Horizontally periodic generalised surface quasigeostrophic patches and layers

We study solutions to the α -SQG equations, which interpolate between the incompressible Euler and surface quasi-geostrophic equations. We extend prior results on existence of bounded patches, proving propagation of H k -regularity of the patch boundary, k ⩾ 3 , for finite time for patches that are periodic in one spatial dimension. Such periodic patches also encompass layers, or two-sided fronts. As the authors have treated the Euler case in prior work, we now primarily focus on the range of α for which α -SQG lies strictly between the Euler and SQG equations.