Direct statistical simulation of low-order dynamo systems

TL;DR

This paper evaluates the effectiveness of direct statistical simulation (DSS) for low-order dynamo models, comparing techniques and developing a Python toolkit for deriving and solving statistical equations, highlighting the robustness of timestepping.

Contribution

It introduces a comprehensive methodology and Python package for deriving and solving cumulant-based statistical equations in low-order dynamo systems, comparing solution techniques.

Findings

Direct detection of fixed points is efficient at second order truncation.

Higher order truncations include many unstable fixed points.

Timestepping is more robust for finding meaningful solutions.

Abstract

In this paper we investigate the effectiveness of direct statistical simulation (DSS) for two low-order models of dynamo action. The first model, which is a simple model of solar and stellar dynamo action, is third-order and has cubic nonlinearities whilst the second has only quadratic nonlinearities and describes the interaction of convection and an aperiodically reversing magnetic field. We show how DSS can be utilised to solve for the statistics of these systems of equations both in the presence and the absence of stochastic terms, by truncating the cumulant hierarchy at either second or third order. We compare two different techniques for solving for the statistics, timestepping -- which is able to locate only stable solutions of the equations for the statistics and direct detection of the fixed points. We develop a complete methodology and symbolic package in Python for deriving the statistical equations governing the Low-order dynamic systems in cumulant expansions. We demonstrate that although direct detection of the fixed points is efficient and accurate for DSS truncated at second order, the addition of higher order terms leads to the inclusion of many unstable fixed points that may be found by direct detection of the fixed point by iterative methods. In those cases timestepping is a more robust protocol for finding meaninful solutions to DSS.