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April  2020, 13(4): 1229-1242. doi: 10.3934/dcdss.2020071

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

Received  December 2017 Revised  August 2018 Published  April 2019

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: Dimension theory, Poincaré recurrences, multifractal analysis.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: François Gay-Balmaz, Darryl D. Holm. Predicting uncertainty in geometric fluid mechanics. Discrete and Continuous Dynamical Systems - S, 2020, 13 (4) : 1229-1242. doi: 10.3934/dcdss.2020071
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.

show all references

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.

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