Hierarchical scale dependence associated with the extension of the nonlinear feedback loop in a seven-dimensional Lorenz model

Abstract. 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 (J(ψ, θ)), where, ψ and θ represent the streamfunction and temperature perturbations, respectively. Fourier modes that represent temperature in the 7DLM can be categorized into three major scales as the primary (the largest scale), secondary, and tertiary (the smallest scale) modes. Further extension of the nonlinear feedback loop within the 7DLM can provide negative nonlinear feedback to stabilize solutions, thus leading to a much larger critical value for the Rayleigh parameter (r_c  ∼  116.9) for the onset of chaos, as compared to an r_c of 42.9 for the 5DLM as well as an r_c of 24.74 for the 3DLM. The r_c is determined by an analysis of ensemble Lyapunov exponents (eLEs) with a Prandtl number (σ) of 10. To examine the dependence of r_c on the value of the Prandtl number, a linear stability analysis is performed near the nontrivial critical point using a wide range of the Rayleigh parameter (40  ≤  r  ≤  195) and the Prandtl number (5  ≤  σ  ≤  25). Then an eLE analysis is conducted using selected values of the Prandtl number. The linear stability analysis is done by solving for the analytical solutions of the critical points, by linearizing the 7DLM with respect to the analytical solutions, and by calculating the eigenvalues of the linearized system. Within the range of (5  ≤  σ  ≤  25), the 7DLM requires a larger r_c for the onset of chaos than the 5DLM.

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 r  =  120, the Pearson correlation coefficients (PCCs) between the primary and secondary modes (i.e., Z and Z₁) and between the secondary and tertiary modes (i.e., Z₁ and Z₂) are 0.988 and 0.998, respectively. Here, Z, Z₁, and Z₂ represent the time-varying amplitudes of the primary, secondary, and tertiary modes, respectively. High PCCs indicate a strong linear relationship among the modes at various scales and a hierarchy of scale dependence. Future work will be undertaken to examine how higher-dimensional LMs may produce a larger critical value for the Rayleigh parameter for the onset of chaos and reveal stronger hierarchical scale dependence.