/[MITgcm]/manual/s_overview/appendix_atmos.tex
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revision 1.1 by adcroft, Wed Aug 8 16:16:18 2001 UTC revision 1.4 by adcroft, Wed Sep 26 17:00:34 2001 UTC
# Line 12  The hydrostatic primitive equations (HPE Line 12  The hydrostatic primitive equations (HPE
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
# 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 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    
# Line 77  For convenience in the model we prefer t Line 78  For convenience in the model we prefer t
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 116  The final form of the HPE's in p coordin Line 118  The final form of the HPE's in p coordin
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}
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