Dynamical property and triadic interaction of Beltrami-type rotating waves

Abstract

In this work, on the one hand, we continue with the study of the dynamic property of Beltrami flows [R. González, “Dynamics of non-axisymmetric Beltrami flows,” Phys. Fluids 26, 114104 (2014)], extending its scopes to non-stationary flows in the rotating system, which allowed us to classify the rotating waves on the basis of their phase velocities. On the other hand, and in accordance with this classification, we study the resonant triadic interaction of these waves. For this purpose, we use the expansion in a Chandrasekhar–Kendall basis for an infinite tube, in an analogous procedure to the one carried out by Waleffe [“The nature of triad interactions in homogeneous turbulence,” Phys. Fluids A 4, 350–363 (1992)], to study the interaction of plane Beltrami waves. Taking an equilibrium of the resulting non-linear equations for the amplitudes of the waves, we consider their linear and non-linear stability. Regarding their linear stability, we see that unlike plane Beltrami waves, their stability depends not only on the relative helicities but also on an interaction factor that depends on the properties and the co-rotating or counter-rotating character of the interacting waves. On the other hand, for non-linear stability dependent on the same parameters as those of linear stability, we find, for one case of analysis, that there is non-linear instability only for some interactions of two co-rotating waves with a counter-rotating one and we exhibit criteria that are sufficient conditions of non-linear stability or non-linear instability.