Geoscience Reference
In-Depth Information
Astartingpointforevaluatingthefunctionalformofthedimensionlessvariables
in (6.5)and (6.6)comes fromthe Blasius solution for momentumand contaminant
exchange in laminar flow over a flat plate (e.g., Incropera and DeWitt 1985). The
analysis expresses the Stanton number (kinematic heat flux divided by the product
offar-fieldvelocityand
δ
T
)in termsofthePrandtlandReynoldsnumber:
Re
−
p
Pr
−
n
St
∝
(6.7)
wheretheexponentsare
p
3forthelaminarcase.
Owen and Thomson (1963) and Yaglom and Kader (1974) adopted the func-
tional form (6.7) in interpreting results from laboratory studies of mass transfer
in turbulent flow over hydraulically rough surfaces. Each analysis considered a
“transition sublayer” in which molecular effects dominated, greatly reducing the
exchange coefficients for scalar quantities compared with momentum. Owen and
Thomsontreated the exponentsas empiricalconstants and suggested that
p
=
1
/
2and
n
=
2
/
=
0
.
45
and
n
8. Yaglom and Kader, on the other hand, assumed a form like (6.7), but
withseveralsmalladditiveconstants.ForlargePrandtl(Schmidt)numbers,theirex-
pressionapproaches(6.7)with
p
=
0
.
=
/
=
/
3.Bothapproachesassumedthat
thedimensionofthetransitionsublayer(thelengthscalein
Re
)wasthesameasthe
heightoftheroughnesselementsinthelaboratoryflows,whichistypically30
1
2and
n
2
×
z
0
.
McPhee et al. (1987) demonstrated that the value for
α
h
inferred from direct mea-
surements of heat flux, friction velocity and far-field temperature and salinity was
about0.0055,onadaywhenmixedlayertemperaturewaswellabovefreezingdur-
ing MIZEX (Section 5.2). This was consistent with (6.7) when the proportionality
constantwas 1.57,which is aboutdoublethatrecommendedby Yaglomand Kader
(1974).Itwasbasedonarelativelylargevalueof
Re
reflectinglargesurfacerough-
nessinthemarginalicezone,whichwepointedoutwasnotnecessarilyappropriate
whenconsideringthetransitionsublayer.
6.3 The “Three-Equation” Interface Solution
Substitutingtheexchangecoefficientparameterization(6.4)into(6.1)and(6.3),the
conservationequationsinthecontrolvolumeare
−
q
+
α
h
u
∗
0
(
T
w
−
T
0
)
−
Q
L
w
0
=
0
(6.8)
α
S
u
∗
0
(
S
w
−
S
0
)+
w
(
S
ice
−
S
0
)=
0
In general, we seek to establish how fast ice is melting or freezing
(
w
0
)
, hence the
w
T
0
and
w
S
0
, in termsof prescribedquantitieswhich
interfacescalar fluxes,
,
,
,
,
,
include
u
∗
0
w
p
and the exchange coefficients. In other words we
seek a solution for three variables
T
w
S
w
q
S
ice
(
,
,
)
. This requires a third equation for
closure, specified by assuming that the interface remains at its salinity determined
freezing temperature:
T
0
=
T
0
S
0
w
0
T
f
(
S
0
)
≈−
mS
0
where
m
is the slope of the “freezing
