Weak and Strong Turbulence in a Class of Complex Ginzburg Landau Equations

Abstract

Engineers tend to be interested mainly in results on real fluids and therefore have less interest in theoretical models other than the Navier Stokes equations. In contrast, physicists and applied mathematicians are more interested in the underlying mechanisms which cause and govern turbulence. This sometimes leads them to study idealised systems which are not wholly physical but which give a degree of insight into the mechanisms behind it. Only in 2 spatial dimensions are the incompressible Navier Stokes equations understood analytically in the sense that there is a rigorous proof of the existence of a finite dimensional global attractor. On a finite periodic domain, if G is the Grashof number, then it turns out^1,2,3 that the dimension of the global attractor for the 2D Navier Stokes equations is bounded above by cG^2/3(1 + logG)^1/3 & below by cG^2/3. Computational methods⁴ are generally good enough to resolve the smallest scale in a 2D flow and, for 2D homogeneous decaying turbulence, the vorticity obeys a maximum principle. No such maximum principle is known to exist in 3D & regularity remains to be proved in this case. Furthermore, numerical resolution of the smallest scale in a 3D flow is still a long way off.