doi:10.5194/npgd-1-519-2014
© Author(s) 2014. This work is distributed under the Creative Commons Attribution 3.0 License. |

Research article

11 Apr 2014

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

Received: 12 Feb 2014 – Accepted for review: 23 Mar 2014 – Discussion started: 11 Apr 2014

Abstract. 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 d^{2} *X*/dτ^{2}+ (*X*^{2}/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 (*X*^{2}) 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 APE_{Y} and APE_{Z}. 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* = (*X*_{t}, *Y*_{t}), *C* = (*X*_{m}, *Y*_{m}), and *D* = (*X*_{t}, -*Y*_{t}). Point *A* is a saddle point. The initial perturbation (*X, Y, Z*) = (0, 1, 0) gives (*X*_{t}, *Y*_{t}) = ( √ 2σr , r) and (*X*_{m}, *Y*_{m}) = (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↑, APE_{Y}↓, APE_{Z}↓}. 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↑, APE_{Y}↑, APE_{Z}↓} (or {KE↓, APE_{Y}↓, APE_{Z}↑}). In Leg *D*–*A*, the solution *X* decreases slowly with {KE↓, APE_{Y}↑, APE_{Z}↑ }. 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* = (-*X*_{t}, -*Y*_{t}), *C* = (-*X*_{m}, 0), and *D* = (-*X*_{t}, *Y*_{t}).

**Citation:**
Shen, B.-W.: On the nonlinear feedback loop and energy cycle of the non-dissipative Lorenz model, Nonlin. Processes Geophys. Discuss., 1, 519-541, doi:10.5194/npgd-1-519-2014, 2014.