Anomalous behaviour of a passive tracer in wave turbulence

We consider the behaviour of a passive tracer in multiscale velocity field, when there is no separation of scales; the energy spectrum of the velocity field extends into the region of long waves and even can be singular there. We suppose that the velocity field is a superposition of random waves. The turbulence of various ocean or atmospheric waves provides examples. We find anomalous diffusion (sub- and super-diffusion), anomalous drift (super-drift), and anomalous spreading of a passive tracer cloud. For the latter we find the existence of two regimes: (i) 'close' passive tracer particles diverge sub- or supper-exponentially in time, and (ii) a 'large' passive tracer cloud spreads as a power-law in time. The exponents, as well as the corresponding pre-factors, are found. The theory is confirmed by numerical simulations.