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

196
22 CHAPTER 5. DYNAMICS OF THE OCEAN
• How is pressure P determined, how does it act?
• Complicated boundary conditions; coastline, surface fluxes ...
• Complicated equation of state (UNESCO 1981)
•
A large part of physical oceanography is in effect dedicated to finding simplifications of the
above equations. In this endeavour it is important to find a balance between simplicity and
accuracy.
How can we simplify these equations? Two important observations:
• The ocean is very very flat: typical depth (H=4km) typical horizontal scale (L=10 000
km)
tel-00545911, version 1 - 13 Dec 2010
• Sea water has only small density differences ∆ρ/ρ ≈ 3 · 10 −3
Z ✻
✻
H
❄
✻η
Figure 5.1: Shallow water configuration
Using this we will try to model the ocean as a shallow homogeneous layer of fluid, and see
how our results compare to observations.
Using the shallowness, equation (5.4) suggests that w/H is of the same order as u h /L,
where u h = √ u 2 + v 2 is the horizontal speed, leading to w ≈ (Hu h )/L and thus w ≪ u h . So
that equation (5.3) reduces to ∂ z P = −gρ which is called the hydrostatic approximation as the
vertical pressure gradient is now independent of the velocity in the fluid.
Using the homogeneity ∆ρ = 0 further suggest that:
X
✲
∂ xz P = ∂ yz P = 0. (5.9)
If we derive equations (5.1) and (5.2) with respect to the vertical direction we can see that if
∂ z u = ∂ z v at some time this propriety will be conserved such that u and v do not vary with
depth. (We have neglected bottom friction). Putting all this together we obtain the following
equations:
∂ t u+ u∂ x u + v∂ y u + 1 ρ ∂ xP = ν∇ 2 u (5.10)
∂ t v+ u∂ x v + v∂ y v + 1 ρ ∂ yP = ν∇ 2 v (5.11)
∂ x u + ∂ y v + ∂ z w = 0 (5.12)
with ∂ z u = ∂ z v = ∂ zz w = 0 (5.13)
+ boundary conditions