3.7.1 On Vortices, Scales, Wavenumbers, and Wave Vectors What are the Small Scales?

The analysis of isotropic turbulence dynamics, as done in this chapter, is usually carried out concurrently in both Fourier and physical space, a very difficult issue being to bridge between these two different approaches.

It is important to emphasize here that several shortcomings usually occur that are misleading. Fourier analysis is based on the use of wave vectors, which are not equivalent to scales, because a wave vector also carries information dealing with orientation. The associated wavenumber, defined as a Euclidian norm of the wave vector, has the dimension of the inverse of a length. A large part of the information is now lost, such as the mode polarity in the helical-mode decomposition denoted by the parameter s in Eq. (2.86).

Another problem is to switch from the scale concept to classical objects of fluid dynamics like vortices. Small scales are very often understood as small vortices, which is wrong. The three reasons are as follows:

1. Neither Fourier analysis, which introduces wave vectors, nor the scale-dependent analysis in the physical space (based on structure functions, scale-dependent increments, etc.) involves the concept of coherent events such as a vortex. It is worth noting that none of the recent definitions of a vortex or a vortex sheet (see Subsection 3.6.1) is based on the the scale concept.

2. Modes in Fourier space are nonlocal in...