Construct an enhancing random-stirring drift on a fixed-energy and fixed-enstrophy manifold

Construct an explicit choice of nonzero coefficients $s_J$ in the drift operator $\mathcal{B}=\sum_{J\in\mathcal{J}}s_JT_J$ such that the relaxation time of averages for the diffusion on the fixed-energy and fixed-enstrophy manifold $\mathcal{M}_{u,v}$ generated by $a\mathcal{B}+\mathcal{A}$ satisfies $\lim_{a\to\infty}\tau_{\rm rel}(a)=0$.

Background

Example 3 studies advection and random stirring on the smooth compact connected manifold $\mathcal{M}_{u,v}$ determined by fixed values of the Euclidean energy and weighted enstrophy. The diffusion operator is built from the vector fields $T_J$ and $R_i$, while the incompressible drift is $\mathcal{B}=\sum_{J\in\mathcal{J}}s_JT_J$.

The paper proves a necessary condition for enhancement: if $\tau_{\rm rel}(a)\to0$ as $a\to\infty$, then the graph formed by the indices with $s_J\neq0$ must be connected and their supports must cover all coordinates. The authors do not establish that any coefficients satisfying this condition actually produce enhancement, leaving the existence of such an example unresolved.