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
+
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