Geoscience Reference
In-Depth Information
2
∗
the stability factor
reducingthe size of the eddies(and the penetrationof turbu-
lent mixing) when interface melting is significant. In the surface layer,
η
λ
increases
with distanceuntil it reaches
λ
max
. Forfreezing(staticallyunstable)conditions,the
situation is more complicated, because in a convective regime turbulence can mix
withlittleornoshear.Ifbuoyancyfluxisstrongenoughthat
2
λ
max
=
η
∗
Λ
∗
u
∗
0
/ |
f
|
is
negative or larger than
c
ml
z
pyc
, it is replaced with
c
ml
z
pyc
where, based
λ
max
=
onLeadExmeasurements,0
4.Inthis way,forsay a gradualtransition
to freezing,the modelis capableof acceptingmild convectiveconditionswithouta
suddenshiftineddysizeandeddyviscosity.
2
Thealgorithmforthedetermining
.
2
<
c
ml
<
0
.
λ
intheupperpycnoclineissimilar,withinput
w
b
p
,and
z
u
∗
p
,
z
p
substituted for the corresponding input values in Fig. 7.2.
Generally, only the right-hand side (stable) would apply, and
−
would normally be
much smaller than in the well mixed layer. However, the model can readily handle
a weak density gradient high in the IOBL (e.g., Fig. 5.17b), in which case
u
∗
p
and
λ
w
b
p
mightbecomparableto theinterfacevalues.
Once a distribution of mixing length is determined, the model calculates eddy
viscosity as the product
and the local turbulent scale velocity also determined
fromtheprevioustimestep.Mostofthetime,thelatteristhesquarerootofthelocal
shearstress
λ
1
/
2
unlessthereisnegativebuoyancyfluxinthedomain,in
whichcase thescalevelocityforthosegridpointsis
w
∗
=(
(
u
∗
=
|
K
u
z
|
)
w
b
)
/
3
.
Instaticallyunstableornearneutrallybuoyantconditions,scalareddydiffusivity
is assumed to be the same as eddy viscosity (Reynold's analogy). In stably strati-
fiedflow,as encounteredintheupperpartofthepycnocline,momentumexchange,
which depends on pressure fluctuations as well as direct mixing, is more efficient
than scalar exchange.For lack of definitive geophysicalmeasurements,the ratio of
eddy diffusivity to eddy viscosity is specified by a formula that approximates lab-
oratory results compiled by Turner (1973) relating the ratio of salt to momentum
transfercoefficients.The
ad hoc
formulais
u
3
1
∗
+
c
ml
d
ml
⎧
⎨
1
Ri
≤
0
.
079
5
√
Ri
K
h
,
S
K
m
=
exp
(
−
1
.
−
0
.
079
)
0
.
079
<
Ri
≤
5
⎩
0
.
039
Ri
>
5
Where
Ri
is thegradientRichardsonnumber.
The reduction of scalar eddy diffusivity relative to eddy viscosity in stratified
flow begsthequestionofwhetherthe halineandthermaldiffusivitiesdiffer.Unfor-
tunately,thereislittlehardevidencefromnaturalboundarylayerflowsinsaltwater
from which to drawconclusionsregardinga ratio of, say,
K
h
/
K
S
. We found during
theupwellingeventdescribedinSections2.6and2.7(seeFigs.2.13and2.14),that
heat was mixed out of the upper pycnocline more rapidly than salt, enough to sig-
nificantly modify the
T/S
properties relative to the ambient surroundings (McPhee
et al. 2005).Intuitively,we suspect that as stability increases and turbulence scales
2
The mixing length scheme also allows the model to function near the equator, although this is
moot for the IOBL.
