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

4.8. ESTIMATION OF FRICTION LAWS AND PARAMETERS IN GRAVITY CURRENTS BY DATA
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A Appendix: Calculation of the Coriolis-Boussinesq parameter for linear
Ekman layers
The Solution for the Ekman spiral of a fluid moving with a constant geostrophic speed of V G
in the y-direction is given by:
ũ = V G exp(−z/δ) sin(z/δ) (19)
ṽ = V G (1 − exp(−z/δ) cos(z/δ)) , (20)
where I have denoted ũ the variable that has a dependence in the z-direction as opposed to
vertically averaged value u = 〈ũ〉 = 1 R h
h 0 ũdz. The vertical average of the velocity component
in the x-direction and its square, using eq. (19), are:
u = 〈ũ〉 = 1 h
u 2 ≠ 〈ũ 2 〉
R h
0 ũdz = V G
h
R h
0 exp(−z/δ) sin(z/δ)dz = V Gδ
2h
= V 2 G
h
R h
0 exp(−2z/δ) sin2 (z/δ)dz = V 2 G δ
8h
= V G
4β
= V 2 G
16β
(21)
(22)
β = 〈ũ2 〉
〈ũ〉 2 = h 2δ
(23)
tel-00545911, version 1 - 13 Dec 2010
Please note that equation (21) shows that the mean velocity in the x-component is only
one quarter of geostrophic velocity divided by β. Linear Ekman layer theory shows, that the
frictional force is equal in both directions, so that in the friction term the u value has to be
multiplied by 4β. The non-linear term in the v-equation is u∂ xv is vanishing to leading order of
β −1 as the u-velocity is concentrated in the Ekman layer, whereas the bulk of the v-momentum
is above it.
I used:
Z
exp(−z/δ) sin(z/δ)dz = −δ (sin(z/δ) + cos(z/δ)) and (24)
2
Z
exp(−2z/δ) sin 2 (z/δ)dz = − exp(−2z/δ) (2sin 2 (z/δ) + sin(z/δ) cos(z/δ) + 1). (25)
8
I emphasise that these calculations are only valid for the unaccelerated laminar Ekman layer.
References
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