Coherent vortex structures and 3D enstrophy cascade

Existence of 2D enstrophy cascade in a suitable mathematical setting, and under suitable conditions compatible with 2D turbulence phenomenology, is known both in the Fourier and in the physical scales. The goal of this paper is to show that the same geometric condition preventing the formation of singularities - 1/2-H\"older coherence of the vorticity direction - coupled with a suitable condition on a modified Kraichnan scale, and under a certain modulation assumption on evolution of the vorticity, leads to existence of 3D enstrophy cascade in physical scales of the flow.Comment: 15 pp; final version -- to appear in CM

Energy cascades and flux locality in physical scales of the 3D Navier-Stokes equations

Rigorous estimates for the total - (kinetic) energy plus pressure - flux in R^3 are obtained from the three dimensional Navier-Stokes equations. The bounds are used to establish a condition - involving Taylor length scale and the size of the domain - sufficient for existence of the inertial range and the energy cascade in decaying turbulence (zero driving force, non-increasing global energy). Several manifestations of the locality of the flux under this condition are obtained. All the scales involved are actual physical scales in R^3 and no regularity or homogeneity/scaling assumptions are made.Comment: 21 pages, 2 figures; accepted to Comm. Math. Phy

Dissipation Length Scale Estimates for Turbulent Flows: A Wiener Algebra Approach

In this paper, a lower bound estimate on the uniform radius of spatial analyticity is established for solutions to the incompressible, forced Navier-Stokes system on an n-torus. This estimate matches previously known estimates provided that a certain bound on the initial data is satisfied. In particular, it is argued that for two-dimensional (2D) turbulent flows, the initial data is guaranteed to satisfy this hypothesized bound on a significant portion of the 2D global attractor, in which case, the estimate on the radius matches the best known one found in Kukavica (1998). A key feature in the approach taken here is the choice of the Wiener algebra as the phase space, i.e., the Banach algebra of functions with absolutely convergent Fourier series, whose structure is suitable for the use of the so-called Gevrey norms. We note that the method can also be applied with other phase spaces such as that of the functions with square-summable Fourier series, in which case the estimate on the radius matches that of Doering and Titi (1995). It can then similarly be shown that for three-dimensional (3D) turbulent flows, this estimate holds on a significant portion of the 3D weak attractor. © 2014 Springer Science+Business Media New York