arXiv subjects

Erwin Luesink

Publications and source records attributed to Erwin Luesink.

At least 19 recordsLinked to original sources

arXiv2026

A new formulation of metriplectic dynamics with an application to quasigeostrophic ocean modeling with advected quantities

A general formulation of metriplectic dynamics is presented, where the metriplectic four-bracket is constructed by multiplying two skew-symmetric brackets. The new formulation is then used to introduce irreversibility in a generalized two-dimensional (2D) quasigeostrophic (QG) upper-ocean model involving advected quantities, with the thermal QG model as a special case. By construction, the resulting dynamics ensure the conservation of internal energy and the generation of entropy, in accordance with the first and second laws of thermodynamics. Our metriplectic dynamics formulation allows for a flexible specification of irreversibility, ranging from a type that results in nearly material conservation of potential vorticity to the representation of realistic forcing and dissipation in 2D QG ocean modeling with advected quantities.

math-ph↗

arXiv2025

Irreversible dynamics on Poisson manifolds

We present a geometric construction of irreversible dynamics on Poisson manifolds that satisfies the axioms of metriplectic mechanics and the GENERIC framework. Our approach relies solely on the underlying Poisson structure and its deformation theory, without requiring any additional metric structure. Specifically, we show that if the second Lichnerowicz-Poisson cohomology group of a Poisson manifold is nontrivial, one can construct a symmetric bracket that generates irreversible dynamics compatible with energy conservation and entropy production. This bracket is derived from a 2-cocycle that deforms the original Poisson structure, thereby modifying the associated Casimir foliation. We illustrate the construction with two finite-dimensional examples and one infinite-dimensional example: the duals of the Lie algebras of the special Euclidean group SE(2), the Galilei group SGal(3) and the group of diffeomorphisms over the circle. These examples demonstrate the applicability of the method in classical mechanics, control theory, and mathematical physics.

math-ph↗

arXiv2025

Symplectic techniques for stochastic differential equations on reductive Lie groups with applications to Langevin diffusions

We show how Langevin diffusions can be interpreted in the context of stochastic Hamiltonian systems with structure-preserving noise and dissipation on reductive Lie groups. Reductive Lie groups provide the setting in which the Lie group structure is compatible with Riemannian structures, via the existence of bi-invariant metrics. This structure allows for the explicit construction of Riemannian Brownian motion via symplectic techniques, which permits the study of Langevin diffusions with noise in the position coordinate as well as Langevin diffusions with noise in both momentum and position.

math.PR↗

arXiv2024

Dual Euler--Poincar\'e/Lie--Poisson formulation of subinertial stratified thermal ocean flow with identification of Casimirs as Noether quantities

This paper investigates the geometric structure of a quasigeostrophic approximation to a recently introduced reduced-gravity thermal rotating shallow-water model that accounts for stratification. Specifically, it considers a low-frequency approximation of a model for flow above the ocean thermocline, governed by primitive equations with buoyancy variations in both horizontal and vertical directions. Like the thermal model, the stratified variant generates circulation patterns reminiscent of submesoscale instabilities visible in satellite images. An improvement is its ability to model mixed-layer restratification due to baroclinic instability. The primary contribution of this paper is to demonstrate that the model is derived from an Euler--Poincar\'e variational principle, culminating in a Kelvin--Noether theorem, previously established solely for the primitive-equation parent model. The model's Lie--Poisson Hamiltonian structure, earlier obtained through direct calculation, is shown to result from a Legendre transform with the associated geometry elucidated by identifying the relevant momentum map. Another significant contribution of this paper is the identification of the Casimirs of the Lie--Poisson system, including a newly found weaker Casimir family forming the kernel of the Lie--Poisson bracket, which results in potential vorticity evolution independent of buoyancy details as it advects under the flow. These conservation laws related to particle relabeling symmetry are explicitly linked to Noether quantities from the Euler--Poincar\'e principle when variations are not constrained to vanish at integration endpoints. The dual Euler--Poincar\'e/Lie--Poisson formalism provides a unified framework for describing quasigeostrophic reduced-gravity stratified thermal flow, mirroring the approach used in the primitive-equation setting.

physics.flu-dyn↗

arXiv2024

Casimir preserving numerical method for global multilayer geostrophic turbulence

Accurate long-term predictions of large-scale flow features on planets are crucial for understanding global atmospheric and oceanic systems, necessitating the development of numerical methods that can preserve essential physical structures over extended simulation periods without excessive computational costs. Recent advancements in the study of global single-layer barotropic models have led to novel numerical methods based on Lie-Poisson discretization that preserve energy, enstrophy and higher-order moments of potential vorticity. This work extends this approach to more complex stratified quasi-geostrophic (QG) systems on the sphere. In this work, we present a formulation of the multi-layer QG equations on the full globe. This allows for extending the Lie-Poisson discretization to multi-layer QG models, ensuring consistency with the underlying structure and enabling long-term simulations without additional regularization. The numerical method is benchmarked through simulations of forced geostrophic turbulence and the long-term behaviour of unforced multi-layered systems. These results demonstrate the structure-preserving properties and robustness of the proposed numerical method, paving the way for a better understanding of the role of high-order conserved quantities in large-scale geophysical flow dynamics.

physics.flu-dyn↗

arXiv2024

Critical latitude in global quasi-geostrophic flow on a rotating sphere

In this paper, we study geostrophic turbulence without external forcing or dissipation, using a Casimir-preserving numerical method. The research examines the formation of large zonal jets, common in geophysical flows, especially in giant gas planets. These jets form due to the east-west stretching of vortices, influenced by the gradient of the Coriolis parameter, leading to a critical latitude beyond which jets do not form. Using a global quasi-geostrophic model with a fully latitude-dependent Coriolis parameter, we investigate this critical latitude, which is theorized to depend only on the product of the Rossby number and the Lamb parameter. By simulating random flow fields, the critical latitude was identified through zonally averaged zonal velocity profiles. Results align with geostrophic theory, especially near typical Rossby and Lamb parameter values for Earth's atmosphere. However, in the regime of weak rotation (high Rossby numbers) and strong stratification (high Lamb values), no clear critical latitude emerges; instead, zonal jet amplitude and width decrease gradually towards the poles. This research paves the way for further study of jet dynamics under a fully latitude-dependent Coriolis parameter.

physics.ao-ph↗

arXiv2024

An exponential map free implicit midpoint method for stochastic Lie-Poisson systems

An integrator for a class of stochastic Lie-Poisson systems driven by Stratonovich noise is developed. The integrator is suited for Lie-Poisson systems that also admit an isospectral formulation, which enables scalability to high-dimensional systems. Its derivation follows from discrete Lie-Poisson reduction of the symplectic midpoint scheme for stochastic Hamiltonian systems. We prove almost sure preservation of Casimir functions and coadjoint orbits under the numerical flow and provide strong and weak convergence rates of the proposed method. The scalability, structure-conservation, and convergence rates are illustrated numerically for the (generalized) rigid body, point vortex dynamics, and the two-dimensional Euler equations on the sphere.

math.NA↗

arXiv2024

Symmetry groups of geodesic equations with applications in water waves

In this work we derive several important equations in water waves and liquid crystals by deriving them as geodesic equations of right-invariant metrics on two infinite-dimensional groups. The equations we obtain this way are the Hopf (inviscid Burgers) equation, the Camassa-Holm equation, the Hunter-Saxton equation and the Korteweg-De Vries equation. We then study the symmetry groups of the equations themselves and show that one can improve the behaviour of the Hopf equation by metric and topological corrections. The symmetry groups of these equations can aid the benchmarking and testing of numerical methods.

math-ph↗

arXiv2024

A flexible numerical tool for large dynamic DC networks

DC networks play an important role within the ongoing energy transition. In this context, simulations of designed and existing networks and their corresponding assets are a core tool to get insights and form a support to decision-making. Hereby, these simulations of DC networks are executed in the time domain. Due to the involved high frequencies and the used controllers, the equations that model these DC networks are stiff and highly oscillatory differential equations. By exploiting sparsity, we show that conventional adaptive time stepping schemes can be used efficiently for the time domain simulation of very large DC networks and that this scales linearly in the computational cost as the size of the networks increase.

eess.SY↗

arXiv2024

Geometric derivation and structure-preserving simulation of quasi-geostrophy on the sphere

We present a geometric derivation of the quasi-geostrophic equations on the sphere, starting from the rotating shallow water equations. We utilise perturbation series methods in vorticity and divergence variables. The derivation employs asymptotic analysis techniques, leading to a global quasi-geostrophic potential vorticity model on the sphere without approximation of the Coriolis parameter. The resulting model forms a closed system for the evolution of potential vorticity with a rich mathematical structure, including Lagrangian and Hamiltonian descriptions. Formulated using the Lie-Poisson bracket reveals the geometric invariants of the quasi-geostrophic model. Motivated by these geometric results, simulations of quasi-geostrophic flow on the sphere are presented based on structure-preserving Lie-Poisson time-integration. We explicitly demonstrate the preservation of Casimir invariants and show that the hyperbolic quasi-geostrophic equations can be simulated in a stable manner over long time. We show the emergence of longitudonal jets, wrapped around the circumference of the sphere in a general direction that is perpendicular to the axis of rotation.

physics.flu-dyn↗

arXiv2023

A continuous data assimilation closure for modeling statistically steady turbulence in large-eddy simulation

A closure model is presented for large-eddy simulation (LES) based on the three-dimensional variational data assimilation algorithm. The approach aims at reconstructing high-fidelity kinetic energy spectra in coarse numerical simulations by including feedback control to represent unresolved dynamics interactions in the flow as stochastic processes. The forcing uses statistics obtained from offline high-fidelity data and requires only few parameters compared to the number of degrees of freedom of LES. This modeling strategy is applied to geostrophic turbulence on the sphere and enables simulating indefinitely at reduced costs. It proves to accurately recover the energy spectra and the zonal velocity profiles in the coarse model, for three generic situations.

physics.flu-dyn↗

arXiv2023

Stochastic Geometric Mechanics for 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. The present chapter describes the mathematical development of the framework of stochastic geometric mechanics in the context of fluid flow and wave dynamics obtained from Lie group-invariant variational principles.

physics.flu-dyn↗

arXiv2023

Intrinsic nonlinear elasticity: An exterior calculus formulation

In this paper we formulate the theory of nonlinear elasticity in a geometrically intrinsic manner using exterior calculus and bundle-valued differential forms. We represent kinematics variables, such as velocity and rate-of-strain, as intensive vector-valued forms while kinetics variables, such as stress and momentum, as extensive covector-valued pseudo-forms. We treat the spatial, material and convective representations of the motion and show how to geometrically convert from one representation to the other. Furthermore, we show the equivalence of our exterior calculus formulation to standard formulations in the literature based on tensor calculus. In addition, we highlight two types of structures underlying the theory. First, the principle bundle structure relating the space of embeddings to the space of Riemannian metrics on the body, and how the latter represents an intrinsic space of deformations. Second, the de Rham complex structure relating the spaces of bundle-valued forms to each other.

math-ph↗

arXiv2023

An explicit method to determine Casimirs in 2D geophysical flows

Conserved quantities in geophysical flows play an important role in the characterisation of geophysical dynamics and aid the development of structure-preserving numerical methods. A significant family of conserved quantities is formed by the Casimirs i.e., integral conservation laws that are in the kernel of the underlying Poisson bracket. The Casimirs hence determine the geometric structure of the geophysical fluid equations among which the enstrophy is well known. Often Casimirs are proposed on heuristic grounds and later verified to be part of the kernel of the Poisson bracket. In this work, we will explicitly construct Casimirs by rewriting the Poisson bracket in vorticity-divergence coordinates thereby providing explicit construction of Casimirs for 2D geophysical flow dynamics.

physics.flu-dyn↗

arXiv2022

Data-driven stochastic Lie transport modelling of the 2D Euler equations

In this paper, we propose and assess several stochastic parametrizations for data-driven modelling of the two-dimensional Euler equations using coarse-grid SPDEs. The framework of Stochastic Advection by Lie Transport (SALT) [Cotter et al., 2019] is employed to define a stochastic forcing that is decomposed in terms of a deterministic basis (empirical orthogonal functions, EOFs) multiplied by temporal traces, here regarded as stochastic processes. The EOFs are obtained from a fine-grid data set and are defined in conjunction with corresponding deterministic time series. We construct stochastic processes that mimic properties of the measured time series. In particular, the processes are defined such that the underlying probability density functions (pdfs) or the estimated correlation time of the time series are retained. These stochastic models are compared to stochastic forcing based on Gaussian noise, which does not use any information of the time series. We perform uncertainty quantification tests and compare stochastic ensembles in terms of mean and spread. Reduced uncertainty is observed for the developed models. On short timescales, such as those used for data assimilation [Cotter et al., 2020], the stochastic models show a reduced ensemble mean error and a reduced spread. Particularly, using estimated pdfs yields stochastic ensembles which rarely fail to capture the reference solution on small time scales, whereas introducing correlation into the stochastic models improves the quality of the coarse-grid predictions with respect to Gaussian noise.

physics.flu-dyn↗

arXiv2022

Casimir preserving spectrum of two-dimensional turbulence

We present predictions of the energy spectrum of forced two-dimensional turbulence obtained by employing a structure-preserving integrator. In particular, we construct a finite-mode approximation of the Navier-Stokes equations on the unit sphere, which, in the limit of vanishing viscosity, preserves the Lie-Poisson structure. As a result, integrated powers of vorticity are conserved in the inviscid limit. We obtain robust evidence for the existence of the double energy cascade, including the formation of the -3 scaling of the inertial range of the direct cascade. We show that this can be achieved at modest resolutions compared to those required by traditional numerical methods.

physics.flu-dyn↗

arXiv2022

Computational modeling for high-fidelity coarsening of shallow water equations based on subgrid data

Small-scale features of shallow water flow obtained from direct numerical simulation (DNS) with two different computational codes for the shallow water equations are gathered offline and subsequently employed with the aim of constructing a reduced-order correction. This is used to facilitate high-fidelity online flow predictions at much reduced costs on coarse meshes. The resolved small-scale features at high resolution represent subgrid properties for the coarse representation. Measurements of the subgrid dynamics are obtained as the difference between the evolution of a coarse grid solution and the corresponding DNS result. The measurements are sensitive to the particular numerical methods used for the simulation on coarse computational grids and can be used to approximately correct the associated discretization errors. The subgrid features are decomposed into empirical orthogonal functions (EOFs), after which a corresponding correction term is constructed. By increasing the number of EOFs in the approximation of the measured values the correction term can in principle be made arbitrarily accurate. Both computational methods investigated here show a significant decrease in the simulation error already when applying the correction based on the dominant EOFs only. The error reduction accounts for the particular discretization errors that incur and are hence specific to the particular simulation method that is adopted. This improvement is also observed for very coarse grids which may be used for computational model reduction in geophysical and turbulent flow problems.

physics.flu-dyn↗

arXiv2021

Casimir preserving stochastic Lie-Poisson integrators

Casimir preserving integrators for stochastic Lie-Poisson equations with Stratonovich noise are developed extending Runge-Kutta Munthe-Kaas methods. The underlying Lie-Poisson structure is preserved along stochastic trajectories. A related stochastic differential equation on the Lie algebra is derived. The solution of this differential equation updates the evolution of the Lie-Poisson dynamics by means of the exponential map. The constructed numerical method conserves Casimir-invariants exactly, which is important for long time integration. This is illustrated numerically for the case of the stochastic heavy top and the stochastic sine-Euler equations.

math.NA↗