[707] | 1 | % ================================================================ |
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| 2 | % Chapter Ñ Appendix B : Diffusive Operators |
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| 3 | % ================================================================ |
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| 4 | \chapter{Appendix B : Diffusive Operators} |
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| 5 | \label{Apdx_B} |
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| 6 | \minitoc |
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| 7 | |
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| 8 | % ================================================================ |
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| 9 | % Horizontal/Vertical 2nd Order Tracer Diffusive Operators |
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| 10 | % ================================================================ |
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| 11 | \section{Horizontal/Vertical 2nd Order Tracer Diffusive Operators} |
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| 12 | \label{Apdx_B_1} |
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| 13 | |
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| 14 | |
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[1223] | 15 | In the $z$-coordinate, the horizontal/vertical second order tracer diffusion operator |
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| 16 | is given by: |
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[817] | 17 | \begin{multline} \label{Apdx_B1} |
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| 18 | D^T = \frac{1}{e_1 \, e_2} \left[ |
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| 19 | \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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| 20 | \left. |
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| 21 | + \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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| 22 | + \frac{\partial }{\partial z}\left( {A^{vT} \;\frac{\partial T}{\partial z}} \right) |
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| 23 | \end{multline} |
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[707] | 24 | |
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[1223] | 25 | In the $s$-coordinate, we defined the slopes of $s$-surfaces, $\sigma_1$ and |
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| 26 | $\sigma_2$ by (!!!A.1!!!) and the vertical/horizontal ratio of diffusion coefficient |
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| 27 | by $\epsilon = A^{vT} / A^{lT}$. The diffusion operator is given by: |
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[707] | 28 | |
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| 29 | \begin{equation} \label{Apdx_B2} |
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[817] | 30 | D^T = \left. \nabla \right|_s \cdot |
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| 31 | \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T \right] \\ |
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| 32 | \;\;\text{where} \;\Re =\left( {{\begin{array}{*{20}c} |
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[707] | 33 | 1 \hfill & 0 \hfill & {-\sigma _1 } \hfill \\ |
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| 34 | 0 \hfill & 1 \hfill & {-\sigma _2 } \hfill \\ |
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| 35 | {-\sigma _1 } \hfill & {-\sigma _2 } \hfill & {\varepsilon +\sigma _1 |
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| 36 | ^2+\sigma _2 ^2} \hfill \\ |
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| 37 | \end{array} }} \right) |
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| 38 | \end{equation} |
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[1223] | 39 | or in expanded form: |
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[707] | 40 | \begin{multline} \label{Apdx_B3} |
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[817] | 41 | D^T=\frac{1}{e_1\,e_2\,e_3 }\;\left[ {\quad \; \; e_2\,e_3\,A^{lT} \;\left. |
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[707] | 42 | {\frac{\partial }{\partial i}\left( {\frac{1}{e_1 }\;\left. {\frac{\partial |
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| 43 | T}{\partial i}} \right|_s -\frac{\sigma _1 }{e_3 }\;\frac{\partial |
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| 44 | T}{\partial s}} \right)} \right|_s } \right. \\ |
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[817] | 45 | +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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| 46 | \;\;+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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| 47 | \shoveright{\;\;\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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[707] | 48 | \end{multline} |
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| 49 | |
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[1223] | 50 | Equation \eqref{Apdx_B2} (or equivalently \eqref{Apdx_B3}) is obtained from |
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| 51 | \eqref{Apdx_B1} without any additional assumption. Indeed, for the special |
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| 52 | case $k=z$ and thus $e_3 =1$, we introduce an arbitrary vertical coordinate |
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| 53 | $s = s (i,j,z)$ as in Appendix~\ref{Apdx_A} and use \eqref{Apdx_A_s_slope} |
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| 54 | and \eqref{Apdx_A_s_chain_rule}. Since no cross horizontal derivative |
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| 55 | $\partial _i \partial _j $ appears in \eqref{Apdx_B1}, the ($i$,$z$) and ($j$,$z$) |
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| 56 | planes are independent. The derivation can then be demonstrated for the |
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| 57 | ($i$,$z$)~$\to$~($j$,$s$) transformation without any loss of generality: |
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[707] | 58 | |
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| 59 | \begin{equation*} |
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[817] | 60 | 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 +\frac{\partial }{\partial z}\left( {A^{vT}\;\frac{\partial T}{\partial z}} \right) \qquad \qquad \qquad \qquad \qquad \qquad \qquad \\ |
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[707] | 61 | \end{equation*} |
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| 62 | \begin{multline*} |
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[817] | 63 | =\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 -\frac{e_1 \,\sigma _1 }{e_3 }\frac{\partial T}{\partial s}} \right)} \right)} \right|_s } \right. \\ |
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| 64 | \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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[707] | 65 | \shoveright{ +\frac{1}{e_3 }\frac{\partial }{\partial s}\left[ {\frac{A^{vT}}{e_3 }\;\frac{\partial T}{\partial s}} \right]} \\ |
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| 66 | \end{multline*} |
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| 67 | \begin{multline*} |
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| 68 | =\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 |
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| 69 | }A^{lT}\;\frac{\partial e_3 }{\partial i}} \right|_s \left. {\frac{\partial T}{\partial i}} \right|_s } \right. \\ |
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| 70 | \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( |
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| 71 | {\frac{e_2 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right) \\ |
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| 72 | \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)\;\;} \right] }\\ |
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| 73 | \end{multline*} |
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| 74 | |
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| 75 | 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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| 76 | |
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| 77 | \begin{multline*} |
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[817] | 78 | =\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. |
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| 79 | -\, {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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| 80 | \qquad -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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[707] | 81 | \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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| 82 | \end{multline*} |
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| 83 | \begin{multline*} |
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| 84 | =\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 |
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| 85 | i}\left( {e_2 \,\sigma _1 A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right. \\ |
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| 86 | \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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| 87 | -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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| 88 | \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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| 89 | \end{multline*} |
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| 90 | using the same remark as just above, it becomes: |
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| 91 | \begin{multline*} |
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| 92 | = \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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| 93 | +\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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| 94 | -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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| 95 | \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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| 96 | \end{multline*} |
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| 97 | |
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[1223] | 98 | Since the horizontal scale factors do not depend on the vertical coordinate, |
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| 99 | the last term of the first line and the first term of the last line cancel, while |
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| 100 | the second line reduces to a single vertical derivative, so it becomes: |
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[817] | 101 | |
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[707] | 102 | \begin{multline*} |
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| 103 | =\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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| 104 | \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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| 105 | \end{multline*} |
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| 106 | |
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[1223] | 107 | in other words, the horizontal Laplacian operator in the ($i$,$s$) plane takes the following form : |
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[707] | 108 | |
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| 109 | \begin{equation*} |
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| 110 | D^T = {\frac{1}{e_1\,e_2\,e_3}} |
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[817] | 111 | \left( {{\begin{array}{*{30}c} |
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[707] | 112 | {\left. {\frac{\partial \left( {e_2 e_3 \bullet } \right)}{\partial i}} \right|_s } \hfill \\ |
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| 113 | {\frac{\partial \left( {e_1 e_2 \bullet } \right)}{\partial s}} \hfill \\ |
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| 114 | \end{array}}}\right) |
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| 115 | \cdot \left[ {A^{lT} |
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[817] | 116 | \left( {{\begin{array}{*{30}c} |
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[707] | 117 | {1} \hfill & {-\sigma_1 } \hfill \\ |
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| 118 | {-\sigma_1} \hfill & {\varepsilon_1^2} \hfill \\ |
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| 119 | \end{array} }} \right) |
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| 120 | \cdot |
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[817] | 121 | \left( {{\begin{array}{*{30}c} |
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[707] | 122 | {\frac{1}{e_1 }\;\left. {\frac{\partial \bullet }{\partial i}} \right|_s } \hfill \\ |
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| 123 | {\frac{1}{e_3 }\;\frac{\partial \bullet }{\partial s}} \hfill \\ |
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| 124 | \end{array}}} |
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| 125 | \right) \left( T \right)} \right] |
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| 126 | \end{equation*} |
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[817] | 127 | |
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[707] | 128 | |
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| 129 | % ================================================================ |
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[817] | 130 | % Isopycnal/Vertical 2nd Order Tracer Diffusive Operators |
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[707] | 131 | % ================================================================ |
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[817] | 132 | \section{Iso/diapycnal 2nd Order Tracer Diffusive Operators} |
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[707] | 133 | \label{Apdx_B_2} |
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| 134 | |
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| 135 | |
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[1223] | 136 | The iso/diapycnal diffusive tensor $\textbf {A}_{\textbf I}$ expressed in the ($i$,$j$,$k$) |
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| 137 | curvilinear coordinate system in which the equations of the ocean circulation model are |
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| 138 | formulated, takes the following form \citep{Redi_JPO82}: |
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[707] | 139 | |
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| 140 | \begin{equation*} |
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| 141 | \textbf {A}_{\textbf I} = \frac{A^{lT}}{\left( {1+a_1 ^2+a_2 ^2} \right)} |
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| 142 | \left[ {{\begin{array}{*{20}c} |
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| 143 | {1+a_1 ^2} \hfill & {-a_1 a_2 } \hfill & {-a_1 } \hfill \\ |
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| 144 | {-a_1 a_2 } \hfill & {1+a_2 ^2} \hfill & {-a_2 } \hfill \\ |
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| 145 | {-a_1 } \hfill & {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\ |
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| 146 | \end{array} }} \right] |
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| 147 | \end{equation*} |
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[817] | 148 | where ($a_1$, $a_2$) are the isopycnal slopes in ($\textbf{i}$, $\textbf{j}$) directions: |
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[707] | 149 | \begin{equation*} |
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| 150 | 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] | 151 | \qquad , \qquad |
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[707] | 152 | a_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}} |
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| 153 | \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1} |
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| 154 | \end{equation*} |
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[817] | 155 | |
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[1223] | 156 | In practice, the isopycnal slopes are generally less than $10^{-2}$ in the ocean, so |
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| 157 | $\textbf {A}_{\textbf I}$ can be simplified appreciably \citep{Cox1987}: |
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[707] | 158 | \begin{equation*} |
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[817] | 159 | {\textbf{A}_{\textbf{I}}} \approx A^{lT} |
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[707] | 160 | \left[ {{\begin{array}{*{20}c} |
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| 161 | 1 \hfill & 0 \hfill & {-a_1 } \hfill \\ |
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| 162 | 0 \hfill & 1 \hfill & {-a_2 } \hfill \\ |
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| 163 | {-a_1 } \hfill & {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\ |
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| 164 | \end{array} }} \right] |
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| 165 | \end{equation*} |
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[817] | 166 | |
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[1223] | 167 | The resulting isopycnal operator conserves the quantity and dissipates its square. |
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| 168 | The demonstration of the first property is trivial as \eqref{Apdx_B2} is the divergence |
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| 169 | of fluxes. Let us demonstrate the second one: |
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[707] | 170 | \begin{equation*} |
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| 171 | \iiint\limits_D T\;\nabla .\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv = -\iiint\limits_D \nabla T\;.\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv |
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| 172 | \end{equation*} |
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| 173 | since |
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[817] | 174 | \begin{align*} |
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[707] | 175 | \nabla T\;.\left( {{\rm {\bf A}}_{\rm {\bf I}} \nabla T} |
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[817] | 176 | \right)&=A^{lT}\left[ {\left( {\frac{\partial T}{\partial i}} \right)^2-2a_1 |
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[707] | 177 | \frac{\partial T}{\partial i}\frac{\partial T}{\partial k}+\left( |
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| 178 | {\frac{\partial T}{\partial j}} \right)^2} \right. \\ |
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[817] | 179 | &\qquad \qquad \qquad \quad |
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| 180 | { \left. -\,{2a_2 \frac{\partial T}{\partial j}\frac{\partial T}{\partial |
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[707] | 181 | k}+\left( {a_1 ^2+a_2 ^2} \right)\left( {\frac{\partial T}{\partial k}} |
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| 182 | \right)^2} \right]} \\ |
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[817] | 183 | &=A_h \left[ {\left( {\frac{\partial T}{\partial i}-a_1 \frac{\partial T}{\partial k}} \right)^2+\left( {\frac{\partial T}{\partial j}-a_2 \frac{\partial T}{\partial k}} \right)^2} \right]\quad \geq 0 |
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| 184 | \end{align*} |
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| 185 | the property becomes obvious. |
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[707] | 186 | |
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[1223] | 187 | The resulting diffusion operator in $z$-coordinate has the following form : |
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[817] | 188 | \begin{multline*} \label{Apdx_B_ldfiso} |
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[707] | 189 | D^T=\frac{1}{e_1 e_2 }\left\{ {\;\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]\;} \right.\;\; \\ |
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| 190 | \;\left. {\;\;\;+\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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| 191 | \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} \right)}{e_3 }\frac{\partial T}{\partial k}} \right)} \right]} \\ |
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| 192 | \end{multline*} |
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| 193 | |
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[1223] | 194 | It has to be emphasised that the simplification introduced, leads to a decoupling |
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| 195 | between ($i$,$z$) and ($j$,$z$) planes. The operator has therefore the same |
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| 196 | expression as \eqref{Apdx_B3}, the diffusion operator obtained for geopotential |
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| 197 | diffusion in the $s$-coordinate. |
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[707] | 198 | |
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| 199 | % ================================================================ |
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| 200 | % Lateral/Vertical Momentum Diffusive Operators |
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| 201 | % ================================================================ |
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| 202 | \section{Lateral/Vertical Momentum Diffusive Operators} |
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| 203 | \label{Apdx_B_3} |
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| 204 | |
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[1223] | 205 | The second order momentum diffusion operator (Laplacian) in the $z$-coordinate |
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| 206 | is found by applying \eqref{Eq_PE_lap_vector}, the expression for the Laplacian |
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| 207 | of a vector, to the horizontal velocity vector : |
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[817] | 208 | \begin{align*} |
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| 209 | \Delta {\textbf{U}}_h |
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| 210 | &=\nabla \left( {\nabla \cdot {\textbf{U}}_h } \right)- |
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| 211 | \nabla \times \left( {\nabla \times {\textbf{U}}_h } \right) \\ |
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| 212 | \\ |
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| 213 | &=\left( {{\begin{array}{*{20}c} |
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[707] | 214 | {\frac{1}{e_1 }\frac{\partial \chi }{\partial i}} \hfill \\ |
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| 215 | {\frac{1}{e_2 }\frac{\partial \chi }{\partial j}} \hfill \\ |
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| 216 | {\frac{1}{e_3 }\frac{\partial \chi }{\partial k}} \hfill \\ |
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| 217 | \end{array} }} \right)-\left( {{\begin{array}{*{20}c} |
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| 218 | {\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}-\frac{1}{e_3 |
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| 219 | }\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial |
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| 220 | u}{\partial k}} \right)} \hfill \\ |
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| 221 | {\frac{1}{e_3 }\frac{\partial }{\partial k}\left( {-\frac{1}{e_3 |
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| 222 | }\frac{\partial v}{\partial k}} \right)-\frac{1}{e_1 }\frac{\partial \zeta |
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| 223 | }{\partial i}} \hfill \\ |
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| 224 | {\frac{1}{e_1 e_2 }\left[ {\frac{\partial }{\partial i}\left( {\frac{e_2 |
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| 225 | }{e_3 }\frac{\partial u}{\partial k}} \right)-\frac{\partial }{\partial |
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| 226 | j}\left( {-\frac{e_1 }{e_3 }\frac{\partial v}{\partial k}} \right)} \right]} |
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| 227 | \hfill \\ |
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| 228 | \end{array} }} \right) |
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[817] | 229 | \\ |
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| 230 | \\ |
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| 231 | &=\left( {{\begin{array}{*{20}c} |
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[707] | 232 | {\frac{1}{e_1 }\frac{\partial \chi }{\partial i}-\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}} \\ |
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| 233 | {\frac{1}{e_2 }\frac{\partial \chi }{\partial j}+\frac{1}{e_1 }\frac{\partial \zeta }{\partial i}} \\ |
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| 234 | 0 \\ |
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| 235 | \end{array} }} \right) |
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| 236 | +\frac{1}{e_3 } |
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| 237 | \left( {{\begin{array}{*{20}c} |
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| 238 | {\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial u}{\partial k}} \right)} \\ |
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| 239 | {\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial v}{\partial k}} \right)} \\ |
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| 240 | {\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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| 241 | \end{array} }} \right) |
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[817] | 242 | \end{align*} |
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[1223] | 243 | Using \eqref{Eq_PE_div}, the definition of the horizontal divergence, the third |
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| 244 | componant of the second vector is obviously zero and thus : |
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[707] | 245 | \begin{equation*} |
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| 246 | \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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| 247 | \end{equation*} |
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| 248 | |
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[1223] | 249 | Note that this operator ensures a full separation between the vorticity and horizontal |
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| 250 | divergence fields (see Appendix~\ref{Apdx_C}). It is only equal to a Laplacian |
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| 251 | applied to each component in Cartesian coordinates, not on the sphere. |
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[707] | 252 | |
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[817] | 253 | The horizontal/vertical second order (Laplacian type) operator used to diffuse |
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[1223] | 254 | horizontal momentum in the $z$-coordinate therefore takes the following form : |
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[817] | 255 | \begin{equation} \label{Apdx_B_Lap_U} |
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| 256 | {\textbf{D}}^{\textbf{U}} = |
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| 257 | \nabla _h \left( {A^{lm}\;\chi } \right) |
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| 258 | - \nabla _h \times \left( {A^{lm}\;\zeta \;{\textbf{k}}} \right) |
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| 259 | + \frac{1}{e_3 }\frac{\partial }{\partial k}\left( {\frac{A^{vm}\;}{e_3 } |
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| 260 | \frac{\partial {\rm {\bf U}}_h }{\partial k}} \right) \\ |
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| 261 | \end{equation} |
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[1223] | 262 | that is, in expanded form: |
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[817] | 263 | \begin{align*} |
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| 264 | D^{\textbf{U}}_u |
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| 265 | & = \frac{1}{e_1} \frac{\partial \left( {A^{lm}\chi } \right)}{\partial i} |
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| 266 | -\frac{1}{e_2} \frac{\partial \left( {A^{lm}\zeta } \right)}{\partial j} |
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| 267 | +\frac{1}{e_3} \frac{\partial u}{\partial k} \\ |
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| 268 | D^{\textbf{U}}_v |
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| 269 | & = \frac{1}{e_2 }\frac{\partial \left( {A^{lm}\chi } \right)}{\partial j} |
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| 270 | +\frac{1}{e_1 }\frac{\partial \left( {A^{lm}\zeta } \right)}{\partial i} |
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| 271 | +\frac{1}{e_3} \frac{\partial v}{\partial k} |
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| 272 | \end{align*} |
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[707] | 273 | |
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[1223] | 274 | Note Bene: introducing a rotation in \eqref{Apdx_B_Lap_U} does not lead to a |
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| 275 | useful expression for the iso/diapycnal Laplacian operator in the $z$-coordinate. |
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| 276 | Similarly, we did not found an expression of practical use for the geopotential |
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| 277 | horizontal/vertical Laplacian operator in the $s$-coordinate. Generally, |
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| 278 | \eqref{Apdx_B_Lap_U} is used in both $z$- and $s$-coordinate systems, that is |
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| 279 | a Laplacian diffusion is applied on momentum along the coordinate directions. |
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