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Abstract
In fluid mechanics and magnetohydrodynamics it is often useful to decompose a vector field into poloidal and toroidal components. In a spherical geometry, the poloidal component contains all of the radial part of the field, while the curl of the toroidal component contains all of the radial current. This paper explores how they work in more general geometries, where space is foliated by nested simply connected surfaces. Vector fields can still be divided into poloidal and toroidal components, but in geometries lacking spherical symmetry it makes sense to further divide the poloidal field into a standard part and a 'shape' term, which in itself behaves like a toroidal field and arises from variations in curvature. The generalised PT decomposition leads to a simple definition of helicity which does not rely on subtracting the helicity of a potential reference field. Instead, the helicity measures the net linking of the standard poloidal field with the toroidal field as well as the new shape field. This helicity is consistent with the relative helicity in spherical and planar geometries. Its time derivative due to motion of field lines in a surface has a simple and intuitively pleasing form.
Original language  English 

Article number  495501 
Journal  Journal of Physics A: Mathematical and Theoretical 
Volume  51 
Issue number  49 
Early online date  23 Oct 2018 
DOIs  
Publication status  Published  12 Nov 2018 
Keywords
 helicity
 magnetic field topology
 poloidal fields
 toroidal fields
ASJC Scopus subject areas
 Statistical and Nonlinear Physics
 Statistics and Probability
 Modelling and Simulation
 Mathematical Physics
 Physics and Astronomy(all)
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Dive into the research topics of 'A generalized PoloidalToroidal decomposition and an absolute measure of helicity'. Together they form a unique fingerprint.Projects
 2 Finished

Complex Magnetic Fields: An Enigma of Solar Plasmas (joint with Durham University)
Hornig, G., Pontin, D. & WilmotSmith, A.
Science and Technology Facilities Council
1/04/13 → 30/06/16
Project: Research