Determining modes for the surface quasi-geostrophic equation

We introduce a determining wavenumber for the surface quasi-geostrophic (SQG) equation defined for each individual trajectory and then study its dependence on the force. While in the subcritical and critical cases this wavenumber has a uniform upper bound, it may blow up when the equation is supercritical. A bound on the determining wavenumber provides determining modes, and measures the number of degrees of freedom of the flow, or resolution needed to describe a solution to the SQG equation.