The poloidal–toroidal decomposition is unique if it is required that the average of the scalar fields Ψ and Φ vanishes on every sphere of radius r.[3]
Cartesian decomposition
A poloidal–toroidal decomposition also exists in Cartesian coordinates, but a mean-field flow has to be included in this case. For example, every solenoidal vector field can be written as
where denote the unit vectors in the coordinate directions.[6]
^Subrahmanyan Chandrasekhar (1961). Hydrodynamic and hydromagnetic stability. International Series of Monographs on Physics. Oxford: Clarendon. See discussion on page 622.
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