Introduction - Turbulence in the Atmosphere - page 21

Earlier work (as summarized by Taylor (1935), for example) had shown that
a turbulent flow has large eddies that interact with the mean flow, contain most
of the kinetic energy of the turbulence, and lie at one end of a wide range of
eddies that interact nonlinearly through mechanisms such as vortex stretching. It
was understood that in equilibrium the rate of working of the fluid against the
viscous stresses in the smallest of these eddies dissipates kinetic energy at the same
mean rate it is extracted from the mean flow by the large eddies. But Taylor ( 1935 )
misidentified the spatial scale of this dissipative microturbulence , as he called it.
He chose the scale (Chapter 7)

νu 2 1 / 2

λ = , (1.42)

but as we discussed in Section 1.6.4 , Kolmogorov argued (and we now accept) that
the dissipative-eddy scale is η

(ν 3 /) 1 / 4 . They are related by

=

λ
η ∼

R 1 / 4 (1.43)

t

(Problem 1.19) ,sothat λ can be considerably larger. We now call λ the Taylor
microscale.
According to Batchelor ( 1996 ), Taylor's “The spectrum of turbulence” (1938)
was his “last paper on turbulence before the needs of World War II took him away
from academic research.” In this paper Taylor connected the power spectral density,
or spectrum, to the two-point correlation function in physical space through the
Fourier transform (Part III) and, in a casual aside, introduced what is now known
as “Taylor's hypothesis” for interpreting a time series at a spatial point as a spatial
record in the upstream direction.
But until Kolmogorov ( 1941 ) there was no unifying view of turbulence dynamics
across the entire scale range. He postulated that for scales beyond the energy-
containing range there are but two governing parameters: themean rate of transfer of
kinetic energy per unit mass from larger eddies to smaller (the energy cascade rate ,
Part III ) and the fluid kinematic viscosity. Therefore (as we showed in Section 1.6.4 )
this extensive scale range yields readily to dimensional analysis. The dimensional
analysis is even simpler in what we now call the inertial subrange, the larger-
scale end of this range. Here the local Reynolds number is large so the viscosity
is not important, and so the turbulence spectrum - the mean-squared amplitude
of velocity fluctuation as a function of spatial wavenumber (inverse scale) - is
then determined solely by the energy cascade rate. Its analytical form, the famous
Kolmogorov spectrum (Chapter 7) , emerges directly from dimensional analysis.
In large- R t turbulence this inertial subrange can be extensive; in the atmospheric
boundary layer it can span four decades.