Optimal dynamical systems of Navier-Stokes equations based on generalized helical-wave bases and the fundamental elements of turbulence

In this paper, we present the theory of constructing optimal generalized helical-wave coupling dynamical systems. Applying the helical-wave decomposition method to Navier-Stokes equations, we derive a pair of coupling dynamical systems based on optimal generalized helical-wave bases. Then with the method of multi-scale global optimization based on coarse graining analysis, a set of global optimal generalized helical-wave bases is obtained. Optimal generalized helical-wave bases retain the good properties of classical helical-wave bases. Moreover, they are optimal for the dynamical systems of Navier-Stokes equations, and suitable for complex physical and geometric boundary conditions. Then we find that the optimal generalized helical-wave vortexes fitted by a finite number of optimal generalized helical-wave bases can be used as the fundamental elements of turbulence, and have important significance for studying physical properties of complex flows and turbulent vortex structures in a deeper level.

Flow structures, nonlinear inertial waves and energy transfer in rotating spheres

We investigate flow structures,nonlinear inertial waves and energy transfer in a rotating fluid sphere,using a Galerkin spectral method based on helical-wave decomposition(HWD).Numerical simulations of flows in a sphere are performed with different system rotation rates,where a large-scale forcing is employed.For the case without system rotation,the intense vortex structures are tube-like.When a weak rotation is introduced,small-scale structures are reduced and vortex tubes tend to align with the rotation axis.As the rotation rate increases,a large-scale anticyclonic vortex structure is formed near the rotation axis.The structure is shown to be led by certain geostrophic modes.When the rotation rate further increases,a cyclone and an anticyclone emerge from the top and bottom of the boundary,respectively,where two quasi-geostrophic equatorially symmetric inertial waves dominate the flow.Based on HWD,effects of spherical confinement on rotating turbulence are systematically studied.It is found that the forward cascade becomes weaker as the rotation increases.When the rotation rate becomes larger than some critical value,dual energy cascades emerge,with an inverse cascade at large scales and a forward cascade at small scales.Finally,the flow behavior near the boundary is studied,where the average boundary layer thickness gets smaller when system rotation increases.The flow behavior in the boundary layer is closely related to the interior flow structures,which create significant mass flux between the boundary layer and the interior fluid through Ekman pumping.