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Dimensional reduction of direct statistical simulation

Published online by Cambridge University Press:  08 July 2020

Altan Allawala
Affiliation:
Department of Physics, Brown University, Providence, RI02912-1843, USA JPMorgan Chase and Co., 237 Park Avenue, New York, NY 10017, USA
S. M. Tobias
Affiliation:
Department of Applied Mathematics, University of Leeds, LeedsLS2 9JT, UK
J. B. Marston*
Affiliation:
Department of Physics, Brown University, Providence, RI02912-1843, USA Brown Theoretical Physics Center, Box S, Brown University, Providence, RI02912, USA
*
Email address for correspondence: marston@brown.edu

Abstract

Direct statistical simulation (DSS) solves the equations of motion for the statistics of turbulent flows in place of the traditional route of accumulating statistics by direct numerical simulation (DNS). That low-order statistics usually evolve slowly compared with instantaneous dynamics is one important advantage of DSS. Depending on the symmetry of the problem and the choice of averaging operation, however, DSS is usually more expensive computationally than DNS because even low-order statistics typically have higher dimension than the underlying fields. Here we show that it is in some cases possible to go much further by using a form of unsupervised learning, proper orthogonal decomposition, to address the ‘curse of dimensionality’. We apply proper orthogonal decomposition directly to DSS in the form of expansions in equal-time cumulants to second order. We explore two averaging operations (zonal and ensemble) and test the approach on two idealized barotropic models of fluid on a rotating sphere (a jet that relaxes deterministically towards an unstable profile and a stochastically driven flow that spontaneously organizes into jets). We show that the method offers the possibility of parameter continuation, in the reduced basis, for the lower-order statistics of the flow. Order-of-magnitude savings in computational cost are sometimes obtained in the reduced basis, potentially enabling access to parameter regimes beyond the reach of DNS.

Type
JFM Papers
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

Ait-Chaalal, F., Schneider, T., Meyer, B. & Marston, J. B. 2016 Cumulant expansions for atmospheric flows. New J. Phys. 18 (2), 025019.Google Scholar
Allawala, A. & Marston, J. B. 2016 Statistics of the stochastically forced Lorenz attractor by the Fokker–Planck equation and cumulant expansions. Phys. Rev. E 94, 052218(9).Google ScholarPubMed
Bakas, N. A. & Ioannou, P. J. 2011 Structural stability theory of two-dimensional fluid flow under stochastic forcing. J. Fluid Mech. 682, 332361.CrossRefGoogle Scholar
Bakas, N. A. & Ioannou, P. J. 2013 Emergence of large scale structure in barotropic 𝛽-plane turbulence. Phys. Rev. Lett. 110 (22), 224501.CrossRefGoogle ScholarPubMed
Bakas, N. A. & Ioannou, P. J. 2014 A theory for the emergence of coherent structures in beta-plane turbulence. J. Fluid Mech. 740, 312341.CrossRefGoogle Scholar
Barkley, D. 2016 Theoretical perspective on the route to turbulence in a pipe. J. Fluid Mech. 803, P1.CrossRefGoogle Scholar
Batchelor, G. K. 1947 Kolmogoroff’s theory of locally isotropic turbulence. In Mathematical Proceedings of the Cambridge Philosophical Society, vol. 43, pp. 533559. Cambridge University Press.Google Scholar
Bauer, P., Thorpe, A. & Brunet, G. 2015 The quiet revolution of numerical weather prediction. Nature 525 (7567), 4755.CrossRefGoogle ScholarPubMed
Bellman, R. E. 1957 Dynamic Programming. Princeton University Press.Google ScholarPubMed
Bellman, R. E. 1961 Adaptive Control Processes: A Guided Tour. Princeton University Press.CrossRefGoogle Scholar
Bergman, L. A. & Spencer, B. F. Jr. 1992 Robust numerical solution of the transient Fokker–Planck equation for nonlinear dynamical systems. In Nonlinear Stochastic Mechanics, pp. 4960. Springer.CrossRefGoogle Scholar
Bouchet, F., Marston, J. B. & Tangarife, T. 2018 Fluctuations and large deviations of Reynolds stresses in zonal jet dynamics. Phys. Fluids 30 (1), 015110–20.CrossRefGoogle Scholar
Bouchet, F. & Simonnet, E. 2009 Random changes of flow topology in two-dimensional and geophysical turbulence. Phys. Rev. Lett. 102 (9), 094504–4.CrossRefGoogle ScholarPubMed
Constantinou, N. C., Farrell, B. F. & Ioannou, P. J. 2014 Emergence and equilibration of jets in beta-plane turbulence: applications of stochastic structural stability theory. J. Atmos. Sci. 71, 18181842.CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 2007 Structure and spacing of jets in barotropic turbulence. J. Atmos. Sci. 64 (10), 36523665.CrossRefGoogle Scholar
Frisch, U. 1995 Turbulence: the Legacy of AN Kolmogorov. Cambridge University Press.CrossRefGoogle Scholar
Frishman, A., Laurie, J. & Falkovich, G. 2017 Jets or vortices – What flows are generated by an inverse turbulent cascade? Phys. Rev. Fluids 2 (3), 032602.CrossRefGoogle Scholar
Ghahramani, Z. 2003 Unsupervised learning. In Advanced Lectures on Machine Learning (ML Summer Schools 2003). Springer.Google Scholar
Hänggi, P. & Talkner, P. 1980 A remark on truncation schemes of cumulant hierarchies. J. Stat. Phys. 22 (1), 6567.CrossRefGoogle Scholar
Herring, J. R. 1963 Investigation of problems in thermal convection. J. Atmos. Sci. 20 (4), 325338.2.0.CO;2>CrossRefGoogle Scholar
Holloway, G. & Hendershott, M. C. 1977 Stochastic closure for nonlinear Rossby waves. J. Fluid Mech. 82 (04), 747765.CrossRefGoogle Scholar
Holmes, P., Lumley, J. L. & Berkooz, G. 1998 Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press.Google Scholar
Huang, H.-P., Galperin, B. & Sukoriansky, S. 2001 Anisotropic spectra in two-dimensional turbulence on the surface of a rotating sphere. Phys. Fluids 13 (1), 225240.CrossRefGoogle Scholar
Kraichnan, R. H. 1980 Realizability inequalities and closed moment equations. Ann. N.Y. Acad. Sci. 357 (1), 3746.CrossRefGoogle Scholar
Kumar, P. & Narayanan, S. 2006 Solution of Fokker–Planck equation by finite element and finite difference methods for nonlinear systems. Sadhana 31 (4), 445461.CrossRefGoogle Scholar
Laurie, J., Boffetta, G., Falkovich, G., Kolokolov, I. & Lebedev, V. 2014 Universal profile of the vortex condensate in two-dimensional turbulence. Phys. Rev. Lett. 113 (25), 254503–5.CrossRefGoogle ScholarPubMed
Laurie, J. & Bouchet, F. 2015 Computation of rare transitions in the barotropic quasi-geostrophic equations. New J. Phys. 17 (1), 125.Google Scholar
Legras, B. 1980 Turbulent phase shift of Rossby waves. Geophys. Astrophys. Fluid Dyn. 15 (1), 253281.CrossRefGoogle Scholar
Lorenz, E. N. 1967 The Nature and Theory of the General Circulation of the Atmosphere, vol. 218. World Meteorological Organization Geneva.Google Scholar
Marston, J. B. 2010 Statistics of the general circulation from cumulant expansions. Chaos 20, 041107.CrossRefGoogle ScholarPubMed
Marston, J. B. 2012 Planetary atmospheres as nonequilibrium condensed matter. Annu. Rev. Condens. Matter Phys. 3 (1), 285310.CrossRefGoogle Scholar
Marston, J. B., Chini, G. P. & Tobias, S. M. 2016 Generalized quasilinear approximation: application to zonal jets. Phys. Rev. Lett. 116 (21), 214501.CrossRefGoogle ScholarPubMed
Marston, J. B., Conover, E. & Schneider, T. 2008 Statistics of an unstable barotropic jet from a cumulant expansion. J. Atmos. Sci. 65, 19551966.CrossRefGoogle Scholar
Marston, J. B., Qi, W. & Tobias, S. M. 2019 Direct Statistical Simulation of a Jet in Zonal Jets: Phenomenology, Genesis, Physics. Cambridge University Press.Google Scholar
Muld, T. W., Efraimsson, G. & Henningson, D. S. 2012 Flow structures around a high-speed train extracted using proper orthogonal decomposition and dynamic mode decomposition. Comput. Fluids 57, 8797.CrossRefGoogle Scholar
Naess, A. & Hegstad, B. K. 1994 Response statistics of van der Pol oscillators excited by white noise. Nonlinear Dyn. 5 (3), 287297.Google Scholar
O’Gorman, P. A. & Schneider, T. 2007 Recovery of atmospheric flow statistics in a general circulation model without nonlinear eddy-eddy interactions. Geophys. Res. Lett. 34, L22801.Google Scholar
Pichler, L., Masud, A. & Bergman, L. A. 2013 Numerical solution of the Fokker–Planck equation by finite difference and finite element methods – a comparative study. In Computational Methods in Stochastic Dynamics, pp. 6985. Springer.CrossRefGoogle Scholar
Resseguier, V., Mémin, E. & Chapron, B. 2015 Stochastic fluid dynamic model and dimensional reduction. In International Symposium on Turbulence and Shear Flow Phenomena (TSFP-9), pp. 16. Begell House.Google Scholar
Salmon, R. 1998 Lectures on Geophysical Fluid Dynamics. Oxford University Press.Google Scholar
Schoeberl, M. R. & Lindzen, R. S. 1984 A numerical simulation of barotropic instability. Part I: wave-mean flow interaction. J. Atmos. Sci. 41 (8), 13681379.2.0.CO;2>CrossRefGoogle Scholar
Skitka, J., Marston, J. B. & Fox-Kemper, B. 2020 Reduced-order quasilinear model of ocean boundary-layer turbulence. J. Phys. Oceanogr. 50 (3), 537558.CrossRefGoogle Scholar
Squire, J. & Bhattacharjee, A. 2015 Generation of large-scale magnetic fields by small-scale dynamo in shear flows. Phys. Rev. Lett. 115 (17), 175003.CrossRefGoogle ScholarPubMed
Tobias, S.2019 The turbulent dynamo. arXiv e-prints, arXiv:1907.03685.Google Scholar
Tobias, S. M., Dagon, K. & Marston, J. B. 2011 Astrophysical fluid dynamics via direct statistical simulation. Astrophys. J. 727 (2), 127.CrossRefGoogle Scholar
Tobias, S. M. & Marston, J. B. 2013 Direct statistical simulation of out-of-equilibrium jets. Phys. Rev. Lett. 110 (10), 104502.CrossRefGoogle ScholarPubMed
Tobias, S. M. & Marston, J. B. 2017 Direct statistical simulation of jets and vortices in 2D flows. Phys. Fluids 29, 111111.CrossRefGoogle Scholar
Varadhan, S. R. S. 1966 Asymptotic probabilities and differential equations. Commun. Pure Appl. Maths 19 (3), 261286.CrossRefGoogle Scholar
von Wagner, U. & Wedig, W. V. 2000 On the calculation of stationary solutions of multi-dimensional Fokker–Planck equations by orthogonal functions. Nonlinear Dyn. 21 (3), 289306.Google Scholar