In this study, we construct a seven-dimensional Lorenz model (7DLM) to
discuss the impact of an extended nonlinear feedback loop on solutions'
stability and illustrate the hierarchical scale dependence of chaotic
solutions. Compared to the 5DLM, the 7DLM includes two additional high
wavenumber modes that are selected based on an analysis of the nonlinear
temperature advection term, a Jacobian term (

In addition to the negative nonlinear feedback illustrated and emulated by
the quasi-equilibrium state solutions for high wavenumber modes, the 7DLM
reveals the hierarchical scale dependence of chaotic solutions. For chaotic
solutions with

In 1963, Prof. Lorenz of MIT published two important papers that first
introduced the concept of finite predictability using an idealized model that
contained three ordinary differential equations (Lorenz, 1963a) and
classified three kinds of predictability (Lorenz, 1963b). The idealized
model, derived from the nonlinear partial differential equations governing
Rayleigh–Benard convection, is known as the three-dimensional Lorenz
model (3DLM). The 3DLM was used to illustrate the sensitive dependence of
numerical solutions on tiny changes in initial conditions. Appearing only in
nonlinear models, this unique feature is known as chaos or the butterfly
effect

Lorenz's studies have made significant impacts on the activities of both
real-world models and idealized models. To minimize the negative impact of
inaccurate initial conditions and to optimize parameters in a dynamical
system, sophisticated data assimilation schemes and systems have been
developed in order to improve forecasts. On the other hand, high-order Lorenz
models

Recent studies (Shen, 2014a, 2015b) based on the five- and six-dimensional Lorenz models (5-D and 6-D LMs) indicated that selections of high wavenumber modes that can properly extend the nonlinear feedback loop of the original 3-D Lorenz model may produce a negative nonlinear feedback to stabilize solutions. Furthermore, the impact of the negative nonlinear feedback was illustrated using revised 3DLMs with one or two parameterized terms that can emulate the effect of the negative feedback (e.g., Shen, 2014a, 2015a). The 5DLM and 6DLM, as well as the revised 3DLMs with parameterizations, require a larger value for the normalized Rayleigh parameter for the onset of chaos. In addition to the negative nonlinear feedback that comes from the nonlinear terms and dissipative terms in association with newly added modes, the destabilizing impact (i.e., positive feedback) of an additional heating term that appears in the 6DLM has been identified and examined by comparing it with the 5DLM. Studies based on the 5-D and 6DLM collectively suggest that the various roles of newly resolved small-scale processes can either stabilize or destabilize solutions, consistent with the impact of butterfly effect as stated by Lorenz (1972). Therefore, in general, to understand the impact of newly added high wavenumber modes on solution stability, it is important to examine the competing/collective impact of small-scale processes. The major similarities and differences between the 5DLM and 6DLM are as follows: (1) both models include negative nonlinear feedback that is associated with the extended nonlinear feedback loop; and (2) the 6DLM includes an additional high wavenumber streamfunction mode that introduces an additional time-dependent equation for its amplitude, an additional heating term, and several nonlinear terms. To improve the stability of high-dimensional LMs, based on the studies with the 5DLM and 6DLM, we suggested that it is important to select new modes to extend the nonlinear feedback loop that can effectively provide negative nonlinear feedback and that it is not critical to include an additional (streamfunction) mode that leads to an additional heating term to provide a positive feedback. Therefore, in this study, we construct a 7DLM using an approach similar to the 5DLM that extends the nonlinear feedback loop without introducing a new heating term.

The long-term goal is to determine under what conditions increasing resolutions can improve the predictions in weather/climate models. To achieve this goal, we first derive the higher-dimensional Lorenz models in order to illustrate the impact of the newly resolved small processes and the additional nonlinear terms (associated with the various mode truncations and model coupling) on system stability. Additionally, the high-dimensional LMs and the revised 3DLM with parameterized terms can be used to test the performance of the numerical methods in calculating the Lyapunov exponents (LEs). These types of studies may help identify an appropriate method for the LE calculation in real-world models. Then, the impact of small-scale processes, resolved by new changes in a model, on the solution stability can be better examined. The paper is organized as follows. Section 2 discusses the governing equations, the seven-dimensional Lorenz model, and the revised three-dimensional Lorenz model with parameterizations, analytical solutions for non-trivial critical points, and numerical approaches. Discussions of numerical solutions and analytical solutions are provided in Sect. 3. A conclusion is provided at the end. Detailed discussion regarding the selection of new modes based on the analysis of the Jacobian term is provided in the Supplement of Shen (2015b).

The governing equation for 2-D (

In this section, we discuss how the 7DLM is constructed using the following
seven Fourier modes:

Note that Shen (2015a) extended the 5DLM into a 6DLM by including the
secondary streamfunction mode (i.e.,

To transform the partial differential equations (Eqs. 1–2) into a set of
ordinary differential equations, a major step is representing the nonlinear
Jacobin terms using the Fourier modes. As discussed in Shen (2014a, 2015b),
the original Lorenz model only includes nonlinear terms from the the Jacobian
term of Eq. (2), which is written as follows:

A schematic diagram of an extended feedback loop which consists of
the downscaling and upscaling processes associated with

After derivations, the following seven equations of the 7DLM can be obtained:

Using Eqs. (14)–(15) of Shen (2015b) for definitions of domain-averaged kinetic
energy (

With Eq. (22) and the 7DLM (Eqs. 11–17), the time derivative of the total
energy becomes

In this section, we discuss how the nonlinear feedback processes resolved by
high wavenumber modes in the 7DLM can be emulated into a revised 3DLM.
Throughout the discussion, the negative nonlinear feedback provided by the
high wavenumber modes will be illustrated. The basic idea is to express
higher wavenumber modes in terms of lower wavenumber modes. To achieve this,
two steps are required. First, an assumption of quasi equilibrium state for
the tertiary modes is made in order to express the teritary modes in terms of
the primary and secondary modes. Simply speaking, an assumption is made that

Although

Here, to examine the linear stability, the analytical solutions of critical
points in the 7DLM are presented. Plugging Eq. (27) into Eq. (13), the seven,
time-independent counterpart of Eqs. (11)–(17) can be reduced to become one,
time-independent equation for the critical point solution of

To obtain numerical solutions for Lorenz models, Fortran codes were
previously developed based on the implementation of the fourth-order
Runge–Kutta scheme (e.g., Shen, 2014a). The codes are modified and used for
calculation of ensemble Lyapunov exponents (e.g.,

The largest ensemble-averaged Lyapunov exponents (eLEs) as a
function of the forcing parameter,

To quantitatively evaluate whether or not the system is chaotic, we calculate
the Lyapunov exponent (LE, e.g., Benettin et al., 1980; Froyland and Alfsen,
1984; Wolf et al., 1985; Nese, 1989; Zeng et al., 1991; Eckhardt and Yao,
1993; Christiansen and Rugh, 1997; Kazantsev, 1999; Sprott, 1997, 2003; Ding
and Li, 2007; Li and Ding, 2011) using the trajectory separation (TS) method
(Sprott, 1997, 2003) and the Gram–Schmidt reorthonormalization (GSR) method
(Wolf et al., 1985; Shen, 2014a, and references therein). Using the given
initial conditions (ICs) and a set of parameters in the LMs, the TS scheme
calculates the largest LE and the GSR scheme produces “

As compared to earlier studies, one unique finding obtained from this study
is a revelation of the scale dependance of chaotic solutions. To determine
the relationship (or the association or dependence) of two variables, we
calculate the Pearson correlation coefficient (PCC) and the Spearman rank
correlation coefficient (SRCC). Both coefficients measure the extent one
variable increases as the other variable increases. The PCC applies an
assumption of a linear relationship, while the SRCC is used as an alternative
for examining the dependence of two variables without an assumption of a
strong linear relationship. For PCCs, a positive high PCC between two
variables indicates a strong direct linear relationship, while a very small
PCC suggests a weak “linear” relationship. A PCC of zero indicates no
linear relationship. In addition to the PCC and SRCC, scatter plots and
linear regression lines are used to qualitatively display the linear
association between two variables. To verify the time lag (or lead) between
two time series (e.g., between the primary mode and tertiary mode), the cross
correlation function (CCF) is calculated. The PCCs (or SRCCs), CCF, and
linear regression lines are calculated using the function “cor” with an
option of “pearson” or “spearman”, the function “ccf”, and the function
“lm”, respectively, as provided by R

The characteristics of various Lorenz models. The “Equations”
column provides a list of the equations used in each specific Lorenz model.
Values for

Figure 2 provides the largest ensemble-averaged Lyapunov exponents (eLEs) as
a function of the forcing parameter

Phase space plots for (

Temperature perturbation in the spatial space (

For the 7DLM and 5DLM, we are able to obtain the analytical solutions of
critical points in the phase space, including Eq. (35) for the 7DLM and
Eq. (19) of Shen (2014a) for the 5DLM. Note that the analytical solutions of
the 3DLM were first solved in Lorenz (1963), and were documented in Eq. (21)
of Shen (2014a). Here, the analytical solutions are employed in Eq. (7) to
obtain the corresponding solution of the temperature perturbation (

The simplicity of the nonlinear analytical solutions may form a good case for testing a more generalized Lorenz model developed using finite difference or finite volume schemes. For example, during the initial model development, the analytical solutions for the 3DLM, 5DLM or 7DLM can be used to verify the numerical solutions from the generalized Lorenz model at comparable resolutions. Using this process, confidence in the performance of the generalized model can be constructed. Then, a much higher-resolution simulation with the generalized model can be used to better address the question of whether or not a higher-resolution model is more stable or more chaotic. Answering this question will be the topic of a future study. Next, a linear stability analysis with the analytical solutions is presented.

The above discussions suggest that the 7DLM with

To examine the impact of negative nonlinear feedback over the range of
Prandtl parameters, a linear stability analysis is performed with respect to
the critical point solutions in Eq. (35). Then, eLE analyses, which require
significant computational resources, are conducted using the selected values
of

Stability analysis for the linearized Lorenz models with

We follow the procedures in Appendix A of Shen (2014a) to linearize the 7DLM
with respect to the critical point solution and construct an eigenvalue
problem using the linear system. Given a pair of

Note that for a

A transition from a stable solution with

Previously, we suggested that the negative nonlinear feedback, which can help
stabilize solutions, can be emulated using the parameterized term

As discussed previously, the 7DLM with

A transition from a stable solution with

Scatter plots for

Scatter plots for the 5DLM and 7DLM with

Scatter plots for the 7DLM with

The Pearson correlation coefficients (PCC) and the Spearman Rank
correlation coefficient (SRCC) between two different variables from the
primary, secondary, and tertiary modes in the 5DLM or 7DLM. For the PCCs,
results obtained from three cases with

The cross-correlation function (CCF) for two time series (

In general, since the 7DLM is a nonlinear model, the secondary (tertiary)
temperature mode

Since the 7DLM includes three major scales containing the primary, secondary,
and tertiary temperature modes, the scale dependance (or correlation) among
them is further analyzed in Fig. 10. The PCCs between the secondary and
tertiary modes, which have a value of 0.967 for

With this study, the impact of an extended nonlinear feedback loop is
discussed and hierarchical scale dependence using the 7-D Lorenz model is
revealed. Based on the analysis of the nonlinear Jacobian term, the 7DLM is
constructed to include seven Fourier modes that possess three major scales,
including primary temperature modes (i.e.,

While the 7DLM produces a chaotic solution with a relatively large

In conclusion, the nonlinear feedback loop can be extended using the new modes that enable the successive downscale and upscale transfer described by the Jacobian term. The 7DLM, constructed by extending the nonlinear feedback loop of the 5DLM, reveals the role of the associated negative nonlinear feedback in stabilizing solutions and the hierarchical scale dependence in chaotic solutions. Future work will examine how higher-dimensional LMs may produce larger critical values for the Rayleigh parameter for the onset of chaos while displaying a stronger hierarchical scale dependence.

In this section, we discuss how the 10 000 initial conditions are generated
for calculation of the ensemble Lyapunov exponents and their impact on
determining the critical value of the Rayleigh number for the onset of chaos.
The 10 000 different ICs are produced as Gaussian white noise with the
center at the trivial critical point (i.e., with a mean value of zero for the
ICs). The method is described by Press et al. (1992) and the Fortran code was
kindly provided by Professor Z. Wu of Florida State University. Figure A1
shows the 100 000 ICs. For the 3-D, 5-D, or 7-D LM with a given

In Fig. 2, we discussed how the critical value of the Rayleigh
number (

10 000 initial conditions for calculation of the ensemble Lyapunov
exponent.

The largest ensemble-averaged Lyapunov exponents (eLEs) as a
function of the forcing parameter,

I thank Y. Wu, one anonymous reviewer, Z. Musielak, J. Duan (Editor),
X. Zeng, R. Pielke Sr., and C. Interlando for valuable comments and
encouragement, and E. K. Yoo for her help in the derivations and verification
of the seven-dimensional Lorenz model. I appreciate R. Carretero's kindness
in inviting me to give a lecture regarding high-order Lorenz models and
providing valuable comments during spring 2015. I am grateful for support
from the College of Science at San Diego State University and the NASA
Advanced Information System Technology (AIST) program. Resources supporting
this work were provided by the NASA High-End Computing (HEC) program and the
NASA Advanced Supercomputing division at Ames Research Center. Finally, I
would like to thank J. Tran for creating lovely animations for the 5DLM and
7DLM, which are available from