Localized asymptotic solutions of the magneto dynamo equation in abc fields
type: Item
instance of: scholarly article
title: Localized asymptotic solutions of the magneto dynamo equation in abc fields
zbMATH Open document ID: 0865.35102
DOI: 10.1007/BF01210420
author: Sergei Yu. Dobrokhotov
author: W. Martinez-Olive
published in: Mathematical Notes
publication date: 1994-11-16T00:00:00Z
review text: One of the problems that arise in the linear theory of a magneto dynamo is to determine whether it is possible for the solution to increase without bound as \(t\to\infty\) for large Reynolds numbers. The mathematical statement of the problem involves investigating the Cauchy problem \[ {\partial H\over\partial t}+(v\cdot\nabla)H=(H\cdot\nabla)v+\nu\Delta H,\;x\in\mathbb{R}^3_x,\;H\bigl|_{t=0}=H_0(x),\text{ div }H_0=0.\tag{1} \] Here, \(v(x,t)\) is a specified vector function (flow rate of liquid), which is determined from the Euler equations and, consequently, \(\text{div }v=0\). The condition \(\text{div }H_0=0\) is, as may be easily verified, preserved by virtue of (1), and the field \(H(x,t)\) is also divergence free. The parameter \(\nu\) is equal to \(1/\text{Re}\), where Re is the Reynolds number, which is usually assumed to be small. We say that there exists a rapid dynamo in \(v\) if there exists a solution of (1) when \(\nu\ll 1\) that is at least exponentially increasing as \(t\to\infty\).
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 76W05
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 687520
zbMATH Keywords: magneto dynamo
zbMATH Keywords: large Reynolds numbers
zbMATH Keywords: Cauchy problem
zbMATH Keywords: Euler equations
MaRDI profile type: Publication
cites work: Q4742487
cites work: A kinematic theory of large magnetic Reynolds number dynamos
cites work: Magnetic field generation by the motion of a highly conducting fluid
cites work: Q3309368
cites work: Kinematic dynamo problem in a linear velocity field
cites work: Q3682920
full work available at URL: https://doi.org/10.1007/bf01210420
OpenAlex ID: W2030080660
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description: scientific article; zbMATH DE number 687520