arXiv subjects

Darryl D. Holm

Publications and source records attributed to Darryl D. Holm.

At least 19 recordsLinked to original sources

arXiv2026

The Geometry of Stochastic Fluid Dynamics

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.

physics.flu-dyn↗

arXiv2026

Numerical comparison of energy- versus circulation-preserving stochastic vortex dynamics

We compare two geometric stochastic frameworks for the two-dimensional Euler equations, being the circulation-preserving stochastic advection by Lie transport (SALT) and the energy-preserving stochastic forcing by Lie transport (SFLT) approaches. While preserving both circulation and energy is ideal, their simultaneous conservation restricts perturbations to a stochastic reparametrization of time. Consequently, a fundamental choice must be made between preserving structure or the kinetic energy. Analysis reveals that SALT is significantly more sensitive to high-frequency flow components, with noise effects scaling by $| \bk |^2$ relative to SFLT. This suggests that SALT acts as a localized perturbation sensitive to sharp gradients, while SFLT behaves as a more regularized global forcing. Numerical experiments on a traveling dipole, vortex merger, and forced-damped turbulence confirm that SALT introduces uncertainty localized near dynamically active vorticity gradients, whereas SFLT produces a more diffuse variance field spread across the domain. These results illustrate how the choice of geometric invariant fundamentally determines scale-sensitivity and spatial distribution of modeled uncertainty in vortex dynamics.

physics.flu-dyn↗

arXiv2026

Variational derivation of a moist thermal rotating shallow water model

We introduce a new energy-conserving, moist shallow water model with thermal stratification and rotation. The model is derived from a variational principle, using a Lagrangian expressed in terms of enthalpy. In this model, the latent heat from phase transitions modifies the buoyancy dynamics, which in turn feeds back to alter the vertically integrated hydrodynamic motion. Finally, we generalise this moisture parameterisation to non-hydrostatic Green-Naghdi equations.

physics.flu-dyn↗

arXiv2025

Surface Wave Solutions in 1D and 2D for the Broer-Kaup-Boussinesq-Kupershmidt (BKBK) System

The BKBK system is a singular perturbation of the classical shallow water equations which modifies their transport velocity to depend on wave elevation slope. This modification introduces backward diffusion terms proportional to a real parameter $κ$. These terms also make BKBK completely integrable as a Hamiltonian system. Remarkably, when $κ=i/2$ the BKBK system may be transformed into the focusing nonlinear Schrödinger (NLS). Thus, the BKBK system with its real parameter $κ$ is complementary to the traditional modulational approach for water waves. We investigate the Lie algebraic and variational properties of the BKBK system in this paper and we study its solution behaviour in certain computational simulations of regularised versions of the 1D and 2D BKBK systems.

math-ph↗

arXiv2025

A comparative numerical study of stochastic Hamiltonian Camassa-Holm equations

We introduce a stochastic perturbation of the Camassa-Holm equation such that, unlike previous formulations, energy is conserved by the stochastic flow. We compare this to a complementary approach which preserves Casimirs of the Poisson bracket. Through an energy preserving numerical implementation of the model, we study the influence of noise on the well-known 'peakon' formation behaviour of the solution. The energy conserving stochastic approach generates an ensemble of solutions which are spread around the deterministic Camassa-Holm solution, whereas the Casimir conserving alternative develops peakons which may propagate away from the deterministic solution more dramatically.

cond-mat.stat-mech↗

arXiv2025

Compound Burgers-KdV Soliton Behaviour: Refraction, Reflection and Fusion

We consider a coupled PDE system between the Burgers equation and the KdV equation to model the interactions between `bore'-like structures and wave-like solitons in shallow water. Two derivations of the resulting Burgers-swept KdV system are presented, based on Lie group symmetry and reduced variational principles. Exact compound soliton solutions are obtained, and numerical simulations show that the Burgers and KdV momenta tend toward a balance at which the coupled system reduces to the integrable Gardner equation. The numerical simulations also reveal rich nonlinear solution behaviours that include refraction, reflection, and soliton fusion, before the balance is finally achieved.

nlin.PS↗

arXiv2025

A thermal Green-Naghdi model with time dependent bathymetry and complete Coriolis force

This paper extends the theoretical Euler-Poincaré framework for modelling ocean mixed layer dynamics. Through a symmetry-broken Lie group invariant variational principle, we derive a generalised Green-Naghdi equation with time dependent bathymetry, a complete Coriolis force, and inhomogeneity of the thermal buoyancy. The nature of the model derived here lends it a potential future application to wave dynamics generated by changes to the bathymetry.

physics.ao-ph↗

arXiv2025

Plasma dynamics in thin domains

In the present work, we study the geometric structures of the Rotating Shallow Water Magnetohydrodynamics (RSW-MHD) equations through a Lie group invariant Euler-Poincaré variational principle. In this geometric framework, we derive new, structure-preserving stochastic RSW-MHD models by introducing stochastic perturbations to the Lie-Poisson structure of the deterministic RSW-MHD equations. The resulting stochastic RSW-MHD equations provide new capabilities for potential application to uncertainty quantification and data assimilation, for example, in space plasma (space weather) and solar physics, particularly in solar tachocline dynamics.

physics.plasm-ph↗

arXiv2024

31 Lectures on Geometric Mechanics

These lecture notes in geometric mechanics are meant to convey insight through clear definitions and workable examples. The lecture format adopted here is intended to convey the immediacy of the taught course and to be useful as a basis for other courses. The lecture notes comprise: AP = Applications of Pure maths, e.g., Noether's theorem: Lie group symmetry of Hamilton's variational principle implies conservation laws for its equations of motion.\smallskip PA = Purifications of Applied maths, e.g., Euler fluid dynamics describes geodesic flow on the manifold of smooth invertible maps acting on the domain of flow. \smallskip Both AP and PA appear here, though the difference is not mentioned. It is left to the reader to decide whether it was AP or PA in each of the lectures containing well over sixty solved exercises. An aspect of modern applications emphasised here is the use of the composition of evolutionary maps for multi-physics, multi-timescale interactions including waves interacting with flows in the Euler--Poincaré framework in geophysical fluid dynamics (GFD) for ocean and atmosphere dynamics, and in magnetohydrodynamics (MHD) for applications in plasma physics such as magnetic confinement fusion (MFC) and astrophysical processes such as Alfvén waves and gravity waves propagating on the Solar tachocline. The topics covered in each lecture can also be gleaned from its table of contents listed at the onset of each lecture.

physics.class-ph↗

arXiv2024

Collisions of Burgers Bores with Nonlinear Waves

This paper treats nonlinear wave current interactions in their simplest form, as an overtaking collision. In one spatial dimension, the paper investigates the collision interaction formulated as an initial value problem of a Burgers bore overtaking solutions of two types of nonlinear wave equations, Korteweg de Vries (KdV) and nonlinear Schrodinger (NLS). The bore wave state arising after the overtaking Burgers-KdV collision in numerical simulations is found to depend qualitatively on the balance between nonlinearity and dispersion in the KdV equation. The Burgers-KdV system is also made stochastic by following the stochastic advection by Lie transport approach (SALT).

physics.flu-dyn↗

arXiv2024

Deterministic and Stochastic Geometric Mechanics for Hall MHD

We derive new models of stochastic Hall magnetohydrodynamics (MHD) by using a symmetry-reduced stochastic Euler-Poincaré variational principle. The new stochastic Hall MHD theory has potential applications for uncertainty quantification and data assimilation in space plasma (space weather) and solar physics. The stochastic geometric mechanics approach we take here produces coordinate-free results which may then be applied in a variety of spatial configurations.

physics.plasm-ph↗

arXiv2024

Geometric theory of perturbation dynamics around non-equilibrium fluid flows

The present work investigates the evolution of linear perturbations of time-dependent ideal fluid flows with advected quantities, expressed in terms of the second order variations of the action corresponding to a Lagrangian defined on a semidirect product space. This approach is related to Jacobi fields along geodesics and several examples are given explicitly to elucidate our approach. Numerical simulations of the perturbation dynamics are also presented.

physics.flu-dyn↗

arXiv2024

Geometric Mechanics of the Vertical Slice Model

The goals of the present work are to: (i) investigate the dynamics of oceanic frontogenesis by taking advantage of the geometric mechanics underlying the class of Vertical Slice Models (VSMs) of ocean dynamics; and (ii) illustrate the versatility and utility of deterministic and stochastic variational approaches by deriving several variants of wave-current interaction models which describe the effects of internal waves propagating within a vertical planar slice embedded in a 3D region of constant horizontal gradient of buoyancy in the direction transverse to the vertical plane.

physics.flu-dyn↗

arXiv2023

Stochastic Geometric Mechanics for Fluid Dynamics

physics.flu-dyn↗

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↗

arXiv2023

On the interactions between mean flows and inertial gravity waves in the WKB approximation

We derive a Wentzel-Kramers-Brillouin (WKB) closure of the generalised Lagrangian mean (GLM) theory by using a phase-averaged Hamilton variational principle for the Euler--Boussinesq (EB) equations. Following Gjaja and Holm 1996, we consider 3D inertial gravity waves (IGWs) in the EB approximation. The GLM closure for WKB IGWs expresses EB wave mean flow interaction (WMFI) as WKB wave motion boosted into the reference frame of the EB equations for the Lagrangian mean transport velocity. We provide both deterministic and stochastic closure models for GLM IGWs at leading order in 3D complex vector WKB wave asymptotics. This paper brings the Gjaja and Holm 1996 paper at leading order in wave amplitude asymptotics into an easily understood short form and proposes a stochastic generalisation of the WMFI equations for IGWs.

physics.flu-dyn↗

arXiv2023

Theoretical analysis and numerical approximation for the stochastic thermal quasi-geostrophic model

This paper investigates the mathematical properties of a stochastic version of the balanced 2D thermal quasigeostrophic (TQG) model of potential vorticity dynamics. This stochastic TQG model is intended as a basis for parametrisation of the dynamical creation of unresolved degrees of freedom in computational simulations of upper ocean dynamics when horizontal buoyancy gradients and bathymetry affect the dynamics, particularly at the submesoscale (250m--10km). Specifically, we have chosen the SALT (Stochastic Advection by Lie Transport) algorithm introduced in [1] and applied in [2,3] as our modelling approach. The SALT approach preserves the Kelvin circulation theorem and an infinite family of integral conservation laws for TQG. The goal of the SALT algorithm is to quantify the uncertainty in the process of up-scaling, or coarse-graining of either observed or synthetic data at fine scales, for use in computational simulations at coarser scales. The present work provides a rigorous mathematical analysis of the solution properties of the thermal quasigeostrophic (TQG) equations with stochastic advection by Lie transport (SALT) [4,5].

math.AP↗

arXiv2023

Lagrangian reduction and wave mean flow interaction

How does one derive models of dynamical feedback effects in multiscale, multiphysics systems such as wave mean flow interaction (WMFI)? We shall address this question for hybrid dynamical systems, whose motion can be expressed as the composition of two or more Lie-group actions. Hybrid systems abound in fluid dynamics. Examples include: the dynamics of complex fluids such as liquid crystals; wind-driven waves propagating with the currents moving on the sea surface; turbulence modelling in fluids and plasmas; and classical-quantum hydrodynamic models in molecular chemistry. From among these examples, the motivating question in this paper is: How do wind-driven waves produce ocean surface currents? The paper first summarises the geometric mechanics approach for deriving hybrid models of multiscale, multiphysics motions in ideal fluid dynamics. It then illustrates this approach for WMFI in the examples of 3D WKB waves and 2D wave amplitudes governed by the nonlinear Schrödinger (NLS) equation propagating in the frame of motion of an ideal incompressible inhomogeneous Euler fluid flow. The results for these examples tell us that the fluid flow in WMFI does not create waves. However, feedback in the opposite direction is possible, since 3D WKB and 2D NLS wave dynamics can indeed create circulatory fluid flow.

math.AP↗