Geoscience Reference
In-Depth Information
7
Kolmogorov scaling, its extensions,
and two-dimensional turbulence
7.1 The inertial subrange
As we saw in
Chapter 6
, turbulent flows of large
R
t
have a wide range of eddies
smaller than
and larger than
η
. Being dominated by inertial forces, these eddies
lie in the
inertial subrange
.
Eddy sizes in the mid regions of a typical daytime atmospheric boundary layer
range from roughly 1000 m to 1 mm, which is six decades. Eddies in the central
three decades, from 3 cm to 30 m, say, contribute negligibly to TKE, fluxes, and
viscous dissipation and might be taken as the inertial subrange. Turbulent flows in
pipes, channels, and jets generally have much smaller
R
t
than geophysical flows
but can have an identifiable, if shorter, inertial subrange.
7.1.1 Energetics
In
Chapter 6
we decomposed the velocity fields in a horizontally homogeneous,
equilibrium boundary layer into resolvable and subfilter-scale parts and derived the
balance equations for their mean kinetic energy per unit mass, TKE
r
and TKE
s
,
respectively. When the scale
that separates r and s eddies is such that
η
,
these TKE equations say
sum of rates of mean advection
,
production
,
turbulent transport
,
and pressure transport of TKE
r
=
rate of loss of TKE
r
by transfer to subfilter scales
,
(7.1)
rate of gain of TKE
s
by transfer from resolvable scales
=
=
.
rate of loss of TKE
s
through viscous dissipation
(7.2)
Thus, the TKE balance within the inertial subrange is simply
rate of gain of TKE by transfer from larger scales
=
rate of loss of TKE by transfer to smaller scales
=
.
(7.3)
145
