Geoscience Reference
In-Depth Information
Figure 4.9 Plot of velocity against time (or x)
showing the definition of the turbulent
component u 0 as the difference between the
instantaneous velocity u and its mean value U.
N otice that with this definition the average
u 0 ¼ 0.
u ( t )
u
U
u = U + u
t
where the stress t m is parallel to the local flow direction and
n is the velocity
gradient normal to the flow. These frictional stresses convert the kinetic energy of the
smallest eddies into heat and prevent energy being transferred to even smaller scales.
Energy may also be consumed by turbulence in mixing a stratified fluid, i.e. by doing
work in moving fluid along vertical density gradients. If the energy supply is insuffi-
cient to supply these demands, turbulence cannot be maintained and will die out.
We will return to this important issue of the competition between mixing and
stratification in Section 4.4 and again in Chapter 6 .
In summary then, turbulence is random in character; it is necessarily dispersive and
brings about the mixing of fluid properties; it is also inherently dissipative and
requires an input of energy to sustain it.
@
u/
@
4.3.2
Turbulent fluxes of scalars
A velocity sensor placed in a turbulent flow in the ocean will record a vector time
series of motion u
ð
t
Þ
which can be regarded as a combination of a mean flow vector
0
u 0 ;
v 0 ;
w 0 Þ
U
. This separ-
ation of the mean and the turbulent flow components, a procedure first introduced by
Osborne Reynolds, is illustrated for motion in the x direction in Fig. 4.9 . Here, the
instantaneous u component consists of the mean U, averaged over some selected time
scale, plus the turbulent component u 0 which, by definition, has zero mean when
averaged over the same period, i.e.
¼ð
U
;
V
;
W
Þ
and a fluctuating turbulent component u
¼ð
u 0 ;
u
¼
U
þ
u 0 ¼
0
:
ð
4
:
29
Þ
Analogous relations for v and w complete the separation of turbulent and mean
components of the flow vector u
. It is also possible to regard the variation of a scalar
property, e.g. salinity s, as consisting of a mean S and fluctuating component s 0 with
zero mean:
s 0 ;
s
¼
S
þ
s 0 ¼
0
:
ð
4
:
30
Þ
The mean flow components U, V and W are the motions described in the basic
dynamical balances in the previous chapter. The turbulent components are some-
thing extra and, as we shall see, are responsible for the enhanced diffusion brought
about by turbulence and contribute to the dynamics by generating frictional stresses.
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