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.