and 
, here, 
is the streamfunction, and 
is the temperature perturbation.. However, only the impact of the latter is included in the 3DLM. In other words, nonlinearity in the 3DLM involves only nonlinear advection of temperature. The nonlinear interaction of two wave modes, which is expressed by the Jacobian term, can generate or impact a third wave mode through a downscale (or upscale) transfer process. The third wave mode can provide feedback to the incipient wave mode(s) via its subsequent upscale (or downscale) transfer process. Downscale and upscale transfer processes associated with the third mode form a nonlinear feedback loop that can be continuously extended when new modes are continuously generated. Appendix B of Shen [2] summarizes the successive downscale and upscale transfer processes via the Jacobian term, 
. Related discussions suggest that an ideal numerical model should include an infinite number of Fourier modes. However, practically, all available numerical models have a finite number of modes. Thus the extension of their nonlinear feedback loop is finite (and incomplete). As a result, mode truncation may impact the degree of nonlinearity and related feedback. Since chaos does appear to be the result of the inclusion of nonlinearity with a limited number of Fourier modes (i.e., a limited degree of nonlinearity), it is important to understand the impact of the increased degrees of nonlinearity in thermodynamic processes (i.e.,
on a solution’s stability. To illustrate this process, the approach, as presented here, was to derive and examine the five- and six-dimensional Lorenz models (5DLM and 6DLM). Newly introduced “nonlinear thermodynamic processes” involve the nonlinear interaction of existing modes (in the 3DLM) and new modes that are associated with either dissipation terms or a heating term. While high-dimensional Lorenz models have been derived and examined in order to understand the impact of mode truncations on a solution’s stability over a period of several years [9-12],efforts have been made to discuss (i) the extension of the nonlinear feedback loop in the 5D or 6DLM; (ii) the role of the nonlinear feedback in stabilizing or destabilizing solutions; (iii) clarifications of the first and second kind of butterfly effect; and (iv) the feasibility of parameterizing negative (or positive) feedbacks into a revised 3DLM with the aim of improving system stability without the need to use higher dimensional LMs, and while understanding how parameterization may change time-varying solutions and the equilibrium state solution.
and 
), as follows:
(1) 
(2) 
(3) 
and 
represent the nonlinear feedback from small-scale processes, derived analytically from a higher-dimensional Lorenz model, e.g., 5DLM or 6DLM. As illustrated in Shen [2], nonlinear terms of the 3DLM, as well as 5DLM and 6DLM, come solely from the advection of temperature perturbations. Thus, only including additional nonlinear terms in Eqs. 2 and 3 is justifiable. Detailed procedures regarding the choices for 
and 
using the 5DLM or 6DLM are provided below.
(4) 
(5) 
(6) 
(7) 
(8) 
and 
are constants (
and 
) and are defined in Shen (2014) and Shen (2015)[2,7]. As compared to 
and 
, 
and 
represent the amplitudes of temperature perturbation modes that are resolved using two additional Fourier modes with high wave-numbers [2,7]. The two modes are referred to as secondary temperature modes. As defined in the previous studies [2,7], the term “feedback” refers to the nonlinear processes that involve the secondary modes (i.e.,
and/or 
). Therefore, Eqs. (4-6) in the 5DLM can be viewed as a 3DLM with the feedback processes (i.e.,
) that result from the secondary modes. Physically, the term 
represents the nonlinear advection of secondary temperature mode that provides feedback to the primary temperature mode.
and 
are first obtained. In Appendix A of Shen [2], 
was shown to be responsible for conversion of domain averaged potential energy between 
and 
mode. Thus the negative nonlinear feedback of 
is also associated with the 
. Next, it is beneficial to express the secondary mode (i.e.,
) in terms of the primary modes (i.e., 
). To achieve this, 
is solved by assuming a steady-state solution for secondary modes in Eqs. (7-8), i.e.,
and 
, as follows: 
(9) 
(10) 
. With 
and 
in Eq. 10, the revised 3DLM (Eqs. 1-3) is referred to as the 3DLM-P5d or 3DLMP5d. Note that Eq. (10) can be simplified, as follows: (1) When 
is small and 
is positive, 
where 
is a tunable parameter. This type of parameterization with 
was first included in a revised 3DLM by Shen [2], who indicated that a larger
is required for the onset of chaos. (2) When 
is large (i.e., 
), 
and is the same as the dissipative term (
) in Eq. (6). Although it is challenging to linearize the revised 3DLM that includes a complicated nonlinear 
term(i.e., Eq. 10), the two simplified cases make it feasible to linearize Eq. (3) and to, thus, calculate Lyapunov vectors using the Gram-Schmidt reorthonormalization procedure discussed in Section 2.4. Additionally, these two simplified versions can be used to illustrate how changes in parameterizations may lead to different equilibrium-state solutions, as discussed below.
) and potential energy (
) are written, as follows: 
(11) 
(12) 
and 
, respectively, one obtains: 
(13) 
(14) 
(15) 
) associated with the primary modes is implicit and is internally responsible for energy conversion, while the nonlinear feedback term involving the secondary modes is represented by 
. The three terms on the right hand side represent energy sinks (or sources), 
and 
. As discussed, 
in Eq. (10), and 
plays a role similar to 
, which adds negative potential energy into the system (Eq. 12) and provides negative thermodynamic feedback. A steady-state solution is reached when the right hand side of Eq. (15) equals zero. The steady-state solution with 
is the critical point solution of the 3DLM (i.e.,
). With the general form of 
in Eq. 10, a steady-state solution appears when the primary modes (
) are replaced by the analytical solutions of the critical points in the 5DLM (Eqs. 19a-c of Shen, 2014), written, as follows: 
(16a) 
(16b) 
(16c) 
) is dominated by 
with 
when 
is relatively small and is dominated by 
when 
is relatively large. The two simplified cases lead to the following different steady-state solutions: 1) 
and 2) 
, respectively. Note that 
has the same form as Eq. (16a) for the 3DLM, the revised 3DLMs, and the 5DLM. Therefore, by choosing a different 
, we can change the equilibrium-state solutions for the revised 3DLM (Eqs. 1-3) (e.g., from the fixed-point solutions of the 3DLM, when 
, to fixed-point solutions of the 5DLM, when 
is described by Eq. 10). Although proper selection of 
(=1/2) in the the first simplified case may lead to the same equilibrium-state solutions as those of the second simplified case, their time-varying solutions are still different. In fact, when 
is small as compared to 
in Eq. (10), this condition poses an upper bound on 
(or 
) as a result of 
. The parameterized term (
) emulates the nonlinear feedback associated with the nonlinear advection of temperature and dissipative processes due to the secondary modes. Thus, any simplifications in 
indicate changes in the nonlinear advection and/or dissipation. The above discussions suggest the dependence for both transient and stead-state solutions on the detailed implementation of a parameterization, including nonlinear advection and/or dissipative processes. Further illustration with numerical results are provided in Figure 4 and discussed in Section 3.
and 
) of the 5DLM, as well as the 6DLM, introduce additional nonlinear and dissipative terms, which, in turn, provide negative nonlinear feedback, the secondary streamfunction mode (
) of the 6DLM (Eqs. 4-5 of Shen (2015)[7]) introduces additional nonlinear terms and adds a heating term. Therefore, these LMs were used to examine the competing impact between an additional heating term and the negative nonlinear feedback [7]. Research has indicated the following: 1) negative nonlinear feedback in association with the secondary temperature modes plays a dominant role in providing feedback for improving the solution stability of the 6DLM; 2) the additional heating term in association with the secondary streamfunction mode may destabilize the solution, but its impact is much smaller than that of the negative nonlinear feedback. Thus, the 6DLM has a comparable but slightly smaller 
(
) for the onset of chaos, as compared to the 5DLM with 
(Table 1). The 5DLM and 6DLM collectively suggest different roles for small-scale processes (i.e., stabilization vs. destabilization).
(17) 
(18) 
, 
and 
are solved and and expressed in terms of (
): 
(19) 
(20) 
(21) 
and 
in Eqs. (17-21) are functions of the primary modes (
). The revised 3DLM (Eqs. 1-3) using Eqs. (17-21) is referred to as the 3DLM-P6d or 3DLMP6d. In Section 2.4, we discuss numerical approaches for the calculation and analysis of solutions in both 3DLMP5d and 3DLMP6d.
) and the dissipative term (i.e.,
) was discussed in Shen (2014) [2]. An important question to be answered is how these processes may be identified in real world models such as climate models and/or weather models [13-15]. While it takes a significant effort to thoroughly address this question, an initial attempt is made in the following discussion. The nonlinear term,
, representing the horizontal and vertical advection of temperature, is commonly included in numerical models (explicitly or implicitly). While the dissipative term may be physically interpreted as an eddy dissipative process, it can be numerically linked to an averaging operator, as discussed below. For example, the dissipative term 
can be simplified, as follows: 
(22a) 
can be approximated by 
(22b) 
and 
represent a grid spacing and time interval, respectively. By choosing 
, we have 
(23) 
), qualitatively suggests that the averaging operator plays a role numerically similar to the Laplacian operator (
) in contributing to dissipative processes.
, is varied, while other parameters, including 
, 
, and 
, are held constant. The dependence of solution’s stability on 
was examined using the 3DLM, 5DLM, 6DLM and a revised 3DLM with 
and 
[2,7]. Below, the ensemble Lyapunov exponents are first discussed then followed by time-dependent solutions obtained from the 5DLM, 3DLMP5d, and 3DLMP6d. Numerical approaches, as discussed by Shen [2], are also briefly summarized.
(24) 
) is 
. The total number of time steps (
) is 500,000. To quantitatively evaluate whether or not the system is chaotic, we calculate the Lyapunov exponent (LE), a measure of the average separation speed of nearby trajectories on the critical point [16-25]. In Shen [2], the two methods implemented and tested are the trajectory separation (TS) method [23]and the Gram-Schmidt reorthonormalization (GSR) procedure[17,21]. Using given initial conditions (ICs) and a set of parameters in the LMs, the TS scheme calculates the largest LE while the GSR scheme produces "n" LEs. Here, "n" is the dimension of the 5D or 6DLM. Obtaining the n LEs using the GSR scheme requires the derivation of the so-called variational equation. Calculations are conducted with 
and 
, yielding 
. To minimize dependence on the ICs, 10,000 ensemble (
) runs with the same model configurations but different ICs are performed, and an ensemble averaged LE (eLE) is obtained from the average of the 10,000 LEs. A large 
and 
are used to understand the long-term average behavior of the solutions in the LMs. The eLEs calculations of the 3DLM, 5DLM and 6DLM, obtained using the two methods above, were previously discussed and compared in Shen [2,7]. A calculation of the Kaplana & Yorke fractal dimension [26]using the (three) leading eLEs from the GSR method was provided in Appendix A of Shen [7], as an additional verification. The results of the 3DLM, obtained using both methods, are presented for a comparison. Only the TS method is applied in the 3DLMP5d and 3DLMP6d because both models introduce complicated nonlinear terms, making it difficult to derive the variational equation for the GSR scheme.
) for the 3DLM, 5DLM, and revised 3DLMs (i.e., 3DLMP5d and 3DLMP6d). Table 1 summaries the critical value (
) for the onset of chaos in these models. As first indicated in Shen [2], the 5DLM requires a larger 
for the onset of chaos than the 3DLM. The critical value (
) of the 5DLM is 42.9 while the 
of the 3DLM is 24.74. It was shown that the negative nonlinear feedback via the 
term can help suppress chaotic responses. 
is present in the 5DLM but not in the 3DLM. Since the 3DLMP5d includes the impact of 
by assuming a steady-state solution for 
and 
in Eqs. (7-8), the critical value (
) is expected to be comparable to that of the 5DLM and larger than that of the 3DLM, as shown in Figure 1 and Table 1. Interestingly, the 
of the 3DLMP5d is 52.1, which is larger than that of the 5DLM. The result suggests that the solution of 3DLMP5d is less chaotic than that of the 5DLM, and that the transient evolution of small-scale processes by the secondary modes, which appear associated with nonzero 
and 
in the 5DLM, may destabilize the solution of the 5DLM, as further discussed using Figure 4.
(41.1) as compared to the 5DLM. The negative nonlinear feedback and the impact of an additional heating term in the 6DLM are emulated in the 3DLMP6d using steady-state solutions for the secondary modes. Therefore, the 3DLMP6d with two parameterized terms is used to ensure the validity of the nonlinear thermodynamic parameterization. In Figure 1 and Table 1, the 
of the 3DLMP6d is 51.4, which is comparable, but slightly smaller, than that of the 3DLMP5d (
). Similarly, the 6DLM has a smaller 
than that of the 5DLM. Figure 2 displays Y-Z plots with 
from the 3DLM, 5DLM, 3DLMP5d, and 3DLMP6d, indicating a chaotic solution for the 3DLM but stable solutions for the rest of the LMs.
, we analyze the terms on the right hand side of Eq. (6), including 
, 
, and 
. Figure 3 indicates that the first two terms are balanced with the 
in order to have a steady-state solution. When the feedback of 
is added into the 3DLMP5d, solutions for the 
and 
(Figure 3b) are comparable to those of the 5DLM, indicating that 
and 
are balanced with 
in the 3DLMP5d. The corresponding solutions of the 3DLMP6 (Figure 3c) are also comparable to those of the 5DLM and 3DLMP5, indicating the validity of the approach in determining and emulating the nonlinear feedback using the 6DLM.
and 
from the 5DLM, 3DLMP5 and 3DLMP6 are examined. As indicated in panels (a)-(b), all of the solutions eventually become steady, while the solution of the 5DLM displays a smaller decay rate. In the zoomed-in view (Figure 4c), the oscillatory solution of 5DLM displays larger amplitudes as compared to those of the 3DLMP5 and 3DLMP6. A comparison of the solutions obtained from the 3DLMP5d and 3DLMP6d indicates that the latter, with an additional term (i.e.,
), produces a solution with a comparable amplitude but a different phase, as compared to the former solution. The result is consistent with that of Shen [7]in that the positive feedback associated with the secondary streamfunction mode is small as compared to the negative feedback largely associated with the secondary temperature modes. To compare the evolution of different solutions, Figure 4c displays the trend mode of the solutions at the early stage (i.e.,
). While transient solutions from the revised 3DLMs with parameterizations (3DLMP5 and 3DLMP6) produce large differences in both phases and amplitudes, their trend modes are very close to that of the 5DLM. The result implies that the revised 3DLMs with parameterizations, which are emulated from the feedback of secondary modes, can produce comparable time averaged solutions as compared to the 5DLM or 6DLM, and lead to the same steady-state solutions when 
is smaller than the 
of the 5DLM (or 6DLM).
) in the 5DLM can be analytically expressed in terms of original Fourier modes within the 3DLM, and that feedbacks can be emulated by introducing an additional term (i.e., 
) into the 3DLM. The term, which emulates the collective impact of the horizontal advection of temperature perturbation and dissipations associated with the high-wavenumber modes, can be analytically determined by solving for a steady-state solution for the high-wavenumber modes within the high-dimensional LM (e.g., 5DLM), as discussed in section 2.1. Using this approach, the revised 3DLM is viewed as the 3DLM with nonlinear feedback parameterized by the additional term. This approach is referred to as the parameterization of nonlinear thermodynamic feedback.
) exceeds a critical value (
), is known as chaos or butterfly effect of the first kind. Although chaotic solutions are associated with the inclusion of nonlinearity (as indicated in a comparison between the nonlinear 3DLM and linearized 3DLM), increased degrees of nonlinearity in higher-dimensional Lorenz models can improve system stability, since both the 5DLM and 6DLM require a larger 
for the onset of chaos [2,7] (Table 1). In this study, we proposed the revised Lorenz model with only one parameterized term (i.e., 3DLMP5d) that can produce more stable solutions as compared to the 3DLM. The new term is added to emulate the nonlinear thermodynamic feedback that was first identified in the 5DLM. By analyzing the nonlinear term,
, using five modes (for the 5DLM), the negative nonlinear feedback of the secondary modes can be mathematically represented by the 
term in Eq. (6). Using the term 
, we demonstrated how to incorporate the negative nonlinear feedback into the revised 3DLM without using additional prognostic equations, based on the following: (1) We added a parameterized term 
, which is equal to 
, into a revised 3DLM. (2) We obtained steady-state solutions for the secondary modes (e.g., 
and 
in the 5DLM) and expressed 
in terms of the primary modes (
). Therefore, the nonlinear feedback processes are emulated using diagnostic equations in the revised 3DLMP5d, while they are explicitly resolved by prognostic equations in the 5DLM. A different simplification of the parameterized 
was found to lead to a different equilibrium state solution, indicating the dependence of the equilibrium state solution on the changes in nonlinear advection associated with the secondary or high-wavenumber modes. For example, the critical point 
in the revised 3DLM using a different 
can be 
, 
, 
, or 
when 
equals 
, 
, 
, or 
(Eq. 10), respectively.
of the 3DLMP5d is 
, which is larger than that of the 3DLM (
). The value is comparable to but larger than the 
of the corresponding 5DLM (
). Similar characteristics are observed when comparing the 3DLMP6d and 6DLM, which yield 
values of 51.4 and 41.1, respectively. The above features suggest that the stability of the 3DMLP5d (3DLMP6d) is overestimated between 
(
), as compared to the 5DLM (or 6DLM). For stable solutions, the revised 3DLM produces the same steady-state (equilibrium-state) solutions as those in the 5DLM (or 6DLM) when parameters are set to be the same in the models (e.g., 
). Although the revised 3DLM (3DLMP5d or 3DLMP6d) is unable to accurately represent the evolution of transient solutions in both amplitudes and phases, as compared to the 5DLM, the trend mode of its transient solution is comparable to that of the 5DLM. The trend mode is determined using the newly developed parallel ensemble empirical mode decomposition (PEEMD) method. The characteristics of solutions such as overestimated stability, lack of detailed evolution, and a comparable trend mode are often found in the comparisons of coarse-resolution climate model and fine-resolution climate (or weather) model simulations.
) and the dissipative term (i.e., 
). A nonlinear feedback loop consists of a pair of downscale and upscale transfer processes via the Jacobian term[2], and can be continuously extended as long as new modes can be continuously generated. The extended nonlinear feedback loop may further stabilize or destabilize system solutions. The mathematical analysis provided in section 2.3 suggests that an averaging operator may contribute to dissipative processes, which play a role numerically similar to the Laplacian operator (
). As nonlinear advection terms and averaging operators are often included in climate models, we hypothesize that the nonlinear feedback of numerical averaging/smoothing may suppress chaotic responses, leading to more stable solutions. This hypothesis will be examined using higher-dimensional Lorenz models and climate/weather models with the TS method in a future study. As numerical averaging may impact the phase and amplitude of transient solutions, posing a challenge for verifying time-varying numerical solutions against observations (or, comparing the transient solution obtained from a low- and high-dimensional models), an alternative method such as the PEEMD for comparing their trend modes is required. Detailed analyses and comparisons of the different Lorenz models and their application to climate model simulations are currently being planned.Editorial Information
Editor-in-Chief
Bianciardi Giorgio
University of Siena
Article Type
Research Article
Publication history
Received date: July 05, 2015
Accepted date: August 15, 2015
Published date: August 19, 2015
Copyright
©2015 Shen BW. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Citation
Shen BW (2015) Parameterization of Negative Nonlinear Feedback using a Five-dimensional Lorenz Model. Fractal Geometry and Nonlinear Anal in Med and Biol 1: doi: 10.15761/FGNAMB.1000109
