  The process of pattern formation in the two dimensional Swift-Hohenberg
equation is examined through numerical and analytic methods. Dynamic scaling
relationships are developed for the collective ordering of convective rolls in
the limit of infinite aspect ratio. The stationary solutions are shown to be
strongly influenced by the strength of noise. Stationary states for small and
large noise strengths appear to be quasi-ordered and disordered respectively.
The dynamics of ordering from an initially inhomogeneous state is very slow in
the former case and fast in the latter. Both numerical and analytic
calculations indicate that the slow dynamics can be characterized by a simple
scaling relationship, with a characteristic dynamic exponent of $1/4$ in the
intermediate time regime.
