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Over the past several years, a series of papers regarding high-dimensional Lorenz models that have applied a different number of Fourier modes have yielded the following generalized Lorenz model (GLM) (Shen 2014-2018, 2019a, b; Shen et al. 2019; Reyes and Shen 2019; Shen et al. 2021a,b; Cui and Shen 2021; Shen et al. 2022a, b, c):

$\frac{d X}{d τ} = σ Y - σ X, (1)$

$\frac{d Y}{d τ} = - X Z + r X - Y, (2)$

$\frac{d Z}{d τ} = X Y - X Y_{1} - b Z, (3)$

$\frac{d Y_{j}}{d τ} = j X Z_{j-1} - (j+1) X Z_{j} - d_{j-1} Y_{j}, (4)$

$\frac{d Z_{j}}{d τ} = (j+1) X Y_{j} - (j+1) X Y_{j+1} - β_{j} Z_{j}, (5)$

$N = \frac{M - 3}{2}; d_{j-1} = \frac{(2 j + 1)^{2} + a^{2}}{1 + a^{2}}; β_{j} = b (j + 1)^{2}; b = \frac{4}{1 + a^{2}} . (6)$

Here, $τ$ is dimensionless time. The three integers j, M, and N are related to the number of additional Fourier modes within higher dimensional Lorenz models (LMs). While M represents the total number of modes (or equations), N indicates the total number of pairs for higher wavenumber modes that do not appear within the original L63 Model. The time-independent parameters, including $σ,$ $r,$ represent the Prandtl number and the normalized Rayleigh number (or the heating parameter), respectively. The heating parameter represents a measure of temperature differences between the bottom and top layers. Parameter "a" is defined as the ratio of the vertical scale of the convection cell to its horizontal scale and a² = 1/2. The last three parameters in Eq. (6) are coefficients for the dissipative terms. Detailed discussions for each of the above terms can be found in the 2-page Supplementary Materials of Shen et al. (2021a). Variable X denotes the amplitude of the Fourier mode for the stream function. Variables (Y, Z), (Y₁, Z₁), (Y₂, Z₂), and (Y₃, Z₃) are referred to as the primary, secondary, tertiary, and quaternary modes, respectively, and represent the amplitudes of the Fourier modes at different wave numbers for temperature. The GLM with M = 5, 7, or 9 is referred to as the 5D-, 7D-, or 9DLM, respectively, and the classical L63 model (referred to as the 3DLM) can be obtained using Eqs. (1-3) without the nonlinear term XY₁, written as follows (Lorenz 1963, 1993):

References

Lorenz, E.N., 1993: The Essence of Chaos; University of Washington Press: Seattle, WA, USA, 1993; 227p.
Lorenz, E.N., 1963: Deterministic nonperiodic flow. J. Atmos. Sci. 1963, 20, 130–141.
Shen, B.-W., 2014: Nonlinear feedback in a five-dimensional Lorenz model. J. Atmos. Sci. 2014, 71, 1701–1723. http://dx.doi.org/10.1175/JAS-D-13-0223.1
Shen, B.-W., 2015: Nonlinear feedback in a six-dimensional Lorenz Model: Impact of an additional heating term. Nonlin. Processes Geophys. 2015, 22, 749–764. https://doi.org/10.5194/npg-23-189-2016
Shen, B.-W., 2016: Hierarchical scale dependence associated with the extension of the nonlinear feedback loop in a seven-dimensional Lorenz model. Nonlin. Processes Geophys. 2016, 23, 189–203.
Shen, B.-W., 2017: On an extension of the nonlinear feedback loop in a nine-dimensional Lorenz model. Chaotic Modeling Simul. 2017, 2, 147–157. (pdf)
Shen, B.-W., 2019a: Aggregated Negative Feedback in a Generalized Lorenz Model. Int. J. Bifurc. Chaos 2019, 29, 1950037. https://doi.org/10.1142/S0218127419500378
Shen, B.-W., 2019b: On the Predictability of 30-Day Global Mesoscale Simulations of African EasterlyWaves during Summer 2006: A View with the Generalized Lorenz Model. Geosciences 2019, 9, 281. https://doi.org/10.3390/geosciences9070281
Shen, B.-W.; Reyes, T.; Faghih-Naini, S., 2019: Coexistence of Chaotic and Non-Chaotic Orbits in a New Nine-Dimensional Lorenz Model. In Proceedings of the 11th Chaotic Modeling and Simulation International Conference, CHAOS 2018, Rome, Italy, 5–8 June 2018; Skiadas, C., Lubashevsky, I., Eds.; Springer Proceedings in Complexity. Springer: Cham, Switzerland, 2019. https://doi.org/10.1007/978-3-030-15297-0_22
Reyes, T. and B.-W. Shen, 2019: A Recurrence Analysis of Chaotic and Non-Chaotic Solutions within a Generalized Nine-Dimensional Lorenz Model. Chaos, Solitons & Fractals. 125 (2019), 1-12. https://doi.org/10.1016/j.chaos.2019.05.003
Cui, J.; Shen, B.-W., 2021: A Kernel Principal Component Analysis of Coexisting Attractors within a Generalized Lorenz Model. Chaos Solitons Fractals 2021, 146, 110865. https://doi.org/10.1016/j.chaos.2021.110865.
Shen, B.-W.; Pielke, R.A., Sr.; Zeng, X.; Baik, J.-J.; Faghih-Naini, S.; Cui, J.; Atlas, R., 2021a: Is Weather Chaotic? Coexistence of Chaos and Order within a Generalized Lorenz Model. Bull. Am. Meteorol. Soc. 2021, 2, E148–E158. https://doi.org/10.1175/BAMS-D-19-0165.1
Shen, B.-W.; Pielke, R.A., Sr.; Zeng, X.; Baik, J.-J.; Faghih-Naini, S.; Cui, J.; Atlas, R.; Reyes, T.A., 2021b: Is Weather Chaotic? Coexisting Chaotic and Non-Chaotic Attractors within Lorenz Models. In Proceedings of the 13th Chaos International Conference CHAOS 2020, Florence, Italy, 9–12 June 2020; Skiadas, C.H., Dimotikalis, Y., Eds.; Springer Proceedings in Complexity. Springer: Cham, Switzerland, 2021. https://doi.org/10.1007/978-3-030-70795-8_57
Shen, B.-W., R. A. Pielke Sr., X. Zeng, J. Cui^#, S. Faghih-Naini^#, W. Paxson^#, A. Kesarkar, X. Zeng, R. Atlas, 2022c: The Dual Nature of Chaos and Order in the Atmosphere. Atmosphere 13, no. 11: 1892. https://doi.org/10.3390/atmos13111892.
Shen, B.-W.^*, R. A. Pielke Sr., X. Zeng, J. Cui, S. Faghih-Naini, W. Paxson, R. Atlas, 2022b: Three Kinds of Butterfly Effects Within Lorenz Models. Encyclopedia 2, no. 3: 1250-1259. https://doi.org/10.3390/encyclopedia2030084
Shen, B.-W.^*, R. A. Pielke Sr., X. Zeng, 2022a: One Saddle Point and Two Types of Sensitivities Within the Lorenz 1963 and 1969 Models. Atmosphere 13, no. 5: 753. https://doi.org/10.3390/atmos13050753

The Generalized Lorenz Model (GLM) and The Lorenz 1963 (L63) Model

References