Geoscience Reference
In-Depth Information
For geophysical boundary layer flows, the advective flux (i.e., how the quantity
γ
is carried with mean and turbulent fluctuations in the flow) almost always domi-
nates, so
F
γ
=
γ
u
. Using the incompressibility conditions,
∇
·
u
=
0
, and ignoring
moleculardiffusion,(2.2)becomes
∂γ
∂
Q
γ
t
+
u
·
∇γ
=
(2.4)
foranarbitraryscalarquantity.Scalarequationsforheatandsaltareofvitalimpor-
tanceintheIOBL, buttheprinciplesholdforothercontaminantsaswell.
Conservation of enthalpy may be expressed as an equation for temperature in
thefluid:
Q
H
(
ρ
∂
T
∂
t
+
u
·
∇
T
=
c
p
)
(2.5)
where
×
10
3
Jkg
−
1
K
−
1
for seawater near freezing), and
Q
H
is the source term, which
typically comprises solar radiation flux divergence in the upper part of the wa-
ter column, but might also include, for example, a phase change associated with
nucleation of frazil crystals in the water away from the immediate interface. The
correspondingsaltequationis
ρ
is density,
c
p
is specific heat at constant pressure (close to 4
Q
S
ρ
∂
S
t
+
u
·
∇
S
=
(2.6)
∂
where
S
issalinityexpressedinunitsofthepracticalsalinityscale(henceforthdesig-
nated
psu
,correspondingcloselytopartsperthousand).Asaboveapossiblesource
Q
S
withinthefluidmightarisefromnucleationoffrazilcrystals.
Substitute vector momentum for the arbitrary property
in (2.4), and interpret
the “momentum source term” as the sum of a pressure gradient in the fluid, and
the acceleration of gravity acting on small density perturbations in the fluid (the
Boussinesq approximation),to arriveat Euler'sequation(essentially Newton's2nd
lawforfluids,ignoringmoleculardiffusion):
γ
p
ρ
−
g
ρ
ρ
∂
u
t
+
u
·
∇
u
=
−
∇
k
(2.7)
∂
The fact that molecular diffusion was ignored in arriving at (2.7) does not exclude
the impact of friction in the fluid, because the nonlinear advective term
u
u
pro-
videsalinkviaturbulencebetweenthelarge-scaleflowanddissipativeprocessesat
smallscales.
·
∇
2.2 Reynolds Fluxes
u
where the angle bracket
denotes an instantaneous ensemble average over some area large compared with
the scale of the “energy containing” turbulent eddies in a flow;
u
is the
deviatory
A local velocity vector may be expressed as
u
=
u
+