Dynamics of non-axisymmetric Beltrami flows

Abstract

In previous works [R. González, G. Sarasua, and A. Costa, “Kelvin waves with helical Beltrami flow structure,” Phys. Fluids 20, 024106 (2008) and R. González, A. Costa, and E. S. Santini, “On a variational principle for Beltrami flows,” Phys. Fluids 22, 074102 (2010)], we analyzed the Beltrami flow structure of Kelvin waves in an ideal fluid. As a result, we were able to show an important feature of Beltrami flows: their stability for Beltrami perturbations with the same eigenvalue as the basic flow. Here, instead, we study the dynamics of Beltrami perturbations by performing a modal analysis. In the first place, we study the modes that are generated by perturbing a uniformly translating and solidly rotating basic flow. In order to simplify the analysis, we consider the non-inertial frame in which this basic flow is at rest. In the second place, we analyze a basic Beltrami flow that is stationary in the non-inertial frame considered and is perturbed with Beltrami modes. We find that the last case is only possible when the perturbation eigenvalue is the same as that of the basic Beltrami flow. This is what we have called dynamical property. In both cases, the dynamics are represented by progressive waves in the moving frame. We apply this analysis to a rotating flow in an infinite cylinder and to an axisymmetric rotating Beltrami flow in a semi-infinite cylinder. In both cases, the development of secondary Beltrami modes is possible due to the dynamical property.