In this talk, we study the emergence of spatially localised coherent structures induced by a compact region of spatial heterogeneity that is motivated by numerical studies into the formation of tornados. While one-dimensional localised patterns induced by spatial heterogeneities have been well studied, proving the existence of fully localised patterns in higher dimensions remains an open problem in pattern formation. We present a general approach to prove the existence of fully localised two-dimensional patterns in partial differential equations containing a compact spatial heterogeneity. This includes patterns with radial and dihedral symmetries, but also extends to patterns beyond these standard rotational symmetry groups. In order to demonstrate the approach, we consider the planar Swift--Hohenberg equation whose linear bifurcation parameter is modified with a radially-symmetric step function. In this case the trivial state is unstable in a compact neighbourhood of the origin and linearly stable outside. The introduction of a spatial heterogeneity results in an infinite family of bifurcation points with finite dimensional kernels, allowing one to prove local and global bifurcation theorems. We prove the existence of local bifurcation branches of fully localised patterns, characterise their stability and bifurcation structure, and then rigorously continue to large amplitude via analytic global bifurcation theory. Notably, the primary (possibly stable) bifurcating branch in the Swift--Hohenberg equation alternates between an axisymmetric spot and a non-axisymmetric `dipole' pattern, depending on the width of the spatial heterogeneity. We also discuss how one can use geometric singular perturbation theory to prove the persistence of the patterns to smooth spatial heterogeneities.

This work is in collaboration with Daniel Hill and Matthew Turner.