| 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). |