Editorial Type:
Article
Article Type:
Research Article

Nonlinear Feedback in a Five-Dimensional Lorenz Model

Bo-Wen Shen Earth System Science Interdisciplinary Center, University of Maryland, College Park, College Park, and NASA Goddard Space Flight Center, Greenbelt, Maryland

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Print Publication:
01 May 2014
DOI:
https://doi.org/10.1175/JAS-D-13-0223.1
Page(s):
1701–1723
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Abstract

In this study, based on the number of modes, the original three-dimensional Lorenz model (3DLM) is generalized with two additional modes [five-dimensional Lorenz model (5DLM)] to examine their role in the predictability of the numerical solutions and to understand the underlying processes that increase the solution stability. As a result of the simplicity of the 5DLM with respect to existing generalized Lorenz models (LMs), the author is able to obtain the analytical solutions of its critical points and identify the role of the major nonlinear term in the solution’s stability, which have previously not been documented in the literature. The nonlinear Jacobian terms of the governing equations are analyzed to highlight the importance of selecting new modes for extending the nonlinear feedback loop of the 3DLM and thus effectively increasing the degree of nonlinearity (i.e., the nonlinear mode–mode interactions) in the 5DLM. It is then shown that numerical solutions in the 5DLM require a larger normalized Rayleigh number r for the onset of chaos and are more predictable than those in the 3DLM when r is between 25 and 40 and the Prandtl number σ is 10. The improved predictability is attributable to the negative nonlinear feedback enabled by the new modes. The role of the (negative) nonlinear feedback is further verified using a revised 3DLM with a parameterized nonlinear eddy dissipative term. The finding of the increased stability in the 5DLM and revised 3DLM with respect to the 3DLM is confirmed with the linear stability analysis and the analysis of the Lyapunov exponents using different values of r and σ. To further understand the impact of an additional heating term, results from the 5DLM and a higher-dimensional LM [e.g., the six-dimensional LM (6DLM)] are analyzed and compared.

Denotes Open Access content.

Corresponding author address: Dr. Bo-Wen Shen, Mesoscale Atmospheric Processes Laboratory, Code 612, NASA Goddard Space Flight Center, Greenbelt, MD 20771. E-mail: bo-wen.shen-1@nasa.gov

Abstract

In this study, based on the number of modes, the original three-dimensional Lorenz model (3DLM) is generalized with two additional modes [five-dimensional Lorenz model (5DLM)] to examine their role in the predictability of the numerical solutions and to understand the underlying processes that increase the solution stability. As a result of the simplicity of the 5DLM with respect to existing generalized Lorenz models (LMs), the author is able to obtain the analytical solutions of its critical points and identify the role of the major nonlinear term in the solution’s stability, which have previously not been documented in the literature. The nonlinear Jacobian terms of the governing equations are analyzed to highlight the importance of selecting new modes for extending the nonlinear feedback loop of the 3DLM and thus effectively increasing the degree of nonlinearity (i.e., the nonlinear mode–mode interactions) in the 5DLM. It is then shown that numerical solutions in the 5DLM require a larger normalized Rayleigh number r for the onset of chaos and are more predictable than those in the 3DLM when r is between 25 and 40 and the Prandtl number σ is 10. The improved predictability is attributable to the negative nonlinear feedback enabled by the new modes. The role of the (negative) nonlinear feedback is further verified using a revised 3DLM with a parameterized nonlinear eddy dissipative term. The finding of the increased stability in the 5DLM and revised 3DLM with respect to the 3DLM is confirmed with the linear stability analysis and the analysis of the Lyapunov exponents using different values of r and σ. To further understand the impact of an additional heating term, results from the 5DLM and a higher-dimensional LM [e.g., the six-dimensional LM (6DLM)] are analyzed and compared.

Denotes Open Access content.

Corresponding author address: Dr. Bo-Wen Shen, Mesoscale Atmospheric Processes Laboratory, Code 612, NASA Goddard Space Flight Center, Greenbelt, MD 20771. E-mail: bo-wen.shen-1@nasa.gov
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  • Anthes, R., cited 2011: Turning the tables on chaos: Is the atmosphere more predictable than we assume? NCAR/UCAR AtmosNews. [Available online at https://www2.ucar.edu/atmosnews/opinion/turning-tables-chaos-atmosphere-more-predictable-we-assume-0.]

  • Atlas, R., and Coauthors, 2005: Hurricane forecasting with the high-resolution NASA finite volume general circulation model. Geophys. Res. Lett., 32, L03801, doi:10.1029/2004GL021513.

    • Search Google Scholar
    • Export Citation

  • Bender, C. M., and S. A. Orszag, 1978: Advanced Mathematical Methods for Scientists and Engineers.McGraw-Hill, 593 pp.

  • Biswas, R., M. J. Aftosmis, C. Kiris, and B.-W. Shen, 2007: Petascale computing: Impact on future NASA missions. Petascale Computing: Architectures and Algorithms, D. Bader, ed., Chapman and Hall/CRC Press, 29–46.

  • Christiansen, F., and H. Rugh, 1997: Computing Lyapunov spectra with continuous Gram-Schmid orthonormalization. Nonlinearity, 10, 1063–1072.

    • Search Google Scholar
    • Export Citation

  • Curry, J. H., 1978: Generalized Lorenz systems. Commun. Math. Phys., 60, 193–204.

  • Curry, J. H., J. R. Herring, J. Loncaric, and S. A. Orszag, 1984: Order and disorder in two- and three-dimensional Benard convection. J. Fluid Mech., 147, 1–38.

    • Search Google Scholar
    • Export Citation

  • Ding, R. O., and J. P. Li, 2007: Nonlinear finite-time Lyapunov exponent and predictability. Phys. Lett., 354A, 396–400.

  • Eckhardt, B., and D. Yao, 1993: Local Lyapunov exponents in chaotic systems. Physica D, 65, 100–108.

  • Froyland, J., and K. H. Alfsen, 1984: Lyapunov-exponent spectra for the Lorenz model. Phys. Rev., 29A, 2928–2931.

  • Gleick, J., 1987: Chaos: Making a New Science. Penguin, 360 pp.

  • Grassberger, P., and I. Procaccia, 1983: Characterization of strange attractors. Phys. Rev. Lett., 5, 346–349.

  • Kaplan, J. L., and J. A. Yorke, 1979: Chaotic behavior of multidimensional difference equations. Functional Differential Equations and the Approximations of Fixed Points, H. O. Peitgen and H. O. Walther, Eds., Lecture Notes in Math, Vol. 730, Springer-Verlag, 204–227.

  • Kazantsev, E., 1999: Local Lyapunov exponents of the quasi-geostrophic ocean dynamics. Appl. Math. Comput., 104, 217–257.

  • Legras, B., and M. Ghil, 1985: Persistent anomalies, blocking, and variations in atmospheric predictability. J. Atmos. Sci., 42, 433–471.

    • Search Google Scholar
    • Export Citation

  • Li, J., and R. Ding, 2011: Temporal-spatial distribution of atmospheric predictability limit by local dynamical analogs. Mon. Wea. Rev., 139, 3265–3283.

    • Search Google Scholar
    • Export Citation

  • Lorenz, E., 1963a: Deterministic nonperiodic flow. J. Atmos. Sci., 20, 130–141.

  • Lorenz, E., 1963b: The predictability of hydrodynamic flow. Trans. N. Y. Acad. Sci.,25, 409–432.

  • Lorenz, E., 1972: Predictability: Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas? Proc. 139th Meeting of AAAS Section on Environmental Sciences, New Approaches to Global Weather: GARP, Cambridge, MA, AAAS, 5 pp. [Available online at http://eaps4.mit.edu/research/Lorenz/Butterfly_1972.pdf.]

  • Musielak, Z. E., D. E. Musielak, and K. S. Kennamer, 2005: The onset of chaos in nonlinear dynamical systems determined with a new fractal technique. Fractals, 13, 19–31.

    • Search Google Scholar
    • Export Citation

  • Nese, J. M., 1989: Quantifying local predictability in phase space. Physica D, 35, 237–250.

  • Nese, J. M., and J. A. Dutton, 1993: Quantifying predictability variations in a low-order ocean atmosphere model: A dynamical systems approach. J. Climate, 6, 185–204.

    • Search Google Scholar
    • Export Citation

  • Nese, J. M., J. A. Dutton, and R. Wells, 1987: Calculated attractor dimensions for low-order spectral models. J. Atmos. Sci., 44, 1950–1972.

    • Search Google Scholar
    • Export Citation

  • Nese, J. M., A. Miller, and J. A. Dutton, 1996: The nature of predictability enhancement in a low-order ocean-atmosphere model. J. Climate, 9, 2167–2172.

    • Search Google Scholar
    • Export Citation

  • Pielke, R., cited2008: The real butterfly effect. [Available online at http://pielkeclimatesci.wordpress.com/2008/04/29/the-real-butterfly-effect/.]

  • Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, 1992: Numerical Recipes in C. Cambridge University Press, 994 pp.

  • Roy, D., and Z. E. Musielak, 2007a: Generalized Lorenz models and their routes to chaos. I. Energy-conserving vertical mode truncations. Chaos Solitons Fractals, 32, 1038–1052.

    • Search Google Scholar
    • Export Citation

  • Roy, D., and Z. E. Musielak, 2007b: Generalized Lorenz models and their routes to chaos. II. Energy-conserving horizontal mode truncations. Chaos Solitons Fractals, 31, 747–756.

    • Search Google Scholar
    • Export Citation

  • Roy, D., and Z. E. Musielak, 2007c: Generalized Lorenz models and their routes to chaos. III. Energy-conserving horizontal and vertical mode truncations. Chaos Solitons Fractals, 33, 1064–1070.

    • Search Google Scholar
    • Export Citation

  • Saltzman, B., 1962: Finite amplitude free convection as an initial value problem. J. Atmos. Sci., 19, 329–341.

  • Shen, B.-W., R. Atlas, J.-D. Chern, O. Reale, S.-J. Lin, T. Lee, and J. Chang, 2006a: The 0.125 degree finite-volume general mesoscale circulation model: Preliminary simulations of mesoscale vortices. Geophys. Res. Lett., 33, L05801, doi:10.1029/2005GL024594.

    • Search Google Scholar
    • Export Citation

  • Shen, B.-W., R. Atlas, O. Reale, S.-J. Lin, J.-D. Chern, J. Chang, C. Henze, and J.-L. Li, 2006b: Hurricane forecasts with a global mesoscale-resolving model: Preliminary results with Hurricane Katrina (2005). Geophys. Res. Lett., 33, L13813, doi:10.1029/2006GL026143.

    • Search Google Scholar
    • Export Citation

  • Shen, B.-W., W.-K. Tao, and B. Green, 2011: Coupling advanced modeling and visualization to improve high-impact tropical weather prediction. IEEE Comput. Sci. Eng., 13 (5), 56–67, doi:10.1109/MCSE.2010.141.

    • Search Google Scholar
    • Export Citation

  • Shen, B.-W., W.-K. Tao, Y.-L. Lin, and A. Laing, 2012: Genesis of twin tropical cyclones as revealed by a global mesoscale model: The role of mixed Rossby gravity waves. J. Geophys. Res., 117, D13114, doi:10.1029/2012JD017450.

    • Search Google Scholar
    • Export Citation

  • Smith, B., J. Boyle, J. Dongarra, B. Garbow, Y. Ikebe, V. Klema, and C. Moler, 1976: Matrix Eigensystem Routines—EISPACK Guide. Lecture Notes in Computer Science, Vol. 6, Springer Verlag, 551 pp.

  • Solomon, S., D. Qin, M. Manning, Z. Chen, M. Marquis, K. Averyt, M. Tignor, and H. L. Miller Jr., Eds., 2007: Climate Change 2007: The Physical Science Basis. Cambridge University Press, 996 pp.

  • Sprott, J. C., 2003: Chaos and Time-Series Analysis. Oxford University Press, 528 pp.

  • Thiffeault, J.-L., and W. Horton, 1996: Energy-conserving truncations for convection with shear flow. Phys. Fluids, 8, 1715–1719.

  • Treve, Y. M., and O. P. Manley, 1982: Energy conserving Galerkin approximations for 2-D hydrodynamic and MHD Benard convection. Physica D, 4, 319–342.

    • Search Google Scholar
    • Export Citation

  • Wolf, A., J. B. Swift, H. L. Swinney, and J. A. Vastano, 1985: Determining Lyapunov exponents from a time series. Physica D, 16, 285–317.

    • Search Google Scholar
    • Export Citation

  • Zeng, X., R. Eykholt, and R. A. Pielke, 1991: Estimating the Lyapunov-exponent spectrum from short time series of low precision. Phys. Rev. Lett., 66, 3229–3232.

    • Search Google Scholar
    • Export Citation

  • Zeng, X., R. A. Pielke, and R. Eykholt, 1992: Estimate of the fractal dimension and predictability of the atmosphere. J. Atmos. Sci., 49, 649–659.

    • Search Google Scholar
    • Export Citation
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