In this study, a six-dimensional Lorenz model (6DLM) is derived, based on a
recent study using a five-dimensional (5-D) Lorenz model (LM), in order to
examine the impact of an additional mode and its accompanying heating term on
solution stability. The new mode added to improve the representation of the
streamfunction is referred to as a secondary streamfunction mode, while the
two additional modes, which appear in both the 6DLM and 5DLM but not in the
original LM, are referred to as secondary temperature modes. Two energy
conservation relationships of the 6DLM are first derived in the
dissipationless limit. The impact of three additional modes on solution
stability is examined by comparing numerical solutions and ensemble Lyapunov
exponents of the 6DLM and 5DLM as well as the original LM. For the onset of
chaos, the critical value of the normalized Rayleigh number (

Fifty years have passed since Lorenz published his breakthrough
modeling study

Lorenz demonstrated the association of the nonlinearity with the existence of
non-trivial critical points and strange attractors in the 3DLM. Shen (2014a,
denoted as Shen14) recently discussed the importance of nonlinearity in both
producing new modes and enabling subsequent negative feedback to improve
solution stability. The feedback loop of the 3DLM was defined by Shen14 as
a pair of downscale and upscale transfer processes associated with the
Jacobian function (in Eq. 2). The feedback loop has been suggested to
stabilize the solution for

In addition to the negative nonlinear feedback, Shen14 indicated that
a conclusion derived from lower-dimensional LMs may not be applicable in all
circumstances in a higher-dimensional LM. For example, although the butterfly
effect (of the first kind) with dependence of solutions on initial conditions
appears in the 3DLM within the range between

In a pioneering study using the generalized LM with a large number of Fourier
modes,

To achieve the goal outlined above, the 3DLM to 5DLM was previously extended in Shen14 by including the two secondary temperature modes. In this study, the 5DLM is extended to the 6DLM by adding an additional mode. The additional mode is included to improve the representation of the streamfunction (e.g., Eqs. 4 and 5), and is, therefore, referred to as the secondary streamfunction mode. While the secondary temperature modes of the 5DLM (as well as the 6DLM) introduce additional nonlinear terms and dissipative terms, which, in turn, provide negative nonlinear feedback, the secondary streamfunction mode of the 6DLM introduces additional nonlinear terms and adds a heating term. The approach, using incremental changes in the number of Fourier modes, can help trace their individual and/or collective impact on solution stability. For example, since the 6DLM also contains the negative nonlinear feedback in association with secondary temperature modes, it becomes feasible to examine the role of the additional heating term in the solution's stability and its competing impact with the negative nonlinear feedback.

The presented work is organized as follows. We describe the governing
equations in Sect. 2.1 and present the derivations of the 6DLM in Sect. 2.2.
We then discuss the energy conservation of the 6DLM in the dissipationless
limit in Sect. 2.3, and numerical approaches for integrations of the LMs and
calculations of ensemble Lyapunov exponents in Sect. 2.4. In Sect. 3.1, we
investigate the potential impact of the additional heating term on the
solution's stability by performing stability analysis near the trivial
critical point. We also illustrate how the feedback loop can be extended
using the secondary streamfunction mode. In Sect. 3.2, numerical results
obtained from the 6DLM are provided and compared to results obtained from the
5DLM. To examine the role of the secondary streamfunction mode and to
identify the major nonlinear feedback term, additional numerical experiments
using the 6DLM and simplified 6DLMs are compared in Sect. 3.3. Then, we
discuss the dependence of the solution's stability on the Prandtl number
(

By assuming 2-D (

To generalize the original Lorenz model, we first use the following six
Fourier modes (which are also listed in Table 1 of Shen14) to derive the
6DLM:

To transform Eqs. (1) and (2) into the “phase” space, a major step is to
calculate the nonlinear Jacobin functions. Calculations indicate that

After derivations, we obtain the 6DLM with the following six
equations:

The 3DLM can be obtained from the 6DLM when terms that involve (

The domain-averaged kinetic energy (

Time evolution of energy conservation laws from the 5D-NLM and
6D-NLM. (

Using the fourth-order Runge–Kutta scheme, the original and higher-order
Lorenz models are integrated forward in time. We vary the value of the
heating parameter

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 (e.g.,

To examine the collective or individual impact of the nonlinear feedback
terms and to identify the major feedback that can improve numerical
predictability in the 5-D and 6-D LMs, we perform additional runs using the
6DLM with additional simplifications. The experiments, as listed in Table 1,
include the following: (1) case 6DLMS1 where three nonlinear terms involving

A list of numerical experiments for different Lorenz models. The
column “Modifications” indicates additional changes in the “Equations”.
The

In the following sections, we discuss the impact of additional modes on
solution stability. In Sect. 3.1, we illustrate the potential role of the

In this section, we first discuss the selection of

The discussions provided above illustrate how the secondary streamfunction
mode (

In this section, we discuss the numerical results of the 6DLM beginning with
energy conservation laws in the dissipationless limit. The non-dissipative
version of the 6DLM (5DLM) is referred to as the 6D-NLM (5D-NLM). Figure 1
provides the time evolution of the total domain-averaged kinetic energy and
available potential energy (

Next, we compare the normalized solutions of (

In the following, we discuss the time evolution of the solutions for the 5DLM
and 6DLM to examine the impact of the secondary modes on solution's stability
and to identify the major feedback associated with these modes. First, we
analyze the

(

Forcing terms of

The largest ensemble-averaged Lyapunov exponents (eLEs) as a
function of the forcing parameter r in different LMs. The eLEs with

Figure 4 provides the analysis, used to determine the critical value of

In this section, we analyze the eLEs of the 6DLM with or without additional
approximations to identify the major feedback term and the impact of

Same as Fig. 4 except for

The

The

The eLEs represent the averaged behavior of the model's solutions over a very
large timescale, so

By comparing the 3DLM and 5DLM, Shen14 suggested that the stability of
solutions in the 3DLM can be improved by the negative nonlinear feedback
through the term (

Previous sections discussed the stability problem only by varying the heating
parameter,

For comparisons, the results obtained from the stability analysis of the 5DLM
and 3DLM in Shen14 are briefly summarized as follows: in Fig. 7, pink and
black lines indicate the contour lines of the

The

Five- and six-dimensional Lorenz models
(5DLM and 6DLM) were derived here and in Shen14 to examine the impact of
additional modes on solution's stability. The 5DLM includes two new Fourier
modes (i.e., the secondary temperature modes

A quantitative comparison of the eLEs from the generalized LMs with or
without additional simplifications suggests the following: (1) the negative
nonlinear feedback, first identified in the 5DLM and represented by

As compared to the 3-D and 5-D LMs in the dissipationless limit, the 6-D
non-dissipative LM also poses two energy conservation relations. One states
the conservation of the total domain-averaged kinetic energy
(

The competing impact of the nonlinearities and the dissipation and
heating terms can be illustrated using Eq. (10) of the 6DLM, as
follows:

The above results provide different impacts associated with various secondary
modes, consistent with Lorenz's statement in 1972, as follows: “If the flap
of a butterfly's wings can be instrumental in generating a tornado, it can
equally well be instrumental in preventing a tornado.” The quote suggests
the appearance of both positive and negative feedbacks (i.e., stabilization
and destabilization) in association with various “small-scale” processes.
Since mode truncation is unavoidable in finite-resolution models, the answer
to the question of whether or not the feedback by new modes is positive or
negative should be made in the proper context. The approach outlined here may
help us understand why some generalized LMs have a larger

The 5DLM and 6DLM share some similarities regarding the system's stability,
but the 6DLM has one additional model. To further our understanding of the
dynamics of chaos, it is required to address whether and where additional
critical points may appear and impact solution's stability in the 6DLM. Due
to increasing difficulties in obtaining the analytical solutions of the
critical points for the 6DLM, it becomes more challenging to perform an
analysis near the critical points. In addition to the analysis for examining
the competing impact between the additional dissipative and heating terms,
the dependence of solution's stability on the timescale (i.e., duration) of
the “forcing” terms deserves additional attention. Results obtained in this
study indicate eLE dependence on the number of modes (i.e., different
resolutions) and resolved processes (i.e., dissipative terms or heating
term). To improve our confidence in the model's long-term climate projections
using high-resolution global weather or climate models
(

Various methods are available for calculating fractal dimensions. There are
several mathematical definitions of different types of fractal dimension
(

The summation of all ensemble-averaged Lyapunov exponents (eLEs) in
the LMs (a) and three leading ensemble-averaged Lyapunov exponents (eLEs) as
a function of the normalized Rayleigh number (

The Kaplan–Yorke fractal dimension of the 3-D, 5-D, and 6-D LMs as
a function of the normalized Rayleigh number (

We thank V. Lucarini, one anonymous reviewer, S. Vannitsem (Editor), Y.-L. Lin, R. Anthes, X. Zeng, and R. Pielke Sr. for valuable comments and encouragement. We are grateful for support from the NASA Advanced Information System Technology (AIST) program of the Earth Science Technology Office (ESTO) and from the NASA Computational Modeling Algorithms and Cyberinfrastructure (CMAC) program. Resources supporting this work were provided by the NASA High-End Computing (HEC) program through the NASA Advanced Supercomputing division at Ames Research Center. Edited by: S. Vannitsem Reviewed by: V. Lucarini and one anonymous referee