Beltrami Fields with a Nonconstant Proportionality Factor are Rare

We consider the existence of Beltrami fields with a nonconstant proportionality factor f in an open subset U of ({mathbb{R}^3}). By reformulating this problem as a constrained evolution equation on a surface, we find an explicit differential equation that f must satisfy whenever there is a nontrivial Beltrami field with this factor. This ensures that there are no nontrivial regular solutions for an open and dense set of factors f in the C^k topology, ({kgeqq 7}). In particular, there are no nontrivial Beltrami fields whenever f has a regular level set diffeomorphic to the sphere. This provides an explanation of the helical flow paradox of Morgulis et al. (Commun Pure Appl Math 48:571–582, 1995).