atmosphere.rst

Atmosphere

In the atmosphere, (see zandp-vert-coord), we interpret:

r = p is the pressure

$$\dot{r}=\frac{Dp}{Dt}=\omega \text{ is the vertical velocity in p coordinates}$$

ϕ = g z is the geopotential height

$$b=\frac{\partial \Pi }{\partial p}\theta \text{ is the buoyancy}$$

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

S = q is the specific humidity

where

T is absolute temperature

p is the pressure

$$\begin{aligned} \begin{aligned} &&z\text{ is the height of the pressure surface} \\\ &&g\text{ is the acceleration due to gravity}\end{aligned} \end{aligned}$$

In the above the ideal gas law, p = ρRT, has been expressed in terms of the Exner function Π(p) given by exner (see also atmos_appendix)

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

where p_c is a reference pressure and κ = 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):

R_fixed = p_top = 0

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

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:

ω = 0 at r = R_fixed (top of the atmosphere)

$$\omega =~\frac{Dp_{s}}{Dt}\text{ at }r=R_{moving}\text{ (bottom of the atmosphere)}$$

Then the (hydrostatic form of) equations horiz-mtm-humidity-salt yields a consistent set of atmospheric equations which, for convenience, are written out in p−coordinates in atmos_appendix - see eqs. atmos-prime-atmos-prime5.