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

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 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 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).