References

NPGD

Nonlinear Processes in Geophysics Discussions

NPGD

Nonlin. Processes Geophys. Discuss.

2198-5634

Göttingen, Germany

10.5194/npgd-1-519-2014

On the nonlinear feedback loop and energy cycle of the non-dissipative Lorenz model

Shen

B.-W.

ESSIC, University of Maryland, College Park, Mesoscale Atmospheric Processes Laboratory, Code 612, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA

11 04 2014

1 1 519 541

2014

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In this study, we discuss the role of the nonlinear terms and linear (heating) term in the energy cycle of the three-dimensional (X–Y–Z) non-dissipative Lorenz model (3D-NLM). (X, Y, Z) represent the solutions in the phase space. We first present the closed-form solution to the nonlinear equation d2 X/dτ2+ (X2/2)X = 0, τ is a non-dimensional time, which was never documented in the literature. As the solution is oscillatory (wave-like) and the nonlinear term (X2) is associated with the nonlinear feedback loop, it is suggested that the nonlinear feedback loop may act as a restoring force. We then show that the competing impact of nonlinear restoring force and linear (heating) force determines the partitions of the averaged available potential energy from Y and Z modes, respectively, denoted as APEY and APEZ. Based on the energy analysis, an energy cycle with four different regimes is identified with the following four points: A(X, Y) = (0,0), B = (Xt, Yt), C = (Xm, Ym), and D = (Xt, -Yt). Point A is a saddle point. The initial perturbation (X, Y, Z) = (0, 1, 0) gives (Xt, Yt) = ( √ 2σr , r) and (Xm, Ym) = (2√  σr , 0). σ is the Prandtl number, and r is the normalized Rayleigh number. The energy cycle starts at (near) point A, A+ = (0, 0+) to be specific, goes through B, C, and D, and returns back to A, i.e., A- = (0,0-). From point A to point B, denoted as Leg A–B, where the linear (heating) force dominates, the solution X grows gradually with { KE↑, APEY↓, APEZ↓}. KE is the averaged kinetic energy. We use the upper arrow (↑) and down arrow (↓) to indicate an increase and decrease, respectively. In Leg B–C (or C–D) where nonlinear restoring force becomes dominant, the solution X increases (or decreases) rapidly with {KE↑, APEY↑, APEZ↓} (or {KE↓, APEY↓, APEZ↑}). In Leg D–A, the solution X decreases slowly with {KE↓, APEY↑, APEZ↑ }. As point A is a saddle point, the aforementioned cycle may be only half of a "big" cycle, displaying the wing pattern of a glasswinged butterfly, and the other half cycle is antisymmetric with respect to the origin, namely B = (-Xt, -Yt), C = (-Xm, 0), and D = (-Xt, Yt).

References 1

Anthes, R.: Turning the tables on chaos: is the atmosphere more predictable than we assume?, UCAR Magazine, spring/summer, available at: <a href="https://www2.ucar.edu/atmosnews/opinion/turning-tables-chaos-atmosphere-more-predictable-we-assume-0">https://www2.ucar.edu/atmosnews/opinion/turning-tables-chaos-atmosphere-more-predictable-we-assume-0</a> (last access: 6 May 2011), 2011.

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

Blender, R. and Lucarini, V.: Nambu representation of an extended Lorenz model with viscous heating, Physica D, 243, 86–91, 2013.

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

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

Floratos, E.: Matrix quantization of turbulence, Int. J. Bifurcat. Chaos, 22, 1250213, <a href="http://dx.doi.org/10.1142/S0218127412502136">https://doi.org/10.1142/S0218127412502136</a>, 2012.

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

Hermiz, K. B., Guzdar, P. N., and Finn, J. M.: Improved low-order model for shear flow driven by Rayleigh–Benard convection, Phys. Rev. E, 51, 325–331, 1995.

Howard, L. N. and Krishnamurti, R. K.: Large-scale flow in turbulent convection: a mathematical model, J. Fluid Mech, 170, 385–410, 1986. \bibitem [Lorenz(1963)]lorenz63 Lorenz, E.: Deterministic nonperiodic flow, J. Atmos. Sci., 20, 130–141, 1963.

Ide, K., Small, D., and Wiggins, S.: Distinguished hyperbolic trajectories in time-dependent fluid flows: analytical and computational approach for velocity fields defined as data sets, Nonlin. Processes Geophys., 9, 237–263, <a href="http://dx.doi.org/10.5194/npg-9-237-2002">https://doi.org/10.5194/npg-9-237-2002</a>, 2002.

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

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

Nambu, Y.: Generalized Hamiltonian dynamics, Phys. Rev. D, 7, 2403, <a href="http://dx.doi.org/10.1103/PhysRevD.7.2405">https://doi.org/10.1103/PhysRevD.7.2405</a>, 1973.

Nevir, P. and Blender, R.: Hamiltonian and Nambu representation of the non-dissipative Lorenz equations, Beitraege zur Physik der Atmosphaere, 67, 133–140, 1994.

Roupas, Z.: Phase space geometry and chaotic attractors in dissipative nambu mechanics, J. Phys. A-Math. Theor., 45, 195101, <a href="http://dx.doi.org/10.1088/1751-8113/45/19/195101">https://doi.org/10.1088/1751-8113/45/19/195101</a>, 2012.

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

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

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

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

Shen, B.-W.: Nonlinear feedback in a five-dimensional Lorenz model, J. Atmos. Sci., <a href="http://dx.doi.org/10.1175/JAS-D-13-0223.1">https://doi.org/10.1175/JAS-D-13-0223.1</a>, in press, 2014a.

Shen, B.-W.: Nonlinear Feedback in a Six-dimensional Lorenz Model: Impact of an Additional Heating Term, J. Atmos. Sci., submitted, 2014b.

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