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Changeset 11543 for NEMO/trunk/doc/latex/NEMO/subfiles/apdx_diff_opers.tex – NEMO

Ignore:
Timestamp:
2019-09-13T15:57:52+02:00 (5 years ago)
Author:
nicolasmartin
Message:

Implementation of convention for labelling references + files renaming
Now each reference is supposed to have the information of the chapter in its name
to identify quickly which file contains the reference (\label{$prefix:$chap_...)

Rename the appendices from 'annex_' to 'apdx_' to conform with the prefix used in labels (apdx:...)
Suppress the letter numbering

File:
1 moved

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

    r11529 r11543  
    55% Chapter Appendix B : Diffusive Operators 
    66% ================================================================ 
    7 \chapter{Appendix B : Diffusive Operators} 
    8 \label{apdx:B} 
     7\chapter{Diffusive Operators} 
     8\label{apdx:DIFFOPERS} 
    99 
    1010\chaptertoc 
    … …  
    1616% ================================================================ 
    1717\section{Horizontal/Vertical $2^{nd}$ order tracer diffusive operators} 
    18 \label{sec:B_1} 
     18\label{sec:DIFFOPERS_1} 
    1919 
    2020\subsubsection*{In z-coordinates} 
    … …  
    2222In $z$-coordinates, the horizontal/vertical second order tracer diffusion operator is given by: 
    2323\begin{align} 
    24   \label{apdx:B1} 
     24  \label{eq:DIFFOPERS_1} 
    2525  &D^T = \frac{1}{e_1 \, e_2}      \left[ 
    2626    \left. \frac{\partial}{\partial i} \left(   \frac{e_2}{e_1}A^{lT} \;\left. \frac{\partial T}{\partial i} \right|_z   \right)   \right|_z      \right. 
    … …  
    3232\subsubsection*{In generalized vertical coordinates} 
    3333 
    34 In $s$-coordinates, we defined the slopes of $s$-surfaces, $\sigma_1$ and $\sigma_2$ by \autoref{apdx:A_s_slope} and 
     34In $s$-coordinates, we defined the slopes of $s$-surfaces, $\sigma_1$ and $\sigma_2$ by \autoref{eq:SCOORD_s_slope} and 
    3535the vertical/horizontal ratio of diffusion coefficient by $\epsilon = A^{vT} / A^{lT}$. 
    3636The diffusion operator is given by: 
    3737 
    3838\begin{equation} 
    39   \label{apdx:B2} 
     39  \label{eq:DIFFOPERS_2} 
    4040  D^T = \left. \nabla \right|_s \cdot 
    4141  \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T  \right] \\ 
    … …  
    5454  \begin{array}{*{20}l} 
    5555    D^T= \frac{1}{e_1\,e_2\,e_3 } & \left\{ \quad \quad \frac{\partial }{\partial i}  \left. \left[  e_2\,e_3 \, A^{lT} 
    56                                \left( \  \frac{1}{e_1}\; \left. \frac{\partial T}{\partial i} \right|_s  
     56                               \left( \  \frac{1}{e_1}\; \left. \frac{\partial T}{\partial i} \right|_s 
    5757                                       -\frac{\sigma_1 }{e_3 } \; \frac{\partial T}{\partial s} \right) \right]  \right|_s  \right. \\ 
    5858        &  \quad \  +   \            \left.   \frac{\partial }{\partial j}  \left. \left[  e_1\,e_3 \, A^{lT} 
    59                                \left( \ \frac{1}{e_2 }\; \left. \frac{\partial T}{\partial j} \right|_s  
     59                               \left( \ \frac{1}{e_2 }\; \left. \frac{\partial T}{\partial j} \right|_s 
    6060                                       -\frac{\sigma_2 }{e_3 } \; \frac{\partial T}{\partial s} \right) \right]  \right|_s  \right. \\ 
    61         &  \quad \  +   \           \left.  e_1\,e_2\, \frac{\partial }{\partial s}  \left[ A^{lT} \; \left(  
    62                      -\frac{\sigma_1 }{e_1 } \; \left. \frac{\partial T}{\partial i} \right|_s  
    63                      -\frac{\sigma_2 }{e_2 } \; \left. \frac{\partial T}{\partial j} \right|_s  
     61        &  \quad \  +   \           \left.  e_1\,e_2\, \frac{\partial }{\partial s}  \left[ A^{lT} \; \left( 
     62                     -\frac{\sigma_1 }{e_1 } \; \left. \frac{\partial T}{\partial i} \right|_s 
     63                     -\frac{\sigma_2 }{e_2 } \; \left. \frac{\partial T}{\partial j} \right|_s 
    6464                          +\left( \varepsilon +\sigma_1^2+\sigma_2 ^2 \right) \; \frac{1}{e_3 } \; \frac{\partial T}{\partial s} \right) \; \right] \;  \right\} . 
    6565  \end{array} 
    … …  
    6767\end{align*} 
    6868 
    69 \autoref{apdx:B2} is obtained from \autoref{apdx:B1} without any additional assumption. 
     69\autoref{eq:DIFFOPERS_2} is obtained from \autoref{eq:DIFFOPERS_1} without any additional assumption. 
    7070Indeed, for the special case $k=z$ and thus $e_3 =1$, 
    71 we introduce an arbitrary vertical coordinate $s = s (i,j,z)$ as in \autoref{apdx:A} and 
    72 use \autoref{apdx:A_s_slope} and \autoref{apdx:A_s_chain_rule}. 
    73 Since no cross horizontal derivative $\partial _i \partial _j $ appears in \autoref{apdx:B1}, 
     71we introduce an arbitrary vertical coordinate $s = s (i,j,z)$ as in \autoref{apdx:SCOORD} and 
     72use \autoref{eq:SCOORD_s_slope} and \autoref{eq:SCOORD_s_chain_rule}. 
     73Since no cross horizontal derivative $\partial _i \partial _j $ appears in \autoref{eq:DIFFOPERS_1}, 
    7474the ($i$,$z$) and ($j$,$z$) planes are independent. 
    7575The derivation can then be demonstrated for the ($i$,$z$)~$\to$~($j$,$s$) transformation without 
    … …  
    160160% ================================================================ 
    161161\section{Iso/Diapycnal $2^{nd}$ order tracer diffusive operators} 
    162 \label{sec:B_2} 
     162\label{sec:DIFFOPERS_2} 
    163163 
    164164\subsubsection*{In z-coordinates} 
    … …  
    170170 
    171171\begin{equation} 
    172   \label{apdx:B3} 
     172  \label{eq:DIFFOPERS_3} 
    173173  \textbf {A}_{\textbf I} = \frac{A^{lT}}{\left( {1+a_1 ^2+a_2 ^2} \right)} 
    174174  \left[ {{ 
    … …  
    193193 
    194194In practice, $\epsilon$ is small and isopycnal slopes are generally less than $10^{-2}$ in the ocean, 
    195 so $\textbf {A}_{\textbf I}$ can be simplified appreciably \citep{cox_OM87}. Keeping leading order terms\footnote{Apart from the (1,0)  
     195so $\textbf {A}_{\textbf I}$ can be simplified appreciably \citep{cox_OM87}. Keeping leading order terms\footnote{Apart from the (1,0) 
    196196and (0,1) elements which are set to zero. See \citet{griffies_bk04}, section 14.1.4.1 for a discussion of this point.}: 
    197197\begin{subequations} 
    198   \label{apdx:B4} 
     198  \label{eq:DIFFOPERS_4} 
    199199  \begin{equation} 
    200     \label{apdx:B4a} 
     200    \label{eq:DIFFOPERS_4a} 
    201201    {\textbf{A}_{\textbf{I}}} \approx A^{lT}\;\Re\;\text{where} \;\Re = 
    202202    \left[ {{ 
    … …  
    210210  and the iso/dianeutral diffusive operator in $z$-coordinates is then 
    211211  \begin{equation} 
    212     \label{apdx:B4b} 
     212    \label{eq:DIFFOPERS_4b} 
    213213    D^T = \left. \nabla \right|_z \cdot 
    214214    \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_z T  \right]. \\ 
    … …  
    216216\end{subequations} 
    217217 
    218 Physically, the full tensor \autoref{apdx:B3} represents strong isoneutral diffusion on a plane parallel to 
     218Physically, the full tensor \autoref{eq:DIFFOPERS_3} represents strong isoneutral diffusion on a plane parallel to 
    219219the isoneutral surface and weak dianeutral diffusion perpendicular to this plane. 
    220220However, 
    221 the approximate `weak-slope' tensor \autoref{apdx:B4a} represents strong diffusion along the isoneutral surface, 
     221the approximate `weak-slope' tensor \autoref{eq:DIFFOPERS_4a} represents strong diffusion along the isoneutral surface, 
    222222with weak \emph{vertical} diffusion -- the principal axes of the tensor are no longer orthogonal. 
    223223This simplification also decouples the ($i$,$z$) and ($j$,$z$) planes of the tensor. 
    224 The weak-slope operator therefore takes the same form, \autoref{apdx:B4}, as \autoref{apdx:B2}, 
     224The weak-slope operator therefore takes the same form, \autoref{eq:DIFFOPERS_4}, as \autoref{eq:DIFFOPERS_2}, 
    225225the diffusion operator for geopotential diffusion written in non-orthogonal $i,j,s$-coordinates. 
    226226Written out explicitly, 
    227227 
    228228\begin{multline} 
    229   \label{apdx:B_ldfiso} 
     229  \label{eq:DIFFOPERS_ldfiso} 
    230230  D^T=\frac{1}{e_1 e_2 }\left\{ 
    231231    {\;\frac{\partial }{\partial i}\left[ {A_h \left( {\frac{e_2}{e_1}\frac{\partial T}{\partial i}-a_1 \frac{e_2}{e_3}\frac{\partial T}{\partial k}} \right)} \right]} 
    … …  
    234234\end{multline} 
    235235 
    236 The isopycnal diffusion operator \autoref{apdx:B4}, 
    237 \autoref{apdx:B_ldfiso} conserves tracer quantity and dissipates its square. 
    238 As \autoref{apdx:B4} is the divergence of a flux, the demonstration of the first property is trivial, providing that the flux normal to the boundary is zero  
     236The isopycnal diffusion operator \autoref{eq:DIFFOPERS_4}, 
     237\autoref{eq:DIFFOPERS_ldfiso} conserves tracer quantity and dissipates its square. 
     238As \autoref{eq:DIFFOPERS_4} is the divergence of a flux, the demonstration of the first property is trivial, providing that the flux normal to the boundary is zero 
    239239(as it is when $A_h$ is zero at the boundary). Let us demonstrate the second one: 
    240240\[ 
    … …  
    256256             j}-a_2 \frac{\partial T}{\partial k}} \right)^2} 
    257257             +\varepsilon \left(\frac{\partial T}{\partial k}\right) ^2\right]      \\ 
    258            & \geq 0 .  
     258           & \geq 0 . 
    259259  \end{array} 
    260260             } 
    … …  
    265265\subsubsection*{In generalized vertical coordinates} 
    266266 
    267 Because the weak-slope operator \autoref{apdx:B4}, 
    268 \autoref{apdx:B_ldfiso} is decoupled in the ($i$,$z$) and ($j$,$z$) planes, 
     267Because the weak-slope operator \autoref{eq:DIFFOPERS_4}, 
     268\autoref{eq:DIFFOPERS_ldfiso} is decoupled in the ($i$,$z$) and ($j$,$z$) planes, 
    269269it may be transformed into generalized $s$-coordinates in the same way as 
    270 \autoref{sec:B_1} was transformed into \autoref{sec:B_2}. 
     270\autoref{sec:DIFFOPERS_1} was transformed into \autoref{sec:DIFFOPERS_2}. 
    271271The resulting operator then takes the simple form 
    272272 
    273273\begin{equation} 
    274   \label{apdx:B_ldfiso_s} 
     274  \label{eq:DIFFOPERS_ldfiso_s} 
    275275  D^T = \left. \nabla \right|_s \cdot 
    276276  \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T  \right] \\ 
    … …  
    295295\] 
    296296 
    297 To prove \autoref{apdx:B_ldfiso_s} by direct re-expression of \autoref{apdx:B_ldfiso} is straightforward, but laborious. 
    298 An easier way is first to note (by reversing the derivation of \autoref{sec:B_2} from \autoref{sec:B_1} ) that 
     297To prove \autoref{eq:DIFFOPERS_ldfiso_s} by direct re-expression of \autoref{eq:DIFFOPERS_ldfiso} is straightforward, but laborious. 
     298An easier way is first to note (by reversing the derivation of \autoref{sec:DIFFOPERS_2} from \autoref{sec:DIFFOPERS_1} ) that 
    299299the weak-slope operator may be \emph{exactly} reexpressed in non-orthogonal $i,j,\rho$-coordinates as 
    300300 
    301301\begin{equation} 
    302   \label{apdx:B5} 
     302  \label{eq:DIFFOPERS_5} 
    303303  D^T = \left. \nabla \right|_\rho \cdot 
    304304  \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_\rho T  \right] \\ 
    … …  
    312312\end{equation} 
    313313Then direct transformation from $i,j,\rho$-coordinates to $i,j,s$-coordinates gives 
    314 \autoref{apdx:B_ldfiso_s} immediately. 
     314\autoref{eq:DIFFOPERS_ldfiso_s} immediately. 
    315315 
    316316Note that the weak-slope approximation is only made in transforming from 
    … …  
    318318The further transformation into $i,j,s$-coordinates is exact, whatever the steepness of the $s$-surfaces, 
    319319in the same way as the transformation of horizontal/vertical Laplacian diffusion in $z$-coordinates in 
    320 \autoref{sec:B_1} onto $s$-coordinates is exact, however steep the $s$-surfaces. 
     320\autoref{sec:DIFFOPERS_1} onto $s$-coordinates is exact, however steep the $s$-surfaces. 
    321321 
    322322 
    … …  
    325325% ================================================================ 
    326326\section{Lateral/Vertical momentum diffusive operators} 
    327 \label{sec:B_3} 
     327\label{sec:DIFFOPERS_3} 
    328328 
    329329The second order momentum diffusion operator (Laplacian) in $z$-coordinates is found by 
    330 applying \autoref{eq:PE_lap_vector}, the expression for the Laplacian of a vector, 
     330applying \autoref{eq:MB_lap_vector}, the expression for the Laplacian of a vector, 
    331331to the horizontal velocity vector: 
    332332\begin{align*} 
    … …  
    371371  }} \right) 
    372372\end{align*} 
    373 Using \autoref{eq:PE_div}, the definition of the horizontal divergence, 
     373Using \autoref{eq:MB_div}, the definition of the horizontal divergence, 
    374374the third component of the second vector is obviously zero and thus : 
    375375\[ 
    376   \Delta {\textbf{U}}_h = \nabla _h \left( \chi \right) - \nabla _h \times \left( \zeta \textbf{k} \right) + \frac {1}{e_3 } \frac {\partial }{\partial k} \left( {\frac {1}{e_3 } \frac{\partial {\textbf{ U}}_h }{\partial k}} \right) .  
     376  \Delta {\textbf{U}}_h = \nabla _h \left( \chi \right) - \nabla _h \times \left( \zeta \textbf{k} \right) + \frac {1}{e_3 } \frac {\partial }{\partial k} \left( {\frac {1}{e_3 } \frac{\partial {\textbf{ U}}_h }{\partial k}} \right) . 
    377377\] 
    378378 
    379379Note that this operator ensures a full separation between 
    380 the vorticity and horizontal divergence fields (see \autoref{apdx:C}). 
     380the vorticity and horizontal divergence fields (see \autoref{apdx:INVARIANTS}). 
    381381It is only equal to a Laplacian applied to each component in Cartesian coordinates, not on the sphere. 
    382382 
    … …  
    384384the $z$-coordinate therefore takes the following form: 
    385385\begin{equation} 
    386   \label{apdx:B_Lap_U} 
     386  \label{eq:DIFFOPERS_Lap_U} 
    387387  { 
    388388    \textbf{D}}^{\textbf{U}} = 
    … …  
    404404\end{align*} 
    405405 
    406 Note Bene: introducing a rotation in \autoref{apdx:B_Lap_U} does not lead to 
     406Note Bene: introducing a rotation in \autoref{eq:DIFFOPERS_Lap_U} does not lead to 
    407407a useful expression for the iso/diapycnal Laplacian operator in the $z$-coordinate. 
    408408Similarly, we did not found an expression of practical use for 
    409409the geopotential horizontal/vertical Laplacian operator in the $s$-coordinate. 
    410 Generally, \autoref{apdx:B_Lap_U} is used in both $z$- and $s$-coordinate systems, 
     410Generally, \autoref{eq:DIFFOPERS_Lap_U} is used in both $z$- and $s$-coordinate systems, 
    411411that is a Laplacian diffusion is applied on momentum along the coordinate directions. 
    412412 
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