An analytical study of the MHD clamshell instability on a sphere

This paper studies the instability of two-dimensional magnetohydrodynamic systems on a sphere using analytical methods. The underlying flow consists of a zonal differential rotation and a toroidal magnetic field is present. Semicircle rules that prescribe the possible domain of the wave velocity in...

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Bibliographic Details
Published in:Journal of fluid mechanics 2022-12, Vol.953, Article A38
Main Authors: Wang, Chen, Gilbert, Andrew D., Mason, Joanne
Format: Article
Language:English
Subjects:
Analytical methods
Asymptotic methods
Asymptotic series
Differential rotation
Eigenvalues
Flow
Flow stability
Fluid flow
Fluid mechanics
Geometry
Hemispheres
Instability
JFM Papers
Magnetic field
Magnetic fields
Magnetism
Magnetohydrodynamics
Mathematical analysis
Physics
Rotating bodies
Rotation
Shear
Shear flow
Solar physics
Stability analysis
Sun
Velocity
Wave velocity
Zonal flow (meteorology)
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Summary:This paper studies the instability of two-dimensional magnetohydrodynamic systems on a sphere using analytical methods. The underlying flow consists of a zonal differential rotation and a toroidal magnetic field is present. Semicircle rules that prescribe the possible domain of the wave velocity in the complex plane for general flow and field profiles are derived. The paper then sets out an analytical study of the ‘clamshell instability’, which features field lines on the two hemispheres tilting in opposite directions (Cally, Sol. Phys., vol. 199, 2001, pp. 231–249). An asymptotic solution for the instability problem is derived for the limit of weak shear of the zonal flow, via the method of matched asymptotic expansions. It is shown that when the zonal flow is solid body rotation, there exists a neutral mode that tilts the magnetic field lines, referred to as the ‘tilting mode’. A weak shear of the zonal flow excites the critical layer of the tilting mode, which reverses the tilting direction to form the clamshell pattern and induces the instability. The asymptotic solution provides insights into properties of the instability for a range of flow and field profiles. A remarkable feature is that the magnetic field affects the instability only through its local behaviour in the critical layer.
ISSN:0022-1120
1469-7645
DOI:10.1017/jfm.2022.973