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Column of water in the sea
Storyboard 
In the case of the ocean, the density of water, depending on its temperature and salinity, varies with depth. For this reason the pressure cannot be calculated with the traditional pressure formula for the water column. It is necessary to consider the effect of the variation in density and calculate by integrating the mass along the column the pressure that occurs at the depth that we wish to estimate.
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Characterization of the ocean layers
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Ekman's transport causes the boundaries between the surface and deepest layers in the ocean to shift. These are characterized by sudden changes in parameters depending on the temperature. In particular there are changes in:
• Temperature (thermocline)
• Salinity (halocline)
• Density (pycnocline)
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Column with variable density
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To calculate the pressure under the sea at a given depth, one must first estimate the mass of a volume element at a certain depth:
The problem in this case is that the density is not constant so the typical relationship of the pressure of the water column cannot be applied.
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Mass element
Equation
A water element of a height
| $ dm = \rho S dz $ |
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Force element
Equation
Con la definici n de la fuerza gravitacional
| $ F_g = m_g g $ |
el aumento de la fuerza en funci n de la masa es
| $ dF = g \, dm $ |
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Variation of force with depth
Equation
Con la variaci n de la masa
| $ dm = \rho S dz $ |
y la variaci n de la fuerza en funci n de la masa
| $ dF = g \, dm $ |
con lo que se obtiene
| $ dF = \rho g S dz $ |
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Pressure element
Equation
Con la definici n de la presi n
| $ p \equiv\displaystyle\frac{ F }{ S }$ |
la presi n aumenta con la fuerza seg n
| $ dp =\displaystyle\frac{ dF }{ S }$ |
en donde se asume que la secci n no varia.
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Pressure increase ratio with depth
Equation
Con la definici n de la presi n
| $ dF = \rho g S dz $ |
el aumento de la fuerza
| $ dp =\displaystyle\frac{ dF }{ S }$ |
lleva a un aumento de la presi n
| $ dp = \rho g dz $ |
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Density modeling
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If you look at the curve of the density of ocean water as a function of depth, you see that it has the shape of an inverted exponential. In other words, the upper part is allowed to compress, reaching a limit where the weight of the column does not lead to greater compression:
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Seawater Density Model
Equation
If you observe the curve of density with depth you can model this with a value for surface density
| $ \rho = \rho_{\infty} - (\rho_{\infty}-\rho_0)e^{-\lambda z }$ |
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Depth pressure calculation
Equation
Con el incremento de la presi n
| $ dp = \rho g dz $ |
se puede mediante integraci n calcular la presi n para cualquier profundidad:
| $ p = p_0 + g\displaystyle\int_0^z \rho\,du$ |
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Pressure as a function of depth
Equation
Si se emplea la funci n de la densidad
| $ \rho = \rho_{\infty} - (\rho_{\infty}-\rho_0)e^{-\lambda z }$ |
en la ecuaci n de la presi n
| $ p = p_0 + g\displaystyle\int_0^z \rho\,du$ |
se obtiene
| $ p = p_0 + \rho_{\infty} g z - \displaystyle\frac{( \rho_{\infty} - \rho_0 ) g }{ \lambda }(1- e^{- \lambda z })$ |
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Video
Video: Column of water in the sea