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.