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Changeset 10354 for NEMO/trunk/doc/latex/NEMO/subfiles/annex_B.tex – NEMO

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Timestamp:
2018-11-21T17:59:55+01:00 (5 years ago)
Author:
nicolasmartin
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Vast edition of LaTeX subfiles to improve the readability by cutting sentences in a more suitable way
Every sentence begins in a new line and if necessary is splitted around 110 characters lenght for side-by-side visualisation,
this setting may not be adequate for everyone but something has to be set.
The punctuation was the primer trigger for the cutting process, otherwise subordinators and coordinators, in order to mostly keep a meaning for each line

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

    r9407 r10354  
    1919 
    2020\subsubsection*{In z-coordinates} 
    21 In $z$-coordinates, the horizontal/vertical second order tracer diffusion operator 
    22 is given by: 
     21In $z$-coordinates, the horizontal/vertical second order tracer diffusion operator is given by: 
    2322\begin{eqnarray} \label{apdx:B1} 
    2423 &D^T = \frac{1}{e_1 \, e_2}      \left[ 
    … …  
    3029 
    3130\subsubsection*{In generalized vertical coordinates} 
    32 In $s$-coordinates, we defined the slopes of $s$-surfaces, $\sigma_1$ and 
    33 $\sigma_2$ by \autoref{apdx:A_s_slope} and the vertical/horizontal ratio of diffusion 
    34 coefficient by $\epsilon = A^{vT} / A^{lT}$. The diffusion operator is given by: 
     31In $s$-coordinates, we defined the slopes of $s$-surfaces, $\sigma_1$ and $\sigma_2$ by \autoref{apdx:A_s_slope} and 
     32the vertical/horizontal ratio of diffusion coefficient by $\epsilon = A^{vT} / A^{lT}$. 
     33The diffusion operator is given by: 
    3534 
    3635\begin{equation} \label{apdx:B2} 
    … …  
    5655\end{subequations} 
    5756 
    58 Equation \autoref{apdx:B2} is obtained from \autoref{apdx:B1} without any 
    59 additional assumption. Indeed, for the special case $k=z$ and thus $e_3 =1$, 
    60 we introduce an arbitrary vertical coordinate $s = s (i,j,z)$ as in \autoref{apdx:A} 
    61 and use \autoref{apdx:A_s_slope} and \autoref{apdx:A_s_chain_rule}. 
    62 Since no cross horizontal derivative $\partial _i \partial _j $ appears in 
    63 \autoref{apdx:B1}, the ($i$,$z$) and ($j$,$z$) planes are independent. 
    64 The derivation can then be demonstrated for the ($i$,$z$)~$\to$~($j$,$s$) 
    65 transformation without any loss of generality: 
     57Equation \autoref{apdx:B2} is obtained from \autoref{apdx:B1} without any additional assumption. 
     58Indeed, for the special case $k=z$ and thus $e_3 =1$, 
     59we introduce an arbitrary vertical coordinate $s = s (i,j,z)$ as in \autoref{apdx:A} and 
     60use \autoref{apdx:A_s_slope} and \autoref{apdx:A_s_chain_rule}. 
     61Since no cross horizontal derivative $\partial _i \partial _j $ appears in \autoref{apdx:B1}, 
     62the ($i$,$z$) and ($j$,$z$) planes are independent. 
     63The derivation can then be demonstrated for the ($i$,$z$)~$\to$~($j$,$s$) transformation without 
     64any loss of generality: 
    6665 
    6766\begin{subequations} 
    … …  
    143142\subsubsection*{In z-coordinates} 
    144143 
    145 The iso/diapycnal diffusive tensor $\textbf {A}_{\textbf I}$ expressed in the ($i$,$j$,$k$) 
    146 curvilinear coordinate system in which the equations of the ocean circulation model are 
    147 formulated, takes the following form \citep{Redi_JPO82}: 
     144The iso/diapycnal diffusive tensor $\textbf {A}_{\textbf I}$ expressed in 
     145the ($i$,$j$,$k$) curvilinear coordinate system in which 
     146the equations of the ocean circulation model are formulated, 
     147takes the following form \citep{Redi_JPO82}: 
    148148 
    149149\begin{equation} \label{apdx:B3} 
    … …  
    155155\end{array} }} \right] 
    156156\end{equation} 
    157 where ($a_1$, $a_2$) are the isopycnal slopes in ($\textbf{i}$, 
    158 $\textbf{j}$) directions, relative to geopotentials: 
     157where ($a_1$, $a_2$) are the isopycnal slopes in ($\textbf{i}$, $\textbf{j}$) directions, relative to geopotentials: 
    159158\begin{equation*} 
    160159a_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1} 
    … …  
    164163\end{equation*} 
    165164 
    166 In practice, isopycnal slopes are generally less than $10^{-2}$ in the ocean, so 
    167 $\textbf {A}_{\textbf I}$ can be simplified appreciably \citep{Cox1987}: 
     165In practice, isopycnal slopes are generally less than $10^{-2}$ in the ocean, 
     166so $\textbf {A}_{\textbf I}$ can be simplified appreciably \citep{Cox1987}: 
    168167\begin{subequations} \label{apdx:B4} 
    169168\begin{equation} \label{apdx:B4a} 
    … …  
    183182 
    184183 
    185 Physically, the full tensor \autoref{apdx:B3} 
    186 represents strong isoneutral diffusion on a plane parallel to the isoneutral 
    187 surface and weak dianeutral diffusion perpendicular to this plane. 
    188 However, the approximate `weak-slope' tensor \autoref{apdx:B4a} represents strong 
    189 diffusion along the isoneutral surface, with weak 
    190 \emph{vertical}  diffusion -- the principal axes of the tensor are no 
    191 longer orthogonal. This simplification also decouples 
    192 the ($i$,$z$) and ($j$,$z$) planes of the tensor. The weak-slope operator therefore takes the same 
    193 form, \autoref{apdx:B4}, as \autoref{apdx:B2}, the diffusion operator for geopotential 
    194 diffusion written in non-orthogonal $i,j,s$-coordinates. Written out 
    195 explicitly, 
     184Physically, the full tensor \autoref{apdx:B3} represents strong isoneutral diffusion on a plane parallel to 
     185the isoneutral surface and weak dianeutral diffusion perpendicular to this plane. 
     186However, 
     187the approximate `weak-slope' tensor \autoref{apdx:B4a} represents strong diffusion along the isoneutral surface, 
     188with weak \emph{vertical} diffusion -- the principal axes of the tensor are no longer orthogonal. 
     189This simplification also decouples the ($i$,$z$) and ($j$,$z$) planes of the tensor. 
     190The weak-slope operator therefore takes the same form, \autoref{apdx:B4}, as \autoref{apdx:B2}, 
     191the diffusion operator for geopotential diffusion written in non-orthogonal $i,j,s$-coordinates. 
     192Written out explicitly, 
    196193 
    197194\begin{multline} \label{apdx:B_ldfiso} 
    … …  
    204201 
    205202The isopycnal diffusion operator \autoref{apdx:B4}, 
    206 \autoref{apdx:B_ldfiso} conserves tracer quantity and dissipates its 
    207 square. The demonstration of the first property is trivial as \autoref{apdx:B4} is the divergence 
    208 of fluxes. Let us demonstrate the second one: 
     203\autoref{apdx:B_ldfiso} conserves tracer quantity and dissipates its square. 
     204The demonstration of the first property is trivial as \autoref{apdx:B4} is the divergence of fluxes. 
     205Let us demonstrate the second one: 
    209206\begin{equation*} 
    210207\iiint\limits_D T\;\nabla .\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv 
    … …  
    229226\end{subequations} 
    230227\addtocounter{equation}{-1} 
    231  the property becomes obvious. 
     228the property becomes obvious. 
    232229 
    233230\subsubsection*{In generalized vertical coordinates} 
    234231 
    235 Because the weak-slope operator \autoref{apdx:B4}, \autoref{apdx:B_ldfiso} is decoupled 
    236 in the ($i$,$z$) and ($j$,$z$) planes, it may be transformed into 
    237 generalized $s$-coordinates in the same way as \autoref{sec:B_1} was transformed into 
    238 \autoref{sec:B_2}. The resulting operator then takes the simple form 
     232Because the weak-slope operator \autoref{apdx:B4}, 
     233\autoref{apdx:B_ldfiso} is decoupled in the ($i$,$z$) and ($j$,$z$) planes, 
     234it may be transformed into generalized $s$-coordinates in the same way as 
     235\autoref{sec:B_1} was transformed into \autoref{sec:B_2}. 
     236The resulting operator then takes the simple form 
    239237 
    240238\begin{equation} \label{apdx:B_ldfiso_s} 
    … …  
    249247\end{equation} 
    250248 
    251 where ($r_1$, $r_2$) are the isopycnal slopes in ($\textbf{i}$, 
    252 $\textbf{j}$) directions, relative to $s$-coordinate surfaces: 
     249where ($r_1$, $r_2$) are the isopycnal slopes in ($\textbf{i}$, $\textbf{j}$) directions, 
     250relative to $s$-coordinate surfaces: 
    253251\begin{equation*} 
    254252r_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1} 
    … …  
    258256\end{equation*} 
    259257 
    260 To prove  \autoref{apdx:B5}  by direct re-expression of \autoref{apdx:B_ldfiso} is 
    261 straightforward, but laborious. An easier way is first to note (by reversing the 
    262 derivation of \autoref{sec:B_2} from \autoref{sec:B_1} ) that the 
    263 weak-slope operator may be \emph{exactly} reexpressed in  
    264 non-orthogonal $i,j,\rho$-coordinates as 
     258To prove \autoref{apdx:B5} by direct re-expression of \autoref{apdx:B_ldfiso} is straightforward, but laborious. 
     259An easier way is first to note (by reversing the derivation of \autoref{sec:B_2} from \autoref{sec:B_1} ) that 
     260the weak-slope operator may be \emph{exactly} reexpressed in non-orthogonal $i,j,\rho$-coordinates as 
    265261 
    266262\begin{equation} \label{apdx:B5} 
    … …  
    273269\end{array} }} \right). 
    274270\end{equation} 
    275 Then direct transformation from $i,j,\rho$-coordinates to 
    276 $i,j,s$-coordinates gives \autoref{apdx:B_ldfiso_s} immediately. 
    277  
    278 Note that the weak-slope approximation is only made in 
    279 transforming from the (rotated,orthogonal) isoneutral axes to the 
    280 non-orthogonal $i,j,\rho$-coordinates. The further transformation 
    281 into $i,j,s$-coordinates is exact, whatever the steepness of 
    282 the  $s$-surfaces, in the same way as the transformation of 
    283 horizontal/vertical Laplacian diffusion in $z$-coordinates, 
     271Then direct transformation from $i,j,\rho$-coordinates to $i,j,s$-coordinates gives 
     272\autoref{apdx:B_ldfiso_s} immediately. 
     273 
     274Note that the weak-slope approximation is only made in transforming from 
     275the (rotated,orthogonal) isoneutral axes to the non-orthogonal $i,j,\rho$-coordinates. 
     276The further transformation into $i,j,s$-coordinates is exact, whatever the steepness of the $s$-surfaces, 
     277in the same way as the transformation of horizontal/vertical Laplacian diffusion in $z$-coordinates, 
    284278\autoref{sec:B_1} onto $s$-coordinates is exact, however steep the $s$-surfaces. 
    285279 
    … …  
    291285\label{sec:B_3} 
    292286 
    293 The second order momentum diffusion operator (Laplacian) in the $z$-coordinate 
    294 is found by applying \autoref{eq:PE_lap_vector}, the expression for the Laplacian 
    295 of a vector,  to the horizontal velocity vector : 
     287The second order momentum diffusion operator (Laplacian) in the $z$-coordinate is found by 
     288applying \autoref{eq:PE_lap_vector}, the expression for the Laplacian of a vector, 
     289to the horizontal velocity vector: 
    296290\begin{align*} 
    297291\Delta {\textbf{U}}_h 
    … …  
    329323\end{array} }} \right) 
    330324\end{align*} 
    331 Using \autoref{eq:PE_div}, the definition of the horizontal divergence, the third 
    332 componant of the second vector is obviously zero and thus : 
     325Using \autoref{eq:PE_div}, the definition of the horizontal divergence, 
     326the third componant of the second vector is obviously zero and thus : 
    333327\begin{equation*} 
    334328\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) 
    335329\end{equation*} 
    336330 
    337 Note that this operator ensures a full separation between the vorticity and horizontal 
    338 divergence fields (see \autoref{apdx:C}). It is only equal to a Laplacian 
    339 applied to each component in Cartesian coordinates, not on the sphere. 
    340  
    341 The horizontal/vertical second order (Laplacian type) operator used to diffuse 
    342 horizontal momentum in the $z$-coordinate therefore takes the following form : 
     331Note that this operator ensures a full separation between 
     332the vorticity and horizontal divergence fields (see \autoref{apdx:C}). 
     333It is only equal to a Laplacian applied to each component in Cartesian coordinates, not on the sphere. 
     334 
     335The horizontal/vertical second order (Laplacian type) operator used to diffuse horizontal momentum in 
     336the $z$-coordinate therefore takes the following form: 
    343337\begin{equation} \label{apdx:B_Lap_U} 
    344338 {\textbf{D}}^{\textbf{U}} = 
    … …  
    360354\end{align*} 
    361355 
    362 Note Bene: introducing a rotation in \autoref{apdx:B_Lap_U} does not lead to a 
    363 useful expression for the iso/diapycnal Laplacian operator in the $z$-coordinate. 
    364 Similarly, we did not found an expression of practical use for the geopotential 
    365 horizontal/vertical Laplacian operator in the $s$-coordinate. Generally, 
    366 \autoref{apdx:B_Lap_U} is used in both $z$- and $s$-coordinate systems, that is 
    367 a Laplacian diffusion is applied on momentum along the coordinate directions. 
     356Note Bene: introducing a rotation in \autoref{apdx:B_Lap_U} does not lead to 
     357a useful expression for the iso/diapycnal Laplacian operator in the $z$-coordinate. 
     358Similarly, we did not found an expression of practical use for 
     359the geopotential horizontal/vertical Laplacian operator in the $s$-coordinate. 
     360Generally, \autoref{apdx:B_Lap_U} is used in both $z$- and $s$-coordinate systems, 
     361that is a Laplacian diffusion is applied on momentum along the coordinate directions. 
    368362\end{document} 
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