Using Fluid Variational Variables to Obtain New Analytic Solutions of Self-Gravitating Flows with Nonzero Helicity

Flow equations being nonlinear are notoriously difficult to solve analytically. In this work we show that, through a three- independent-functions variational formalism for steady barotropic flows, new analytical solutions of the flow equations can be obtained. A family of flows on predetermined toroidal Bernoulli surfaces is constructed. These flows have nonzero helicity and may be maintained by a suitable irrotational force distribution. For a particular density distribution, this force field can be provided approximately by self-gravitation.