Global parametric solutions of scalar transport

Passive scalar transport involves complex interactions between advection and diffusion, where the global transport rate depends upon scalar diffusivity and the values of the (possibly large) set of parameters controlling the advective flow. Although computation of a single solution of the advection–diffusion equation (ADE) is simple, in general it is prohibitively expensive to compute the parametric variation of solutions over the full parameter space QQ, even though this is crucial for, e.g. optimization, parameter estimation, and elucidating the global structure of transport. By decomposing the flows within QQ so as to exploit symmetries, we derive a spectral method that solves the ADE over QQ three orders of magnitude faster than other methods of similar accuracy. Solutions are expressed in terms of the exponentially decaying natural periodic patterns of the ADE, sometimes called “strange eigenmodes”. We apply the method to the experimentally realisable rotated arc mixer chaotic flow, both to establish numerical properties and to calculate the fine-scale structure of the global solution space for transport in this chaotic flow. Over 105 solutions within QQ are resolved, and spatial pattern locking, a symmetry breaking transition to disordered spatial patterns, and fractally distributed optima in transport rate are observed. The method exhibits exponential convergence, and efficiency increases with resolution of QQ.