1. Introduction
Since the 1960s, when Lorenz presented the sensitive dependence of numerical solutions on initial conditions (ICs) with simplified governing equations describing a two-dimensional, forced, dissipative Rayleigh–Benard convection, it has been widely recognized that perfect deterministic weather predictions are impossible. In his breakthrough modeling study where three spatial Fourier modes were used to represent the streamfunction and temperature perturbations of the convection, Lorenz (1963a) showed that numerical results become chaotic, with sensitivities to ICs, when a normalized Rayleigh number r exceeds a critical number (e.g., rc = 24.74 for a constant Prandtl number σ = 10). This model is referred to as a three-dimensional Lorenz model (3DLM) in the present study. Lorenz associated the chaotic behavior with the inclusion of the nonlinearity. Subsequent to his follow-up presentation in 1972 (Lorenz 1972), the term “butterfly effect” was introduced to describe the sensitive dependence on ICs; later this became a metaphor (or symbol) for indicating that small-scale perturbations can make a huge impact on large-scale flows. In this study, the former and latter definitions are referred to as the butterfly effect of the first and second kind, respectively. The studies by Lorenz laid the foundation for chaos theory, which was viewed as the third scientific revolution of the twentieth century after relativity and quantum mechanics and is being applied in various fields including earth science, mathematics, philosophy, and physics (e.g., Gleick 1987; Anthes 2011).
Since the publications of Lorenz (1963a, 1972), views regarding the predictability of weather and climate have been significantly influenced by the butterfly effect (of the first and second kinds) or chaos theory (Solomon et al. 2007, 96–97; Pielke 2008). It is well accepted that weather is chaotic with only a finite predictability, and it is believed that the source of chaos in the 3DLM is the nonlinearities. Based on this understanding, one might expect that solutions to the equations with more nonlinear modes would become more chaotic, equivalent to stating that the appearance of small-scale features and their nonlinear effects, resolved by the additional modes, may make the system more chaotic. Since high-resolution global modeling approaches (e.g., Atlas et al. 2005; Shen et al. 2006a,b), which require tremendous computing resources, have become a current trend for weather prediction and climate projection, it becomes important to understand the role of the increased resolutions in the solution’s stability (or predictability) of the models. Three kinds of predictability that were proposed by Lorenz (1963b) include 1) intrinsic predictability that is dependent only on a flow itself, 2) attainable predictability that is limited by the imperfect initial conditions, and 3) practical predictability that shows dependence on (mathematical) formulas. The last type is discussed in this study by deriving a generalized Lorenz model (LM) and comparing its predictability with that of the 3DLM. In the literature, the term “a generalized LM” has been used to refer to the model that has modes more than the 3DLM.
Previous studies with the inclusion of additional Fourier modes have suggested that a larger r is required for the onset of chaos in generalized LMs [e.g., rc of approximately 43.5 in the generalized LM with 14 modes (Curry 1978)]. As compared to the aforementioned studies, a more systematic study for examining the resolution dependence of chaotic solutions was conducted numerically by Curry et al. (1984). They observed an irregular change in the degree of chaos as the resolution increased from a low resolution (i.e., three Fourier modes) and obtained a steady-state solution with sufficiently high resolution. However, as the resolution of the numerical weather models is finite and has always been increasing incrementally, it is important to understand the role of the incremental degree of nonlinearity in the solution’s stability. The term “degree of nonlinearity” is loosely defined as the degree of mode–mode interactions and is introduced to emphasize that the nonlinearities in numerical models such as the 3DLM are truncated (or finite) as a result of mode truncation. A more specific definition is given in section 3a. In a recent study of the routes to chaos in generalized LMs, Roy and Musielak (2007a) emphasized the importance in selecting modes that can conserve the system’s energy in the dissipationless limit. Furthermore, by analyzing the onset of chaos in the 3DLM and different generalized LMs with five and up to nine modes, Roy and Musielak (2007a,b,c) reported that some generalized LMs required a larger r (rc ~ 40) for the onset of chaos, but others displayed a comparable (e.g., rc ~ 24.74 in one of their LMs with six modes) or even a smaller rc (e.g., rc ~ 22 in their LM with five modes). The aforementioned studies give an inconclusive answer to the question of whether higher-dimensional LMs are more stable (predictable). A possible reason for this discrepancy among existing generalized LMs is presumably related to the various truncations of modes, leading to different degrees of nonlinearity. This is addressed using the following question in this study: under which conditions could the increased degree of nonlinearity improve solution stability? We will address this question with generalized LMs in this study.
In this study, we extend the 3DLM to the five-dimensional LM (5DLM) by including two additional Fourier modes with two additional vertical wavenumbers. In a companion paper (B.–W. Shen 2013, unpublished manuscript), we extend the 5DLM to the six-dimensional LM (6DLM) with an additional mode. Although the nonlinear mode–mode interactions in the 5DLM (6DLM) are still much less complicated than those in global weather models (e.g., Shen et al. 2006a,b) or the model used by Curry et al. (1984), they can be analyzed analytically to trace their impact on solution stability, illustrating the importance of proper selection in the new modes that can effectively increase the degree of nonlinearity. For example, we will discuss how additional nonlinear and damping terms, which are introduced in the 5DLM, can provide negative nonlinear feedback for improving the solution stability. The term “improvement of solution stability” is defined as the disappearance of a positive Lyapunov exponent (LE; e.g., Wolf et al. 1985) or appearance of a stable nontrivial critical point in a generalized LM that has the same system parameters (e.g., the normalized Rayleigh number) as the 3DLM. In a companion paper (B.-W. Shen 2013, unpublished manuscript) with the 6DLM, we further examine the competing impact of an additional heating term as compared to the dissipative and nonlinear terms that are first introduced in the 5DLM. In sections 2a and 2b of this study, we describe the governing equations and present the derivations of the 5DLM. In section 2c, we propose a revised 3D Lorenz model with a “parameterized” term (denoted 3DLMP) that can effectively emulate the impact of the negative nonlinear feedback that is explicitly resolved in the higher-dimensional (5D) LMs. In section 2d, we present the analytical solutions of the critical points in the 5DLM, 3DLM, and 3DLMP. Numerical approaches for the integrations of these models and the calculations of the Lyapunov exponents are discussed in section 2e. In sections 3a and 3b, we use mathematical equations to illustrate the nonlinear feedback loop in the 3DLM and discuss how the feedback loop can be extended with proper selection of new modes in the 5DLM. We then refer to the degree of the extension of the feedback loop, which depends on the number of modes and their hierarchical-scale interactions (i.e., interconnectivity or interactions of interactions) in the LMs, as the degree of nonlinearity. Then, we present the numerical results of the 3DLM, 5DLM, and 3DLMP in sections 3c and 3d. In section 3e, we discuss the dependence of the solution’s stability on the Prandtl number in the (σ, r) space. Conclusions appear at the end.
2. The generalized Lorenz models and numerical methods
a. The governing equations
b. The 5D Lorenz model
The six modes and their derivatives. Here l and m, representing the horizontal and vertical wavenumbers, are defined as πa/H and π/H, respectively. Also, H is the vertical scale of the convection, and a is a ratio of the vertical scale to the horizontal scale. The term 
c. A revised 3DLM with parameterized feedback
d. Analytical solutions of critical points
Based on the previous discussions, one may wonder if a four-dimensional LM (4DLM) can be obtained with additional simplifications. By ignoring Z1 (i.e., XZ1) in Eq. (13), the nontrivial critical point solution of Xc for Eqs. (10)–(13) can be obtained: 
e. Numerical approaches
The 3DLM, 3DLMP, and 5DLM are integrated forward in time with the fourth-order Runge–Kutta scheme. Since our main goal is to understand the impact of the additional modes on the representation of the advection of the temperature perturbation and the subsequent nonlinear interaction, we vary the value of r with other parameters kept constant, including σ = 10, 
Note that small Δτ and large N and En are used simply because our goal is to understand the long-term-averaged behavior of the solutions of the LMs. Unless stated otherwise, in this study, the eLEs as a function of r are discussed. Here we mainly analyze the results from the TS scheme (in Figs. 7, 8, 10) and use the results with the GSR scheme for verification (in Fig. 7a). To verify the performance of the GSR scheme, its leading eLEs are being used to estimate the Kaplan–Yorke fractal dimension (Kaplan and Yorke 1979). For a given r = 28 and σ = 10, the estimated fractal dimension with the GSR scheme is 2.061 272 08, which is very close to the 2.063 reported in Nese et al. (1987, p. 1957), and close to the 2.062 reported by Professor Sprott (http://sprott.physics.wisc.edu/chaos/lorenzle.htm). Additional experiments indicate that a larger Δτ and smaller N and En could also produce similar results with no impact on the conclusions of this study.
3. Nonlinear feedback of additional modes and their impacts on system stability
In the following sections, we discuss the impact of additional modes on the degree of nonlinearity and solution stability. We analyze the Jacobian term J(ψ, θ) to illustrate the nonlinear feedback loop of the 3DLM in section 3a and discuss how the feedback loop is extended by the proper selection of the M5 and M6 modes in section 3b. Then, we present numerical results from the 5DLM to examine the impact of the nonlinear feedback processes enabled by the two new modes in section 3c. The results of the revised 3DLMP with a parameterized term are analyzed in section 3d to verify the role of the negative feedback in improving solution stability.
a. The nonlinear feedback loop in the 3DLM
The Jacobian functions for the nonlinear interactions of the six modes. Coef indicates the coefficient corresponding to the specific Jacobian function. The crossed-out symbol indicates the negligence of a term that involves the crossed-out term. For example, 
A schematic diagram of the feedback loop that consists of the downscale and upscale transfer processes associated with J(M1, Mj), where j = 2, 3, 5, or 6. For a given Mj mode, J(M1, Mj) may lead to a downscale transfer process indicated by a downward arrow and an upscale transfer process indicated by an upward arrow. While scale interactions in the 3DLM forms a feedback loop (pink arrows), additional interactions in the 5DLM extend the feedback loop (blue arrows). The original feedback loop (pink arrows) and extended feedback loops (blue arrows) may be viewed as the main trunk and branches, respectively. A number in parentheses is the coefficient of the specific mode. The M3(ml) in the leftmost column represents that the M3 mode with a coefficient of ml is generated or influenced by a downscale transfer process from J(M1, M2). The terms −XZ and −XY1, which appear in Eqs. (11) and (12), are associated with the upscale transfer process of the J(M1, M3) (by the pink upward arrow) and J(M1, M5) (by the blue upward arrow), respectively. The characters ⊗3D and ⊗5D indicate the end of downscale transfer due to mode truncation in the 3DLM and 5DLM, respectively.
Citation: Journal of the Atmospheric Sciences 71, 5; 10.1175/JAS-D-13-0223.1
b. The extended nonlinear feedback loop in the 5DLM
Equations (31) and (35) indicate the differences in the representation of the nonlinear J(M1, M3) for the 3DLM and 5DLM. The missing M5 in Eq. (31) is equivalent to replacing the [sin(3mz) + sin(−mz)] by sin(−mz). As indicated by a simple comparison between the two terms in Fig. 2, the inclusion of the new mode leads to finer representation of J(ψ, θ) near the top and bottom boundaries. Specifically, the solutions in Eqs. (31) and (35) have different signs in layers of (0 < z < H/4) and (3H/4 < z < H), suggesting opposite phases. The differences are presumably related to the rapid changes in the sign of the solutions in the presence of chaos, which, however, is beyond the scope of the present study.
A comparison between [sin(3mz) + sin(−mz)] (blue curve) and sin(−mz) (red curve). The first represents the interaction of M1 and M3 via J(M1, M3), and the second provides an approximation to J(M1, M3) by neglecting sin(3mz) in the original 3DLM. The areas shaded with blue lines indicate opposite phases between these two modes at 0 ≤ z ≤ H/4 and 3H/4 ≤ z ≤ H.
Citation: Journal of the Atmospheric Sciences 71, 5; 10.1175/JAS-D-13-0223.1
Mathematically, Eqs. (35) and (36) collectively represent a “forcing” term, J[M1, J(M1, M3)], in d2Z/dτ2 that can be derived by taking the time derivative of Eq. (2). With the inclusion of M5, J(M1, M5) provides not only upscaling feedback to the M3 mode but also a downscale energy transfer to a smaller-scale wave mode that, in turn, requires the inclusion of the M6 mode (Fig. 1; Table 2). As discussed in appendix A, the M6 mode is required to conserve the total energy in the dissipationless limit. The term XY1 is responsible for the transfer of the domain-averaged potential energy at different scales (between the M3 and M6 modes). The feedback loop is further extended to M5 → M6 → M5 through J(M1, M5) and J(M1, M6). In summary, while the inclusion of M3 forms a feedback loop by introducing Eq. (17) in the 3DLM, the inclusion of M5 and M6 extends the feedback loop by introducing Eqs. (13) and (14) where additional dissipative terms are included. In the next section, we examine whether the feedback of the aforementioned nonlinear processes is positive or negative and show that the −XY1 term can provide the negative feedback to stabilize solutions.
c. Numerical results of the 5DLM
Figure 3 shows the normalized solutions of (Y, Z) and (Y1, Z1) using the 3DLM and 5DLM with three different values of r. The scales for normalization are the critical points (e.g., Yc and Zc) as defined in Eqs. (19) and (21) for the 5DLM and 3DLM, respectively. As first shown in Lorenz (1963a), when the 3D system (in the 3DLM) becomes chaotic at a large r (r > rc, rc = 24.74), the solution never reaches a steady state but oscillates irregularly with time around the nontrivial critical points. This feature can be seen in Fig. 3a from the 3DLM with r = 25. It has been shown that the solution is sensitive to initial conditions, which are referred to as the butterfly effect (of the first kind). As compared to the 3DLM, the 5DLM with the same r value of 25 produces a steady-state solution, as indicated by the converged trajectory that approaches a critical point at (Y/Yc, Z/Zc) = (−1, 1) in Fig. 3b. The 5DLM continues to generate steady-state solutions until r is beyond 43 (which will be discussed in Fig. 7). For an r value of 43.5, the 5DLM produces a chaotic solution with a butterfly pattern in the Y–Z space (Fig. 3c). The corresponding solutions for Y1 and Z1 are shown in Fig. 3d and have low values when Y rapidly changes its signs.
Phase space plots in the (a) 3DLM and (b)–(d) 5DLM. (a) 
Citation: Journal of the Atmospheric Sciences 71, 5; 10.1175/JAS-D-13-0223.1
Numerical results that display temporal fluctuations near the critical points are analyzed in Figs. 4 and 5. For stable cases in both the 3DLM and 5DLM (Figs. 4a,c), the solutions oscillate at small time scales and their envelopes decay at large time scales. The decay rate that leads to steady states is larger in the 5DLM than in the 3DLM. For chaotic cases shown in Figs. 4b and 4d, from the 3DLM with r = 25 to the 5DLM with r = 43.5, respectively, the solutions oscillate in the beginning and gradually grow with time. Chaos appears subsequently; its onset can be identified by rapid changes in the signs of X (or Y).
Time series plots for the (a),(b) 3DLM and (c),(d) 5DLM. The X, Y, and Z plots are in orange, green, and black, respectively. The Y1 and Z1 plots are in purple and blue, respectively. The three modes in the 3DLM with (a) r = 20 and (b) r = 25. The five modes in the 5DLM with (c) r = 25 and (d) r = 43.5.
Citation: Journal of the Atmospheric Sciences 71, 5; 10.1175/JAS-D-13-0223.1
The r–time diagram of the normalized numerical solutions from the 5DLM. The term r ranges from 25 to 50 with Δr = 0.5. (a) Z/Zc, (b) Z1/Z1c, (c) −Y/Yc, and (d) −Y1/Y1c. The terms Zc, Z1c, Yc, and Y1c are defined in Eqs. (19a)–(19d), respectively. The black line indicates the constant value of r = 43, which is close to rc = 42.9 for the 5DLM.
Citation: Journal of the Atmospheric Sciences 71, 5; 10.1175/JAS-D-13-0223.1
By calculating the numerical solutions of the 5DLM over a wide range of r and normalizing them using the corresponding critical points in Eq. (19), we show that the r–time diagram of the normalized solutions is useful in displaying stable and chaotic regions, providing a qualitative method of determining the rc for the onset of chaos. In Fig. 5 where (Z/Zc, Z1/Z1c, −Y/Yc, and −Y1/Y1c) are shown, white areas display the normalized values of 1 ± 0.01. For r = 25–43, the appearance of stable critical points is indicated by the white areas with a sufficient long period of time. In contrast, a chaotic regime can be identified as r > rc (where rc ~ 43) by rapid changes in both the sign and magnitude of the normalized solutions. This critical value is consistent with the analysis of the Lyapunov exponent (discussed later with Fig. 7b). Other than the above, this figure is able to monitor the transient processes and suggests a longer time for solutions to become steady (chaotic) when r gets closer to rc, consistent with the analysis of the eLEs that are close to zero as r ~ rc.
To examine the improved stability of the solutions in the 5DLM, we analyze the time evolution of each term on the right-hand side of Eqs. (10)–(12). Results from Eq. (12) are compared with those from Eq. (17) to illustrate the major difference between the 5DLM and 3DLM. For a stable case in the 3DLM (e.g., r = 20 in Fig. 6a), a steady-state solution exists in association with a balance between the nonlinear term (XY) and the dissipative term (bZ). However, at a large r (e.g., r = 25 in Fig. 6b), both of the terms evolve with time at a different growth rate and the solutions appear chaotic. The analysis seemingly supports the understanding that the source of chaos is the nonlinearity, as XY appears as a forcing term with respect to the other term bZ for the M3 mode in Eq. (17). However, by contrast, the 5DLM using the same normalized Rayleigh number (r = 25) produces a steady-state solution that corresponds to the balanced state achieved by the three terms XY, bZ, and XY1 (Fig. 6c). The second nonlinear term (XY1) has a magnitude comparable to bZ but is missing in the 3DLM. A similar balanced state can be found in the case with r = 35 (Fig. 6d). The comparison between Figs. 6b and 6c suggests the importance of XY1 in stabilizing the solution with r = 25, indicating the importance of an increased degree of nonlinearity. As discussed earlier, the feedback of XY1 to the dZ/dτ for the M3 mode [Eq. (12)] can be mathematically illustrated using a pair of Jacobian functions, J(M1, M3) and J(M1, M5), depicting the nonlinear processes of downscale transfer and subsequent upscale transfer that extend the feedback loop. From a macroscopic view discussed in appendix A, XY is responsible for the transfer of the domain-averaged kinetic energy and potential energy; XY1 is responsible for the transfer of the domain-averaged potential energy at different scales, which provides a path for dissipation via the 4bZ1 term in Eq. (14).
Forcing terms of dZ/dτ, which are from Eq. (17) of the 3DLM and Eq. (12) of the 5DLM. Results from the 3DLM with (a) r = 20 and (b) r = 25. Results from the 5DLM with (c) r = 25 and (d) r = 35. The black and orange lines represent XY and bZ, respectively, while the blue line represents XY1. In the 3DLM, XY and bZ are balanced to reach a steady state. In the 5DLM, the additional term XY1 is required to reach a steady state.
Citation: Journal of the Atmospheric Sciences 71, 5; 10.1175/JAS-D-13-0223.1
To quantitatively measure the degree of chaotic responses in the LMs with the goal of understanding the system’s predictability, we calculate the eLE using the TS and GSR numerical methods that were discussed in section 2e. Figure 7a shows the eLEs of the 3DLM and 5DLM as a function of r with 20 ≤ r ≤ 120 and an increment of one (Δr = 1), while Fig. 7b shows the eLEs of the 5DLM with 35 ≤ r ≤ 50 and Δr = 0.1. For the 3DLM, the eLEs using the TS scheme, as shown in a pink curve, suggest the appearance of chaos as r > rc, and rc is approximately 23.7. This rc is slightly smaller than the (linear) theoretical value of 24.74 proposed by Lorenz (1963a) using the stability analysis of the linearized 3DLM. Note that the accuracy of the rc depends on many factors, including the values of the system’s parameters (e.g., σ, b, and/or do), different initial conditions, numerical schemes, and so on. As our goal is to illustrate the (negative) nonlinear feedback associated with the new modes in the generalized LM, we made no attempt at searching for a precise rc. We use Δr = 0.1 to identify the rc, which is defined as the lowest value of r when the eLE becomes positive from negative. In addition to the transition from stable regions (eLEs < 0) to chaotic regions (eLEs > 0), two of the so-called window regions where the LEs are nearly zero can be identified in the vicinity of r = 93 and r = 100 in the pink curve. The results of the 3DLM, which display a relatively smaller rc and indicate the appearance of windows, are in good agreement with previous studies [e.g., Fig. 1 of Froyland and Alfsen (1984)]. To understand the sensitivity of the eLE calculations to a specific scheme, a comparison of the eLEs using the TS procedure and GSR scheme (e.g., the orange circle in Fig. 7a) was made, showing insignificant differences except near the window regions (e.g., the green curve in Fig. 7a).
The largest eLEs as a function of r in different LMs. (a) The eLEs of the 3DLM with Δr = 1 using different numerical schemes, including the TS scheme (pink) and the GSR procedure (orange). The green line represents the 5-times differences of the results from the two schemes, and the black line indicates the eLEs of the 5DLM. (b) The eLEs with Δr = 0.1 for the 5DLM. The appearance of chaotic solutions is indicated by positive eLEs. Note that the theoretical critical value of r for the onset of chaos in the original 3DLM is 24.74.
Citation: Journal of the Atmospheric Sciences 71, 5; 10.1175/JAS-D-13-0223.1
As compared to the 3DLM, the eLEs of the 5DLM (the black curve in Figs. 7a and 7b) indicate the following: (i) that a larger r (rc ~ 42.9) is required for the onset of the chaos; (ii) that one window exists but appears at a slightly larger r (i.e., r = 107); and (iii) that eLEs are comparable to the corresponding ones of 3DLM for 44 < r < 80 and display large differences when r < 44 and r > 80 (e.g., near window regions).
d. Results of the revised 3DLM
The previous discussions indicated that the XY1 plays a role in stabilizing the solutions in the 5DLM with 25 ≤ r ≤ 40, and the XY1 is the only difference between the first three equations of the 5DLM [Eqs. (10)–(12)] and the 3DLM [Eqs. (15)–(17)]. In section 2c, we proposed to emulate the XY1 using qX2 with a tunable parameter q in the revised 3DLM, as shown in Eqs. (15), (16), and (18). The range of q within 0–0.5 can be roughly estimated by the following relation 
As in Fig. 7, but for 3DLMP with q = 0.15 (green), 0.17 (red), or 0.19 (orange).
Citation: Journal of the Atmospheric Sciences 71, 5; 10.1175/JAS-D-13-0223.1
In addition to the long-term-averaged behavior of the solutions represented by the eLEs, we examine the time evolution of the solutions (X, Y, Z) from the revised 3DLM, normalized by the critical points of the original 3DLM [Eq. (21)]. Figure 9a displays the normalized solutions for the case using r = 35 and q = 0.17 that initially oscillate and later approach a steady state after τ = 40. The steady-state solutions of the nondimensional 
(a) Time series plots for the revised 3DLMP with r = 35 and q = 0.17. The orange, green, and black lines show 
Citation: Journal of the Atmospheric Sciences 71, 5; 10.1175/JAS-D-13-0223.1
The above experiments suggested that although the 3DLM becomes chaotic at 25 ≤ r ≤ 40, an additional nonlinear dissipative term that emulates the negative feedback, explicitly resolved in a higher-dimensional (5D) LM, can effectively and realistically stabilize the solutions of the revised 3DLM, leading to a (stable) steady-state solution. Using a given set of system parameters, the critical points (steady-state solutions) in a revised 3DLMP are not exactly the same as those in the original 3DLM. However, the differences between the former and the latter can be remained within 10% if a value of q is properly selected (i.e., q ≤ 0.19).
e. Stability analysis in the (σ, r) space
The previous sections discussed the stability problem by varying r. Here we examine the dependence of solution stability on σ and address the question of whether the 5DLM still requires a larger r for the onset of chaos when different values of σ are used. Although a task-level parallelism was implemented in the schemes for the eLE calculation, it is still computationally intensive for obtaining eLEs over a wide range of values for both σ and r (i.e., 5 ≤ σ ≤ 25, 20 ≤ r ≤ 50). Therefore, to achieve our goal efficiently, we begin with the stability analysis of the linearized LMs at a nontrivial critical point and conduct the eLE analysis using selected values of σ. The former is to examine the local predictability, while the latter is to give a measure of the total predictability of the system.
Numerical procedures for the local (or linear) stability analysis in the (σ, r) space are discussed in appendix C and briefly summarized as follows. To perform a stability analysis of the 3DLM, 3DLMP, or 5DLM, we linearize each of these LMs with respect to one of its nontrivial critical points [e.g., Eqs. (C2)–(C6)], obtain its characteristic or eigensystem [i.e., Eq. (C8)], and solve for their eigenvalues. The analytical solutions of critical points for the 5DLM [Eq. (19)], 3DLM [Eq. (21)], and revised 3DLMP [Eq. (22)] are used for the analysis. An eigenvalue λ can be a real or complex number, and its real part is denoted Re(λ). The appearance of a positive Re(λ) suggests an unstable solution near the critical point. In the following, we examine the solution stability by checking whether the largest Re(λ) is positive or negative.
Figure 10a shows the contour lines of the Re(λ) = 0 in the (σ, r) space, each of which describes the critical value 
Stability analysis of the linearized Lorenz models with Δσ = 0.01 and Δr = 0.01. (a) The leading eigenvalue Re(λ) as a function of σ and r. The pink, red, and black lines indicate a constant contour of Re(λ) = 0 for the linearized 3DLM, 3DLMP, and 5DLM, respectively. The solid circles with the same color scheme indicate rc determined by the eLEs analysis with Δr = 0.1 in the corresponding nonlinear LM. The green multiplication sign shows the relation r = σ(σ + b + 3)/(σ − b − 1) for λ = 0 in the linearized 3DLM. (b) Re(λ) as a function of r and a given σ = 10 for 3DLM (pink), 3DLMP (red), and 5DLM (black). It shows that the critical value of r with Re(λ) = 0 is 24.74, 43.54, and 45.94 for the linearized 3DLM, 3DLMP, and 5DLM, respectively.
Citation: Journal of the Atmospheric Sciences 71, 5; 10.1175/JAS-D-13-0223.1
Numerical experiments with different Lorenz models. The column “=q=” indicates additional information in the equations. The column “Figures” lists the figures that include the solutions from each of the Lorenz models. The rc and 
In realizing the stability dependence on σ from the linear analysis, we perform additional eLE calculations using our LMs with σ = 13, 16, 19, 22, and 25 and plot the rc values as solid circles. In each of the selected runs, the eLE analysis produces a slightly smaller critical value with respect to the linear stability analysis, that is, 
4. Concluding remarks
In this study, we derived the generalized 5D Lorenz model (LM) to investigate the impact of two higher-wavenumber modes on the numerical predictability. The domain-averaged total energy of the 5DLM is conserved in the dissipationless limit. Distinct from other studies with generalized LMs, we provided physical justification for the choices of additional modes that can improve solution stability and focused on the interpretation of the nonlinear-scale interactions (i.e., increased degree of nonlinearity) enabled by these additional modes. We first illustrated the nonlinear feedback loop in the 3DLM and emphasized the importance of properly selecting new modes to extend the feedback loop and thus improving the degree of nonlinearity in the 5DLM. By comparing with other generalized LMs, we found that the 5DLM might serve as the lowest-order generalized LM with increased system stability. The inclusion of new modes introduces both nonlinear terms and dissipative terms that have collective impact on the increase of solution stability. The additional nonlinear terms are mainly associated with the improved vertical advection of temperature. The mathematical simplicity of the 5DLM with respect to existing generalized LMs makes it easier to obtain the analytical solutions of its critical points, identify the major feedback process and its role in the solutions’ stability of the generalized LMs (e.g., 5DLM and 6DLM), and perform (linear) local stability analysis near the critical points over a wide range of parameters (σ, r). The analyses of both local stability and ensemble-averaged Lyapunov exponents (eLEs) show that the 5DLM requires a larger normalized Rayleigh number r for the appearance of chaotic solutions than the 3DLM. While Lorenz demonstrated the association of the nonlinearity with the existence of the nontrivial critical points and strange attractors in the 3DLM, we emphasized the importance of the nonlinearity in both producing new modes and enabling subsequent negative feedback to improve solution stability. More details are given below.
Through the mathematical analysis of the 3DLM, we discussed the feedback loop that includes the nonlinear terms J(M1, M2) and J(M1, M3) [Eqs. (30) and (31)]. The inclusion of the M3 mode in the 3DLM enables the appearance of the stable nontrivial critical points when 1 ≤ r < 24.74 but leads to chaotic solutions when r > 24.74. In comparison, the inclusion of the M5 mode in the 5DLM can improve the representation of J(M1, M3) by enabling a downscale transfer process and provide feedback to the M3 mode via an upscale transfer process J(M1, M5), which adds the −XY1 term in dZ/dτ [Eq. (12)]. Therefore, the nonlinear loop is extended through the Jacobian terms J(M1, M3) and J(M1, M5) [Eqs. (35) and (36)] and is further extended through J(M1, M5) and J(M1, M6) in the 5DLM, as shown in Fig. 1. Based on the eLE calculations, the critical value rc for the 5DLM with σ = 10 is approximately 42.9. The rc value of the 5DLM is comparable to the one determined by the local stability analysis of the linearized 5DLM that gives 45.94. Both the eLE analysis and the local (or linear) stability analysis suggest that the 5DLM still produces stable steady-state solutions when r ranges from 25 to 42, while the solution of the 3DLM becomes chaotic.
To understand the differences in the predictability between the 3DLM and 5DLM, the competing impact of the nonlinear term XY1 against other nonlinear and dissipation terms was illustrated with the use of Eq. (12) dZ/dτ = XY − XY1 − bZ. While the first nonlinear term (XY) and the linear term (bZ) act as a forcing term and dissipative term, respectively, the second nonlinear term (XY1) may work as an additional dissipative term. Therefore, chaotic responses that appear in the 3DLM can be suppressed further by the additional modes in the 5DLM, producing stable solutions such as 1 ≤ r < 42.9. However, we would like to emphasize that the negative feedback by the term −XY1 comes from the collective effects of the nonlinear and dissipative terms associated with the new modes and that it is not trivial to separate them. A macroscopic view suggests that XY1 enables the transfer of domain-averaged potential energy at different scales, which in turn enables the feedback associated with the dissipation of the M6 mode [i.e., 4bZ1 in Eq. (14)]. Although chaos may appear in the presence of nonlinearity as well as a heating term in the 3DLM, the increased degree of nonlinearity with additional dissipative terms (i.e., the extension of nonlinear feedback loop) in the 5DLM can reduce chaotic responses. Simply speaking, the appearance of small-scale processes that involve the nonlinear interactions with damping terms may help stabilize solutions. The role of the negative nonlinear feedback by −XY1 was further demonstrated by parameterizing its effect into the revised 3DLMP. Based on the analysis of the analytical solutions for the critical points of the 5DLM, the negative nonlinear feedback process through −XY1 is emulated by a nonlinear eddy dissipation term (−qX2, q ≥ 0). As the revised 3DLMP produces stable solutions as 25 ≤ r ≤ 40, it is suggested that the predictability (or chaos) of the 3DLM can be improved (or suppressed) by the nonlinear dissipation term.
Since numerical solutions with the 5DLM display sensitive dependence on ICs after r is greater than 42.9, the butterfly effect of the first kind exists. As the 5DLM (3DLM) contains only one horizontal and four (two) vertical wave modes, the predetermined nonlinear mode–mode interactions among the selected modes cannot generate any new modes and thus limit their spatial-scale interactions and upscale energy transfer. In addition, the inclusion of new modes could impact (i.e., increase) the stability of solutions in the 5DLM. Therefore, it is suggested that the appearance of the butterfly effect of the first kind cannot directly lead to the conclusion that small perturbations can alter large-scale structure, namely, the butterfly effect of the second kind, because 1) it requires further upscale transfer of energy by additional low-wavenumber modes and 2) the inclusion of new modes may have a significant impact on the solution stability (i.e., an extremely large r for the onset of chaos).
While chaotic solutions (associated with the butterfly effect of the first kind) occur in the low-dimensional LMs (e.g., 3DLM and 5DLM) that include very limited nonlinear-scale interactions (i.e., limited degree of nonlinearity), it was reported that stable solutions could be obtained in the “sufficiently high-resolution” model by Curry et al. (1984). Therefore, it is hypothesized that solution stability in high-dimensional LMs can be further increased through additional negative nonlinear feedback with additional modes in numerical modeling. However, the nonexistence of a nontrivial critical point in the 4DLM (as r > do + 1) may indicate the importance of proper mode truncation in improving the solution stability of the nonlinear system that has a finite degree of nonlinearity. Specifically, a comparison among the 3DLM, 4DLM, and 5DLM suggests that the inclusion of only the M5 (e.g., Y1) mode cannot effectively improve stability; while the inclusion of both the M5 and M6 modes can improve stability, the latter requires the former to help provide its feedback to the 3DLM through the new feedback loops, namely, M3 → M5 → M3 and M5 → M6 → M5. In addition, M6 is required to conserve the domain-averaged total energy in the dissipationless limit. Therefore, we suggest that an incremental change in the degree of nonlinearity (e.g., with only M5 mode) may not be a sufficient condition for improving stability particularly in the low-dimensional LMs. We will continue to examine this feature by incrementally increasing the number of modes in generalized LMs.
To achieve the above goals, we have derived a 6DLM with the inclusion of the M4 mode 
Acknowledgments
We thank anonymous reviewers, Drs. Y.-L. Lin, R. Anthes, X. Zeng, R. Pielke, J. Dutton, A. Molod, H.-M. H. Juang, S. Cheung, C.-L. Shie, and Y.-L. Wu for their valuable comments and encouragement, Professor Z. Wu for providing the code for the calculation of Gaussian white noise, and Ms. J. Dunbar and K. Massaro for proofreading this manuscript. 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. Special thanks are due to the library personnel of NASA GSFC Library for helping obtain a copy of several classical journal articles.
APPENDIX A
Energy Conservation in the 5DLM
APPENDIX B
Downscale and Upscale Transfer Processes in the Nonlinear Feedback Loop: A Simple Illustration
APPENDIX C
Numerical Method of the Stability Analysis near a Critical Point
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