Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-25T14:48:00.690Z Has data issue: false hasContentIssue false
  • English
  • Français

Direct control of the small-scale energy balance in two-dimensional fluid dynamics

Published online by Cambridge University Press:  07 October 2015

Jason Frank ,
Benedict Leimkuhler  and
Keith W. Myerscough
Jason Frank
Affiliation:
Mathematical Institute, Utrecht University, PO Box 80010, 3508 TA Utrecht, The Netherlands
Benedict Leimkuhler
Affiliation:
School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, James Clerk Maxwell Building, King’s Buildings, Edinburgh EH9 3JZ, UK
Keith W. Myerscough*
Affiliation:
Centrum Wiskunde and Informatica, PO Box 94079, 1090 GB Amsterdam, The Netherlands
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We explore the direct modification of the pseudo-spectral truncation of two-dimensional, incompressible fluid dynamics to maintain a prescribed kinetic energy spectrum. The method provides a means of simulating fluid states with defined spectral properties, for the purpose of matching simulation statistics to given information, arising from observations, theoretical prediction or high-fidelity simulation. In the scheme outlined here, Nosé–Hoover thermostats, commonly used in molecular dynamics, are introduced as feedback controls applied to energy shells of the Fourier-discretized Navier–Stokes equations. As we demonstrate in numerical experiments, the dynamical properties (quantified using autocorrelation functions) are only modestly perturbed by our device, while ensemble dispersion is significantly enhanced compared with simulations of a corresponding truncation incorporating hyperviscosity.

JFM classification

Geophysical and Geological Flows: Atmospheric flows Mathematical Foundations: Computational methods Turbulent Flows: Homogeneous turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 782 , 10 November 2015 , pp. 240 - 259
Check for updates
Copyright
© 2015 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramov, R. V. & Majda, A. J. 2003 Statistically relevant conserved quantities for truncated quasigeostrophic flow. Proc. Natl Acad. Sci. USA 100 (7), 3841–3846.CrossRefGoogle ScholarPubMed
Allen, M. P. & Tildesley, D. J. 1989 Computer Simulation of Liquids. (Oxford Science Publications) , vol. 57. Oxford University Press.Google Scholar
Batchelor, G. K. 1969 Computation of the energy spectrum in homogeneous two-dimensional turbulence. Phys. Fluids 12 (12), II–233.CrossRefGoogle Scholar
Berner, J., Shutts, G. J., Leutbecher, M. & Palmer, T. N. 2009 A spectral stochastic kinetic energy backscatter scheme and its impact on flow-dependent predictability in the ECMWF ensemble prediction system. J. Atmos. Sci. 66 (3), 603–626.CrossRefGoogle Scholar
Bofetta, G. & Musacchio, S. 2010 Evidence for the double cascade scenario in two-dimensional turbulence. Phys. Rev. E 82 (1), 016307.Google Scholar
Bouchet, F. & Venaille, A. 2012 Statistical mechanics of two-dimensional and geophysical flows. Phys. Rep. 515 (5), 227–295.CrossRefGoogle Scholar
Bracco, A., McWilliams, J., Murante, G., Provenzale, A. & Weiss, J. B. 2000 Revisiting freely decaying two-dimensional turbulence at millennial resolution. Phys. Fluids 12 (11), 2931–2941.CrossRefGoogle Scholar
Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 2006 Spectral Methods: Fundamentals in Single Domains. Springer.CrossRefGoogle Scholar
Danilov, S. D. & Gurarie, D. 2000 Quasi-two-dimensional turbulence. Phys. Uspekhi 43 (9), 863–900.CrossRefGoogle Scholar
Delong, S., Griffith, B. E., Vanden-Eijnden, E. & Donev, A. 2013 Temporal integrators for fluctuating hydrodynamics. Phys. Rev. E 87 (3), 033302.CrossRefGoogle Scholar
DelSole, T. 2000 A fundamental limitation of Markov models. J. Atmos. Sci. 57, 2158–2168.2.0.CO;2>CrossRefGoogle Scholar
Domaradzki, J. A. & Saiki, E. M. 1997 Backscatter models for large-eddy simulations. Theor. Comput. Fluid Dyn. 9 (2), 75–83.CrossRefGoogle Scholar
Donev, A., Vanden-Eijnden, E., Garcia, A. & Bell, J. 2010 On the accuracy of finite-volume schemes for fluctuating hydrodynamics. Commun. Appl. Maths Comput. Sci. 5 (2), 149–197.CrossRefGoogle Scholar
Dubinkina, S. & Frank, J. 2007 Statistical mechanics of Arakawa’s discretizations. J. Comput. Phys. 227 (2), 1286–1305.CrossRefGoogle Scholar
Dubinkina, S. & Frank, J. 2010 Statistical relevance of vorticity conservation in the hamiltonian particle-mesh method. J. Comput. Phys. 229 (7), 2634–2648.CrossRefGoogle Scholar
E, W., Mattingly, J. C. & Sinai, Y. 2001 Gibbsian dynamics and ergodicity for the stochastically forced Navier–Stokes equation. Commun. Math. Phys. 224 (1), 83–106.CrossRefGoogle Scholar
E, W. & Mattingly, J. C. 2001 Ergodicity for the Navier–Stokes equation with degenerate random forcing: finite-dimensional approximation. Commun. Pure Appl. Math. 54 (11), 1386–1402.CrossRefGoogle Scholar
Evensen, G. 2009 Data Assimilation: the Ensemble Kalman Filter, 2nd edn. Springer.CrossRefGoogle Scholar
Farazmand, M. M., Kevlahan, N. K.-R. & Protas, B. 2011 Controlling the dual cascade of two-dimensional turbulence. J. Fluid Mech. 668, 202–222.CrossRefGoogle Scholar
Farge, M., Kevlahan, N., Perrier, V. & Goirand, E. 1996 Wavelets and turbulence. Proc. IEEE 84 (4), 639–669.CrossRefGoogle Scholar
Fatkullin, I. & Vanden-Eijnden, E. 2004 A computational strategy for multiscale systems with applications to lorenz 96 model. J. Comput. Phys. 200 (2), 605–638.CrossRefGoogle Scholar
Frank, J. & Gottwald, G. 2011 The Langevin limit of the Nosé–Hoover–Langevin thermostat. J. Stat. Phys. 143 (4), 715–724.CrossRefGoogle Scholar
Frenkel, D. & Smit, B. 2002 Understanding Molecular Simulation: From Algorithms to Applications. Springer.Google Scholar
Frisch, U. 1995 Turbulence: the Legacy of A. N. Kolmogorov. Cambridge University Press.CrossRefGoogle Scholar
Fröhlich, J. & Schneider, K. 1999 Computation of decaying turbulence in an adaptive wavelet basis. Physica D 134 (3), 337–361.CrossRefGoogle Scholar
Gotoh, T. 1998 Energy spectrum in the inertial and dissipation ranges of two-dimensional steady turbulence. Phys. Rev. E 57 (3), 2984.CrossRefGoogle Scholar
Grooms, I. & Majda, A. J. 2013 Efficient stochastic superparameterization for geophysical turbulence. Proc. Natl Acad. Sci. USA 110 (12), 4464–4469.CrossRefGoogle ScholarPubMed
Hairer, E., Lubich, C. & Wanner, G. 2006 Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, vol. 31. Springer.Google Scholar
Hasselmann, K. 1976 Stochastic climate models part I. Theory. Tellus 28 (6), 473–485.Google Scholar
Hoover, W. G. 1985 Canonical dynamics: equilibrium phase-space distributions. Phys. Rev. A 31 (3), 1695.CrossRefGoogle ScholarPubMed
Jansen, M. F. & Held, I. M. 2014 Parameterizing subgrid-scale eddy effects using energetically consistent backscatter. Ocean Model. 80, 36–48.CrossRefGoogle Scholar
Jones, A. & Leimkuhler, B. 2011 Adaptive stochastic methods for sampling driven molecular systems. J. Chem. Phys. 135 (8), 084125.CrossRefGoogle ScholarPubMed
Kent, J., Thuburn, J. & Wood, N. 2012 Assessing implicit large eddy simulation for two-dimensional flow. Q. J. R. Meteorol. Soc. 138 (663), 365–376.CrossRefGoogle Scholar
Kraichnan, R. H. 1967 Inertial ranges in two-dimensional turbulence. Phys. Fluids 10, 1417–1423.CrossRefGoogle Scholar
Kraichnan, R. H. 1971 Inertial-range transfer in two- and three-dimensional turbulence. J. Fluid Mech. 47 (03), 525–535.CrossRefGoogle Scholar
Leimkuhler, B. & Matthews, C. 2013 Rational construction of stochastic numerical methods for molecular sampling. Appl. Math. Res. Express 2013 (1), 34–56.Google Scholar
Leimkuhler, B., Noorizadeh, E. & Penrose, O. 2011 Comparing the efficiencies of stochastic isothermal molecular dynamics models. J. Stat. Phys. 143 (5), 921–942.CrossRefGoogle Scholar
Leimkuhler, B., Noorizadeh, E. & Theil, F. 2009 A gentle thermostat for molecular dynamics. J. Stat. Phys. 135 (2), 261–277.CrossRefGoogle Scholar
Leimkuhler, B. & Reich, S. 2004 Simulating Hamiltonian Dynamics. Cambridge University Press.Google Scholar
Leith, C. E. 1968 Diffusion approximation for two-dimensional turbulence. Phys. Fluids 11 (3), 671–672.CrossRefGoogle Scholar
Lindborg, E. 1999 Can the atmospheric kinetic energy spectrum be explained by two-dimensional turbulence? J. Fluid Mech. 388, 259–288.CrossRefGoogle Scholar
Majda, A. J., Timofeyev, I. & Vanden Eijnden, E. 2001 A mathematical framework for stochastic climate models. Commun. Pure Appl. Maths 54 (8), 891–974.CrossRefGoogle Scholar
McLachlan, R. I. & Quispel, G. R. W. 2002 Splitting methods. Acta Numerica 11, 341–434.CrossRefGoogle Scholar
Miller, J. 1990 Statistical mechanics of euler equations in two dimensions. Phys. Rev. Lett. 65 (17), 2137.CrossRefGoogle ScholarPubMed
Mori, H. 1965 Transport, collective motion, and brownian motion. Prog. Theor. Phys. 33 (3), 423–455.CrossRefGoogle Scholar
Nastrom, G. D. & Gage, K. S. 1985 A climatology of atmospheric wavenumber spectra of wind and temperature observed by commercial aircraft. J. Atmos. Sci. 42 (9), 950–960.2.0.CO;2>CrossRefGoogle Scholar
Nastrom, G. D., Gage, K. S. & Jasperson, W. H. 1984 Kinetic energy spectrum of large- and mesoscale atmospheric processes. Nature 310, 36–38.CrossRefGoogle Scholar
Nosé, S. 1984a A molecular dynamics method for simulations in the canonical ensemble. Mol. Phys. 52 (2), 255–268.CrossRefGoogle Scholar
Nosé, S. 1984b A unified formulation of the constant temperature molecular dynamics methods. J. Chem. Phys. 81 (1), 511–519.CrossRefGoogle Scholar
Robert, R. 1991 A maximum-entropy principle for two-dimensional perfect fluid dynamics. J. Stat. Phys. 65 (3–4), 531–553.CrossRefGoogle Scholar
Robert, R. & Sommeria, J. 1991 Statistical equilibrium states for two-dimensional flows. J. Fluid Mech. 229, 291–310.CrossRefGoogle Scholar
Saffman, P. G. 1971 On the spectrum and decay of random two-dimensional vorticity distributions at large Reynolds number. Stud. Appl. Maths 50, 377–383.CrossRefGoogle Scholar
Samoletov, A. A., Dettmann, C. P. & Chaplain, M. A. J. 2007 Thermostats for ‘slow’ configurational modes. J. Stat. Phys. 128 (6), 1321–1336.CrossRefGoogle Scholar
Shutts, G. 2005 A kinetic energy backscatter algorithm for use in ensemble prediction systems. Q. J. R. Meteorol. Soc. 131 (612), 3079–3102.CrossRefGoogle Scholar
Sukoriansky, S., Galperin, B. & Chekhlov, A. 1999 Large scale drag representation in simulations of two-dimensional turbulence. Phys. Fluids 11 (10), 3043–3053.CrossRefGoogle Scholar
Thuburn, J., Kent, J. & Wood, N. 2014 Cascades, backscatter and conservation in numerical models of two-dimensional turbulence. Q. J. R. Meteorol. Soc. 140 (679), 626–638.CrossRefGoogle Scholar
Tulloch, R. & Smith, K. S. 2006 A theory for the atmospheric energy spectrum: depth-limited temperature anomalies at the tropopause. Proc. Natl Acad. Sci. USA 103 (40), 14690–14694.CrossRefGoogle ScholarPubMed
Xing, Y., Majda, A. J. & Grabowski, W. W. 2009 New efficient sparse space-time algorithms for superparameterization on mesoscales. Mon. Weath. Rev. 137 (12), 4307–4324.CrossRefGoogle Scholar
Zwanzig, R. 1961 Memory effects in irreversible thermodynamics. Phys. Rev. 124 (4), 983.CrossRefGoogle Scholar

Frank et al. supplementary movie

A movie showing the evolution and chaotic divergence of the turbulent simulations. Both rows feature simulation results using (left) the model with added Hyperviscosity, (centre) the Reference simulation, and (right) the Nos ́e-Hoover method as indicated with † in table 1. The initial conditions are the same for all of the simulations in one row, but slightly different between top and bottom rows. Compare with Figure 4 in the paper.

Download Frank et al. supplementary movie(Video)
Video 4.4 MB