The Boussinesq approximation for flows with fast rotation or strong swirl

For flows with small density variations, it is convenient to approximate them as incompressible whilst retaining the leading order effects due to the density variations, thus avoiding issues associated with acoustic waves. The classical approach is the Boussinesq approximation which was originally motivated by the desire to account for gravitational buoyancy effects. There are many situations of interest where relatively fast rotations or swirls are present in flows with mild density variations which could benefit from such an approximation to an essentially incompressible system. In problems where a distinguished rotating frame of reference is readily identified, Boussinesq-type approximations accounting for centrifugal buoyancy are well-known. However, when such a frame of reference is not uniquely identified, for example due to differential rotations, we provide a straightforward approach to arrive at a consistent Boussinesq-type approximation.