Chaotic advection in three dimensions

In three-dimensional flows the velocity field cannot be defined through a streamfunction, therefore the advection of fluid elements does not have the simple Hamiltonian structure as in two dimensions. One significant result on mixing in three dimensions is related to the existence of invariant surfaces in steady inviscid flows (Arnold, 1965). The velocity field of such flows is a solution of the time-independent Euler equation 

An example of a Beltrami flow is the ABC flow, named after Arnold, Beltrami and Childress. It is defined by the velocity field 

However, the effect of a small perturbation in action-action-angle type flows is quite different. The two-parameter family of invariant cycles coalesce into invariant tori that are connected by resonant sheets defined by the u(h,l2) = 0 condition. The consequence of this is that contrary to action-angle-angle flows in this case a trajectory can cover the whole phase space and no transport barriers exist. Thus, in this type of flows global uniform mixing can be achieved for arbitrarily small perturbations. This type of resonance induced dispersion has been demonstrated numerically in a low-Reynolds number Couette flow between two rotating spheres by Cartwright et al. 

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