Waves, turbulent motions and mixing - Introduction to the Physical and Biological Oceanography of Shelf Seas - page 117

where, in the last step, we have written the turbulent flux of density in terms of the
eddy diffusivity for density as in Equation (4.39) . If turbulence is to continue, this
energy demand must be met by a power source which in many cases is to be found in
the mean flow.
Let us assume that the mean flow consists of a steady, vertically sheared flow in the
x direction with a mean velocity profile U(z). Consider the frictional stresses t zx
acting on the upper and lower faces of a small cuboid in the fluid shown in Fig. 4.13b .
The rate of working on the cuboid by the stress on the lower face is given by the force
t zx dxdy times the velocity U, i.e. t zx Udxdy. This differs from the rate of working at
the upper face by:

P 1 ¼ @ð

tU Þ

dzdydz : ð 4 : 50 Þ

@ z

where we are omitting the subscripts from T.
The quantity P 1 is the net rate of input of energy to the cuboid. Not all of this power
is, however, available to produce turbulence since some of the work done contributes
to the energetics of the mean flow, i.e. the frictional forces may increase or decrease
the kinetic energy of the mean flow U(z). As we saw in (3.2.3), there is a net force on
a cuboid due to the frictional stress of ð@ t =@ z Þ dxdydz which works on the mean

flow at a rate of:

P 2 ¼ @

t

: ð : Þ

z Udxdydz

4 51

@

The difference between the rates of working P 1 and P 2 is:

dzdydz

P 2 ¼ @ð

tU

z þ @

Þ t

t @

U
@

P 1

z U

¼

z dzdydz

@ @

ð 4 : 52 Þ

t @

U
@

¼ per unit volume :

z

This is the power available to fuel turbulence. Relating the stress to the velocity shear
by the eddy viscosity as in Equation (4.38) , the available power per unit volume can
then be written as:

2
: t @

U
@

N z @

U
@

P a ¼

z ¼

ð 4 : 53 Þ

z

In order to maintain turbulence, the power supply P a must exceed the

demand, i.e.:

2

N z @

U
@

gK z @
@

ð 4 : 54 Þ

z z