  We construct a determining form for the surface quasi-geostrophic (SQG)
equation with subcritical dissipation. In particular, we show that the global
attractor for this equation can be embedded in the long-time dynamics of an
ordinary differential equation (ODE) called a determining form. Indeed, there
is a one-to-one correspondence between the trajectories in the global attractor
of the SQG equation and the steady state solutions of the determining form. The
determining form is a true ODE in the sense that its vector field is Lipschitz.
This is shown by combining De Giorgi techniques and elementary harmonic
analysis. Finally, we provide elementary proofs of the existence of
time-periodic solutions, steady state solutions, as well as the existence of
finitely many determining parameters for the SQG equation.
