Changeset 11331


Ignore:
Timestamp:
2019-07-23T15:02:57+02:00 (13 months ago)
Author:
davestorkey
Message:

Documentation: small corrections and tidy up of Appendix B.

File:
1 edited

Legend:

Unmodified
Added
Removed
  • NEMO/trunk/doc/latex/NEMO/subfiles/annex_B.tex

    r11151 r11331  
    6262\end{align*} 
    6363 
    64 Equation \autoref{apdx:B2} is obtained from \autoref{apdx:B1} without any additional assumption. 
     64\autoref{apdx:B2} is obtained from \autoref{apdx:B1} without any additional assumption. 
    6565Indeed, for the special case $k=z$ and thus $e_3 =1$, 
    6666we introduce an arbitrary vertical coordinate $s = s (i,j,z)$ as in \autoref{apdx:A} and 
     
    9494    & =\frac{1}{e_1\,e_2\,e_3 }\left[ {\left. {\;\;\;\frac{\partial }{\partial i}\left( {\frac{e_2\,e_3 }{e_1}\,A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)} \right|_s \left. -\, {e_3 \frac{\partial }{\partial i}\left( {\frac{e_2 \,\sigma_1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right. \\ 
    9595    & \qquad \qquad \quad -e_2 A^{lT}\;\frac{\partial \sigma_1 }{\partial s}\left. {\frac{\partial T}{\partial i}} \right|_s -e_1 \,\sigma_1 \frac{\partial }{\partial s}\left( {\frac{e_2 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right) \\ 
    96     & \qquad \qquad \quad\shoveright{ \left. { +e_1 \,\sigma_1 \frac{\partial }{\partial s}\left( {\frac{e_2 \,\sigma_1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right)+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 }{e_3 }A^{vT}\;\frac{\partial T}{\partial z}} \right)\;\;\;} \right] }\\ 
     96    & \qquad \qquad \quad\shoveright{ \left. { +e_1 \,\sigma_1 \frac{\partial }{\partial s}\left( {\frac{e_2 \,\sigma_1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right)+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 }{e_3 }A^{vT}\;\frac{\partial T}{\partial s}} \right)\;\;\;} \right] }\\ 
    9797    \\ 
    9898    &=\frac{1}{e_1 \,e_2 \,e_3 } \left[ {\left. {\;\;\;\frac{\partial }{\partial i} \left( {\frac{e_2 \,e_3 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)} \right|_s \left. {-\frac{\partial }{\partial i}\left( {e_2 \,\sigma_1 A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right. \\ 
     
    105105  \begin{array}{*{20}l} 
    106106    % 
    107     \intertext{using the same remark as just above, it becomes:} 
     107    \intertext{Using the same remark as just above, it becomes:} 
    108108    % 
    109109    &= \frac{1}{e_1 \,e_2 \,e_3 } \left[ {\left. {\;\;\;\frac{\partial }{\partial i} \left( {\frac{e_2 \,e_3 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s -e_2 \,\sigma_1 A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right.\;\;\; \\ 
     
    117117    % 
    118118    \intertext{Since the horizontal scale factors do not depend on the vertical coordinate, 
    119     the last term of the first line and the first term of the last line cancel, while 
    120     the second line reduces to a single vertical derivative, so it becomes:} 
     119    the two terms on the second line cancel, while 
     120    the third line reduces to a single vertical derivative, so it becomes:} 
    121121  % 
    122122    & =\frac{1}{e_1 \,e_2 \,e_3 }\left[ {\left. {\;\;\;\frac{\partial }{\partial i}\left( {\frac{e_2 \,e_3 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s -e_2 \,\sigma_1 \,A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right. \\ 
    123123    & \qquad \qquad \quad \shoveright{ \left. {+\frac{\partial }{\partial s}\left( {-e_2 \,\sigma_1 \,A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s +A^{lT}\frac{e_1 \,e_2 }{e_3 }\;\left( {\varepsilon +\sigma_1 ^2} \right)\frac{\partial T}{\partial s}} \right)\;\;\;} \right]} \\ 
    124124    % 
    125     \intertext{in other words, the horizontal/vertical Laplacian operator in the ($i$,$s$) plane takes the following form:} 
    126   \end{array} 
    127   } \\ 
     125    \intertext{In other words, the horizontal/vertical Laplacian operator in the ($i$,$s$) plane takes the following form:} 
     126  \end{array} 
     127  } \\  
    128128  % 
    129129  {\frac{1}{e_1\,e_2\,e_3}} 
     
    169169  \left[ {{ 
    170170        \begin{array}{*{20}c} 
    171           {1+a_1 ^2} \hfill & {-a_1 a_2 } \hfill & {-a_1 } \hfill \\ 
    172           {-a_1 a_2 } \hfill & {1+a_2 ^2} \hfill & {-a_2 } \hfill \\ 
    173           {-a_1 } \hfill & {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\ 
     171          {1+a_2 ^2 +\varepsilon a_1 ^2} \hfill & {-a_1 a_2 (1-\varepsilon)} \hfill & {-a_1 (1-\varepsilon) } \hfill \\ 
     172          {-a_1 a_2 (1-\varepsilon) } \hfill & {1+a_1 ^2 +\varepsilon a_2 ^2} \hfill & {-a_2 (1-\varepsilon)} \hfill \\ 
     173          {-a_1 (1-\varepsilon)} \hfill & {-a_2 (1-\varepsilon)} \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\ 
    174174        \end{array} 
    175175      }} \right] 
     
    182182  \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1} 
    183183\] 
    184  
    185 In practice, isopycnal slopes are generally less than $10^{-2}$ in the ocean, 
    186 so $\textbf {A}_{\textbf I}$ can be simplified appreciably \citep{cox_OM87}: 
     184and, as before, $\epsilon = A^{vT} / A^{lT}$. 
     185 
     186In practice, $\epsilon$ is small and isopycnal slopes are generally less than $10^{-2}$ in the ocean, 
     187so $\textbf {A}_{\textbf I}$ can be simplified appreciably \citep{cox_OM87}. Keeping leading order terms\footnote{Apart from the (1,0)  
     188and (0,1) elements which are set to zero. See \citet{griffies_bk04}, section 14.1.4.1 for a discussion of this point.}: 
    187189\begin{subequations} 
    188190  \label{apdx:B4} 
     
    284286\] 
    285287 
    286 To prove \autoref{apdx:B5} by direct re-expression of \autoref{apdx:B_ldfiso} is straightforward, but laborious. 
     288To prove \autoref{apdx:B_ldfiso_s} by direct re-expression of \autoref{apdx:B_ldfiso} is straightforward, but laborious. 
    287289An easier way is first to note (by reversing the derivation of \autoref{sec:B_2} from \autoref{sec:B_1} ) that 
    288290the weak-slope operator may be \emph{exactly} reexpressed in non-orthogonal $i,j,\rho$-coordinates as 
     
    306308the (rotated,orthogonal) isoneutral axes to the non-orthogonal $i,j,\rho$-coordinates. 
    307309The further transformation into $i,j,s$-coordinates is exact, whatever the steepness of the $s$-surfaces, 
    308 in the same way as the transformation of horizontal/vertical Laplacian diffusion in $z$-coordinates, 
     310in the same way as the transformation of horizontal/vertical Laplacian diffusion in $z$-coordinates in 
    309311\autoref{sec:B_1} onto $s$-coordinates is exact, however steep the $s$-surfaces. 
    310312 
     
    316318\label{sec:B_3} 
    317319 
    318 The second order momentum diffusion operator (Laplacian) in the $z$-coordinate is found by 
     320The second order momentum diffusion operator (Laplacian) in $z$-coordinates is found by 
    319321applying \autoref{eq:PE_lap_vector}, the expression for the Laplacian of a vector, 
    320322to the horizontal velocity vector: 
     
    361363\end{align*} 
    362364Using \autoref{eq:PE_div}, the definition of the horizontal divergence, 
    363 the third componant of the second vector is obviously zero and thus : 
     365the third component of the second vector is obviously zero and thus : 
    364366\[ 
    365367  \Delta {\textbf{U}}_h = \nabla _h \left( \chi \right) - \nabla _h \times \left( \zeta \right) + \frac {1}{e_3 } \frac {\partial }{\partial k} \left( {\frac {1}{e_3 } \frac{\partial {\textbf{ U}}_h }{\partial k}} \right) 
     
    386388  & = \frac{1}{e_1} \frac{\partial \left( {A^{lm}\chi   } \right)}{\partial i} 
    387389    -\frac{1}{e_2} \frac{\partial \left( {A^{lm}\zeta } \right)}{\partial j} 
    388     +\frac{1}{e_3} \frac{\partial u}{\partial k}      \\ 
     390    +\frac{1}{e_3} \frac{\partial}{\partial k} \left( \frac{A^{vm}}{e_3} \frac{\partial u}{\partial k} \right)      \\ 
    389391  D^{\textbf{U}}_v 
    390392  & = \frac{1}{e_2 }\frac{\partial \left( {A^{lm}\chi   } \right)}{\partial j} 
    391393    +\frac{1}{e_1 }\frac{\partial \left( {A^{lm}\zeta } \right)}{\partial i} 
    392     +\frac{1}{e_3} \frac{\partial v}{\partial k} 
     394    +\frac{1}{e_3} \frac{\partial}{\partial k} \left( \frac{A^{vm}}{e_3} \frac{\partial v}{\partial k} \right)   
    393395\end{align*} 
    394396 
Note: See TracChangeset for help on using the changeset viewer.