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1.3.2 Atmosphere

In the atmosphere, (see figure 1.15), we interpret:

$\displaystyle r=p$ is the pressure (1.10)

$\displaystyle \dot{r}=\frac{Dp}{Dt}=\omega$    is the vertical velocity in $\displaystyle p$ coordinates (1.11)

$\displaystyle \phi =g\,z$ is the geopotential height (1.12)

$\displaystyle b=\frac{\partial \Pi }{\partial p}\theta$    is the buoyancy (1.13)

$\displaystyle \theta =T(\frac{p_{c}}{p})^{\kappa }$ is potential temperature (1.14)

$\displaystyle S=q,$ is the specific humidity (1.15)

where

$\displaystyle T$ is absolute temperature    

$\displaystyle p$ is the pressure    


    $\displaystyle z$ is the height of the pressure surface  
    $\displaystyle g$ is the acceleration due to gravity  

In the above the ideal gas law, $ p=\rho RT$, has been expressed in terms of the Exner function $ \Pi (p)$ given by (see Appendix Atmosphere)

$\displaystyle \Pi (p)=c_{p}(\frac{p}{p_{c}})^{\kappa }$ (1.16)

where $ p_{c}$ is a reference pressure and $ \kappa =R/c_{p}$ with $ R$ the gas constant and $ c_{p}$ the specific heat of air at constant pressure.

At the top of the atmosphere (which is `fixed' in our $ r$ coordinate):

$\displaystyle R_{fixed}=p_{top}=0$    

In a resting atmosphere the elevation of the mountains at the bottom is given by

$\displaystyle R_{moving}=R_{o}(x,y)=p_{o}(x,y)$    

i.e. the (hydrostatic) pressure at the top of the mountains in a resting atmosphere.

The boundary conditions at top and bottom are given by:


    $\displaystyle \omega =0~$at $\displaystyle r=R_{fixed}$    (top of the atmosphere) (1.17)
$\displaystyle \omega$ $\displaystyle =$ $\displaystyle \frac{Dp_{s}}{Dt}$; at $\displaystyle r=R_{moving}$ (bottom of the atmosphere) (1.18)

Then the (hydrostatic form of) equations (1.1-1.6) yields a consistent set of atmospheric equations which, for convenience, are written out in $ p$ coordinates in Appendix Atmosphere - see eqs(1.59).


next up previous contents
Next: 1.3.3 Ocean Up: 1.3 Continuous equations in Previous: 1.3.1 Kinematic Boundary conditions   Contents
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