Scaling and predictability in surface quasi-geostrophic turbulence

Turbulent flows are strongly chaotic and unpredictable, with a Lyapunov exponent that increases with the Reynolds number. Here, we study the chaoticity of the surface quasi-geostrophic system, a two-dimensional model for geophysical flows that displays a direct cascade similar to that of three-dimensional turbulence. Using high-resolution direct numerical simulations, we investigate the dependence of the Lyapunov exponent on the Reynolds number and find an anomalous scaling exponent larger than that predicted by dimensional arguments. We also study the finite-time fluctuation of the Lyapunov exponent by computing the Cramér function associated with its probability distribution. We find that the Cramér function attains a self-similar form at large $\textit{Re}$ .