An infinity of microscales for turbulence

It is has long been accepted that the Kolmogorov microscale $\eta = (\nu^3/\varepsilon)^{1/4}$ is the smallest dynamically significant length scale of turbulence (e.g.,[1]), where $\nu$ is the kinematic viscosity and $\varepsilon$ is the dissipation. Following George [2] it is argued that there are an infinity of smaller scales, say $\eta_n =(\nu^{n+3}/\varepsilon_n)^{1/(2n+4)}$ where $\varepsilon_1$ is the dissipation of the dissipation, $\varepsilon_2$ is the dissipation of the dissipation of the dissipation, etc. Each of these is equal to a spectral moment in homogeneous turbulence, $(2 \nu)^{n+1}\int_0^\infty k^{2n+2} E(k) dk$. Time scales can be similarly defined. It is demonstrated how these play an important role, especially in non-stationary turbulence where Kolmogorov's equilibrium hypothesis is invalid.\\[4pt] [1] Tennekes and Lumley (1972) A First Course in Turbulence, MIT Press.\\[0pt] [2] George, W.K (2012) Asymp. Effect of Initial and Upstream Conditions on Turbulence, J. Fluids Engr, 134, 1061203-1--27.