doc/overview/atmosphere.rst

f67abf1ee3 Jeff*0001 Atmosphere
                0002 ----------
                0003
                0004 In the atmosphere, (see :numref:`zandp-vert-coord`), we interpret:
                0005
                0006 .. math:: r=p\text{  is the pressure}
                0007    :label: atmos-r
                0008
                0009 .. math::
                0010    \dot{r}=\frac{Dp}{Dt}=\omega \text{  is the vertical velocity in p coordinates}
                0011    :label: atmos-omega
                0012
                0013 .. math:: \phi =g\,z\text{  is the geopotential height}
                0014    :label: atmos-phi
                0015
                0016 .. math::
                0017    b=\frac{\partial \Pi }{\partial p}\theta \text{  is the buoyancy}
                0018    :label: atmos-b
                0019
                0020 .. math::
0bad585a21 Navi*0021    \theta =T \left( \frac{p_{c}}{p} \right)^{\kappa} \text{  is potential temperature}
f67abf1ee3 Jeff*0022    :label: atmos-theta
                0023
                0024 .. math:: S=q \text{  is the specific humidity}
                0025    :label: atmos-s
                0026
                0027 where
                0028
                0029 .. math:: T\text{ is absolute temperature}
                0030
                0031 .. math:: p\text{ is the pressure}
                0032
                0033 .. math::
                0034    \begin{aligned}
                0035    &&z\text{ is the height of the pressure surface} \\
                0036    &&g\text{ is the acceleration due to gravity}\end{aligned}
                0037
                0038 In the above the ideal gas law, :math:`p=\rho RT`, has been expressed in
                0039 terms of the Exner function :math:`\Pi (p)` given by :eq:`exner`
                0040 (see also :numref:`atmos_appendix`)
                0041
0bad585a21 Navi*0042 .. math:: \Pi (p)=c_{p} \left( \frac{p}{p_{c}} \right)^{\kappa},
f67abf1ee3 Jeff*0043    :label: exner
                0044
0bad585a21 Navi*0045 where :math:`p_{c}` is a reference pressure and :math:`\kappa = R/c_{p}`
f67abf1ee3 Jeff*0046 with :math:`R` the gas constant and :math:`c_{p}` the specific heat of
                0047 air at constant pressure.
                0048
                ** Warning **Wide character in print at /usr/local/share/lxr/source line 1030, <$git> line 50.


0049 At the top of the atmosphere (which is ‘fixed’ in our :math:`r`
                0050 coordinate):
                0051
0bad585a21 Navi*0052 .. math:: R_{\rm fixed}=p_{\rm top}=0.
f67abf1ee3 Jeff*0053
                0054 In a resting atmosphere the elevation of the mountains at the bottom is
                0055 given by
                0056
0bad585a21 Navi*0057 .. math:: R_{\rm moving}=R_{o}(x,y)=p_{o}(x,y) ,
f67abf1ee3 Jeff*0058
                0059 i.e. the (hydrostatic) pressure at the top of the mountains in a
                0060 resting atmosphere.
                0061
                0062 The boundary conditions at top and bottom are given by:
                0063
                0064 .. math::
0bad585a21 Navi*0065    \omega =0~\text{at }r=R_{\rm fixed} \text{ (top of the atmosphere)}
f67abf1ee3 Jeff*0066    :label: fixed-bc-atmos
                0067
                0068 .. math::
0bad585a21 Navi*0069    \omega =~\frac{Dp_{s}}{Dt}\text{ at }r=R_{\rm moving}\text{ (bottom of the atmosphere)}
f67abf1ee3 Jeff*0070    :label: moving-bc-atmos
                0071
                0072 Then the (hydrostatic form of) equations
                0073 :eq:`horiz-mtm`-:eq:`humidity-salt` yields a consistent set of
                0074 atmospheric equations which, for convenience, are written out in
                0075 :math:`p-`\coordinates in :numref:`atmos_appendix` - see
                0076 eqs. :eq:`atmos-prime`-:eq:`atmos-prime5`.
                0077