/[MITgcm]/manual/s_overview/appendix_ocean.tex
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revision 1.1 by adcroft, Wed Aug 8 16:16:19 2001 UTC revision 1.2 by cnh, Tue Sep 11 14:39:38 2001 UTC
# Line 11  non-Boussinesq equations for oceanic mot Line 11  non-Boussinesq equations for oceanic mot
11  \begin{eqnarray}  \begin{eqnarray}
12  \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}}%
13  _{h}+\frac{1}{\rho }\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}  _{h}+\frac{1}{\rho }\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}
14  \label{eq-zns-hmom} \\  \\
15  \epsilon _{nh}\frac{Dw}{Dt}+g+\frac{1}{\rho }\frac{\partial p}{\partial z}  \epsilon _{nh}\frac{Dw}{Dt}+g+\frac{1}{\rho }\frac{\partial p}{\partial z}
16  &=&\epsilon _{nh}\mathcal{F}_{w}  \label{eq-zns-hydro} \\  &=&\epsilon _{nh}\mathcal{F}_{w}  \\
17  \frac{1}{\rho }\frac{D\rho }{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}%  \frac{1}{\rho }\frac{D\rho }{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}%
18  _{h}+\frac{\partial w}{\partial z} &=&0  \label{eq-zns-cont} \\  _{h}+\frac{\partial w}{\partial z} &=&0  \\
19  \rho &=&\rho (\theta ,S,p)  \label{eq-zns-eos} \\  \rho &=&\rho (\theta ,S,p)  \\
20  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq-zns-heat} \\  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \\
21  \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq-zns-salt}  \frac{DS}{Dt} &=&\mathcal{Q}_{s}
22    \label{eq:non-boussinesq}
23  \end{eqnarray}  \end{eqnarray}
24  These equations permit acoustics modes, inertia-gravity waves,  These equations permit acoustics modes, inertia-gravity waves,
25  non-hydrostatic motions, a geostrophic (Rossby) mode and a thermo-haline  non-hydrostatic motions, a geostrophic (Rossby) mode and a thermo-haline
# Line 194  equations: Line 195  equations:
195  \begin{eqnarray}  \begin{eqnarray}
196  \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}}%
197  _{h}+\frac{1}{\rho _{c}}\mathbf{\nabla }_{z}p^{\prime } &=&\vec{\mathbf{%  _{h}+\frac{1}{\rho _{c}}\mathbf{\nabla }_{z}p^{\prime } &=&\vec{\mathbf{%
198  \mathcal{F}}}  \label{eq-zpe-hmom} \\  \mathcal{F}}}  \label{eq:ocean-mom} \\
199  \epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho ^{\prime }}{\rho _{c}}+\frac{1}{\rho  \epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho ^{\prime }}{\rho _{c}}+\frac{1}{\rho
200  _{c}}\frac{\partial p^{\prime }}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}  _{c}}\frac{\partial p^{\prime }}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}
201  \label{eq-zpe-hydro} \\  \label{eq:ocean-wmom} \\
202  \mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z}  \mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z}
203  &=&0  \label{eq-zpe-cont} \\  &=&0  \label{eq:ocean-cont} \\
204  \rho ^{\prime } &=&\rho (\theta ,S,p_{o}(z))-\rho _{c}  \label{eq-zpe-eos} \\  \rho ^{\prime } &=&\rho (\theta ,S,p_{o}(z))-\rho _{c}  \label{eq:ocean-eos} \\
205  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq-zpe-heat} \\  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq:ocean-theta} \\
206  \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq-zpe-salt}  \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq:ocean-salt}
207  \end{eqnarray}  \end{eqnarray}
208  Note that the hydrostatic pressure of the resting fluid, including that  Note that the hydrostatic pressure of the resting fluid, including that
209  associated with $\rho _{c}$, is subtracted out since it has no effect on the  associated with $\rho _{c}$, is subtracted out since it has no effect on the
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