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