We now return to ideal MHD, so that E + V × B = 0. The magnetic flux through and closed circuit C is where S is any surface bounded by C. Since ∇ · B = 0, we can write B = ∇ × A, where A is the vector potential. Then the flux can also be written as Now consider the volume defined by all field lines enclosed by the curve C. This volume V defines a flux tube. The flux Φ within V is constant because B is everywhere tangent to its boundary. We know that, since ∇ · B = 0, the tube thus defined either closes on itself or fills space ergodically. Any finite volume V 0 contains an infinite number of such flux tubes.
The chain of destiny can only be grasped one link at a time.
Winston Churchill
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Schnack, D.D. (2009). The Wöltjer Invariants of Ideal MHD, Topological Invariance, Magnetic and Cross-Helicity. In: Lectures in Magnetohydrodynamics. Lecture Notes in Physics, vol 780. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00688-3_12
Download citation
DOI: https://doi.org/10.1007/978-3-642-00688-3_12
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-00687-6
Online ISBN: 978-3-642-00688-3
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)