Topological entropy of stationary two-dimensional turbulence

Deformation of material lines drives transport and dissipation in many industrial and natural flows. Here, we report an exact Eulerian formula for the stretching rate of a material line, also known as the topological entropy, in a prototype two-dimensional fluid. The only requirement is a distribution of eigenvalues of the strain rate tensor and their decorrelation time. This eliminates the need for Lagrangian tracking in experimental turbulence where particle trajectories are entangled, and thus poorly resolved. Numerical simulations reveal an excellent agreement between our Eulerian estimate and the stretching rate of a Lagrangian material line, over a wide range of Reynolds number.