# CMS-Flow:Wind Pressure

## Wind Surface Stress

The wind surface stress is calculated as

 $\tau_{si} = \rho_a C_D W W_i$ (1)

where

$\rho_a$ = air density at sea level [~1.2 kg/m3]
$C_D$ = wind drag coefficient [-]
$W_i$ = 10-m wind speed [m/s]
$W = \sqrt{W_i W_i}$= 10-m wind velocity magnitude [m/s].

The wind speed is calculated using either an Eulerian or Lagrangian reference frame as

 $W_i = W_i^E - \gamma_W U_i$ (2)

where:

$W_i^E$ = 10-m atmospheric wind speed relative to the solid earth (Eulerian wind speed) [m/s]
$\gamma_W$= equal to zero for the Eulerian reference frame or one for the Lagrangian reference frame
$U_i$= current velocity [m/s].

Using the Lagrangian reference frame or relative wind speed is more accurate and realistic for field applications (Bye 1985; Pacanowski 1987; Dawe and Thompson 2006), but the option to use the Eulerian wind speed is provided for idealized cases and backward compatibility. The drag coefficient is calculated using the formula of Hsu (1988) and modified for high wind speeds based on field data by Powell et al. (2003) (Figure 2):

 C_d = \left\{ \begin{align} &\left(\frac{\kappa}{14.56 - 2 \ln W} \right)^2 &for\ W\leq 30\ m/s \\ &10^{-3}\ max(3.86-0.04W,1.5) &for\ W > 30\ m/s \end{align} \right. (3)

Figure 2. Modified Hsu (1988) wind drag coefficient.

Powell et al. (2003) speculate that the reason for the decrease in drag coefficient with higher wind speeds is due to increasing foam coverage leading to the formation of a slip surface at the air-sea interface.

Wind measurements taken at heights other than 10 m are converted to 10-m wind speeds using the 1/7 rule (Shore Protection Manual (SPM) 1984; Coastal Engineering Manual (CEM) 2002):

 $W = W_i^{z} \biggl ( \frac{10}{z} \biggr )^{1/7}$ (4)

where z is the elevation above the sea surface of the wind measurement, and Wiz is the wind velocity at height z.

## Pressure

The atmospheric pressure is included in the momentum equations as an additional pressure gradient term

 $\frac{h}{\rho_0} \frac{\partial p_a }{\partial x_j}$ (5)

## Symbol Index

Symbol Description Units
$\tau_{W}$ Wind shear stress Pa
$\rho_a$ Atmospheric density (~1.2 kg/m^3) kg/m^3
$C_d$ Wind drag coefficient none
$W$ 10-min averaged wind speed at 10 m above the sea surface m/sec
$\theta$ Wind direction m/sec

## References

• Bye, J. A. T. 1985. Large-scale momentum exchange in the coupled atmosphere–ocean, coupled ocean–atmosphere models, 51–61. Amsterdam: Elsevier Science Publishers.
• Coastal Engineering Manual (CEM). 2002. Engineer Manual 1110-2-1100. Washington, DC: US Army Corps of Engineers.
• Dawe, J. T., and L. Thompson. 2006. Effect of ocean surface currents on wind stress, heat flux, and wind power input to the ocean. Geophysical Research Letters (3):3, L09604.
• Hsu, S.A. (1988) Coastal meteorology. Academic Press, San Diego, CA.
• Pacanowski, R. C. 1987. Effect of equatorial currents on surface wind stress. Journal of Physical Oceanography (17):833–838.
• Powell, M. D., P. J., Vickery, and T. A. Reinhold. 2003. Reduced drag coefficient for high wind speeds in tropical cyclones. Nature (422):279–283.
• Shore Protection Manual (1984) 4th ed., 2nd Vol, U.S. Army Engineer Waterways Experiment Station, U.S. Government Printing Office, Washington, DC.