[6997] | 1 | \documentclass[NEMO_book]{subfiles} |
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| 2 | \begin{document} |
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[707] | 3 | % ================================================================ |
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| 4 | % Chapter Ñ Appendix B : Diffusive Operators |
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| 5 | % ================================================================ |
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| 6 | \chapter{Appendix B : Diffusive Operators} |
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| 7 | \label{Apdx_B} |
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| 8 | \minitoc |
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| 9 | |
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[2282] | 10 | |
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| 11 | \newpage |
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| 12 | $\ $\newline % force a new ligne |
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| 13 | |
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[707] | 14 | % ================================================================ |
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| 15 | % Horizontal/Vertical 2nd Order Tracer Diffusive Operators |
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| 16 | % ================================================================ |
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| 17 | \section{Horizontal/Vertical 2nd Order Tracer Diffusive Operators} |
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| 18 | \label{Apdx_B_1} |
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| 19 | |
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[3294] | 20 | \subsubsection*{In z-coordinates} |
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| 21 | In $z$-coordinates, the horizontal/vertical second order tracer diffusion operator |
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[1223] | 22 | is given by: |
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[2282] | 23 | \begin{eqnarray} \label{Apdx_B1} |
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[3294] | 24 | &D^T = \frac{1}{e_1 \, e_2} \left[ |
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| 25 | \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. |
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[2282] | 26 | \left. |
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[3294] | 27 | + \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] |
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[817] | 28 | + \frac{\partial }{\partial z}\left( {A^{vT} \;\frac{\partial T}{\partial z}} \right) |
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[2282] | 29 | \end{eqnarray} |
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[707] | 30 | |
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[3294] | 31 | \subsubsection*{In generalized vertical coordinates} |
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| 32 | In $s$-coordinates, we defined the slopes of $s$-surfaces, $\sigma_1$ and |
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| 33 | $\sigma_2$ by \eqref{Apdx_A_s_slope} and the vertical/horizontal ratio of diffusion |
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[2282] | 34 | coefficient by $\epsilon = A^{vT} / A^{lT}$. The diffusion operator is given by: |
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[707] | 35 | |
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| 36 | \begin{equation} \label{Apdx_B2} |
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[3294] | 37 | D^T = \left. \nabla \right|_s \cdot |
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[817] | 38 | \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T \right] \\ |
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| 39 | \;\;\text{where} \;\Re =\left( {{\begin{array}{*{20}c} |
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[707] | 40 | 1 \hfill & 0 \hfill & {-\sigma _1 } \hfill \\ |
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| 41 | 0 \hfill & 1 \hfill & {-\sigma _2 } \hfill \\ |
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[3294] | 42 | {-\sigma _1 } \hfill & {-\sigma _2 } \hfill & {\varepsilon +\sigma _1 |
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[707] | 43 | ^2+\sigma _2 ^2} \hfill \\ |
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| 44 | \end{array} }} \right) |
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| 45 | \end{equation} |
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[1223] | 46 | or in expanded form: |
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[2282] | 47 | \begin{subequations} |
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[3294] | 48 | \begin{align*} {\begin{array}{*{20}l} |
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| 49 | D^T=& \frac{1}{e_1\,e_2\,e_3 }\;\left[ {\ \ \ \ e_2\,e_3\,A^{lT} \;\left. |
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[2282] | 50 | {\frac{\partial }{\partial i}\left( {\frac{1}{e_1}\;\left. {\frac{\partial T}{\partial i}} \right|_s -\frac{\sigma _1 }{e_3 }\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right. \\ |
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| 51 | &\qquad \quad \ \ \ +e_1\,e_3\,A^{lT} \;\left. {\frac{\partial }{\partial j}\left( {\frac{1}{e_2 }\;\left. {\frac{\partial T}{\partial j}} \right|_s -\frac{\sigma _2 }{e_3 }\;\frac{\partial T}{\partial s}} \right)} \right|_s \\ |
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[3294] | 52 | &\qquad \quad \ \ \ + e_1\,e_2\,A^{lT} \;\frac{\partial }{\partial s}\left( {-\frac{\sigma _1 }{e_1 }\;\left. {\frac{\partial T}{\partial i}} \right|_s -\frac{\sigma _2 }{e_2 }\;\left. {\frac{\partial T}{\partial j}} \right|_s } \right. |
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| 53 | \left. {\left. {+\left( {\varepsilon +\sigma _1^2+\sigma _2 ^2} \right)\;\frac{1}{e_3 }\;\frac{\partial T}{\partial s}} \right)\;\;} \right] |
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| 54 | \end{array} } |
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[2282] | 55 | \end{align*} |
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| 56 | \end{subequations} |
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[707] | 57 | |
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[3294] | 58 | Equation \eqref{Apdx_B2} is obtained from \eqref{Apdx_B1} without any |
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| 59 | additional assumption. Indeed, for the special case $k=z$ and thus $e_3 =1$, |
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| 60 | we introduce an arbitrary vertical coordinate $s = s (i,j,z)$ as in Appendix~\ref{Apdx_A} |
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| 61 | and use \eqref{Apdx_A_s_slope} and \eqref{Apdx_A_s_chain_rule}. |
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| 62 | Since no cross horizontal derivative $\partial _i \partial _j $ appears in |
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| 63 | \eqref{Apdx_B1}, the ($i$,$z$) and ($j$,$z$) planes are independent. |
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| 64 | The derivation can then be demonstrated for the ($i$,$z$)~$\to$~($j$,$s$) |
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[2282] | 65 | transformation without any loss of generality: |
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[707] | 66 | |
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[3294] | 67 | \begin{subequations} |
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| 68 | \begin{align*} {\begin{array}{*{20}l} |
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| 69 | D^T&=\frac{1}{e_1\,e_2} \left. {\frac{\partial }{\partial i}\left( {\frac{e_2}{e_1}A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_z } \right)} \right|_z |
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[2282] | 70 | +\frac{\partial }{\partial z}\left( {A^{vT}\;\frac{\partial T}{\partial z}} \right) \\ |
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[3294] | 71 | \\ |
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| 72 | % |
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| 73 | &=\frac{1}{e_1\,e_2 }\left[ {\left. {\;\frac{\partial }{\partial i}\left( {\frac{e_2}{e_1}A^{lT}\;\left( {\left. {\frac{\partial T}{\partial i}} \right|_s |
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| 74 | -\frac{e_1\,\sigma _1 }{e_3 }\frac{\partial T}{\partial s}} \right)} \right)} \right|_s } \right. \\ |
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| 75 | & \qquad \qquad \left. { -\frac{e_1\,\sigma _1 }{e_3 }\frac{\partial }{\partial s}\left( {\frac{e_2 }{e_1 }A^{lT}\;\left. {\left( {\left. {\frac{\partial T}{\partial i}} \right|_s -\frac{e_1 \,\sigma _1 }{e_3 }\frac{\partial T}{\partial s}} \right)} \right|_s } \right)\;} \right] |
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| 76 | \shoveright{ +\frac{1}{e_3 }\frac{\partial }{\partial s}\left[ {\frac{A^{vT}}{e_3 }\;\frac{\partial T}{\partial s}} \right]} \qquad \qquad \qquad \\ |
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| 77 | \\ |
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| 78 | % |
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| 79 | &=\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{e_2 }{e_1}A^{lT}\;\frac{\partial e_3 }{\partial i}} \right|_s \left. {\frac{\partial T}{\partial i}} \right|_s } \right. \\ |
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[2282] | 80 | & \qquad \qquad \quad \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 -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) \\ |
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[3294] | 81 | & \qquad \qquad \quad \shoveright{ -e_1 \,\sigma _1 \frac{\partial }{\partial s}\left( {-\frac{e_2 \,\sigma _1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right)\;\,\left. {+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 }{e_3 }A^{vT}\;\frac{\partial T}{\partial s}} \right)\quad} \right] }\\ |
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[2282] | 82 | \end{array} } \\ |
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[3294] | 83 | % |
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[2282] | 84 | {\begin{array}{*{20}l} |
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| 85 | \intertext{Noting that $\frac{1}{e_1} \left. \frac{\partial e_3 }{\partial i} \right|_s = \frac{\partial \sigma _1 }{\partial s}$, it becomes:} |
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| 86 | % |
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[3294] | 87 | & =\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. \\ |
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| 88 | & \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) \\ |
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| 89 | & \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] }\\ |
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[2282] | 90 | \\ |
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[3294] | 91 | &=\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. \\ |
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| 92 | & \qquad \qquad \quad \left. {+\frac{e_2 \,\sigma _1 }{e_3}A^{lT}\;\frac{\partial T}{\partial s} \;\frac{\partial e_3 }{\partial i}} \right|_s -e_2 A^{lT}\;\frac{\partial \sigma _1 }{\partial s}\left. {\frac{\partial T}{\partial i}} \right|_s \\ |
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| 93 | & \qquad \qquad \quad-e_2 \,\sigma _1 \frac{\partial}{\partial s}\left( {A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 \,\sigma _1 ^2}{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right) \\ |
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| 94 | & \qquad \qquad \quad\shoveright{ \left. {-\frac{\partial \left( {e_1 \,e_2 \,\sigma _1 } \right)}{\partial s} \left( {\frac{\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]} |
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| 95 | \end{array} } \\ |
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| 96 | {\begin{array}{*{20}l} |
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[2282] | 97 | % |
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| 98 | \intertext{using the same remark as just above, it becomes:} |
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| 99 | % |
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[3294] | 100 | &= \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.\;\;\; \\ |
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| 101 | & \qquad \qquad \quad+\frac{e_1 \,e_2 \,\sigma _1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}\;\frac{\partial \sigma _1 }{\partial s} - \frac {\sigma _1 }{e_3} A^{lT} \;\frac{\partial \left( {e_1 \,e_2 \,\sigma _1 } \right)}{\partial s}\;\frac{\partial T}{\partial s} \\ |
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| 102 | & \qquad \qquad \quad-e_2 \left( {A^{lT}\;\frac{\partial \sigma _1 }{\partial s}\left. {\frac{\partial T}{\partial i}} \right|_s +\frac{\partial }{\partial s}\left( {\sigma _1 A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)-\frac{\partial \sigma _1 }{\partial s}\;A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right) \\ |
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| 103 | & \qquad \qquad \quad\shoveright{\left. {+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 \,\sigma _1 ^2}{e_3 }A^{lT}\;\frac{\partial T}{\partial s}+\frac{e_1 \,e_2}{e_3 }A^{vT}\;\frac{\partial T}{\partial s}} \right)\;\;\;} \right] } |
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| 104 | \end{array} } \\ |
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| 105 | {\begin{array}{*{20}l} |
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[2282] | 106 | % |
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[3294] | 107 | \intertext{Since the horizontal scale factors do not depend on the vertical coordinate, |
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| 108 | the last term of the first line and the first term of the last line cancel, while |
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[2282] | 109 | the second line reduces to a single vertical derivative, so it becomes:} |
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| 110 | % |
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[3294] | 111 | & =\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. \\ |
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| 112 | & \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]} |
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| 113 | \\ |
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[2282] | 114 | % |
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[3294] | 115 | \intertext{in other words, the horizontal/vertical Laplacian operator in the ($i$,$s$) plane takes the following form:} |
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| 116 | \end{array} } \\ |
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[2282] | 117 | % |
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[3294] | 118 | {\frac{1}{e_1\,e_2\,e_3}} |
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[817] | 119 | \left( {{\begin{array}{*{30}c} |
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[707] | 120 | {\left. {\frac{\partial \left( {e_2 e_3 \bullet } \right)}{\partial i}} \right|_s } \hfill \\ |
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| 121 | {\frac{\partial \left( {e_1 e_2 \bullet } \right)}{\partial s}} \hfill \\ |
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| 122 | \end{array}}}\right) |
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| 123 | \cdot \left[ {A^{lT} |
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[817] | 124 | \left( {{\begin{array}{*{30}c} |
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[707] | 125 | {1} \hfill & {-\sigma_1 } \hfill \\ |
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[3294] | 126 | {-\sigma_1} \hfill & {\varepsilon + \sigma_1^2} \hfill \\ |
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[707] | 127 | \end{array} }} \right) |
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[3294] | 128 | \cdot |
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[817] | 129 | \left( {{\begin{array}{*{30}c} |
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[707] | 130 | {\frac{1}{e_1 }\;\left. {\frac{\partial \bullet }{\partial i}} \right|_s } \hfill \\ |
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| 131 | {\frac{1}{e_3 }\;\frac{\partial \bullet }{\partial s}} \hfill \\ |
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[2282] | 132 | \end{array}}} \right) \left( T \right)} \right] |
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| 133 | \end{align*} |
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| 134 | \end{subequations} |
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[3294] | 135 | \addtocounter{equation}{-2} |
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[707] | 136 | |
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| 137 | % ================================================================ |
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[817] | 138 | % Isopycnal/Vertical 2nd Order Tracer Diffusive Operators |
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[707] | 139 | % ================================================================ |
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[817] | 140 | \section{Iso/diapycnal 2nd Order Tracer Diffusive Operators} |
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[707] | 141 | \label{Apdx_B_2} |
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| 142 | |
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[3294] | 143 | \subsubsection*{In z-coordinates} |
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[707] | 144 | |
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[3294] | 145 | The iso/diapycnal diffusive tensor $\textbf {A}_{\textbf I}$ expressed in the ($i$,$j$,$k$) |
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| 146 | curvilinear coordinate system in which the equations of the ocean circulation model are |
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[1223] | 147 | formulated, takes the following form \citep{Redi_JPO82}: |
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[707] | 148 | |
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[3294] | 149 | \begin{equation} \label{Apdx_B3} |
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[707] | 150 | \textbf {A}_{\textbf I} = \frac{A^{lT}}{\left( {1+a_1 ^2+a_2 ^2} \right)} |
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| 151 | \left[ {{\begin{array}{*{20}c} |
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| 152 | {1+a_1 ^2} \hfill & {-a_1 a_2 } \hfill & {-a_1 } \hfill \\ |
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| 153 | {-a_1 a_2 } \hfill & {1+a_2 ^2} \hfill & {-a_2 } \hfill \\ |
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| 154 | {-a_1 } \hfill & {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\ |
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| 155 | \end{array} }} \right] |
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[3294] | 156 | \end{equation} |
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| 157 | where ($a_1$, $a_2$) are the isopycnal slopes in ($\textbf{i}$, |
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| 158 | $\textbf{j}$) directions, relative to geopotentials: |
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[707] | 159 | \begin{equation*} |
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| 160 | a_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1} |
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[817] | 161 | \qquad , \qquad |
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[3294] | 162 | a_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}} |
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[707] | 163 | \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1} |
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| 164 | \end{equation*} |
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[817] | 165 | |
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[3294] | 166 | In practice, isopycnal slopes are generally less than $10^{-2}$ in the ocean, so |
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[1223] | 167 | $\textbf {A}_{\textbf I}$ can be simplified appreciably \citep{Cox1987}: |
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[3294] | 168 | \begin{subequations} \label{Apdx_B4} |
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| 169 | \begin{equation} \label{Apdx_B4a} |
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| 170 | {\textbf{A}_{\textbf{I}}} \approx A^{lT}\;\Re\;\text{where} \;\Re = |
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[707] | 171 | \left[ {{\begin{array}{*{20}c} |
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| 172 | 1 \hfill & 0 \hfill & {-a_1 } \hfill \\ |
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| 173 | 0 \hfill & 1 \hfill & {-a_2 } \hfill \\ |
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| 174 | {-a_1 } \hfill & {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\ |
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[3294] | 175 | \end{array} }} \right], |
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| 176 | \end{equation} |
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| 177 | and the iso/dianeutral diffusive operator in $z$-coordinates is then |
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| 178 | \begin{equation}\label{Apdx_B4b} |
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| 179 | D^T = \left. \nabla \right|_z \cdot |
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| 180 | \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_z T \right]. \\ |
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| 181 | \end{equation} |
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| 182 | \end{subequations} |
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[817] | 183 | |
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[3294] | 184 | |
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| 185 | Physically, the full tensor \eqref{Apdx_B3} |
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| 186 | represents strong isoneutral diffusion on a plane parallel to the isoneutral |
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| 187 | surface and weak dianeutral diffusion perpendicular to this plane. |
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| 188 | However, the approximate `weak-slope' tensor \eqref{Apdx_B4a} represents strong |
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| 189 | diffusion along the isoneutral surface, with weak |
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| 190 | \emph{vertical} diffusion -- the principal axes of the tensor are no |
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| 191 | longer orthogonal. This simplification also decouples |
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| 192 | the ($i$,$z$) and ($j$,$z$) planes of the tensor. The weak-slope operator therefore takes the same |
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| 193 | form, \eqref{Apdx_B4}, as \eqref{Apdx_B2}, the diffusion operator for geopotential |
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| 194 | diffusion written in non-orthogonal $i,j,s$-coordinates. Written out |
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| 195 | explicitly, |
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| 196 | |
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| 197 | \begin{multline} \label{Apdx_B_ldfiso} |
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| 198 | D^T=\frac{1}{e_1 e_2 }\left\{ |
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| 199 | {\;\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]} |
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| 200 | {+\frac{\partial}{\partial j}\left[ {A_h \left( {\frac{e_1}{e_2}\frac{\partial T}{\partial j}-a_2 \frac{e_1}{e_3}\frac{\partial T}{\partial k}} \right)} \right]\;} \right\} \\ |
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| 201 | \shoveright{+\frac{1}{e_3 }\frac{\partial }{\partial k}\left[ {A_h \left( {-\frac{a_1 }{e_1 }\frac{\partial T}{\partial i}-\frac{a_2 }{e_2 }\frac{\partial T}{\partial j}+\frac{\left( {a_1 ^2+a_2 ^2+\varepsilon} \right)}{e_3 }\frac{\partial T}{\partial k}} \right)} \right]}. \\ |
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| 202 | \end{multline} |
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| 203 | |
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| 204 | |
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| 205 | The isopycnal diffusion operator \eqref{Apdx_B4}, |
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| 206 | \eqref{Apdx_B_ldfiso} conserves tracer quantity and dissipates its |
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| 207 | square. The demonstration of the first property is trivial as \eqref{Apdx_B4} is the divergence |
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[1223] | 208 | of fluxes. Let us demonstrate the second one: |
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[707] | 209 | \begin{equation*} |
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[3294] | 210 | \iiint\limits_D T\;\nabla .\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv |
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| 211 | = -\iiint\limits_D \nabla T\;.\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv, |
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[707] | 212 | \end{equation*} |
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[3294] | 213 | and since |
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| 214 | \begin{subequations} |
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| 215 | \begin{align*} {\begin{array}{*{20}l} |
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| 216 | \nabla T\;.\left( {{\rm {\bf A}}_{\rm {\bf I}} \nabla T} |
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| 217 | \right)&=A^{lT}\left[ {\left( {\frac{\partial T}{\partial i}} \right)^2-2a_1 |
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| 218 | \frac{\partial T}{\partial i}\frac{\partial T}{\partial k}+\left( |
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| 219 | {\frac{\partial T}{\partial j}} \right)^2} \right. \\ |
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[2282] | 220 | &\qquad \qquad \qquad |
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[3294] | 221 | { \left. -\,{2a_2 \frac{\partial T}{\partial j}\frac{\partial T}{\partial k}+\left( {a_1 ^2+a_2 ^2+\varepsilon} \right)\left( {\frac{\partial T}{\partial k}} \right)^2} \right]} \\ |
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| 222 | &=A_h \left[ {\left( {\frac{\partial T}{\partial i}-a_1 \frac{\partial |
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| 223 | T}{\partial k}} \right)^2+\left( {\frac{\partial T}{\partial |
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| 224 | j}-a_2 \frac{\partial T}{\partial k}} \right)^2} |
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| 225 | +\varepsilon \left(\frac{\partial T}{\partial k}\right) ^2\right] \\ |
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[2282] | 226 | & \geq 0 |
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[3294] | 227 | \end{array} } |
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[817] | 228 | \end{align*} |
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[2282] | 229 | \end{subequations} |
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[3294] | 230 | \addtocounter{equation}{-1} |
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| 231 | the property becomes obvious. |
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[707] | 232 | |
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[3294] | 233 | \subsubsection*{In generalized vertical coordinates} |
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[707] | 234 | |
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[3294] | 235 | Because the weak-slope operator \eqref{Apdx_B4}, \eqref{Apdx_B_ldfiso} is decoupled |
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| 236 | in the ($i$,$z$) and ($j$,$z$) planes, it may be transformed into |
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| 237 | generalized $s$-coordinates in the same way as \eqref{Apdx_B_1} was transformed into |
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| 238 | \eqref{Apdx_B_2}. The resulting operator then takes the simple form |
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[707] | 239 | |
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[3294] | 240 | \begin{equation} \label{Apdx_B_ldfiso_s} |
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| 241 | D^T = \left. \nabla \right|_s \cdot |
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| 242 | \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T \right] \\ |
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| 243 | \;\;\text{where} \;\Re =\left( {{\begin{array}{*{20}c} |
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| 244 | 1 \hfill & 0 \hfill & {-r _1 } \hfill \\ |
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| 245 | 0 \hfill & 1 \hfill & {-r _2 } \hfill \\ |
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| 246 | {-r _1 } \hfill & {-r _2 } \hfill & {\varepsilon +r _1 |
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| 247 | ^2+r _2 ^2} \hfill \\ |
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| 248 | \end{array} }} \right), |
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| 249 | \end{equation} |
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| 250 | |
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| 251 | where ($r_1$, $r_2$) are the isopycnal slopes in ($\textbf{i}$, |
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| 252 | $\textbf{j}$) directions, relative to $s$-coordinate surfaces: |
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| 253 | \begin{equation*} |
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| 254 | r_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1} |
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| 255 | \qquad , \qquad |
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| 256 | r_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}} |
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| 257 | \right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1}. |
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| 258 | \end{equation*} |
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| 259 | |
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| 260 | To prove \eqref{Apdx_B5} by direct re-expression of \eqref{Apdx_B_ldfiso} is |
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| 261 | straightforward, but laborious. An easier way is first to note (by reversing the |
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| 262 | derivation of \eqref{Apdx_B_2} from \eqref{Apdx_B_1} ) that the |
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| 263 | weak-slope operator may be \emph{exactly} reexpressed in |
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| 264 | non-orthogonal $i,j,\rho$-coordinates as |
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| 265 | |
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| 266 | \begin{equation} \label{Apdx_B5} |
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| 267 | D^T = \left. \nabla \right|_\rho \cdot |
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| 268 | \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_\rho T \right] \\ |
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| 269 | \;\;\text{where} \;\Re =\left( {{\begin{array}{*{20}c} |
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| 270 | 1 \hfill & 0 \hfill &0 \hfill \\ |
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| 271 | 0 \hfill & 1 \hfill & 0 \hfill \\ |
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| 272 | 0 \hfill & 0 \hfill & \varepsilon \hfill \\ |
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| 273 | \end{array} }} \right). |
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| 274 | \end{equation} |
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| 275 | Then direct transformation from $i,j,\rho$-coordinates to |
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| 276 | $i,j,s$-coordinates gives \eqref{Apdx_B_ldfiso_s} immediately. |
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| 277 | |
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| 278 | Note that the weak-slope approximation is only made in |
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| 279 | transforming from the (rotated,orthogonal) isoneutral axes to the |
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| 280 | non-orthogonal $i,j,\rho$-coordinates. The further transformation |
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| 281 | into $i,j,s$-coordinates is exact, whatever the steepness of |
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| 282 | the $s$-surfaces, in the same way as the transformation of |
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| 283 | horizontal/vertical Laplacian diffusion in $z$-coordinates, |
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| 284 | \eqref{Apdx_B_1} onto $s$-coordinates is exact, however steep the $s$-surfaces. |
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| 285 | |
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| 286 | |
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[707] | 287 | % ================================================================ |
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| 288 | % Lateral/Vertical Momentum Diffusive Operators |
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| 289 | % ================================================================ |
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| 290 | \section{Lateral/Vertical Momentum Diffusive Operators} |
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| 291 | \label{Apdx_B_3} |
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| 292 | |
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[3294] | 293 | The second order momentum diffusion operator (Laplacian) in the $z$-coordinate |
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| 294 | is found by applying \eqref{Eq_PE_lap_vector}, the expression for the Laplacian |
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| 295 | of a vector, to the horizontal velocity vector : |
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[817] | 296 | \begin{align*} |
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[3294] | 297 | \Delta {\textbf{U}}_h |
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[817] | 298 | &=\nabla \left( {\nabla \cdot {\textbf{U}}_h } \right)- |
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| 299 | \nabla \times \left( {\nabla \times {\textbf{U}}_h } \right) \\ |
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| 300 | \\ |
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| 301 | &=\left( {{\begin{array}{*{20}c} |
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[707] | 302 | {\frac{1}{e_1 }\frac{\partial \chi }{\partial i}} \hfill \\ |
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| 303 | {\frac{1}{e_2 }\frac{\partial \chi }{\partial j}} \hfill \\ |
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| 304 | {\frac{1}{e_3 }\frac{\partial \chi }{\partial k}} \hfill \\ |
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| 305 | \end{array} }} \right)-\left( {{\begin{array}{*{20}c} |
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[3294] | 306 | {\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}-\frac{1}{e_3 |
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| 307 | }\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial |
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[707] | 308 | u}{\partial k}} \right)} \hfill \\ |
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[3294] | 309 | {\frac{1}{e_3 }\frac{\partial }{\partial k}\left( {-\frac{1}{e_3 |
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| 310 | }\frac{\partial v}{\partial k}} \right)-\frac{1}{e_1 }\frac{\partial \zeta |
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[707] | 311 | }{\partial i}} \hfill \\ |
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[3294] | 312 | {\frac{1}{e_1 e_2 }\left[ {\frac{\partial }{\partial i}\left( {\frac{e_2 |
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| 313 | }{e_3 }\frac{\partial u}{\partial k}} \right)-\frac{\partial }{\partial |
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| 314 | j}\left( {-\frac{e_1 }{e_3 }\frac{\partial v}{\partial k}} \right)} \right]} |
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[707] | 315 | \hfill \\ |
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| 316 | \end{array} }} \right) |
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[817] | 317 | \\ |
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| 318 | \\ |
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| 319 | &=\left( {{\begin{array}{*{20}c} |
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[707] | 320 | {\frac{1}{e_1 }\frac{\partial \chi }{\partial i}-\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}} \\ |
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| 321 | {\frac{1}{e_2 }\frac{\partial \chi }{\partial j}+\frac{1}{e_1 }\frac{\partial \zeta }{\partial i}} \\ |
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| 322 | 0 \\ |
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| 323 | \end{array} }} \right) |
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| 324 | +\frac{1}{e_3 } |
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| 325 | \left( {{\begin{array}{*{20}c} |
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| 326 | {\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial u}{\partial k}} \right)} \\ |
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| 327 | {\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial v}{\partial k}} \right)} \\ |
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| 328 | {\frac{\partial \chi }{\partial k}-\frac{1}{e_1 e_2 }\left( {\frac{\partial ^2\left( {e_2 \,u} \right)}{\partial i\partial k}+\frac{\partial ^2\left( {e_1 \,v} \right)}{\partial j\partial k}} \right)} \\ |
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| 329 | \end{array} }} \right) |
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[817] | 330 | \end{align*} |
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[3294] | 331 | Using \eqref{Eq_PE_div}, the definition of the horizontal divergence, the third |
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[1223] | 332 | componant of the second vector is obviously zero and thus : |
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[707] | 333 | \begin{equation*} |
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| 334 | \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) |
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| 335 | \end{equation*} |
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| 336 | |
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[3294] | 337 | Note that this operator ensures a full separation between the vorticity and horizontal |
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| 338 | divergence fields (see Appendix~\ref{Apdx_C}). It is only equal to a Laplacian |
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[1223] | 339 | applied to each component in Cartesian coordinates, not on the sphere. |
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[707] | 340 | |
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[3294] | 341 | The horizontal/vertical second order (Laplacian type) operator used to diffuse |
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[1223] | 342 | horizontal momentum in the $z$-coordinate therefore takes the following form : |
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[817] | 343 | \begin{equation} \label{Apdx_B_Lap_U} |
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[3294] | 344 | {\textbf{D}}^{\textbf{U}} = |
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[817] | 345 | \nabla _h \left( {A^{lm}\;\chi } \right) |
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| 346 | - \nabla _h \times \left( {A^{lm}\;\zeta \;{\textbf{k}}} \right) |
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| 347 | + \frac{1}{e_3 }\frac{\partial }{\partial k}\left( {\frac{A^{vm}\;}{e_3 } |
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[3294] | 348 | \frac{\partial {\rm {\bf U}}_h }{\partial k}} \right) \\ |
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[817] | 349 | \end{equation} |
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[1223] | 350 | that is, in expanded form: |
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[817] | 351 | \begin{align*} |
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[3294] | 352 | D^{\textbf{U}}_u |
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[817] | 353 | & = \frac{1}{e_1} \frac{\partial \left( {A^{lm}\chi } \right)}{\partial i} |
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| 354 | -\frac{1}{e_2} \frac{\partial \left( {A^{lm}\zeta } \right)}{\partial j} |
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| 355 | +\frac{1}{e_3} \frac{\partial u}{\partial k} \\ |
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[3294] | 356 | D^{\textbf{U}}_v |
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[817] | 357 | & = \frac{1}{e_2 }\frac{\partial \left( {A^{lm}\chi } \right)}{\partial j} |
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| 358 | +\frac{1}{e_1 }\frac{\partial \left( {A^{lm}\zeta } \right)}{\partial i} |
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| 359 | +\frac{1}{e_3} \frac{\partial v}{\partial k} |
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| 360 | \end{align*} |
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[707] | 361 | |
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[3294] | 362 | Note Bene: introducing a rotation in \eqref{Apdx_B_Lap_U} does not lead to a |
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| 363 | useful expression for the iso/diapycnal Laplacian operator in the $z$-coordinate. |
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| 364 | Similarly, we did not found an expression of practical use for the geopotential |
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| 365 | horizontal/vertical Laplacian operator in the $s$-coordinate. Generally, |
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| 366 | \eqref{Apdx_B_Lap_U} is used in both $z$- and $s$-coordinate systems, that is |
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[1223] | 367 | a Laplacian diffusion is applied on momentum along the coordinate directions. |
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[6997] | 368 | \end{document} |
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