Beltrami fields with a nonconstant proportionality factor are rare

Abstract

We consider the existence of Beltrami fields with a nonconstant proportionality factor $f$ in an open subset $U$ of ${\mathbf{R}}^3$. 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 solutions for an open and dense set of factors $f$ in the $C^k$ topology. 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, Yudovich and Zaslavsky.

13 pages

References in corpus (1)

• Dissipative continuous Euler flows

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