Solitary Waves, Homoclinic Orbits, and Nonlinear Oscillations Within the Non-dissipative Lorenz Model, the Inviscid Pedlosky Model, and the KdV Equation

Abstract

In contrast to the conventional view that weather is chaotic, a revised view on the dual nature of chaos and order in weather and climate has recently been proposed. The revised view is based on the findings of attractor coexistence using the classical and generalized Lorenz models, as well as promising 30 day simulations using a high-resolution global model. To provide additional support, this study further illustrates mathematical universalities between the Lorenz and Pedlosky models whose solutions represent very different physical processes, including small-scale convection and large-scale quasi-geostrophic baroclinic waves. A comparison amongst the non-dissipative Lorenz model, the inviscid Pedlosky model, the KdV equation, and other systems is additionally provided in order to reveal the same form of these solutions for solitary waves and homoclinic orbits, and to propose a generic conservative system with two families of oscillatory solutions. The generic system is then applied to illustrate the role of nonlinearity in producing stable critical points for stabilizing the system. An analysis of the generic system (i.e., the non-dissipative Lorenz model) and the Lorenz model that retains one or three dissipative terms reiterates the collective impact of multiple dissipative terms with nonlinearity (as well as the heating term) on the appearance of unstable critical points and irregular and chaotic responses. The results suggest the need for a systematic approach for examining the impact of new (stable) components on the local and global stability of the new coupled system.

Appendices

Appendix A: The Pedlosky Model and a Two Layer Quasi-geostrophic System

Here, I provide a brief introduction to the Pedlosky model that is based on the following two-layer quasi-geostrophic (QG) system for studying weakly nonlinear baroclinic wave-mean interactions [31,32,33]:

$$\begin{aligned}&\left[ {\partial \over \partial t} + {\partial \varPsi _1 \over \partial x} {\partial \over \partial y} - {\partial \varPsi _1 \over \partial y} {\partial \over \partial x} \right] \left( \nabla ^2 \varPsi _1 - F (\varPsi _1- \varPsi _2) \right) = \gamma \nabla ^2 \varPsi _1 \end{aligned}$$

(28)

$$\begin{aligned}&\left[ {\partial \over \partial t} + {\partial \varPsi _2 \over \partial x} {\partial \over \partial y} - {\partial \varPsi _2 \over \partial y} {\partial \over \partial x} \right] \left( \nabla ^2 \varPsi _2 - F (\varPsi _2- \varPsi _1) \right) = \gamma \nabla ^2 \varPsi _2 \end{aligned}$$

(29)

The above system is applied to a three-dimensional domain that consists of a channel in the two-dimensional $x-y$ plane and two layers with an initially equal depth in the vertical direction. The 2D channel is periodic in the x direction and has a finite width of L in the y direction. Two vertical layers are denoted by different subscripts (i.e., 1 and 2). A constant but different density is applied in each of the layers. The difference of densities (i.e., the vertical gradient of density) leads to a baroclinicity that may act as an energy source for unstable solutions. $\varPsi _1$ and $\varPsi _2$ represent streamfunctions that define velocities in the x and y directions for layers 1 and 2, respectively. F is a rotational Froude number that is a function of the Coriolis parameter, (reduced) gravitational acceleration, the length of the domain, and the depth of the layer, all of which are constants. In this study, an assumption of a constant Coriolis parameter is made in order to simplify discussions. The two terms that involves $\gamma$ and appear on the right hand side represent dissipative terms.

In a series of studies by Prof. Pedlosky, the following Pedlosky model was derived (e.g., Eqs. 2.12a, b of [32]):

$$\begin{aligned} {d^2 R \over d \tau ^2} + \alpha \eta {d R\over d \tau } - (D+1) R + R^3 = 0, \end{aligned}$$

(30)

$$\begin{aligned} {d D \over d \tau } + \eta D + \beta \eta R^2 = 0. \end{aligned}$$

(31)

The above system includes one second-order ODE and one first-order ODE for two state variables are R and D. R represents the scaled amplitude of the streamfunction (or wave solution). While the mathematical definition of the state variable D was given in Eq. 2.13 of [32], its physical impact on the system’s solution is provided below in order to facilitate discussions. Three time-independent parameters include $\alpha$ and $\beta$, which are functions of horizontal wavenumbers of the solutions, and $\eta$ that is a rescaled dissipative coefficient. As a result, three terms that involve $\alpha \eta$, $\eta$, and $\beta \eta$ represent the dependence of dissipation on scales.

1.1 An Alternative Form of the Lorenz Model by Marzec and Spiegel

Marzec and Spiegel [25] transformed the Lorenz model into the following system with 2nd-order and 1st-order ODEs [6]:

$$\begin{aligned} {d^2 X \over d \tau ^2 } + \left( \sigma +1 \right) {d X\over d \tau } + \varLambda X + {1 \over 2 } X^3 = 0, \end{aligned}$$

(32)

$$\begin{aligned} {d \varLambda \over d \tau } = - b \varLambda - b\sigma (r-1) + (\sigma - {1\over 2} b) X^2, \end{aligned}$$

(33)

where $\varLambda =\sigma (Z-r+1) - {1\over 2} X^2$. By comparing this system to Eqs. (30)–(33), we define $D = {1\over 2} X^2 - \sigma Z$ and $R=X/\sqrt{2}$, leading to $\varLambda = - (D+ \sigma r -\sigma )$. Furthermore, Eqs. (32) and (33) become:

$$\begin{aligned} {d^2 R \over d \tau ^2 } + \left( \sigma +1 \right) {d R\over d \tau } - (D+ \sigma r -\sigma ) R + R^3 = 0, \end{aligned}$$

(34)

$$\begin{aligned} {d D \over d \tau } + bD + (2\sigma -b) R^2 = 0. \end{aligned}$$

(35)

Under a proper choice of parameters (i.e., $b=\eta$, $(2\sigma -b)=\beta \eta$, $(\sigma +1)=\alpha \eta$, and $(\sigma r -\sigma )=1$), also listed in Table 1, the Pedlosky model with Eqs. (30) and (31) is mathematically identical to the Lorenz model with Eqs. (34) and (35). On the other hand, the model requires different intervals of parameters for the onset of chaos (e.g., a smaller $\sigma$). Additionally, a special case with $b=2\sigma$ decouples Eq. (35) from Eq. (34) and yields an exponential decaying solution of $D = D_0 exp(-b\tau )$, here $D_0$ is an initial condition. This special case is further discussed in Sect. 3.3.

Following the interpretation of [25], the solution of Eq. (30) may be viewed as oscillations of a damped spring whose time varying stiffness $-(D+1)$ is governed by Eq. (31). Here, the time evolving stiffness, appearing as the coefficient of the linear term in Eq. (30), may be positive or negative, indicating the competing impact of forcing, dissipation, and nonlinearity, as discussed in the main text.

Appendix B: A Nonlinear Pendulum Equation

Here, I provide a brief introduction to the nonlinear pendulum equation and a short note on the stability of a stable equilibrium point in the system. Figure 6 displays a pendulum consisting of a weightless rod of length L with an attached bob with a mass of m. The other end of the rod is supported at a point on a wall. The point of support is denoted as the origin. As a result, the mass of the pendulum is free to oscillate or rotate. The time varying position of the mass is determined by the angle, $\theta$, between the rod and the downward vertical direction. As shown in Fig. 6, such an angle is measured in the counterclockwise direction. As a result, the natural position is $\theta =0$ and the inverted position where the pendulum bob is vertical, with the weight in the up position, is $\theta =\pi$. The two positions are referred to as the lower and upper equilibrium point, respectively, and classified as either stable or unstable equilibrium points, respectively.

Fig. 6

A pendulum consisting of a weightless rod of length L and a bob with a mass of m. The bob and the point of support are marked with a red and black dot, respectively. The parameter “g” denotes the gradational force. The angle $\theta$ is measured in the counterclockwise direction. Stable and unstable equilibrium points appear at $\theta = 0$ and $\theta = \pi$, respectively

The time varying angle ($\theta$) of the nonlinear pendulum with a dissipative term is governed by the following equation:

$$\begin{aligned} {d^2 \theta \over d \tau ^2 } + k {d \theta \over d \tau } + sin(\theta ) = 0; \end{aligned}$$

(36)

here, k represents the dissipative coefficient. For the non-dissipative system (i.e., $k=0$), the above system has a stable and unstable point at $\theta =2n \pi$ and $\theta =(2n+1)\pi$, respectively. Here n is an integer.

Fig. 7

Potential energy functions for the full and simplified pendulum equations near the stable equilibrium point in Eqs. (37)–(38)

While the stability of the unstable critical point at $\theta =\pi$ is analyzed in the main text, the stability of the stable critical point at $\theta =0$ is discussed below. Using a Taylor series expansion with $sin(\theta ) =\theta$ or $sin(\theta ) =\theta -\theta ^3/6$, Eq. (36) can be approximated by:

$$\begin{aligned} {d^2 \theta \over d \tau ^2 } + \theta = 0, \end{aligned}$$

(37)

or

$$\begin{aligned} {d^2 \theta \over d \tau ^2 } + \theta - {\theta ^3 \over 6} = 0. \end{aligned}$$

(38)

Within Eq. (37) that contains a linear term, only one critical point at $(\theta , d\theta /d\tau )=(0,0)$, which is a center, is found. The critical point is stable and is associated with a simple harmonic oscillation. In comparison, within Eq. (38) that includes a cubic nonlinear term, two additional critical points appear at $(\theta , d\theta /d\tau )=(\pm \sqrt{6},0)$. These non-trivial critical points are saddle points that are unstable. As a result, the simplified nonlinear system that has a limited degree of nonlinearity may be locally stable but globally unstable when initial kinetic energy is sufficiently strong. The features are illustrated in Fig. 7. Global instability in the simplified nonlinear system (Eq. (38)) is not consistent with that in the full pendulum system in Eq. (36), since the successive appearance of stable and unstable critical points within the full system (i.e., at $\theta =2n\pi$ and $\theta =(2n+1)\theta$) can still limit growth of the unstable solution, yielding global stability. However, such an issue with global instability may appear within a complicated system (e.g., climate models).