  This paper is concerned with the asymptotic behavior of solutions of the
two-dimensional Navier-Stokes equations with both non-autonomous deterministic
and stochastic terms defined on unbounded domains. We first introduce a
continuous cocycle for the equations and then prove the existence and
uniqueness of tempered random attractors. We also characterize the structures
of the random attractors by complete solutions. When deterministic forcing
terms are periodic, we show that the tempered random attractors are also
periodic. Since the Sobolev embeddings on unbounded domains are not compact, we
establish the pullback asymptotic compactness of solutions by Ball's idea of
energy equations.
