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

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58 CHAPTER 10. PENETRATION OF SURFACE FLUXES
Where Q ≈ 200Wm −2 , c p = 4000JK −1 kg −1 , ρ = 1000kgm −3 if we take τ to be one day get:
T A ≈ 20K and L = 5.2cm, this means that the surface temperature in the ocean varies by 40K
in one day and the heat only penetrates a few centimeters. If τ is one year, considering the
seasonal cycle, T A ≈ 400K and L ≈ 1m. This means that the surface temperature in the ocean
varies by 800K in one year and the heat only penetrates about one meter. This does not at all
correspond to observation!
When using the molecular viscosity, ν we can also calculate the thickness of the Ekman
layer δ = √ 2ν/f which is found to be a few centimeters. The observed thickness of the Ekman
layer in the ocean is however over 100 times larger.
This shows that molecular diffusion can not explain the vertical heat transport, and molecular
viscosity can not explain the vertical transport of momentum! But what else can?
10.2 Turbulent Transport
tel-00545911, version 1 - 13 Dec 2010
In the early 20th century fluid dynamicists as L. Prandtl suggested that small scale turbulent
motion mixes scalars and momentum very much like the molecular motion does, only that
the turbulent mixing coefficients are many orders of magnitude larger than their molecular
counterparts. This is actually something that can easily be verified by gently poring a little
milk into a mug of coffee. Without stirring the coffee will be cold before the milk has spread
evenly in the mug, with a little stirring the coffee and milk are mixed in less than a second.
In this section we like to have a quantitative look at the concept of eddy viscosity in a
very simplified frame work that nevertheless contains all the important pieces. The starting
point of our investigation are the Navier-Stokes equations (5.1 –5.4). We start by considering
the two dimensional motion in the x − z-plane. Motion and dependence in the y direction
are neglected only to simplify the algebra, and do not lead to important changes. We further
suppose that the large scale velocity field is only directed in the x-direction and depends only
on the z-direction U(z). The x and z component of the small scale turbulent motion is given
by u ′ and w ′ , respectively.
( ) ( )
u U(z) + u
′
=
w w ′ (10.5)
with U(z) = 〈u〉 x .
The 〈.〉 x operator denotes the average over a horizontal slice:
A(y) = 〈a(x,y)〉 x = 1 ∫
a(x,y)dx (10.6)
L L
(10.7)
In the sequel we will use the following rules:
and
〈∂ x a(x,z)〉 x =
1
x2 − x1
〈λ(z)a〉 x = λ(z)〈a〉 x (10.8)
〈∂ z a〉 x = ∂ z 〈a〉 x (10.9)
∫ x2
x1
∂ x a(x,z)dx = a(x 2,z) − a(x 1 ,z)
, (10.10)
x 2 − x1
which vanishes if a(x,z) is bounded and we take the limit of the averaging interval L = (x2 −
x1) → ∞. But of course:
〈ab〉 x ≠ 〈a〉 x 〈b〉 x . (10.11)