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In the former case the stratification tends to be destroyed
by turbulent mixing, in the latter it remains there and the
turbulence itself is dissipated. In both cases an inhibition
against the tendency for turbulent mixing exists and a loss
of turbulent energy results.
mix the fluid, turbulent accelerations in the boundary layer
have to overcome resistance due to buoyancy (Fig. 4.26).
He imagined a situation where the density contrast is very
large; try as it might, a turbulent flow may never mix the
denser layer upward. Conversely a low negative buoyancy
is more easily overcome. Appropriate dimensions of resisting
buoyancy force to applied inertial force per unit volume
are -
4.6.1 Criteria for shear stability
of density-stratified flow
u
2
/
l
for the latter. Taking
the ratio of these, all dimensions cancel (Fig. 4.26) and we
have the simplest possible form of the bulk
Richardson
Number, Ri
. The smaller the value the more likely it is that
any stratified shear flow will undergo mixing and homoge-
nization. Although this derivation has the correct basic
physical principles, it somewhat ignores the physical situa-
tion envisaged, that of a shear flow with a
continuous
stable
vertical variation of density (Fig. 4.27). The former is
characterized by a negative velocity gradient, the latter by
a negative density gradient. The two combine to give the
gradient Richardson number, Ri
g
.
g
for the former and
Richardson noted that stratified fluid undergoing shear has
a negative upward gradient of density and that in order to
Richardson
4.6.2 Stratification and the phenomenon of
double diffusion
The dual control of density by temperature and salinity in
ocean waters leads to an interesting scenario because adja-
cent water masses in thermal contact lose thermal contrast
much more quickly than they can lose salinity contrast.
This is because the molecular diffusion (conduction) of
heat is
c
.10
2
faster than the molecular diffusion of salinity.
We imagine a scenario of
metastable stratification
of water
layers, for example, an upper salty, warm layer has an initial
density,
Fig. 4.26
Any stratified turbulent flow has an accelerative (inertial)
tendency to mix or destabilize any stratified fluid. As in the
Reynolds number argument, call this force the
inertial term
and
give its order of magnitude per unit volume as
u
2
l
1
.
The stratified flow also has a buoyancy force that may act to stabilize
the flow or to destabilize it. A stabilizing force involves density
decreasing upward. For a mean density contrast of
across the
flow the stabilizing force per unit volume acting is
g
.
The ratio of the stabilizing buoyancy force to the destabilizing iner-
tial force is the dimensionless bulk
Richardson number, Ri
. Negative
Ri
corresponds to a destabilizing buoyancy force, positive
Ri
to a
stabilizing force. To check for dimensions:
2
.
Such a scenario is to be widely expected in the oceans as a
consequence of summer evaporation and warming of sur-
face layers, or to the inflow and outflow of contrasting
water masses like that of the well-known western
1
, less than that of a lower cool, fresh layer,
(
u
2
l
1
)
(
g
)
(ML
3
L
2
T
2
L
1
)
(L
3
M
1
L
1
T
2
)
0
u
1
r
1
Schematic representation of a stably stratified flow undergo-
ing shear in its boundary layer
Here the
Ri
condition is given by the ratio of density gradient
times
g
over density times velocity gradient squared in
symbols:
r
0
>
r
1
u
0
<
u
1
d
r
/d
z
= -ve
du/dz = -ve
(d
u
/d
z
)
2
This is the gradient Richardson Number,
Ri
g
.
Negative
Ri
g
corresponds to a destabilizing buoyancy force, positive
Ri
g
to
a stabilizing force
z
-(
g
d
r
/d
z
) /
r
u
0
r
0
x
Fig. 4.27
We need to consider gradients of density and velocity in order to fully utilize the Richardson criterion.
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