The Geometry of Stochastic Fluid Dynamics

Darryl D. Holm

Home/Archive/arXiv

arXiv · 2608.15321

The Geometry of Stochastic Fluid Dynamics

Darryl D. Holm

View original source ↗Read source document ↗Export metadata ↓

Abstract

Stochastic geometric mechanics (SGM) is known for its potential utility in quantifying uncertainty in global climate modelling of the Earth's ocean and atmosphere while also preserving the fundamental advective transport properties of ideal fluid flow. This paper is a pedagogical review of the recent developments of the mathematical framework of stochastic geometric mechanics obtained from Lie group-invariant stochastic variational principles in the context of model building for upper ocean dynamics, The paper is divided into the following five parts. Part I discusses the origins of geometric mechanics applications in deterministic fluid dynamics. Part II focuses on the example of the deterministic 3D Euler Boussinesq (EB) equations. Part III adds stochastic transport to the 3D Euler Boussinesq (EB) and derives its SALT equations. (SALT is the abbreviation of Stochastic Advection by Lie Transport.) Part IV focuses on Lagrangian Averaged Stochastic Lie Transport, abbreviated as LA-SALT. LA-SALT treats atmospheric `climate' as the ensemble expectation, while the atmospheric `weather' is treated as a field of pathwise fluctuations, as discussed in Ed Lorenz's famous 1995 lecture. Part V applies SALT and LA-SALT to create stochastic Ocean--Atmosphere Models, abbreviated as SOAM.. The SOAM approach brings us back to Hasselmann's 1976 paradigm, which decomposes a general climate model into its deterministic and stochastic parts.

Explore related subjects

physics.flu-dyn math-ph math.MP

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Darryl D. Holm. 2026-08-15. The Geometry of Stochastic Fluid Dynamics. https://arxiv.org/abs/2608.15321

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Mathematical modeling on peristaltic flow of a Prandtl fluid with effects of slip conditions and inclined magnetic field

The manuscript provides a description of a theoretical analysis of a non-Newtonian Prandtl fluid subject to peristaltic flow through an inclined asymmetric channel. We explore the effect of an inclined magnetic field on the peristaltic flow. This is relevant for applications involving fluid flow in narrow, inclined (tilted) tubes similar to blood vessels or the digestive system. The model also includes thermodynamic aspects such as heat diffusion (the Soret effect) and viscous dissipation resulting from wall-fluid slip conditions, which may help optimize medical devices such as lab-on-a-chip systems and dialysis machines. In this study, the concentration of a generic chemical, temperature, and fluid velocity are taken into account through mass, heat, and momentum balances, respectively. The solution is approximated using numerical techniques suitable for long wavelengths (low frequency) and low Reynolds numbers. The study also discusses trapping phenomena, which are crucial from a clinical point of view. The developed insights can improve the understanding of physiological flows in the gastrointestinal tract and blood vessels. By understanding how the fluid moves and how particles are trapped, these insights may contribute to the design of improved medical pumps and artificial organs. Graphical visualizations are provided for the fluid velocity profile, temperature distribution, and concentration of a generic chemical. Furthermore, the numerical results are validated through comparison with a closed-form solution from a benchmark problem.

physics.flu-dyn↗

arXiv2026

Discovery of a dispersion model at high Peclet numbers

Peclet number characterises the transition from classical Taylor-Aris dispersion to convection-dominated longitudinal solute transport, with the classical model becoming inadequate at extremely high radial Peclet number $Pe_r$. We develop a novel explicit-closure one-dimensional (1-D) effective dispersion model for this high-$Pe_r$ regime by introducing two closure coefficients, $θ_u$ and $θ_d$, whose functional structures are identified using low-frequency transfer-function matching and a modified Kolmogorov-Arnold network (KAN). The resulting model captures the transition from classical Taylor-Aris dispersion at low $Pe_r$ to convection-dominated dispersion at high $Pe_r$. Analysis reveals that, in the high-$Pe_r$ regime, axial transport is redistributed between the effective convection flux and the dispersive flux, resulting in a reduced macroscopic convection velocity. Numerical validation demonstrates close agreement with the convection-diffusion model over the investigated high-$Pe_r$ conditions, while the classical Taylor-Aris model exhibits substantial deviations. Application of the proposed model to averaged flow velocity inversion further demonstrates improved velocity estimation, particularly in the high-$Pe_r$ regime. These results highlight the importance of accounting for non-classical dispersion for reliable contrast-agent-based arterial blood flow velocimetry and provide new insight into high-$Pe_r$ mass transport.

physics.flu-dyn↗

arXiv2026

Optimization of fluid mixing by reinforcement learning using limit cycles of a dynamical system

We propose a method to overcome the difficulties encountered when applying reinforcement learning to fluid mixing processes. The proposed method has two main features: (i) it does not require detailed measurements of the flow state, and (ii) by effectively exploiting a stable limit cycle of a two-dimensional dynamical system (the Li'enard system), it can stably perform optimization without imposing explicit constraints on the control parameters. As an illustrative example, we optimize a process in which a fluid contained in a cylindrical vessel is mixed by periodically rotating the vessel. The resulting optimal vessel motion is physically reasonable: it reverses its direction of rotation before a solid-body rotation state is established. Furthermore, even when the fluid viscosity increases with time during the mixing process, the method can continuously adapt the control parameters to the changing viscosity.

physics.flu-dyn↗