Guillaume: 1 revision

2008-12-03T10:39:08Z

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192.168.20.110 at 17:25, 10 October 2006

2006-10-10T17:25:58Z

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Upwelling is quite well described [http://www.pfeg.noaa.gov/products/PFEL/modeled/indices/upwelling/NA/what_is_upwell.html here].

==Upwelling index ==
The upwelling index (''UI'') is defined as the ''Ekman Transport per 100 m of coastline'' induced by wind parallel to the coast. In the Western Iberian coast. This corresponds to the zonal component of the wind.

The [[#The Ekman transport|''volume Ekman transport'']] is defined in ''Kundu'' as the momentum of a steady-state horizontally homogeneous viscid geophysical flow integrated over depth. Thus, assuming a steady state constant zonal wind stress <math>\tau</math>

<math>UI=-100\,\frac{\tau}{\rho\,f}</math> m3/s/100m.

===Wind stress===
To calculate the wind stress <math>\tau</math> from the wind velocity modulus <math>U_{10}</math> (air speed at 10 m height from the free surface) ''Ekman'' used the bulk formula:

<math>\tau=\rho_{air}C_D\,U_{10}^2</math>

<math>\rho_{air}=1.25</math> ''kg/m3'' is an usual value for the air density.

<math>C_D</math> is an air drag coefficient. It is mainly dependent from surface rugosity length ''z0'' and from the [[Richardson number]]. However a large case of linear parameterizations is available throughout the litterature, with no well defined standard:

<math>C_D = a + b\,U_{10}</math>

*''Large & Pond 1981'' use ''CD = .44 + .63 U10'',
*''Smith & Banke 1975'' use ''CD = .63 + .66U10''.

==Viscid, horizontally homogeneous, steady-state, geophysical fluid equations==
The flow equations of motion are given in ''Kundu'' by

<math>-fv=\nu \frac{d^2u}{d z^2},</math>

<math>fu=\nu \frac{d^2v}{d z^2}.</math>

==The Ekman spiral==
''Kundu'' shows that when acted by a constant wind stress <math>\tau</math> along the ''x''-direction, the solution to the above equations yield

<math>u=\frac{\tau/\rho}{\sqrt{f\nu}}e^{z/\delta}\cos\left(-\frac{z}{8}+\frac{\pi}{4}\right),</math>

<math>v=-\frac{\tau/\rho}{\sqrt{f\nu}}e^{z/\delta}\sin\left(-\frac{z}{8}+\frac{\pi}{4}\right),</math>

where <math>\delta\equiv\sqrt{\frac{2 \nu}{f}}.</math>

The velocity vector rotates clockwise (looking down) with dept. Its magnitude decays exponentially as well with decay constant <math>\delta</math>. This spiral shaped flow is called the ''Ekman spiral''.
In the works of Ekman, Ekman defined a thickness of decay of the spiraled flow based on the flow geometry. This thickness later defined the ''Ekman layer'' and is usually around 50 m for the open ocean(''Stewart''). ''Kundu'' however defines the thickness of the Ekman layer simply by its decay constant <math>\delta</math>.

==The Ekman transport==
When acted by a constant wind stress <math>\tau</math> along the ''x''-direction, the volume transport in the Ekman layer is given by

<math>\int_{-\infty}^{0}u\,dz=0</math>,

<math>\int_{-\infty}^{0}v\,dz=-\frac{\tau}{\rho\,f}</math>.

Thus the ''net transport is to the right of the wind stress in the northern hemisphere'' and is independent of viscosity. This is due because near the surface <math>\rho \nu \frac{du}{dz}=\tau,\quad z=0</math>.
Over statistical flow averages, the ''Ekman transport'' calculation fits very well with experimental data and, thus, is considered a reliable calculation.

==References==
#''Kundu''
#''Stewart''
#''Pietrzak2002''
#''Large & Pond 1981''
#''Smith & Banke 1975''

[[Category:Science]]

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192.168.20.110 at 17:25, 10 October 2006