  We develop a new and general method to prove the the existence of the random
attractor (strong attractor) for the primitive equations (PEs) of large-scale
ocean and atmosphere dynamics under $non$-$periodic$ boundary conditions and
driven by infinite-dimensional additive fractional Wiener processes. In
contrast to our new method, the common method, compact Sobolev embedding
theorem, is to obtain the uniform $a$ $priori$ estimates in some Sobolev space
whose regularity is high enough. But this is very complicated for the 3D
stochastic PEs with the $non$-$periodic$ boundary conditions. Therefore, the
existence of universal attractor ( weak attractor) was established in previous
work.
  The main idea of our method is that we first derive that $\mathbb{P}$-almost
surely the solution operator of stochastic PEs is compact. Then we construct a
compact absorbing set by virtue of the compact property of the the solution
operator and the existence of a absorbing set. We should point out that our
method has some advantages over the common method of using compact Sobolev
embedding theorem, i.e., if the random attractor in some Sobolev space do exist
in view of the common method, our method would then further implies the
existence of random attractor in this space. The present work provides a
general way for proving the existence of random attractor for common classes of
dissipative stochastic partial differential equations and improves the existing
results concerning random attractor of stochastic PEs. In a forth coming paper,
we use this new method to prove the existence of strong attractor for the
stochastic moist primitive equations, improving the results, the existence of
weak (universal) attractor of the deterministic model.
