===== Bulk Formula =====

Author: Anna Muntwyler\\
Edited by Gabriel Zehnder
====Surface wind stress====
In physical oceanography and fluid dynamics, the wind stress is the shear stress exerted by the wind on the surface of large bodies of water. It is the force component parallel to the surface, per unit area, as applied by the wind on the water surface. The wind stress is affected by the wind speed, the shape of the wind waves and the atmospheric stratification. It is one of the components of the air–sea interaction.

The magnitude of this shear force per unit contact area is estimated through wind-shear or wind-drag formulas.
These formulas parametrize the shear stress as a function of the wind speed at a certain height z above the surface. 
The height at which the wind speed is referred to in wind-drag formulas is usually 10 meters above the water surface.
The quantity $ {\tau}$ is a matrix, the Stress Tensor. The stress tensor is a material property of the fluid.\\

\\

**Figure 1 and 2:**

In the first of the following two figures is the decrease of shear stress with the depth of the fluid shown.\\
The second one illustrates how different fluids react with increasing shear stress. The Ocean reacts like a Newtonian fluid, that means the shear stress increases continuously relating to the shear rate. \\
This makes it easier to calculate the bulk formula. 

{{:lecture7:shear_stress.png?400|}}    {{:lecture7:scherrate.png?580|}}






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If wind blows over water it exerts a stress on the water surface which could be written as

\begin{equation*}
\ {\tau} = \rho_{air}×u'w' = \rho_{air}×u^*^2 = \rho_{air}×C(z)×u^2(z)
\end{equation*}
  
$ \rho_{air}$ = density  of  air [$\frac{kg}{m^3}$]\\
$ u'w'$ = derivation of wind speed in direction u and w\\
$ C(z)$ = drag  coefficient (German:  Windschubspannungskoeffizient), it is a constant which is unit-less.\\
$C(z)$ is a derived from the [[lecture7:lawofthewall|Law of the Wall]]. It is assumed to be constant for a constant roughness length $z_0$, a fixed height $z=10m$, and a certain range of $u$.

Typical values for $C(z)$ at $z = 10m$ are:

 if $ u(z) < 7 \frac{m}{s} \ \rightarrow \ C(z) = 1.1 \times 10^{-3}$\\
 if $ u(z) > 25 \frac{m}{s} \ \rightarrow \ C(z) = 2.3 \times 10^{-3}$
         

====Why do we need bulk formulas?====

To quantify the distribution of fluxes into and out of the ocean, that is:

•	fluxes of sensible heat

•	fluxes of latent heat 

•	momentum at the sea surface,

the Turbulence Sensor System (“Gust Probe”) is used on low-flying aircraft or offshore platforms.
The use of such gust-probes is very expensive, and radiometers must be carefully maintained. Neither can be used to obtain long-term, global values of fluxes. To calculate these fluxes from practical measurements, observed correlations are used between fluxes and variables that can be measured globally.

====Which bulk formulas exist?====


The following formulas for fluxes of sensible and latent heat and momentum are all known as bulk formulas
\begin{equation*}
\ {\tau} = {\rho}*C(z)*u^2(z)
\end{equation*}
\begin{equation*}
\ QS = {\rho}*CP*CS*u(z)*(Ts-Ta)
\end{equation*}
\begin{equation*}
\ QL = {\rho}*LE*CL*u(z)*(qs-qa)
\end{equation*}
  
**Parameters**:  
  
$ {\tau}$ =     Vector for Wind stress [Pa]. It describes the influence of the turbulence on the mean flow. 
 The form and strength depends on the turbulent flow and not on the fluid. As a symmetric 3-dimensional matrix it has independent components.

$ CL$ =    Latent  heat  transfer  coefficient [1.35 × 10-3]\\

$ CP$ =    Specific  heat  capacity  of  air [J/kg/K]\\

$ CS$ =    Sensible  heat  transfer  coefficient [0.9 × 10-3]\\

$ LE$ =    Latent  heat  of  evaporation [2.5 × 106 J/kg]\\

$ qa$ =    Specific  humidity  of  air  10m  above  the  sea [kg(water vapor)/kg (air)] \\

$ qs$ =    Specific  humidity  of  air  at  the  sea  surface [kg(water vapor)/kg (air)]\\

$ QL$ =    Latent  heat  flux [W/m²]\\

$ QS $=    Sensible  heat  flux [W/m²]\\ 

$ Ta$ =    Air temperature measured with thermometers on ships\\ 

$ Ts$ =    Air temperature measured from space\\

====Sources====

Münnich, M. (2014): Introduction to Physical Oceanography: Lecture 7

http://oceanworld.tamu.edu/resources/ocng_textbook/chapter05/chapter05_04.htm (Retrieved 09.06.2014)

http://oceanworld.tamu.edu/resources/ocng_textbook/chapter05/chapter05_03.htm (Retrieved 09.06.2014)

http://en.wikipedia.org/wiki/Wind_stress (Retrieved 13.06.2014)

====Comments====