  In this article, we prove that data assimilation by feedback nudging can be
achieved for the three-dimensional quasi-geostrophic equation in a simplified
scenario using only large spatial scale observables on the dynamical boundary.
On this boundary, a scalar unknown (buoyancy or surface temperature of the
fluid) satisfies the surface quasi-geostrophic equation. The feedback nudging
is done on this two-dimensional model, yet ultimately synchronizes the
streamfunction of the three-dimensional flow. The main analytical difficulties
are due to the presence of a nonlocal dissipative operator in the surface
quasi-geostrophic equation. This is overcome by exploiting a suitable partition
of unity, the modulus of continuity characterization of Sobolev space norms,
and the Littlewood-Paley decomposition to ultimately establish various
boundedness and approximation-of-identity properties for the observation
operators.

Analysis of PDEs