Advanced Search
Article Contents
Article Contents

Predicting uncertainty in geometric fluid mechanics

  • * Corresponding author: Darryl D. Holm

    * Corresponding author: Darryl D. Holm
Abstract Full Text(HTML) Related Papers Cited by
  • 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.

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


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] S. Albeverio, A. B. Cruzeiro and D. D. Holm, Stochastic Geometric Mechanics, Springer, 2017.
    [2] A. ArnaudonA. 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. CruzeiroD. 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.
  • 加载中

Article Metrics

HTML views(666) PDF downloads(343) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint