Optimal unstirred state of a passive scalar

Given a passive tracer distribution $f(x,y)$ , what is the simplest unstirred pattern that may be reached under incompressible advection? This question partially motivated by recent studies of three-dimensional (3-D) magnetic reconnection, in which patterns topological invariant called field line helicity greatly simplify until reaching relaxed state. We test two approaches: variational method with minimal constraints, and relaxation scheme where velocity determined explicitly $f$ . Both methods achieve similar convergence for simple cases. However, guarantees monotonic decrease Dirichlet seminorm numerically more robust. therefore apply latter to complex mixed modelled on 3-D braids. The unstirring separates into small number large-scale regions initial topology, well preserved during computation. Interestingly, found have same topology as Similarity simplification empirically reconnection simulations supports idea advection an important principle evolution.