1
|
Tyranowski TM. Stochastic variational principles for the collisional Vlasov-Maxwell and Vlasov-Poisson equations. Proc Math Phys Eng Sci 2021; 477:20210167. [PMID: 35153571 PMCID: PMC8385346 DOI: 10.1098/rspa.2021.0167] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/23/2021] [Accepted: 07/16/2021] [Indexed: 11/24/2022] Open
Abstract
In this work, we recast the collisional Vlasov-Maxwell and Vlasov-Poisson equations as systems of coupled stochastic and partial differential equations, and we derive stochastic variational principles which underlie such reformulations. We also propose a stochastic particle method for the collisional Vlasov-Maxwell equations and provide a variational characterization of it, which can be used as a basis for a further development of stochastic structure-preserving particle-in-cell integrators.
Collapse
Affiliation(s)
- Tomasz M. Tyranowski
- Max-Planck-Institut für Plasmaphysik, Boltzmannstraße 2, Garching 85748, Germany
- Technische Universität München, Zentrum Mathematik, Boltzmannstraße 3, Garching 85748, Germany
| |
Collapse
|
2
|
Hofmanová M, Leahy JM, Nilssen T. On a rough perturbation of the Navier–Stokes system and its vorticity formulation. ANN APPL PROBAB 2021. [DOI: 10.1214/20-aap1603] [Citation(s) in RCA: 7] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
|
3
|
Abstract
Spearheaded by the recent efforts to derive stochastic geophysical fluid dynamics models, we present a general framework for introducing stochasticity into variational principles through the concept of a semi-martingale driven variational principle and constraining the component variables to be compatible with the driving semi-martingale. Within this framework and the corresponding choice of constraints, the Euler–Poincaré equation can be easily deduced. We show that the deterministic theory is a special case of this class of stochastic variational principles. Moreover, this is a natural framework that enables us to correctly characterize the pressure term in incompressible stochastic fluid models. Other general constraints can also be incorporated as long as they are compatible with the driving semi-martingale.
Collapse
Affiliation(s)
- O. D. Street
- Department of Mathematics, Imperial College London, London, UK
| | - D. Crisan
- Department of Mathematics, Imperial College London, London, UK
| |
Collapse
|
4
|
Abstract
In this paper we propose a stochastic model reduction procedure for deterministic equations from geophysical fluid dynamics. Once large-scale and small-scale components of the dynamics have been identified, our method consists in modelling stochastically the small scales and, as a result, we obtain that a transport-type Stratonovich noise is sufficient to model the influence of the small scale structures on the large scales ones. This work aims to contribute to motivate the use of stochastic models in fluid mechanics and identifies examples of noise of interest for the reduction of complexity of the interaction between scales. The ideas are presented in full generality and applied to specific examples in the last section.
Collapse
|
5
|
Holm DD. Stochastic modelling in fluid dynamics: Itô versus Stratonovich. Proc Math Phys Eng Sci 2020; 476:20190812. [PMID: 32518504 PMCID: PMC7277131 DOI: 10.1098/rspa.2019.0812] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/21/2019] [Accepted: 04/09/2020] [Indexed: 11/25/2022] Open
Abstract
Suppose the observations of Lagrangian trajectories for fluid flow in some physical situation can be modelled sufficiently accurately by a spatially correlated Itô stochastic process (with zero mean) obtained from data which is taken in fixed Eulerian space. Suppose we also want to apply Hamilton’s principle to derive the stochastic fluid equations for this situation. Now, the variational calculus for applying Hamilton’s principle requires the Stratonovich process, so we must transform from Itô noise in the data frame to the equivalent Stratonovich noise. However, the transformation from the Itô process in the data frame to the corresponding Stratonovich process shifts the drift velocity of the transformed Lagrangian fluid trajectory out of the data frame into a non-inertial frame obtained from the Itô correction. The issue is, ‘Will non-inertial forces arising from this transformation of reference frames make a difference in the interpretation of the solution behaviour of the resulting stochastic equations?’ This issue will be resolved by elementary considerations.
Collapse
Affiliation(s)
- Darryl D Holm
- Mathematics Department, Imperial College London, London, UK
| |
Collapse
|
6
|
Pinier B, Mémin E, Laizet S, Lewandowski R. Stochastic flow approach to model the mean velocity profile of wall-bounded flows. Phys Rev E 2019; 99:063101. [PMID: 31330641 DOI: 10.1103/physreve.99.063101] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/06/2018] [Indexed: 11/07/2022]
Abstract
There is no satisfactory model to explain the mean velocity profile of the whole turbulent layer in canonical wall-bounded flows. In this paper, a mean velocity profile expression is proposed for wall-bounded turbulent flows based on a recently proposed stochastic representation of fluid flows dynamics. This original approach, called modeling under location uncertainty, introduces in a rigorous way a subgrid term generalizing the eddy-viscosity assumption and an eddy-induced advection term resulting from turbulence inhomogeneity. This latter term gives rise to a theoretically well-grounded model for the transitional zone between the viscous sublayer and the turbulent sublayer. An expression of the small-scale velocity component is also provided in the viscous zone. Numerical assessments of the results are provided for turbulent boundary layer flows, pipe flows and channel flows at various Reynolds numbers.
Collapse
Affiliation(s)
- Benoît Pinier
- Inria/IRMAR/U. Rennes I, Campus universitaire de Beaulieu, 35042 Rennes Cedex, France
| | - Etienne Mémin
- Inria/IRMAR/U. Rennes I, Campus universitaire de Beaulieu, 35042 Rennes Cedex, France
| | - Sylvain Laizet
- Department of Aeronautics, Imperial College London, South Kensington campus, London SW7 2AZ, United Kingdom
| | - Roger Lewandowski
- IRMAR/U. Rennes I, Campus universitaire de Beaulieu, 35042 Rennes Cedex, France
| |
Collapse
|
7
|
Abstract
We use a Hamiltonian interacting particle system to derive a stochastic mean field system whose McKean–Vlasov equation yields the incompressible Navier–Stokes equation. Since the system is Hamiltonian, the particle relabeling symmetry implies a Kelvin Circulation Theorem along stochastic Lagrangian paths. Moreover, issues of energy dissipation are discussed and the model is connected to other approaches in the literature.
Collapse
|
8
|
Balasuriya S, Gottwald GA. Estimating stable and unstable sets and their role as transport barriers in stochastic flows. Phys Rev E 2018; 98:013106. [PMID: 30110781 DOI: 10.1103/physreve.98.013106] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/02/2018] [Indexed: 11/07/2022]
Abstract
We consider the situation of a large-scale stationary flow subjected to small-scale fluctuations. Assuming that the stable and unstable manifolds of the large-scale flow are known, we quantify the mean behavior and stochastic fluctuations of particles close to the unperturbed stable and unstable manifolds and their evolution in time. The mean defines a smooth curve in physical space, while the variance provides a time- and space-dependent quantitative estimate where particles are likely to be found. This allows us to quantify transport properties such as the expected volume of mixing as the result of the stochastic fluctuations of the transport barriers. We corroborate our analytical findings with numerical simulations in both compressible and incompressible flow situations. We moreover demonstrate the intimate connection of our results with finite-time Lyapunov exponent fields, and with spatial mixing regions.
Collapse
Affiliation(s)
- Sanjeeva Balasuriya
- School of Mathematical Sciences, University of Adelaide, Adelaide SA 5005, Australia
| | - Georg A Gottwald
- School of Mathematics and Statistics, University of Sydney, Sydney NSW 2006, Australia
| |
Collapse
|
9
|
Holm DD, Tyranowski TM. New variational and multisymplectic formulations of the Euler-Poincaré equation on the Virasoro-Bott group using the inverse map. Proc Math Phys Eng Sci 2018; 474:20180052. [PMID: 29887752 DOI: 10.1098/rspa.2018.0052] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/29/2018] [Accepted: 04/09/2018] [Indexed: 11/12/2022] Open
Abstract
We derive a new variational principle, leading to a new momentum map and a new multisymplectic formulation for a family of Euler-Poincaré equations defined on the Virasoro-Bott group, by using the inverse map (also called 'back-to-labels' map). This family contains as special cases the well-known Korteweg-de Vries, Camassa-Holm and Hunter-Saxton soliton equations. In the conclusion section, we sketch opportunities for future work that would apply the new Clebsch momentum map with 2-cocycles derived here to investigate a new type of interplay among nonlinearity, dispersion and noise.
Collapse
Affiliation(s)
- Darryl D Holm
- Mathematics Department, Imperial College London, London SW7 2AZ, UK
| | - Tomasz M Tyranowski
- Mathematics Department, Imperial College London, London SW7 2AZ, UK.,Max-Planck-Institut für Plasmaphysik, Boltzmannstraße 2, 85748 Garching, Germany
| |
Collapse
|
10
|
Gay-Balmaz F, Holm DD. Stochastic Geometric Models with Non-stationary Spatial Correlations in Lagrangian Fluid Flows. JOURNAL OF NONLINEAR SCIENCE 2018; 28:873-904. [PMID: 29769757 PMCID: PMC5943459 DOI: 10.1007/s00332-017-9431-0] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/16/2017] [Accepted: 12/02/2017] [Indexed: 05/29/2023]
Abstract
Inspired by spatiotemporal observations from satellites of the trajectories of objects drifting near the surface of the ocean in the National Oceanic and Atmospheric Administration's "Global Drifter Program", this paper develops data-driven stochastic models of geophysical fluid dynamics (GFD) with non-stationary spatial correlations representing the dynamical behaviour of oceanic currents. Three models are considered. Model 1 from Holm (Proc R Soc A 471:20140963, 2015) is reviewed, in which the spatial correlations are time independent. Two new models, called Model 2 and Model 3, introduce two different symmetry breaking mechanisms by which the spatial correlations may be advected by the flow. These models are derived using reduction by symmetry of stochastic variational principles, leading to stochastic Hamiltonian systems, whose momentum maps, conservation laws and Lie-Poisson bracket structures are used in developing the new stochastic Hamiltonian models of GFD.
Collapse
Affiliation(s)
- François Gay-Balmaz
- Laboratoire de Météorologie Dynamique, CNRS and École Normale Supérieure de Paris, 24 Rue Lhomond, 75005 Paris, France
| | - Darryl D. Holm
- Department of Mathematics, Imperial College, London, SW7 2AZ UK
| |
Collapse
|
11
|
Holm DD, Putkaradze V. Dynamics of non-holonomic systems with stochastic transport. Proc Math Phys Eng Sci 2018; 474:20170479. [DOI: 10.1098/rspa.2017.0479] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/13/2017] [Accepted: 12/01/2017] [Indexed: 11/12/2022] Open
Abstract
This paper formulates a variational approach for treating observational uncertainty and/or computational model errors as stochastic transport in dynamical systems governed by action principles under non-holonomic constraints. For this purpose, we derive, analyse and numerically study the example of an unbalanced spherical ball rolling under gravity along a stochastic path. Our approach uses the Hamilton–Pontryagin variational principle, constrained by a stochastic rolling condition, which we show is equivalent to the corresponding stochastic Lagrange–d’Alembert principle. In the example of the rolling ball, the stochasticity represents uncertainty in the observation and/or error in the computational simulation of the angular velocity of rolling. The influence of the stochasticity on the deterministically conserved quantities is investigated both analytically and numerically. Our approach applies to a wide variety of stochastic, non-holonomically constrained systems, because it preserves the mathematical properties inherited from the variational principle.
Collapse
Affiliation(s)
- D. D. Holm
- Department of Mathematics, Imperial College, London SW7 2AZ, UK
| | - V. Putkaradze
- Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G2G1
| |
Collapse
|
12
|
Arnaudon A, De Castro AL, Holm DD. Noise and Dissipation on Coadjoint Orbits. JOURNAL OF NONLINEAR SCIENCE 2017; 28:91-145. [PMID: 29367809 PMCID: PMC5756579 DOI: 10.1007/s00332-017-9404-3] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/11/2017] [Accepted: 06/12/2017] [Indexed: 06/01/2023]
Abstract
We derive and study stochastic dissipative dynamics on coadjoint orbits by incorporating noise and dissipation into mechanical systems arising from the theory of reduction by symmetry, including a semidirect product extension. Random attractors are found for this general class of systems when the Lie algebra is semi-simple, provided the top Lyapunov exponent is positive. We study in details two canonical examples, the free rigid body and the heavy top, whose stochastic integrable reductions are found and numerical simulations of their random attractors are shown.
Collapse
Affiliation(s)
- Alexis Arnaudon
- Department of Mathematics, Imperial College, London, SW7 2AZ UK
| | - Alex L. De Castro
- Department of Mathematics, Imperial College, London, SW7 2AZ UK
- Departamento de Matemática, PUC-Rio, Rio de Janeiro, 22451-900 Brazil
| | - Darryl D. Holm
- Department of Mathematics, Imperial College, London, SW7 2AZ UK
| |
Collapse
|