https://doi.org/10.5194/npgd-1-519-2014
https://doi.org/10.5194/npgd-1-519-2014
Status: this preprint was under review for the journal NPG but the revision was not accepted.

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

B.-W. Shen

Abstract. In this study, we discuss the role of the nonlinear terms and linear (heating) term in the energy cycle of the three-dimensional (XYZ) 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 AB, 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 BC (or CD) where nonlinear restoring force becomes dominant, the solution X increases (or decreases) rapidly with {KE↑, APEY↑, APEZ↓} (or {KE↓, APEY↓, APEZ↑}). In Leg DA, 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).

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B.-W. Shen

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Status: closed
AC: Author comment | RC: Referee comment | SC: Short comment | EC: Editor comment - Printer-friendly version - Supplement
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Status: closed
Status: closed
AC: Author comment | RC: Referee comment | SC: Short comment | EC: Editor comment - Printer-friendly version - Supplement
• RC C94:
• RC C160:
• AC C469:
• AC C537:
• AC C483:

B.-W. Shen

B.-W. Shen

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