Extreme events and instantons in Lagrangian passive scalar turbulence models

The advection and mixing of a scalar quantity by fluid flow is an important problem in engineering and natural sciences. The statistics of the passive scalar exhibit complex behavior even in the presence of a Gaussian velocity field. This paper is concerned with two Lagrangian turbulence models that are based on the recent fluid deformation model, but adding a passive scalar field with uniform mean gradient. For a range of Reynolds numbers, these models can reproduce the statistics of passive scalar turbulence. For these models, we demonstrate how events of extreme passive scalar gradients can be recovered by computing the instanton, i.e., the saddle-point configuration of the associated stochastic field theory. It allows us to both reproduce the heavy-tailed statistics associated with passive scalar turbulence, and recover the most likely mechanism leading to such extreme events. We further demonstrate that events of large negative strain in these models undergo spontaneous symmetry breaking.