Li, K, Marston, JB and Tobias, SM (2021) Direct statistical simulation of low-order dynamo systems. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 477 (2254). ISSN 1364-5021
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 while the second has only quadratic nonlinearities and describes the interaction of convection and an aperiodically reversing magnetic field. We show how DSS can be used 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 meaningful solutions to DSS.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2021 The Author(s) Published by the Royal Society. All rights reserved. This is an author produced version of an article, published in Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | direct statistical simulation, solar dynamo, geodynamo, chaos |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Applied Mathematics (Leeds) |
Funding Information: | Funder Grant number EU - European Union 786780 Simons Foundation Not Known |
Depositing User: | Symplectic Publications |
Date Deposited: | 20 Sep 2021 11:07 |
Last Modified: | 17 Jan 2022 14:59 |
Status: | Published |
Publisher: | The Royal Society |
Identification Number: | 10.1098/rspa.2021.0427 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:178312 |