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Changeset 10406 for NEMO/trunk/doc/latex/NEMO/subfiles/chap_model_basics.tex – NEMO

Ignore:
Timestamp:
2018-12-18T11:25:09+01:00 (5 years ago)
Author:
nicolasmartin
Message:

Edition of math environments

Location:
NEMO/trunk/doc/latex
Files:
4 edited

Legend:

Unmodified
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  • NEMO/trunk/doc/latex

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  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_model_basics.tex

    r10354 r10406  
    6868            +\frac{1}{2}\nabla \left( {{\rm {\bf U}}^2} \right)}    \right]_h 
    6969 -f\;{\rm {\bf k}}\times {\rm {\bf U}}_h  
    70 -\frac{1}{\rho _o }\nabla _h p + {\rm {\bf D}}^{\rm {\bf U}} + {\rm {\bf F}}^{\rm {\bf U}} 
     70-\frac{1}{\rho_o }\nabla _h p + {\rm {\bf D}}^{\rm {\bf U}} + {\rm {\bf F}}^{\rm {\bf U}} 
    7171  \end{equation} 
    7272  \begin{equation}     \label{eq:PE_hydrostatic} 
    … …  
    570570   -   \frac{1}{2\,e_1}           \frac{\partial}{\partial i} \left(  u^2+v^2   \right)  
    571571   -   \frac{1}{e_3    }  w     \frac{\partial u}{\partial k}      &      \\ 
    572    -   \frac{1}{e_1    }            \frac{\partial}{\partial i} \left( \frac{p_s+p_h }{\rho _o}    \right)     
     572   -   \frac{1}{e_1    }            \frac{\partial}{\partial i} \left( \frac{p_s+p_h }{\rho_o}    \right)     
    573573   &+   D_u^{\vect{U}}  +   F_u^{\vect{U}}      \\ 
    574574\\ 
    … …  
    577577       -   \frac{1}{2\,e_2 }        \frac{\partial }{\partial j}\left(  u^2+v^2  \right)    
    578578       -   \frac{1}{e_3     }   w  \frac{\partial v}{\partial k}     &      \\ 
    579        -   \frac{1}{e_2     }        \frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho _o}  \right)     
     579       -   \frac{1}{e_2     }        \frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho_o}  \right)     
    580580    &+  D_v^{\vect{U}}  +   F_v^{\vect{U}} 
    581581\end{split} \end{equation} 
    … …  
    595595      +        \frac{\partial \left( {e_1 \,v\,u} \right)}{\partial j}  \right) 
    596596                 - \frac{1}{e_3 }\frac{\partial \left( {         w\,u} \right)}{\partial k}    \\ 
    597 -   \frac{1}{e_1 }\frac{\partial}{\partial i}\left( \frac{p_s+p_h }{\rho _o}   \right) 
     597-   \frac{1}{e_1 }\frac{\partial}{\partial i}\left( \frac{p_s+p_h }{\rho_o}   \right) 
    598598+   D_u^{\vect{U}} +   F_u^{\vect{U}} 
    599599\end{multline} 
    … …  
    607607      +        \frac{\partial \left( {e_1 \,v\,v} \right)}{\partial j}  \right) 
    608608                 - \frac{1}{e_3 } \frac{\partial \left( {        w\,v} \right)}{\partial k}    \\ 
    609 -   \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho _o}    \right) 
     609-   \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho_o}    \right) 
    610610+  D_v^{\vect{U}} +  F_v^{\vect{U}}  
    611611\end{multline} 
    … …  
    771771   -   \frac{1}{2\,e_1} \frac{\partial}{\partial i} \left(  u^2+v^2   \right)  
    772772   -   \frac{1}{e_3} \omega \frac{\partial u}{\partial k}       \\ 
    773    -   \frac{1}{e_1} \frac{\partial}{\partial i} \left( \frac{p_s + p_h}{\rho _o}    \right)     
    774    +  g\frac{\rho }{\rho _o}\sigma _1  
     773   -   \frac{1}{e_1} \frac{\partial}{\partial i} \left( \frac{p_s + p_h}{\rho_o}    \right)     
     774   +  g\frac{\rho }{\rho_o}\sigma _1  
    775775   +   D_u^{\vect{U}}  +   F_u^{\vect{U}} \quad 
    776776\end{multline} 
    … …  
    780780   -   \frac{1}{2\,e_2 }\frac{\partial }{\partial j}\left(  u^2+v^2  \right)         
    781781   -   \frac{1}{e_3 } \omega \frac{\partial v}{\partial k}         \\ 
    782    -   \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho _o}  \right)  
    783     +  g\frac{\rho }{\rho _o }\sigma _2    
     782   -   \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho_o}  \right)  
     783    +  g\frac{\rho }{\rho_o }\sigma _2    
    784784   +  D_v^{\vect{U}}  +   F_v^{\vect{U}} \quad 
    785785\end{multline} 
    … …  
    796796      +        \frac{\partial \left( {e_1 \, e_3 \, v\,u} \right)}{\partial j}   \right) 
    797797   - \frac{1}{e_3 }\frac{\partial \left( { \omega\,u} \right)}{\partial k}    \\ 
    798    - \frac{1}{e_1} \frac{\partial}{\partial i} \left( \frac{p_s + p_h}{\rho _o}    \right)     
    799    +  g\frac{\rho }{\rho _o}\sigma _1  
     798   - \frac{1}{e_1} \frac{\partial}{\partial i} \left( \frac{p_s + p_h}{\rho_o}    \right)     
     799   +  g\frac{\rho }{\rho_o}\sigma _1  
    800800   +   D_u^{\vect{U}}  +   F_u^{\vect{U}} \quad 
    801801\end{multline} 
    … …  
    809809      +        \frac{\partial \left( {e_1 \; e_3  \,v\,v} \right)}{\partial j}   \right) 
    810810                 - \frac{1}{e_3 } \frac{\partial \left( { \omega\,v} \right)}{\partial k}    \\ 
    811    -   \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho _o}  \right)  
    812     +  g\frac{\rho }{\rho _o }\sigma _2    
     811   -   \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho_o}  \right)  
     812    +  g\frac{\rho }{\rho_o }\sigma _2    
    813813   +  D_v^{\vect{U}}  +   F_v^{\vect{U}} \quad 
    814814\end{multline} 
    … …  
    896896$\textit{z*} = 0$ and  $\textit{z*} = -H$ respectively. 
    897897Also the divergence of the flow field is no longer zero as shown by the continuity equation: 
    898 \begin{equation*}  
     898\[  
    899899\frac{\partial r}{\partial t} = \nabla_{\textit{z*}} \cdot \left( r \; \rm{\bf U}_h \right) 
    900900      \left( r \; w\textit{*} \right) = 0  
    901 \end{equation*}  
     901\]  
    902902%} 
    903903 
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