced in the sense of It\^o, Stratonovich or a limit of Wong-Zakai type. We examine this comparison in the wider context of discretising stochastic differential equations with and without the L\'evy area. From the numerical viewpoint, we demonstrate performing higher order discretisations with the use of a L\'evy area can lead to the loss of conserved area and angle quantities. Such behaviour is not physically expected in the Stratonovich model. Conversely, we study Stochastic Advection by Lie Transport and its derivation from homogenisation theory, which introduces drift corrections of the same class naturally. From the viewpoint of homogenisation, the qualitative properties of the Wong-Zakai anomaly are physically motivated as arising due to correlations from a fast and mean scale fluid decomposition.

math.DS↗

arXiv2023

Variational Principles on Geometric Rough Paths and the Lévy Area Correction

In this paper, we describe two effects of the Lévy area correction on the invariant measure of stochastic rigid body dynamics on geometric rough paths. From the viewpoint of dynamics, the Lévy area correction introduces an additional deterministic torque into the rigid body motion equation on geometric rough paths. When the dynamics is driven by coloured noise, and for rigid body dynamics with double-bracket dissipation, theoretical and numerical results show that this additional deterministic torque shifts the centre of the probability distribution function by shifting the Hamiltonian function in the exponent of the Gibbsian invariant measure.

nlin.CD↗