What makes a steady flow to favour kinematic magnetic field generation: A statistical analysis

TL;DR

This study investigates the statistical properties of steady, incompressible flows to understand their ability to generate magnetic fields via kinematic dynamo action, using numerical simulations and statistical analysis.

Contribution

It combines classical dynamo theory with statistical methods to analyze flow properties and explores the potential of data-driven approaches for dynamo recognition.

Findings

No correlation between vorticity, kinetic helicity, and dynamo capability.

Large ensemble of flows shows traditional quantities are insufficient predictors.

Suggests need for advanced data-driven methods like neural networks.

Abstract

To advance our understanding of the magnetohydrodynamic (MHD) processes in liquid metals, in this paper we propose an approach combining the classical methods in the dynamo theory based on numerical simulations of the partial differential equations governing the evolution of the magnetic field with the statistical methods. In this study, we intend to answer the following ``optimization'' question: Can we find a statistical explanation what makes a flow to favour magnetic field generation in the linear regime (i.e. the kinematic dynamo is considered), where the Lorenz force is neglected? The flow is assumed to be steady and incompressible, and the magnetic field generation is governed by the magnetic induction equation. The behaviour of its solution is determined by the dominant (i.e. with the largest real part) eigenvalue of the magnetic induction operator. Considering an ensemble of 2193 randomly generated flows, we solved the kinematic dynamo problem and performed an attempt to find a correlation between the dominant eigenvalue and the standard quantities used in hydrodynamics -- vorticity and kinetic helicity. We have found that there is no visible relation between the property of the flow to be a kinematic dynamo and these quantities. This enables us to conclude that the problem requires a more elaborated approach to ``recognize'' if the flow is a dynamo or not; we plan to solve it using contemporary data-driven approach based on deep neural networks.