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line,”determinedfromtheUNESCOformulaforfreezingtemperatureasafunction
of
T
and
S
forvaluesneartheinterface(e.g.,Gill 1982).Combiningthiswith (6.8)
thenprovidesa quadraticequationfor,e.g.,
S
0
,in termsofthreetemperaturescales
mS
0
+(
T
H
+
T
L
−
mS
ice
+
T
p
)
S
0
−
(
T
H
+
T
P
)
S
ice
−
T
L
S
w
=
0
(6.9)
where
q
α
h
u
∗
0
T
L
=
α
S
Q
L
w
p
Q
L
α
h
u
∗
0
T
H
=
T
w
−
;
α
h
;
T
P
=
fromwhich
w
0
and
T
0
follow,alongwiththeinterfacefluxes.Normallythepercola-
tionvelocityissmallenoughthatthelastscale isinconsequential.
Double diffusion
is a term coined to describe what occurs when scalar contam-
inants in a fluid diffuse at different rates. In cold seawater, for example, molecular
thermal diffusivity is about 200 times greater than salt diffusivity. So if one were
to place a parcel of relatively warm, saline water next to a cold, fresh parcel in a
quiescentsetting,at some distancefromthe initialboundary,thechangein temper-
ature with time relative to the initial differencein temperaturewould be far greater
than the change in salinity relative to its initial difference. In most oceanographic
usage, double diffusion refers to the fact that since seawater density depends on
both temperature and salt concentration, different diffusion rates can lead to den-
sity differencesthat will cause fluid motionand mixing,even in the absence of any
external forces besides gravity. In a different context, double diffusion in the thin
water layer adjacent to the ice/ocean boundary is potentially quite important for
the heat and mass balance of sea ice, because the relatively small observed melt
rates described above imply that the Prandtl and Schmidt numbers, which depend
onmoleculardiffusivities,controltheexchangesofheatandsalt. We stress thatthe
developmenthere concentrates on diffusion across a thin fluid sublayer on the liq-
uid side of the interface, as opposed to how heat and salt are diffused within the
ice crystal lattice. Although perhaps not obvious at the outset, mechanisms for the
latter appearsto quite differentformelting versusfreezing(see, e.g.,Feltham et al.
2006),andthisturnsouttohaveimportantconsequencestheexchangecoefficients,
asdescribedbelow.
Note from (6.9) that if
q
is negligibly small, as it often is in the summer ice
with rapid melting conditions, then the quadratic solution depends on the ratio
of exchange coefficients
R
=
α
h
/
α
S
, which is a measure of the strength of dou-
ble diffusion at the interface. If
R
1, there is no double diffusion and salt plays
a passive role in the heat exchange. As
R
increases, heat transfer increases rela-
tive to salt transfer. Returning to the control volume artifice, if
R
=
=
1thenjust
enough salt will enter the control volume to keep
T
0
≈
T
f
(
S
w
)
and heat transfer
will continue unabated. For
R
1, the downward flux of water freshened by melt
is inhibited, so that
T
0
>
and the thermal driving is reduced. In this way,
meltingisratelimitedbydouble-diffusiveeffectsinthetransitionsublayersforheat
andsalt.
T
f
(
S
w
)
