/[MITgcm]/manual/s_overview/appendix_atmos.tex
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revision 1.4 by adcroft, Wed Sep 26 17:00:34 2001 UTC revision 1.5 by cnh, Thu Sep 27 01:57:17 2001 UTC
# Line 10  coordinates} Line 10  coordinates}
10    
11  The hydrostatic primitive equations (HPEs) in p-coordinates are:  The hydrostatic primitive equations (HPEs) in p-coordinates are:
12  \begin{eqnarray}  \begin{eqnarray}
13  \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}%  \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}
14  _{h}+\mathbf{\nabla }_{p}\phi &=&\vec{\mathbf{\mathcal{F}}}  _{h}+\mathbf{\nabla }_{p}\phi &=&\vec{\mathbf{\mathcal{F}}}
15  \label{eq:atmos-mom} \\  \label{eq:atmos-mom} \\
16  \frac{\partial \phi }{\partial p}+\alpha &=&0  \label{eq-p-hydro-start} \\  \frac{\partial \phi }{\partial p}+\alpha &=&0  \label{eq-p-hydro-start} \\
17  \mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{%  \mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{
18  \partial p} &=&0  \label{eq:atmos-cont} \\  \partial p} &=&0  \label{eq:atmos-cont} \\
19  p\alpha &=&RT  \label{eq:atmos-eos} \\  p\alpha &=&RT  \label{eq:atmos-eos} \\
20  c_{v}\frac{DT}{Dt}+p\frac{D\alpha }{Dt} &=&\mathcal{Q}    c_{v}\frac{DT}{Dt}+p\frac{D\alpha }{Dt} &=&\mathcal{Q}  
# Line 24  where $\vec{\mathbf{v}}_{h}=(u,v,0)$ is Line 24  where $\vec{\mathbf{v}}_{h}=(u,v,0)$ is
24  surfaces) component of velocity,$\frac{D}{Dt}=\vec{\mathbf{v}}_{h}\cdot  surfaces) component of velocity,$\frac{D}{Dt}=\vec{\mathbf{v}}_{h}\cdot
25  \mathbf{\nabla }_{p}+\omega \frac{\partial }{\partial p}$ is the total  \mathbf{\nabla }_{p}+\omega \frac{\partial }{\partial p}$ is the total
26  derivative, $f=2\Omega \sin lat$ is the Coriolis parameter, $\phi =gz$ is  derivative, $f=2\Omega \sin lat$ is the Coriolis parameter, $\phi =gz$ is
27  the geopotential, $\alpha =1/\rho $ is the specific volume, $\omega =\frac{Dp%  the geopotential, $\alpha =1/\rho $ is the specific volume, $\omega =\frac{Dp
28  }{Dt}$ is the vertical velocity in the $p-$coordinate. Equation \ref  }{Dt}$ is the vertical velocity in the $p-$coordinate. Equation \ref
29  {eq-p-firstlaw} is the first law of thermodynamics where internal energy $%  {eq-p-firstlaw} is the first law of thermodynamics where internal energy $
30  e=c_{v}T$, $T$ is temperature, $Q$ is the rate of heating per unit mass and $%  e=c_{v}T$, $T$ is temperature, $Q$ is the rate of heating per unit mass and $
31  p\frac{D\alpha }{Dt}$ is the work done by the fluid in compressing.  p\frac{D\alpha }{Dt}$ is the work done by the fluid in compressing.
32    
33  It is convenient to cast the heat equation in terms of potential temperature  It is convenient to cast the heat equation in terms of potential temperature
# Line 36  Differentiating \ref{eq-p-eos-start} we Line 36  Differentiating \ref{eq-p-eos-start} we
36  \[  \[
37  p\frac{D\alpha }{Dt}+\alpha \frac{Dp}{Dt}=R\frac{DT}{Dt}  p\frac{D\alpha }{Dt}+\alpha \frac{Dp}{Dt}=R\frac{DT}{Dt}
38  \]  \]
39  which, when added to the heat equation \ref{eq-p-firstlaw} and using $%  which, when added to the heat equation \ref{eq-p-firstlaw} and using $
40  c_{p}=c_{v}+R$, gives:  c_{p}=c_{v}+R$, gives:
41  \begin{equation}  \begin{equation}
42  c_{p}\frac{DT}{Dt}-\alpha \frac{Dp}{Dt}=\mathcal{Q}  c_{p}\frac{DT}{Dt}-\alpha \frac{Dp}{Dt}=\mathcal{Q}
# Line 55  The following relations will be useful a Line 55  The following relations will be useful a
55  the Exner function:  the Exner function:
56  \[  \[
57  c_{p}T=\Pi \theta \;\;;\;\;\frac{\partial \Pi }{\partial p}=\frac{\kappa \Pi  c_{p}T=\Pi \theta \;\;;\;\;\frac{\partial \Pi }{\partial p}=\frac{\kappa \Pi
58  }{p}\;\;;\;\;\alpha =\frac{\kappa \Pi \theta }{p}=\frac{\partial \ \Pi }{%  }{p}\;\;;\;\;\alpha =\frac{\kappa \Pi \theta }{p}=\frac{\partial \ \Pi }{
59  \partial p}\theta \;\;;\;\;\frac{D\Pi }{Dt}=\frac{\partial \Pi }{\partial p}%  \partial p}\theta \;\;;\;\;\frac{D\Pi }{Dt}=\frac{\partial \Pi }{\partial p}
60  \frac{Dp}{Dt}  \frac{Dp}{Dt}
61  \]  \]
62  where $b=\frac{\partial \ \Pi }{\partial p}\theta $ is the buoyancy.  where $b=\frac{\partial \ \Pi }{\partial p}\theta $ is the buoyancy.
# Line 72  and on substituting into (\ref{eq-p-heat Line 72  and on substituting into (\ref{eq-p-heat
72  \end{equation}  \end{equation}
73  which is in conservative form.  which is in conservative form.
74    
75  For convenience in the model we prefer to step forward (\ref{theta-equation}%  For convenience in the model we prefer to step forward (\ref{theta-equation}
76  ) rather than (\ref{eq-p-firstlaw}).  ) rather than (\ref{eq-p-firstlaw}).
77    
78  \subsubsection{Boundary conditions}  \subsubsection{Boundary conditions}
# Line 116  _{o}(p_{o})=g~Z_{topo}$, defined: Line 116  _{o}(p_{o})=g~Z_{topo}$, defined:
116    
117  The final form of the HPE's in p coordinates is then:  The final form of the HPE's in p coordinates is then:
118  \begin{eqnarray}  \begin{eqnarray}
119  \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}%  \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}
120  _{h}+\mathbf{\nabla }_{p}\phi ^{\prime } &=&\vec{\mathbf{\mathcal{F}}}  _{h}+\mathbf{\nabla }_{p}\phi ^{\prime } &=&\vec{\mathbf{\mathcal{F}}}
121                    \\                    \\
122  \frac{\partial \phi ^{\prime }}{\partial p}+\alpha ^{\prime } &=&0  \frac{\partial \phi ^{\prime }}{\partial p}+\alpha ^{\prime } &=&0
123                     \\                     \\
124  \mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{%  \mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{
125  \partial p} &=&0                    \\  \partial p} &=&0                    \\
126  \frac{\partial \Pi }{\partial p}\theta ^{\prime } &=&\alpha ^{\prime }  \frac{\partial \Pi }{\partial p}\theta ^{\prime } &=&\alpha ^{\prime }
127                    \\                    \\
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