  In this paper we show that, with probability 1, a random Beltrami field
exhibits chaotic regions that coexist with invariant tori of complicated
topologies. The motivation to consider this question, which arises in the study
of stationary Euler flows in dimension 3, is V.I. Arnold's 1965 conjecture that
a typical Beltrami field exhibits the same complexity as the restriction to an
energy hypersurface of a generic Hamiltonian system with two degrees of
freedom. The proof hinges on the obtention of asymptotic bounds for the number
of horseshoes, zeros, and knotted invariant tori and periodic trajectories that
a Gaussian random Beltrami field exhibits, which we obtain through a nontrivial
extension of the Nazarov--Sodin theory for Gaussian random monochromatic waves
and the application of different tools from the theory of dynamical systems,
including KAM theory, Melnikov analysis and hyperbolicity. Our results hold
both in the case of Beltrami fields on $\mathbf{R}^3$ and of high-frequency
Beltrami fields on the 3-torus.
