Determine whether the ABC flow is a fast kinematic dynamo

Determine whether the ABC velocity field constitutes a fast kinematic dynamo; equivalently, establish whether an exponentially growing magnetic-field mode persists as the magnetic Reynolds number R_m tends to infinity.

Background

The paper introduces the kinematic dynamo problem for the magnetic advection–diffusion equation and discusses the ABC flow as a canonical chaotic velocity field investigated numerically by Arnold and Korkina. The leading eigenvalue of the advection–diffusion operator is associated with the fastest-growing magnetic-field mode, but the behavior of this mode in the high-magnetic-Reynolds-number limit is not settled.

The unresolved issue is whether exponential growth survives as magnetic diffusivity tends to zero, or, equivalently, whether the ABC flow supports fast kinematic dynamo action. The paper develops finite-element approximations for spectra and pseudospectra but does not resolve this underlying analytical dynamo question.

References

It is still unknown whether this field is a fast kinematic dynamo, e.g., whether an exponentially growing mode of $\bm{B}$ survives as $R_m \to \infty$.

— Finite element exterior calculus for spectra and pseudospectra of advection-diffusion of differential forms  (2610.01501 - Boffi et al., 1 Oct 2026) in Introduction, quotation attributed to Arnold and Khesin; equation (arnold-khesin)