/[MITgcm]/manual/s_overview/appendix_atmos.tex
ViewVC logotype

Diff of /manual/s_overview/appendix_atmos.tex

Parent Directory Parent Directory | Revision Log Revision Log | View Revision Graph Revision Graph | View Patch Patch

revision 1.1.1.1 by adcroft, Wed Aug 8 16:16:18 2001 UTC revision 1.6 by cnh, Wed Oct 10 16:48:38 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-p-hmom-start} \\  \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-p-cont-start} \\  \partial p} &=&0  \label{eq:atmos-cont} \\
19  p\alpha &=&RT  \label{eq-p-eos-start} \\  p\alpha &=&RT  \label{eq:atmos-eos} \\
20  c_{v}\frac{DT}{Dt}+p\frac{D\alpha }{Dt} &=&\mathcal{Q}  \label{eq-p-firstlaw}  c_{v}\frac{DT}{Dt}+p\frac{D\alpha }{Dt} &=&\mathcal{Q}  
21    \label{eq:atmos-heat}
22  \end{eqnarray}  \end{eqnarray}
23  where $\vec{\mathbf{v}}_{h}=(u,v,0)$ is the `horizontal' (on pressure  where $\vec{\mathbf{v}}_{h}=(u,v,0)$ is the `horizontal' (on pressure
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 35  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 43  c_{p}\frac{DT}{Dt}-\alpha \frac{Dp}{Dt}= Line 44  c_{p}\frac{DT}{Dt}-\alpha \frac{Dp}{Dt}=
44  \end{equation}  \end{equation}
45  Potential temperature is defined:  Potential temperature is defined:
46  \begin{equation}  \begin{equation}
47  \theta =T(\frac{p_{c}}{p})^{\kappa }  \label{eq-potential-temp}  \theta =T(\frac{p_{c}}{p})^{\kappa }  \label{eq:potential-temp}
48  \end{equation}  \end{equation}
49  where $p_{c}$ is a reference pressure and $\kappa =R/c_{p}$. For convenience  where $p_{c}$ is a reference pressure and $\kappa =R/c_{p}$. For convenience
50  we will make use of the Exner function $\Pi (p)$ which defined by:  we will make use of the Exner function $\Pi (p)$ which defined by:
# Line 54  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 67  c_{p}\frac{DT}{Dt}=\frac{D(\Pi \theta )} Line 68  c_{p}\frac{DT}{Dt}=\frac{D(\Pi \theta )}
68  \]  \]
69  and on substituting into (\ref{eq-p-heat-interim}) gives:  and on substituting into (\ref{eq-p-heat-interim}) gives:
70  \begin{equation}  \begin{equation}
71  \Pi \frac{D\theta }{Dt}=\mathcal{Q}  \label{theta-equation}  \Pi \frac{D\theta }{Dt}=\mathcal{Q}  \label{eq:potential-temperature-equation}
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}
79    
80  The upper and lower boundary conditions are :  The upper and lower boundary conditions are :
81  \begin{eqnarray*}  \begin{eqnarray}
82  \mbox{at the top:}\;\;p=0 &&\text{, }\omega =\frac{Dp}{Dt}=0 \\  \mbox{at the top:}\;\;p=0 &&\text{, }\omega =\frac{Dp}{Dt}=0 \\
83  \mbox{at the surface:}\;\;p=p_{s} &&\text{, }\phi =\phi _{topo}=g~Z_{topo}  \mbox{at the surface:}\;\;p=p_{s} &&\text{, }\phi =\phi _{topo}=g~Z_{topo}
84  \end{eqnarray*}  \label{eq:boundary-condition-atmosphere}
85    \end{eqnarray}
86  In $p$-coordinates, the upper boundary acts like a solid boundary ($\omega  In $p$-coordinates, the upper boundary acts like a solid boundary ($\omega
87  =0 $); in $z$-coordinates and the lower boundary is analogous to a free  =0 $); in $z$-coordinates and the lower boundary is analogous to a free
88  surface ($\phi $ is imposed and $\omega \neq 0$).  surface ($\phi $ is imposed and $\omega \neq 0$).
# Line 93  example, the hydrostatic geopotential as Line 95  example, the hydrostatic geopotential as
95  is not dynamically relevant and can therefore be subtracted from the  is not dynamically relevant and can therefore be subtracted from the
96  equations. The equations written in terms of perturbations are obtained by  equations. The equations written in terms of perturbations are obtained by
97  substituting the following definitions into the previous model equations:  substituting the following definitions into the previous model equations:
98  \begin{eqnarray*}  \begin{eqnarray}
99  \theta &=&\theta _{o}+\theta ^{\prime } \\  \theta &=&\theta _{o}+\theta ^{\prime } \label{eq:atmos-ref-prof-theta} \\
100  \alpha &=&\alpha _{o}+\alpha ^{\prime } \\  \alpha &=&\alpha _{o}+\alpha ^{\prime }  \label{eq:atmos-ref-prof-alpha}\\
101  \phi &=&\phi _{o}+\phi ^{\prime }  \phi &=&\phi _{o}+\phi ^{\prime } \label{eq:atmos-ref-prof-phi}
102  \end{eqnarray*}  \end{eqnarray}
103  The reference state (indicated by subscript ``0'') corresponds to  The reference state (indicated by subscript ``0'') corresponds to
104  horizontally homogeneous atmosphere at rest ($\theta _{o},\alpha _{o},\phi  horizontally homogeneous atmosphere at rest ($\theta _{o},\alpha _{o},\phi
105  _{o}$) with surface pressure $p_{o}(x,y)$ that satisfies $\phi  _{o}$) with surface pressure $p_{o}(x,y)$ that satisfies $\phi
# Line 114  _{o}(p_{o})=g~Z_{topo}$, 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  \label{eq-p-hmom} \\                    \\
122  \frac{\partial \phi ^{\prime }}{\partial p}+\alpha ^{\prime } &=&0  \frac{\partial \phi ^{\prime }}{\partial p}+\alpha ^{\prime } &=&0
123  \label{eq-p-hydro} \\                     \\
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  \label{eq-p-cont} \\  \partial p} &=&0                    \\
126  \frac{\partial \Pi }{\partial p}\theta ^{\prime } &=&\alpha ^{\prime }  \frac{\partial \Pi }{\partial p}\theta ^{\prime } &=&\alpha ^{\prime }
127  \label{eq-p-eos} \\                    \\
128  \frac{D\theta }{Dt} &=&\frac{\mathcal{Q}}{\Pi }  \label{eq-p-heat}  \frac{D\theta }{Dt} &=&\frac{\mathcal{Q}}{\Pi }  \label{eq:atmos-prime}
129  \end{eqnarray}  \end{eqnarray}
Legend:
Removed from v.1.1.1.1  
changed lines
  Added in v.1.6
  ViewVC Help
Powered by ViewVC 1.1.22