Changeset 10406 for NEMO/trunk/doc/latex/NEMO/subfiles/annex_B.tex
- Timestamp:
- 2018-12-18T11:25:09+01:00 (5 years ago)
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NEMO/trunk/doc/latex/NEMO/subfiles/annex_B.tex
r10354 r10406 20 20 \subsubsection*{In z-coordinates} 21 21 In $z$-coordinates, the horizontal/vertical second order tracer diffusion operator is given by: 22 \begin{ eqnarray} \label{apdx:B1}22 \begin{align} \label{apdx:B1} 23 23 &D^T = \frac{1}{e_1 \, e_2} \left[ 24 24 \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. … … 26 26 + \left. \frac{\partial}{\partial j} \left( \frac{e_1}{e_2}A^{lT} \;\left. \frac{\partial T}{\partial j} \right|_z \right) \right|_z \right] 27 27 + \frac{\partial }{\partial z}\left( {A^{vT} \;\frac{\partial T}{\partial z}} \right) 28 \end{ eqnarray}28 \end{align} 29 29 30 30 \subsubsection*{In generalized vertical coordinates} … … 156 156 \end{equation} 157 157 where ($a_1$, $a_2$) are the isopycnal slopes in ($\textbf{i}$, $\textbf{j}$) directions, relative to geopotentials: 158 \ begin{equation*}158 \[ 159 159 a_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1} 160 160 \qquad , \qquad 161 161 a_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}} 162 162 \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1} 163 \ end{equation*}163 \] 164 164 165 165 In practice, isopycnal slopes are generally less than $10^{-2}$ in the ocean, … … 204 204 The demonstration of the first property is trivial as \autoref{apdx:B4} is the divergence of fluxes. 205 205 Let us demonstrate the second one: 206 \ begin{equation*}206 \[ 207 207 \iiint\limits_D T\;\nabla .\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv 208 208 = -\iiint\limits_D \nabla T\;.\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv, 209 \ end{equation*}209 \] 210 210 and since 211 211 \begin{subequations} … … 249 249 where ($r_1$, $r_2$) are the isopycnal slopes in ($\textbf{i}$, $\textbf{j}$) directions, 250 250 relative to $s$-coordinate surfaces: 251 \ begin{equation*}251 \[ 252 252 r_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1} 253 253 \qquad , \qquad 254 254 r_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}} 255 255 \right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1}. 256 \ end{equation*}256 \] 257 257 258 258 To prove \autoref{apdx:B5} by direct re-expression of \autoref{apdx:B_ldfiso} is straightforward, but laborious. … … 325 325 Using \autoref{eq:PE_div}, the definition of the horizontal divergence, 326 326 the third componant of the second vector is obviously zero and thus : 327 \ begin{equation*}327 \[ 328 328 \Delta {\textbf{U}}_h = \nabla _h \left( \chi \right) - \nabla _h \times \left( \zeta \right) + \frac {1}{e_3 } \frac {\partial }{\partial k} \left( {\frac {1}{e_3 } \frac{\partial {\textbf{ U}}_h }{\partial k}} \right) 329 \ end{equation*}329 \] 330 330 331 331 Note that this operator ensures a full separation between
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