Waves, turbulent motions and mixing - Introduction to the Physical and Biological Oceanography of Shelf Seas - page 116

(a)

w 3

z

w 1

r 1 r 2 r 3 r 4

r ( z )

w 4

z p

w 2

Reference level

(b)

z

U ( z )

t B

B

d z

t A

d y

A

d x

Figure 4.13 The energetics of turbulence in a stratified fluid. (a) Particles moving vertically
with turb ulen t velocity w 0 and turbulent density deviation r 0 in a stratified fluid. The average
product g 0 w 0 is the rate of increase of the potential energy in vertical mixing. (b) In order
to maintain turbulence, the energy consumed in mixing must be supplied by the shear flow.
To determine the energy available to drive turbulence we first compare the rate of working by
the shear stress t on the cuboid surfaces in planes A and B and then subtract the rate of
working in the mean flow.

D

in Fig 4.13a . A particle of volume V p will be changing its potential energy (PE)at

a rate given by:

d ð PE Þ

dt ¼

V p ð þ 0 Þ gw 0

ð 4 : 48 Þ

where 0 ¼

is the density difference from the mean density in the horizontal

¼

plane and we are assuming that the mean vertical velocity W 0. The average rate of

PE change per unit volume for all particles crossing a plane is then:

1 d ð PE Þ

gK z @
@

K z 0 N 2

¼ 0 w 0 ¼

z ¼

ð : Þ

g 4 49

V p dt