Journal cover Journal topic
Nonlinear Processes in Geophysics An interactive open-access journal of the European Geosciences Union
Journal topic

Journal metrics

Journal metrics

  • IF value: 1.699 IF 1.699
  • IF 5-year value: 1.559 IF 5-year
    1.559
  • CiteScore value: 1.61 CiteScore
    1.61
  • SNIP value: 0.884 SNIP 0.884
  • IPP value: 1.49 IPP 1.49
  • SJR value: 0.648 SJR 0.648
  • Scimago H <br class='hide-on-tablet hide-on-mobile'>index value: 52 Scimago H
    index 52
  • h5-index value: 21 h5-index 21
Discussion papers
https://doi.org/10.5194/npg-2016-40
© Author(s) 2016. This work is distributed under
the Creative Commons Attribution 3.0 License.
https://doi.org/10.5194/npg-2016-40
© Author(s) 2016. This work is distributed under
the Creative Commons Attribution 3.0 License.

Research article 23 Sep 2016

Research article | 23 Sep 2016

Review status
This discussion paper is a preprint. It has been under review for the journal Nonlinear Processes in Geophysics (NPG). The revised manuscript was not accepted.

On periodic solutions associated with the nonlinear feedback loop in the non-dissipative Lorenz model

B.-W. Shen B.-W. Shen
  • Department of Mathematics and Statistics, San Diego State University, 5500 Campanile Drive, San Diego, CA, 92182, USA

Abstract. In this study, we discuss the role of the linear heating term and nonlinear terms associated with a nonlinear feedback loop in the energy cycle of the three-dimensional (XYZ) non-dissipative Lorenz model (3D-NLM), where (X, Y, Z) represent the solutions in the phase space. Using trigonometric functions, we first present the closed-form solution of the nonlinear equation d2X/dτ2 + (X2/2)X = 0 without the heating term (i.e., rX), (where τ is a non-dimensional time and r is the normalized Rayleigh number), a solution that has not been previously documented. Since the solution of the simplified 3D-NLM is oscillatory (wave-like) and since the nonlinear term (X3) is associated with the nonlinear feedback loop, here, we suggest that the nonlinear feedback loop may act as a restoring force. When the heating term is considered, the system yields three critical points. A linear analysis suggests that the origin (i.e., the trivial critical point) is a saddle point and that the other two non-trivial critical points are stable. Here, we provide an analysis for three types of solutions that are associated with these critical points. Two of the solutions represent closed curves that travel around one non-trivial critical point or all three critical points. The third type of solution, appearing to connect the stable and unstable manifolds of the saddle point, is called the homoclinic orbit. Using the solution that contains one non-trivial critical point, here, we show that the competing impact of the nonlinear restoring force and the linear (heating) force determines the partitions of the averaged available potential energy from the Y and Z modes. Based on the energy analysis, an energy cycle with four different regimes is identified. The cycle is only half of a "large" cycle, displaying the wing pattern of a glasswinged butterfly. The other half cycle is antisymmetric with respect to the origin. The two types of oscillatory solutions with either a small cycle or a large cycle are orbitally stable. As compared to the oscillatory solutions, the homoclinic orbit is not periodic because it "takes forever" to reach the origin. Two trajectories with starting points near the homoclinic orbit may be diverged because one moves with a small cycle and the other moves with a large cycle. Therefore, the homoclinic orbit is not orbitally stable. In a future study, dissipation and/or additional nonlinear terms will be included in order to determine how their interactions with the original nonlinear feedback loop and the heating term may change the periodic orbits (as well as homoclinic orbits) to become quasi-periodic orbits and chaotic solutions.

B.-W. Shen
Interactive discussion
Status: closed
Status: closed
AC: Author comment | RC: Referee comment | SC: Short comment | EC: Editor comment
Printer-friendly Version - Printer-friendly version Supplement - Supplement
Interactive discussion
Status: closed
Status: closed
AC: Author comment | RC: Referee comment | SC: Short comment | EC: Editor comment
Printer-friendly Version - Printer-friendly version Supplement - Supplement
B.-W. Shen
B.-W. Shen
Viewed  
Total article views: 1,462 (including HTML, PDF, and XML)
HTML PDF XML Total BibTeX EndNote
838 557 67 1,462 29 73
  • HTML: 838
  • PDF: 557
  • XML: 67
  • Total: 1,462
  • BibTeX: 29
  • EndNote: 73
Views and downloads (calculated since 23 Sep 2016)
Cumulative views and downloads (calculated since 23 Sep 2016)
Viewed (geographical distribution)  
Total article views: 1,383 (including HTML, PDF, and XML) Thereof 1,381 with geography defined and 2 with unknown origin.
Country # Views %
  • 1
1
 
 
 
 
Cited  
Saved  
No saved metrics found.
Discussed  
No discussed metrics found.
Latest update: 25 Jun 2019
Publications Copernicus
Short summary
Using a nondissipative Lorenz model, we show that the nonlinear feedback loop alone may act a restoring forcing and the heating term alone can produce a saddle point. We illustrate that the nonlinear feedback loop and heating term collectively lead to three critical points and three types of solutions, which may contain a small or big energy cycle. The existence of the homoclinic orbit and two types of oscillatory solutions suggests the appearance of diverged trajectories.
Using a nondissipative Lorenz model, we show that the nonlinear feedback loop alone may act a...
Citation