By assuming 2-D (x, z), incompressible and Boussinesq flow, the following equations were used by and : ∂∂t∇2ψ=-Jψ,∇2ψ+ν∇4ψ+gα∂θ∂x,∂θ∂t=-J(ψ,θ)+ΔTH∂ψ∂x+κ∇2θ. Here ψ is the streamfunction that gives the u=-ψz and w=ψx, which, respectively, represent the horizontal and vertical velocities; θ is the temperature perturbation; and ΔT represents the temperature difference at the bottom and top boundaries. The constants g, α, ν, and κ denote the acceleration of gravity, the coefficient of thermal expansion, the kinematic viscosity, and the thermal conductivity, respectively. The Jacobian of two arbitrary functions is defined as J(A,B)=(∂A/∂x)∂B/∂z-∂A/∂z∂B/∂x. Additionally, ∇4ψ=∂/∂x∇2∂ψ/∂x+∂/∂z∇2∂ψ/∂z. Based on the above partial differential equations, introduced a system of three ordinary differential equations to illustrate the characteristics of chaotic solutions. This system is a simplified version of the one derived by . For the reader's convenience, the same symbols as those in and are used here.

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: M1=2sin⁡(lx)sin⁡(mz),M2=2cos⁡(lx)sin⁡(mz),M3=sin⁡(2mz),M4=2sin⁡(lx)sin⁡(3mz),M5=2cos⁡(lx)sin⁡(3mz),M6=sin⁡(4mz). Here l and m are defined as πa/H and π/H, representing the horizontal and vertical wavenumbers, respectively, and a is a ratio of the vertical scale of the convection cell to its horizontal scale, i.e., a=l/m. The term H is the domain height, and 2H/a represents the domain width. Using these modes, ψ and θ can be represented as follows: ψ=C1(XM1+X1M4),θ=C2YM2+Y1M5-ZM3-Z1M6,C1=κ(1+a2)a,C2=ΔTπRcRa,Rc=π4a2(1+a2)3,Ra-1=νκgαH3ΔT, where C1 and C2 are constants, Ra is the Rayleigh number and Rc is its critical value for the free-slip Rayleigh–Benard problem. Using Eqs. (5) and (6), solutions within the 6DLM are represented by the six spatial modes M1 to M6 (Eqs. 3, 4) and their corresponding time-varying amplitudes (X, Y, Z, X1, Y1, Z1), respectively. By comparison, Eq. (3) was used to derived the 3DLM, and Eqs. (3) and (4) without M4 were used to derive the 5DLM. While the 3DLM and 6DLM (5DLM) have one horizontal wavenumber, they contain two and four vertical wavenumbers, respectively. In the text below, to facilitate discussions, M1 and M4 are referred to as primary and secondary streamfunction modes, respectively, M2 and M3 are referred to as primary temperature modes, and M5 and M6 are referred to as secondary temperature modes. Here, the reader should note that an implicit limitation of this approach is that nonlinear interactions among the selected modes cannot generate (impact) any new (other) modes that are not pre-selected, suggesting limited (spatial) scale interactions. While the impact of the secondary temperature modes (i.e., Y1 and Z1) on the solution's stability was discussed by Shen14 with the 5DLM, the impact of the secondary streamfunction mode (i.e., X1), which introduces a heating term (rX1), is the focus of the 6DLM provided here.

To transform Eqs. (1) and (2) into the “phase” space, a major step is to calculate the nonlinear Jacobin functions. Calculations indicate that J(ψ,∇2ψ) in Eq. (1) does not lead to any explicit term in the final 6DLM, or the 3DLM or the 5DLM. Here, the Jacobian term of Eq. (2), which is written as follows, is discussed: J(ψ,θ)=C1C2(XYJ(M1,M2)-XZJ(M1,M3)+XY1J(M1,M5)-XZ1J(M1,M6)+X1YJ(M4,M2)-X1ZJ(M4,M3)+X1Y1J(M4,M5)-X1Z1J(M4,M6)). Note that the 3DLM only contains the first two terms on the right-hand side of Eq. (7), namely XYJ(M1,M2) and -XZJ(M1,M3), while the 5DLM includes the first four terms.

After derivations, we obtain the 6DLM with the following six equations: dXdτ=-σX+σY,dYdτ=-XZ+X1Z-2X1Z1+rX-Y,dZdτ=XY-XY1-X1Y-bZ,dX1dτ=-doσX1+σdoY1,dY1dτ=XZ-2XZ1+rX1-doY1,dZ1dτ=2XY1+2X1Y-4bZ1. Here, τ=κ(1+a2)(π/H)2t (dimensionless time), σ=ν/κ (the Prandtl number), r=Ra/Rc (the normalized Rayleigh number, or the heating parameter), b=4/(1+a2), and do=(9+a2)/(1+a2). After deriving the 6DLM in the fall of 2011, the 6DLM outlined here was compared with the work by ), who obtained the same 6DLM. A more detailed analysis regarding how the system conserves energy in the dissipationless limit, as well as a comparison with the 3DLM and 5DLM, is provided in the following discussion.

The 3DLM can be obtained from the 6DLM when terms that involve (X1, Y1, Z1) are neglected. Alternatively, Eqs. (8)–(10) can be viewed as a 3DLM with the feedback processes that result from the three additional modes. Therefore, the 6DLM can be viewed as a coupled system that consists of the 3DLM (Eqs. 8–10) and a forced dissipative system with an additional heating term (e.g., Eqs. 11–13). Here, and in Shen14, unless otherwise stated, the term “feedback” refers to the nonlinear process that involves the secondary modes, namely (X1, Y1, and/or Z1). The 5DLM in Shen14 can be also obtained by ignoring the X1 and dX1/dτ in the 6DLM. As a result, the 6DLM can be viewed as a coupled system that consists of the 5DLM and an additional equation (i.e., Eq. 11) that introduces nonlinear feedback associated with an additional heating term (i.e., Eq. 12).

The domain-averaged kinetic energy (KE‾), available potential energy (APE‾), and potential energy (PE‾) are defined e.g.,, as follows: KE‾=12∫02H/a∫0H(u2+w2)dzdx,APE‾=-gαH2ΔT∫02H/a∫0H(θ)2dzdx,PE‾=-∫02H/a∫0Hgα(zθ)dzdx. Through straightforward derivations, we obtain the following equations: KE‾=Co2(X2+doX12),KEp‾=Co2X2. Here Co=π2κ21+a2a3. KEp‾ contains only a portion of the total KE‾ of the 6DLM from the primary streamfunction mode X, but represents the total KE‾ in the 5DLM and 3DLM. In a similar manner, as follows, APE‾=-Co2σr(Y2+Z2+Y12+Z12),PE‾=-Coσ(Z+Z1/2). Equations (17a) and (18) yield the following: KE‾+APE‾=Co2X2+doX12-σrY2+Z2+Y12+Z12=C3, while Eqs. (17b) and (19) lead to the following: KEp‾+PE‾=CoX22-σZ+Z12=C4. With Eqs. (8)–(13) in the dissipationless limit, the time derivatives of both Eqs. (20) and (21) are zero, so both C3 and C4 are constants. Therefore, Eqs. (20) and (21) indicate two energy conservation laws, including the conservation of the total KE‾ and APE‾ (i.e., Eq. 20). However, it should be noted that, as follows, KE‾+PE‾=CoX22+doX122-σZ+Z12≠constant. By comparison, the two energy conservation laws of the 5DLM are written as follows: KE5-D‾+APE5-D‾=Co2X2-σrY2+Z2+Y12+Z12=C5, KE5-D‾+PE5-D‾=CoX22-σZ+Z12=C6. It can been shown that both C5 and C6 are constants. Therefore, in the 5DLM, in addition to the conservation of the KE‾ and APE‾, the KE‾ and PE‾ are also conserved.

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 r but keep other parameters as constants, including σ=10, a=1/2, b=8/3, do=19/3, and a minimum value for Rc=27π4/4. In Figs. 1, 2, 3 and 6, the initial conditions are given as follows: (X,Y,Z,X1,Y1,Z1)=(0,1,0,0,0,0). The dimensionless time interval (△τ) is 0.0001. The total number of time steps (N) is 1 000 000 in Fig. 1 and 500 000 in Figs. 2, 3, and 6, yielding a total dimensionless time (τ) of 100 and 50, respectively. In Figs. 2 and 6, the solutions of the 3DLM and 5DLM are rescaled by the analytical solutions of their critical points (i.e., Eqs. 21 and 19 of Shen14). The solutions of the 6DLM are rescaled by the critical points of the 5DLM. In Sect. 3.4, the dependence of solution stability on the Prandtl number (σ) is discussed with selected values of σ.

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., ). In Shen14, the two methods implemented and tested are the trajectory separation (TS) method (e.g., ); and the Gram–Schmidt reorthonormalization (GSR) procedure e.g.,. Here, a brief summary of how LEs are calculated using the two methods is provided. Using given initial conditions (ICs) and a set of parameters in the LMs, the TS scheme calculates the largest LE, and the GSR scheme produces “n” LEs; here “n” is the dimension of the 5-D or 6-D LM. Calculations are conducted with △τ=0.0001 and N=10 000 000, yielding τ=1000. To minimize the dependence on the ICs, 10 000 ensemble (En=10 000) 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 N and En are used to understand the long-term average behavior of the solutions of the LMs and simplified LMs where some terms are ignored. While eLEs calculations using the above two methods were previously discussed and compared in Shen14, here, a calculation of the Kaplan–Yorke fractal dimension using the (three) leading eLEs from the GSR method is provided in Appendix A as an additional verification. Unless stated otherwise in the main text, the largest ensemble-averaged LE (eLE) for a given r is obtained from the TS method.

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 X1 are neglected and only one feedback term (XY1) is retained in Eqs. (9) and (10), (2) case 6DLMS2 where only XY1 is ignored in Eq. (10), and (3) case 6DLMS3 where rX1 is removed from Eq. (12). Results from these simplified 6DLMs are presented in Sect. 3.3.

In this section, we first discuss the selection of M4 and then its impact. As indicated in Shen14 and discussed in the Supplement, the inclusion of M5 and M6 modes is based on the analysis of the Jacobian term, J(ψ,θ), and can improve the representations of the temperature perturbation and the nonlinear advection of temperature. The appearance of ∂M5/∂x associated with the linear term ∂θ/∂x of Eq. (1) requires the inclusion of an M4 mode and the ∂M4/∂x associated with △T∂ψ/∂x of Eq. (2) provides feedback to the M5 mode (in Table 1 of Shen14). The M4 mode shares the same horizontal and vertical wavenumbers as the M5, but has a different phase (i.e., sin⁡(lx) vs. cos⁡(lx) in Eq. 4). Alternatively, via the ∂θ/∂x and △T∂ψ/∂x, the M4 and M5 modes are linked as follows: dX1dτ∝-doσX1+σdoY1,dY1dτ∝rX1-doY1, which can be derived by linearizing Eqs. (11) and (12) at the trivial critical point. The linearized equations are decoupled with the rest of the equations on the 6DLM, suggesting that the heating term (rX1) can impact other modes as well as the stability of the nonlinear 6DLM via nonlinear feedback, as discussed below. The above equations are reduced to the following: d2Y1dτ2+do(σ+1)dY1dτ-σdor-do3=0. By assuming the solution Y1∝exp⁡(βτ), we obtain the following two roots for β: β±(r)=-do(σ+1)±do2(σ+1)2+4σr-do3/do2. Here, β+ (β-) represents the larger (smaller) root. An unstable normal mode with β+>0 appears when r>do3. When do=1, the result in Eq. (29) can be applied to the linearized 3DLM. As do=19/3 and r<do3(∼254) in this study, both β+ and β- are negative and ∂β/∂r is positive. The focus is β+ because the corresponding mode dominates the solution as a result of a smaller decay rate as compared to β-. β+ has a minimum (i.e., the largest decay rate) as r=0, and increases as r increases (up to 254), leading to a decreasing decay rate. In the limit of r=0 and σ≥1, the minima of Eq. (29) can be written as follows: β+(r=0)=-doandβ-(r=0)=-doσ. The β+=-do provides the same decay rate as the one derived directly from Eq. (27) with r=0 (i.e., the removal of rX1). The simple analysis indicates that the inclusion of M4, as a result of β+<0 and |β+(r≠0)|<|β+(r=0)|, can lead to a solution component with a smaller decay rate. In other words, the inclusion of rX1 effectively reduces the dissipative impact of -doY1 in Eq. (27). Here, the reader should note that the relative impact of r with respect to σ can be estimated using the ratio between the first and second arguments of the radical in Eq. (29), written as 4σ(r-do3)/(σ+1)2/do3. The result suggests that rX1 becomes less important when a larger σ is used.

The discussions provided above illustrate how the secondary streamfunction mode (M4) may impact the growth rate of Y1 via the linear heating term (rX1). Additionally, M4 can also provide its nonlinear feedback by extending the nonlinear feedback loop of the 5DLM (as well as the 3DLM), as follows (also see Table 2 of Shen 2014a): J(M4,M2)=2mlM6-mlM3,J(M4,M3)=mlM2,J(M4,M6)=-2mlM2. While Eqs. (31) and (32) form a feedback loop with M2→M3→M2, Eqs. (31) and (33) enable another feedback loop with M2→M6→M2. Equations (32) and (33) only contain the vertical advection of temperature due to ∂M3/∂x=∂M6/∂x=0. The two equations suggest that both M3 and M6 can provide upscaling feedback to M2 through their interaction with M4, leading to two terms in Eq. (9), i.e., dY/dτ∝X1Z-2X1Z1. When Z1 is close to Z/2, their collective impact may become insignificant, X1(Z-2Z1)∼0, as compared to the other terms in Eq. (9). Since the former criterion can be met near the stable critical points of the 5DLM (e.g., Eq. 20b of Shen14) and since the 6DLM shares some similarities with the 5DLM, X1Z and -2X1Z1 are neglected in the 6DLMS1, whose results are discussed in Sect. 3.3. In the next section, we first compare the numerical results of the 5DLM and 6DLM.

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 (KE‾+APE‾) for both the 6D-NLM (blue) and 5D-NLM (red). While the total domain-averaged kinetic energy and potential energy (KE‾+PE‾) is shown in pink for the 5D-NLM, the kinetic energy of the primary streamfunction mode and the potential energy (KEp‾+PE‾) is shown in green for the 6D-NLM. Using the initial conditions in Eq. (25), the initial values of the normalized KE‾+APE‾ for the 6D-NLM (Eq. 20) and the 5D-NLM (Eq. 23) are given as C3/Co and C5/Co, respectively, and equal to -σ/2r. C3/Co (or C5/Co) is -0.2 for r=25 and -0.11 for r=45. The initial values of the normalized KEp‾+PE‾ for the 6D-NLM (Eq. 21) and the KE‾+PE‾ for the 5D-NLM (Eq. 24) are given as C4/Co and C6/Co, respectively, and both are zero. To effectively illustrate the conservation properties of the four quantities above, the time evolution of their deviations from the corresponding initial values produce four lines when plotted. Each of the lines may be shifted by a constant. For example, while the red line in Fig. 1 represents the time evolution of the deviation for KE‾+APE‾ in the 5D-NLM, (i.e., KE5-D‾(τ)+APE5-D‾(τ)-KE5-D‾(0)-APE5-D‾(0)), the blue line with a constant shift of 0.02 represents the time evolution of the deviation for KE‾+PE‾ in the 6D-NLM (i.e., KE‾(τ)+APE‾(τ)-KE‾(0)-APE‾(0)+0.02). As indicated in Fig. 1, each of the four quantities is conservative.

Next, we compare the normalized solutions of (Y, Z) in the 3DLM, 5DLM, and 6DLM with two different values of r. Normalization scales are defined by the critical points listed in Table 1. Figure 2a and b display the solutions from the 3DLM and 6DLM with r=35. Although the critical value (rc) for the onset of chaos is rc=24.74 in the 3DLM , a larger value is chosen for comparison with the 6DLM. The solution of the 3DLM never reaches a steady state but oscillates irregularly with time surrounding the non-trivial critical points. In contrast, as indicated by the converged trajectory that approaches a critical point that is close to (Y/Yc, Z/Zc)=(-1,1), the 6DLM yields a steady-state solution. Note that the normalization scales, Yc and Zc, are the critical points of the 5DLM, because it is difficult to obtain the analytical solution of the critical points in the 6DLM and the former and latter share similarities as discussed later. The 6DLM continues to generate steady-state solutions until r is beyond 41.1 (as discussed in Fig. 4). With an r value of 42.0, the 6DLM leads to a chaotic solution with a “butterfly” pattern in Y–Z space (Fig. 2d), while the 5DLM still produces a stable solution (Fig. 2c).

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 dZ/dτ (e.g., Eq. (10) for the 6DLM and Eq. (12) of Shen14 for the 5DLM) for the cases using r=35 that have steady-state solutions. Figure 3 indicates that all of the terms with the exception of X1Y, in the dZ/dτ of the 6DLM, yield comparable results to their counterparts in the 5DLM, indicating that XY1 also plays an important role in stabilizing the solution of the 6DLM as compared to the 5DLM. While the negative feedback by XY1 was verified by parameterizing its impact as a nonlinear eddy dissipation term into the 3DLM in Shen14, further verification using the 6DLM is provided in the following section. Due to a small value of X1, the X1Y is small as compared to other terms. A small value of X1 could also be inferred from the steady-state solution to Eq. (11), giving X1=Y1/d2≪Y1 as do=19/3. Additionally, the time evolution of the XY suggests that a steady state in the 5DLM is reached earlier than it is in the 6DLM, consistent with the decay rate analysis in Sect. 3.1.

Figure 4 provides the analysis, used to determine the critical value of r for the onset of chaos for both the 5DLM and 6DLM, of the eLEs as a function of the normalized Rayleigh parameter r. Both models produce similar distributions of the eLEs for 35≤r≤50, with the following features: (1) within the stable region (as eLEs <0), the magnitude of the eLEs is relatively smaller in the 6DLM; (2) the 6DLM requires a slightly smaller r (rc∼41.1) for the onset of chaos than the 5DLM (rc∼42.9); and (3) in fully chaotic regions (e.g., r>44), the eLEs of the 5DLM and 6DLM are in good agreement, with very small differences. The first two results are consistent with the stability analysis provided in Sect. 3.1, suggesting that inclusion of the M4 mode in the 6DLM may reduce the dissipative impact associated with the M5 mode.

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 M4 in the 6DLM. While the 6DLM has four nonlinear feedback terms (X1Z and -2X1Z1 in Eq. 9, and -XY1 and -X1Y in Eq. 10), the 5DLM only has one term, -XY1. Nonlinear feedback terms are defined as the nonlinear terms involving the secondary modes (X1, Y1, and Z1). Therefore, comparable eLEs between these two LMs suggest that -XY1 may play the most significant role in providing feedback for stabilizing solutions in the 6DLM. To verify this hypothesis, additional experiments are performed with the following simplified 6DLMs: 6DLMS1, 6DLMS2 and 6DLMS3, as introduced in Sect. 2.4 and listed in Table 1. While the 6DLMS1 case retains only one nonlinear feedback term, XY1, the 6DLMS2 case only neglects this term. By comparison, the 6DLMS3 case is designed to examine the role of the linear heating term (rX1) in Eq. (12). The corresponding eLEs are shown in Fig. 5. The eLEs of the 6DLMS2 resemble those of the 3DLM (Fig. 5a) with the exception of the window regions, indirectly indicating the importance of XY1 in stabilizing the solutions in the 6DLM. With the exception of the transition regions from eLEs<0 to eLEs>0 over a small range of r (i.e., r∼ 41–43), the eLEs of the 6DLMS1 and 6DLMS3 are close to those in the 6DLM and 5DLM. The rc of these two cases are determined to be 42.3 and 42.1, respectively, which are slightly larger (smaller) than rc=41.1 (rc=42.9) for the 6DLM (5DLM), as shown in Fig. 5b. In addition, the magnitudes of the LEs in the stable regions are determined to be relatively larger (smaller) than those in the 6DLM (5DLM). Since the 6DLMS1 ignores the nonlinear feedback terms associated with the X1 and since the 6DLMS3 neglects the rX1 term, the features of the 6DLMS1 and 6DLMS3, as compared to the 6DLM, also indicate that the impact of the M4 may slightly destabilize solutions.

The eLEs represent the averaged behavior of the model's solutions over a very large timescale, so N=10 000 000 and T=N△t=1000 (e.g., the T in Eq. (23) of Shen14 should approach infinity) are used. Since some of the terms in the simplified LMs (e.g., 6DLMS1-3) are ignored, it is important to check the time evolution of the solutions on a finite timescale in order to understand whether and how the solutions approach a stable critical point, or oscillate rapidly between (unstable) non-trivial critical points. To this end, we examine the r time diagram of the normalized solutions in Fig. 6, which displays the primary mode, -Y/Yc, and secondary mode, -Y1/Y1c, from the 6DLM, 6DLMS1, and 6DLMS3. Here, Yc and Y1c are the analytical solutions of the critical points from the 5DLM. Using this approach, the deviation of the normalized solutions from 1 (i.e., -Y/Yc-1) indicates the impact of the M4 mode that is missing in the 5DLM. In Fig. 6, the sharp gradient of the solutions with dense contour lines near the constant value of r=43 (in black) roughly indicates the critical value of r for the onset of chaos, consistent with the analysis of the eLEs in Fig. 5 (see Table 1). In stable regions, the primary mode, -Y/Yc, evolves with time and comes within 1±0.01 in each of the three cases (Fig. 6a, c, e). For the 6DLMS1 that only includes one nonlinear feedback term (XY1), the values of the secondary mode, -Y1/Y1c, in stable regions are also within 1±0.01 (Fig. 6d). By comparison, the normalized solutions (-Y1/Y1c) for the 6DLM and 6DLMS3 are within 1 and 0.9 in the steady state, suggesting a deviation within 10 % from the corresponding critical point of the 5DLM. If we view the stable solutions of the 5DLM as the results of the control run, the 6DLM provides approximate steady-state solutions that have derivations of only around 1 % in Y and approximately 10 % in Y1. The above results indicate that the nonlinear terms associated with the X1 (i.e., M4 mode) may produce larger relative deviations in the secondary mode Y1 (a high-wavenumber mode) than in the primary mode Y (a low wave-number mode).

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 (-XY1), enabled by the secondary temperature modes (Y1 and Z1) in the 5DLM. The result motivated an examination of whether or not a higher-dimensional model is more stable or less chaotic (i.e., a larger critical value of r) than a lower-dimensional model. In this study, the comparison of the 5DLM and 6DLM indicates that the additional mode (M4) in the 6DLM does not help increase but slightly decreases the critical value of r for the onset of chaos. In other words, the inclusion of M4 provides positive feedback that destabilizes the solutions through the heating term (e.g., rX1 in Eq. 12) and/or through its nonlinear interaction with other modes. Based on the results obtained from the 5DLM and 6DLM, we have demonstrated the roles of secondary modes (i.e., small-scale processes) in stabilizing and destabilizing system solutions. In addition, the collective impact of these secondary modes on the improvement of solution stability has been examined. Since the aforementioned results are obtained from the LMs with a fixed value of σ=10, the dependence of the stability in the 6DLM on various values of σ is discussed in the next section.

Previous sections discussed the stability problem only by varying the heating parameter, r. Here, we examine the dependence of solution stability on the parameter σ, and address the question of whether or not the 6DLM still requires a smaller (larger) rc for the onset of chaos than the 5DLM (3DLM) when different values of σ are used. To efficiently achieve the goal, we conduct the eLE analysis for the 6DLM using selected values of σ, and compare it with that from the 5DLM. The dependence of the 5DLM's stability on σ was previously examined by Shen14 by performing both linear stability and eLE analyses.

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 Re(λ)=0 in the (σ, r) space for the linearized 3DLM and 5DLM, respectively. Since λ is the largest eigenvalue, each line describes the critical value rcl as a function of σ, where the superscript “l” of rcl indicates the local (or linear) analysis. Following each of the contour lines in the direction of increasing σ, its right (or left) hand side contains areas with negative (or positive) values of Re(λ), suggesting stable (or unstable) solutions. Therefore, unstable solutions (Re(λ)>0) appear as rcl<r. Solid circles with the same color scheme indicate the rc determined using the eLE analysis with selected values of σ: σ=10, 13, 16, 19, 22 and 25. Given a σ, rc is, in general, smaller than rcl in both the 3DLM and 5DLM, as previously documented (see Shen14 for additional details).

The rc of the 6DLM, with the eLE analysis, is shown in Fig. 7 with blue multiplication signs. For all of the selected cases, the critical value rc in the 6DLM is larger than that in the 3DLM, suggesting that over the range between σ=10∼25, the 6DLM requires a larger r for the onset of chaos than the 3DLM. By comparison, in each of the selected cases with σ=10, 13, 16, and 19, the critical value (rc) in the 6DLM is (slightly) smaller than the one in the 5DLM. As a result, the 6DLM is less stable than the 5DLM as 10≤r<22. However, for the case with σ=22 (or σ=25), the rc of the 6DLM is comparable (or slightly larger), as compared to that of the 5DLM. The results may indicate a different role for the M4 mode between σ<22 and σ>22, or suggest the importance of increasing the ensemble members and/or increasing the coverage of the initial conditions for the calculations of the eLEs, all of which are subject to future study.