Etudes et évaluation de processus océaniques par des hiérarchies ...

234
60 CHAPTER 10. PENETRATION OF SURFACE FLUXES
where −ν eddy is the proportionality coefficient. Looking at fig. 10.1 this choice seems reasonable:
firstly the coefficient should be negative as upward moving fluid transport a fluid parcel that
originates from an area with a lower average velocity in the x-direction to an area with a higher
average velocity in the x-direction, such that u ′ is likely to be negative. The reverse is true for
downward transport. Such that 〈w ′ u ′ 〉 x is likely to be negative. Secondly, a higher gradient is
likely to increase |u ′ | and such also −〈w ′ u ′ 〉 x .
Using the K-closure we obtain:
∂ t U + F = (ν + ν ′ eddy)∂ zz U, (10.19)
tel-00545911, version 1 - 13 Dec 2010
which is identical to eq. (10.12) except for the increased effective viscosity ν eddy = ν + ν ′ eddy
called the eddy viscosity
Exercise 58: perform the calculations without neglecting the motion and dependence in the
y-direction.
Exercise 59: perform the calculations for a passive scalar (a scalar quantity that diffuses and
is transported by the fluid without acting on the velocity field).
10.3 Convection
Oceanic convection is the buoyancy driven vertical mixing of water masses. Convection occurs
when the water column is unstable, that is, heavier water is lying above lighter water. In the
ocean this typically occurs when the surface waters are either cooled by atmospheric forcing or
their salinity is increased by evaporation. (Note that for waters with a salinity above 25PSU
density always decreases when temperature increases). If a isothermal ocean of depth H is
subject to a heat flux of Q its temperature change is given by:
∂ t T =
Q
c p ρH
(10.20)
In extreme cooling events in polar oceans the heat flux can reach Q = −10 3 Wm −2 . A typical
value of the heat capacity of sea water is c p = 4000.Jkg −1 K −1 .
10.4 Richardson Number
✛
✛
✛
✛
U
✲
✲
✲
✲
ρ
A
✻
δz
❄
B
Figure 10.2: Exchanging volumes A and B in a sheared stably stratified flow.
When considering the vertical mixing in the ocean we usually have a large scale horizontal
flow that has a vertical shear ∂ z U which has a tendency to destabilize the flow and generate
turbulence. On the other hand the flow usually has a stable stratification that suppresses