Predicting uncertainty in geometric fluid mechanics

François Gay-Balmaz; Darryl D. Holm

doi:10.3934/dcdss.2020071

Article Contents

2020, Volume 13, Issue 4: 1229-1242. Doi: 10.3934/dcdss.2020071

This issue Previous Article On the geometry of twisted prolongations, and dynamical systems Next Article Sub-Riemannian geometry and finite time thermodynamics Part 1: The stochastic oscillator

Predicting uncertainty in geometric fluid mechanics

François Gay-Balmaz^1, and
Darryl D. Holm^2, ,

1.
CNRS - LMD - IPSL, École Normale Supérieure de Paris, 24 Rue Lhomond, 75005, Paris, France
2.
Department of Mathematics, Imperial College, London SW7 2AZ, UK, Springfield, MO 65810, USA

^* Corresponding author: Darryl D. Holm
^* Corresponding author: Darryl D. Holm

Received: November 30, 2017

Revised: July 31, 2018

Published: April 2020

Abstract Full Text(HTML) Related Papers Cited by

Abstract

We review opportunities for stochastic geometric mechanics to incorporate observed data into variational principles, in order to derive data-driven nonlinear dynamical models of effects on the variability of computationally resolvable scales of fluid motion, due to unresolvable, small, rapid scales of fluid motion.

Keywords:

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	S. Albeverio, A. B. Cruzeiro and D. D. Holm, Stochastic Geometric Mechanics, Springer, 2017.
[2]	A. Arnaudon, A. L. de Castro and D. D. Holm, Noise and dissipation on coadjoint orbits, J. Nonlin. Sci., 28 (2018), 91-145. doi: 10.1007/s00332-017-9404-3.
[3]	V. I. Arnol'd, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Annales de l'institut Fourier, 16 (1966), 319-361. doi: 10.5802/aif.233.
[4]	V. I. Arnol'd, Mathematical Methods of Classical Mechanics, volume 60 of Graduate Texts in Mathematics, Springer-Verlag, New York
[5]	J.-M. Bismut, Mécanique aléatoire, In Tenth Saint Flour Probability Summer School—1980 (Saint Flour, 1980), volume 929 of Lecture Notes in Math., pages 1–100, Springer, Berlin-New York, 1982.
[6]	N. Bou-Rabee and H. Owhadi, Stochastic variational integrators, IMA J. Numer. Anal., 29 (2009), 421-443. doi: 10.1093/imanum/drn018.
[7]	C. J. Cotter, G. A. Gottwald and D. D. Holm, Stochastic partial differential fluid equations as a diffusive limit of deterministic Lagrangian multi-time dynamics, Proc. Roy. Soc. A, 473 (2017), 20170388, 10pp. doi: 10.1098/rspa.2017.0388.
[8]	C. J. Cotter, D. Crisan, D. D. Holm, W. Pan and I. Shevchenko, Modelling uncertainty using circulation-preserving stochastic transport noise in a 2-layer quasi-geostrophic model, arXiv preprint, arXiv: 1802.05711.
[9]	D. Crisan, F. Flandoli and D. Holm, Solution properties of a 3d stochastic Euler fluid equation, arXiv: 1704.06989, [math-ph], 2017. doi: 10.1007/s00332-018-9506-6.
[10]	A. B. Cruzeiro, D. D. Holm and T. S. Ratiu, Momentum maps and stochastic Clebsch action principles, Commun. in Math. Phys., 357 (2018), 873-912. doi: 10.1007/s00220-017-3048-x.
[11]	F. Gay-Balmaz and D. D. Holm, Stochastic geometric models with non-stationary spatial correlations in Lagrangian fluid flows, J. Nonlin. Sci., 28 (2018), 873-904. doi: 10.1007/s00332-017-9431-0.
[12]	F. Gay-Balmaz and V. Putkaradze, On noisy extensions of nonholonomic constraints, J. Nonlin. Sci., 26 (2016), 1571-1613. doi: 10.1007/s00332-016-9313-x.
[13]	D. D. Holm, Variational principles for stochastic fluid dynamics, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 471 (2015), 20140963, 19 pp. doi: 10.1098/rspa.2014.0963.
[14]	J. E. Marsden and A. Weinstein, Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Physica D: Nonlinear Phenomena, 7 (1983), 305-323. doi: 10.1016/0167-2789(83)90134-3.
[15]	J. McWilliams, A note on a consistent quasigeostrophic model in a multiply connected domain, Dynam. Atmos. Ocean, 1 (1977), 427-441. doi: 10.1016/0377-0265(77)90002-1.
[16]	H. Yoshimura and F. Gay-Balmaz, Hamilton–Pontryagin principle for incompressible ideal fluids, AIP Conference Proceedings, 1376 (2011), 645-647. doi: 10.1063/1.3652002.
[17]	H. Yoshimura and J. E. Marsden, Dirac structures in Lagrangian mechanics. I. Implicit Lagrangian systems, J. Geom. Phys., 57 (2006), 133-156. doi: 10.1016/j.geomphys.2006.02.009.