Direct statistical simulation of the Lorenz63 system

Abstract

We use direct statistical simulation to find the low-order statistics of the well-known dynamical system, the Lorenz63 model. Instead of accumulating statistics from numerical simulation of the dynamical system or solving the Fokker–Planck equation for the full probability distribution of the dynamical system, we directly solve the equations of motion for the low-order statistics after closing them by making several different choices for the truncation. Fixed points of the statistics are obtained either by time evolving or by iterative methods. The stability and statistical realizability of the fixed points of the statistics are analyzed, and the statistics so obtained are compared to those found by the traditional approach. Low-order statistics of the chaotic Lorenz63 system can be obtained from cumulant expansions more efficiently than by accumulation via direct numerical simulation or by solution of the Fokker–Planck equation.

Improved assessment of the statistical stability of turbulent flows using extended Orr–Sommerfeld stability analysis

“…Applying a closure at next order so C (4) is some assumed function of the lower cumulants or simply ignored (termed CE3) is less straightforward as the ensuing positive definiteness of the second cumulant is not automatic (Marston, Qi & Tobias 2019). This difficulty explains the popularity of the lower-order CE2 approximation where, for example, the existence and stability of steady solutions has recently been investigated for ordinary differential equation (ODE) systems (Li, Marston & Tobias 2021;Li et al 2022).…”

“…This difficulty explains the popularity of the lower-order CE2 approximation where, for example, the existence and stability of steady solutions has recently been investigated for ordinary differential equation (ODE) systems (Li, Marston & Tobias 2021; Li et al. 2022).…”

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“…Li et al. 2022) and therefore it is prudent to begin with CE2. The CE2 has been shown to be effective for simple systems where tightly coupled correlations control the dynamics and driving is either stochastic (e.g.…”

“…Whilst higher-order DSS models do succeed in reproducing results that CE2 does not (see , for a recent review), it is not clear a priori which model is required for a given system (even for simple systems, e.g. Li et al 2022) and therefore it is prudent to begin with CE2. The CE2 has been shown to be effective for simple systems where tightly coupled correlations control the dynamics and driving is either stochastic (e.g.…”