Waves, turbulent motions and mixing - Introduction to the Physical and Biological Oceanography of Shelf Seas - page 110

where suffices 1, 2, 3 correspond to x, y, z components. (e.g. u 1 0 ¼ w 0 )

Here the first index of t, i, indicates the direction which is normal to the plane
in which the stress oper ates , wh ile th e second, j, indicates the direction of the stress.
So, for example, t zx ¼

u 0 ; u 2 0 ¼ v 0 ; u 0 3 ¼

u 0 3 u 0 1 is the component of stress acting in the x-y plane
along the x direction. Since t ij ¼

w 0 u 0 ¼

t ji , only six of the nine components are independ-
ent. The three stress components with i u 0 1 u 0 1 ) represent normal

¼ j (e.g. t xx ¼

stresses (i.e. components of pressure) while those with i j are tangential stresses.

These Reynolds stress components can be shown to arise more formally by
inserting the velocity decomposition (Equation 4.29) into the equations of motion
and averaging. When this is done in a rigorous way (see for example Kundu, 1990 )
with the inclusion of the forces associated with molecular viscosity, the turbulent
stresses appear in parallel with the viscous stresses. In the ocean, however, the
Reynolds stresses are almost always much greater than the viscous stresses, so that
the latter can frequently be omitted from the equations.
As we saw in the Ekman theory of Section 3.5 , to solve problems involving
frictional stresses, we need to know how the Reynolds stresses relate to the properties
of the mean flow. Generally, however, we have rather little knowledge of these
stresses and the way in which they vary with the properties of the mean flow.
To get around this problem, it is commonly assumed that a stress component can
be related to the velocity gradient by analogy with the relation for viscous stresses in
Equation (4.28) . For example, we define an eddy viscosity N z which relates the
horizontal stresses to the corresponding vertical velocity gradients according to:

6¼

N z @

U
@

N z @

V
@

t zx ¼

z ;

t zy ¼

z :

ð 4 : 38 Þ

So we are making an assumption, without any sound theoretical basis, that turbulent
stresses act in a similar fashion to molecular stresses, but with a generally higher
value for the appropriate viscosity because large turbulent eddies are mediating the
transfers of momentum rather than small molecular collisions. Although mathemat-
ically convenient and widely used, this assumption reflects the immense difficulties in
observing turbulent fluxes and it still leaves us with the question of how to determine
the appropriate numerical value of N z for each flow situation.

4.3.5 Fickian diffusion

There is, of course, the same problem in specifying the scalar fluxes due to turbulence.
We noted above that molecular diffusion is described by Fick's law in which the
property flux is proportional to the gradient of concentration (Equation 4.27) .
A proportionality similar to that underlying Equation (4.38) is again often assumed
for the turbulent scalar flux components so that we write, for example, the x
component of the turbulent diffusive flux as:

K x @

S

u 0 s 0 ¼ ð 4 : 39 Þ

@ x