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Each component of the velocity field $u$ is determined by the active scalar $\\theta$ through $\\mathcal{R} \\Lambda^{-1} P(\\Lambda) \\theta$ where $\\mathcal{R}$ denotes a Riesz transform, $\\Lambda=(-\\Delta)^{1/2}$ and $P(\\Lambda)$ represents a family of Fourier multiplier operators. The 2D Navier-Stokes vorticity equations correspond to the special case $P(\\Lambda)=I$ while the surface quasi-geostrophic (SQG) equation to $P(\\Lambda) =\\Lambda$. 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