[10414] | 1 | \documentclass[../main/NEMO_manual]{subfiles} |
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| 2 | |
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[6997] | 3 | \begin{document} |
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[707] | 4 | % ================================================================ |
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[10414] | 5 | % Chapter Appendix B : Diffusive Operators |
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[707] | 6 | % ================================================================ |
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| 7 | \chapter{Appendix B : Diffusive Operators} |
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[9407] | 8 | \label{apdx:B} |
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[10414] | 9 | |
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[707] | 10 | \minitoc |
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| 11 | |
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[2282] | 12 | \newpage |
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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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[9393] | 17 | \section{Horizontal/Vertical $2^{nd}$ order tracer diffusive operators} |
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[9407] | 18 | \label{sec:B_1} |
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[707] | 19 | |
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[3294] | 20 | \subsubsection*{In z-coordinates} |
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[10414] | 21 | |
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[10354] | 22 | In $z$-coordinates, the horizontal/vertical second order tracer diffusion operator is given by: |
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[10414] | 23 | \begin{align} |
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| 24 | \label{apdx:B1} |
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| 25 | &D^T = \frac{1}{e_1 \, e_2} \left[ |
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| 26 | \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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| 27 | \left. |
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| 28 | + \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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| 29 | + \frac{\partial }{\partial z}\left( {A^{vT} \;\frac{\partial T}{\partial z}} \right) |
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[10406] | 30 | \end{align} |
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[707] | 31 | |
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[3294] | 32 | \subsubsection*{In generalized vertical coordinates} |
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[10414] | 33 | |
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[10354] | 34 | In $s$-coordinates, we defined the slopes of $s$-surfaces, $\sigma_1$ and $\sigma_2$ by \autoref{apdx:A_s_slope} and |
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| 35 | the vertical/horizontal ratio of diffusion coefficient by $\epsilon = A^{vT} / A^{lT}$. |
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| 36 | The diffusion operator is given by: |
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[707] | 37 | |
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[10414] | 38 | \begin{equation} |
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| 39 | \label{apdx:B2} |
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| 40 | D^T = \left. \nabla \right|_s \cdot |
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| 41 | \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T \right] \\ |
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| 42 | \;\;\text{where} \;\Re =\left( {{ |
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| 43 | \begin{array}{*{20}c} |
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| 44 | 1 \hfill & 0 \hfill & {-\sigma_1 } \hfill \\ |
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| 45 | 0 \hfill & 1 \hfill & {-\sigma_2 } \hfill \\ |
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| 46 | {-\sigma_1 } \hfill & {-\sigma_2 } \hfill & {\varepsilon +\sigma_1 |
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| 47 | ^2+\sigma_2 ^2} \hfill \\ |
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| 48 | \end{array} |
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| 49 | }} \right) |
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[707] | 50 | \end{equation} |
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[1223] | 51 | or in expanded form: |
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[10414] | 52 | \begin{align*} |
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| 53 | { |
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| 54 | \begin{array}{*{20}l} |
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| 55 | D^T=& \frac{1}{e_1\,e_2\,e_3 }\;\left[ {\ \ \ \ e_2\,e_3\,A^{lT} \;\left. |
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| 56 | {\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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| 57 | &\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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| 58 | &\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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| 59 | \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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| 60 | \end{array} |
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| 61 | } |
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[2282] | 62 | \end{align*} |
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[707] | 63 | |
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[11331] | 64 | \autoref{apdx:B2} is obtained from \autoref{apdx:B1} without any additional assumption. |
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[10354] | 65 | Indeed, for the special case $k=z$ and thus $e_3 =1$, |
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| 66 | we introduce an arbitrary vertical coordinate $s = s (i,j,z)$ as in \autoref{apdx:A} and |
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| 67 | use \autoref{apdx:A_s_slope} and \autoref{apdx:A_s_chain_rule}. |
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| 68 | Since no cross horizontal derivative $\partial _i \partial _j $ appears in \autoref{apdx:B1}, |
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| 69 | the ($i$,$z$) and ($j$,$z$) planes are independent. |
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| 70 | The derivation can then be demonstrated for the ($i$,$z$)~$\to$~($j$,$s$) transformation without |
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| 71 | any loss of generality: |
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[707] | 72 | |
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[10414] | 73 | \begin{align*} |
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| 74 | { |
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| 75 | \begin{array}{*{20}l} |
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| 76 | 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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| 77 | +\frac{\partial }{\partial z}\left( {A^{vT}\;\frac{\partial T}{\partial z}} \right) \\ \\ |
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| 78 | % |
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| 79 | &=\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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| 80 | -\frac{e_1\,\sigma_1 }{e_3 }\frac{\partial T}{\partial s}} \right)} \right)} \right|_s } \right. \\ |
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| 81 | & \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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| 82 | \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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| 83 | % |
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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{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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| 85 | & \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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| 86 | & \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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| 87 | \end{array} |
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| 88 | } \\ |
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| 89 | % |
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| 90 | { |
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| 91 | \begin{array}{*{20}l} |
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| 92 | \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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| 93 | % |
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| 94 | & =\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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| 95 | & \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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[11331] | 96 | & \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 s}} \right)\;\;\;} \right] }\\ |
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[10414] | 97 | \\ |
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| 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 } \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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| 99 | & \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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| 100 | & \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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| 101 | & \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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| 102 | \end{array} |
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| 103 | } \\ |
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| 104 | { |
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| 105 | \begin{array}{*{20}l} |
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| 106 | % |
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[11331] | 107 | \intertext{Using the same remark as just above, it becomes:} |
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[10414] | 108 | % |
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| 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+\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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| 111 | & \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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| 112 | & \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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| 113 | \end{array} |
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| 114 | } \\ |
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| 115 | { |
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| 116 | \begin{array}{*{20}l} |
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| 117 | % |
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| 118 | \intertext{Since the horizontal scale factors do not depend on the vertical coordinate, |
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[11331] | 119 | the two terms on the second line cancel, while |
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| 120 | the third line reduces to a single vertical derivative, so it becomes:} |
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[10414] | 121 | % |
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| 122 | & =\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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| 123 | & \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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| 124 | % |
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[11331] | 125 | \intertext{In other words, the horizontal/vertical Laplacian operator in the ($i$,$s$) plane takes the following form:} |
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[10414] | 126 | \end{array} |
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[11331] | 127 | } \\ |
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[10414] | 128 | % |
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| 129 | {\frac{1}{e_1\,e_2\,e_3}} |
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| 130 | \left( {{ |
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| 131 | \begin{array}{*{30}c} |
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| 132 | {\left. {\frac{\partial \left( {e_2 e_3 \bullet } \right)}{\partial i}} \right|_s } \hfill \\ |
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| 133 | {\frac{\partial \left( {e_1 e_2 \bullet } \right)}{\partial s}} \hfill \\ |
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| 134 | \end{array}}} |
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| 135 | \right) |
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| 136 | \cdot \left[ {A^{lT} |
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| 137 | \left( {{ |
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| 138 | \begin{array}{*{30}c} |
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| 139 | {1} \hfill & {-\sigma_1 } \hfill \\ |
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| 140 | {-\sigma_1} \hfill & {\varepsilon + \sigma_1^2} \hfill \\ |
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| 141 | \end{array} |
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| 142 | }} \right) |
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| 143 | \cdot |
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| 144 | \left( {{ |
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| 145 | \begin{array}{*{30}c} |
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| 146 | {\frac{1}{e_1 }\;\left. {\frac{\partial \bullet }{\partial i}} \right|_s } \hfill \\ |
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| 147 | {\frac{1}{e_3 }\;\frac{\partial \bullet }{\partial s}} \hfill \\ |
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| 148 | \end{array} |
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| 149 | }} \right) \left( T \right)} \right] |
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[2282] | 150 | \end{align*} |
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[10414] | 151 | %\addtocounter{equation}{-2} |
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[707] | 152 | |
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| 153 | % ================================================================ |
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[817] | 154 | % Isopycnal/Vertical 2nd Order Tracer Diffusive Operators |
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[707] | 155 | % ================================================================ |
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[9393] | 156 | \section{Iso/Diapycnal $2^{nd}$ order tracer diffusive operators} |
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[9407] | 157 | \label{sec:B_2} |
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[707] | 158 | |
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[3294] | 159 | \subsubsection*{In z-coordinates} |
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[707] | 160 | |
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[10354] | 161 | The iso/diapycnal diffusive tensor $\textbf {A}_{\textbf I}$ expressed in |
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| 162 | the ($i$,$j$,$k$) curvilinear coordinate system in which |
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| 163 | the equations of the ocean circulation model are formulated, |
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[11123] | 164 | takes the following form \citep{redi_JPO82}: |
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[707] | 165 | |
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[10414] | 166 | \begin{equation} |
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| 167 | \label{apdx:B3} |
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| 168 | \textbf {A}_{\textbf I} = \frac{A^{lT}}{\left( {1+a_1 ^2+a_2 ^2} \right)} |
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| 169 | \left[ {{ |
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| 170 | \begin{array}{*{20}c} |
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[11331] | 171 | {1+a_2 ^2 +\varepsilon a_1 ^2} \hfill & {-a_1 a_2 (1-\varepsilon)} \hfill & {-a_1 (1-\varepsilon) } \hfill \\ |
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| 172 | {-a_1 a_2 (1-\varepsilon) } \hfill & {1+a_1 ^2 +\varepsilon a_2 ^2} \hfill & {-a_2 (1-\varepsilon)} \hfill \\ |
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| 173 | {-a_1 (1-\varepsilon)} \hfill & {-a_2 (1-\varepsilon)} \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\ |
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[10414] | 174 | \end{array} |
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| 175 | }} \right] |
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[3294] | 176 | \end{equation} |
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[10354] | 177 | where ($a_1$, $a_2$) are the isopycnal slopes in ($\textbf{i}$, $\textbf{j}$) directions, relative to geopotentials: |
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[10406] | 178 | \[ |
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[10414] | 179 | a_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1} |
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| 180 | \qquad , \qquad |
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| 181 | a_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}} |
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| 182 | \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1} |
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[10406] | 183 | \] |
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[11331] | 184 | and, as before, $\epsilon = A^{vT} / A^{lT}$. |
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[817] | 185 | |
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[11331] | 186 | In practice, $\epsilon$ is small and isopycnal slopes are generally less than $10^{-2}$ in the ocean, |
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| 187 | so $\textbf {A}_{\textbf I}$ can be simplified appreciably \citep{cox_OM87}. Keeping leading order terms\footnote{Apart from the (1,0) |
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| 188 | and (0,1) elements which are set to zero. See \citet{griffies_bk04}, section 14.1.4.1 for a discussion of this point.}: |
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[10414] | 189 | \begin{subequations} |
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| 190 | \label{apdx:B4} |
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| 191 | \begin{equation} |
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| 192 | \label{apdx:B4a} |
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| 193 | {\textbf{A}_{\textbf{I}}} \approx A^{lT}\;\Re\;\text{where} \;\Re = |
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| 194 | \left[ {{ |
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| 195 | \begin{array}{*{20}c} |
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| 196 | 1 \hfill & 0 \hfill & {-a_1 } \hfill \\ |
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| 197 | 0 \hfill & 1 \hfill & {-a_2 } \hfill \\ |
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| 198 | {-a_1 } \hfill & {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\ |
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| 199 | \end{array} |
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| 200 | }} \right], |
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| 201 | \end{equation} |
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| 202 | and the iso/dianeutral diffusive operator in $z$-coordinates is then |
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| 203 | \begin{equation} |
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| 204 | \label{apdx:B4b} |
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| 205 | D^T = \left. \nabla \right|_z \cdot |
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| 206 | \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_z T \right]. \\ |
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| 207 | \end{equation} |
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[3294] | 208 | \end{subequations} |
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[817] | 209 | |
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[10354] | 210 | Physically, the full tensor \autoref{apdx:B3} represents strong isoneutral diffusion on a plane parallel to |
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| 211 | the isoneutral surface and weak dianeutral diffusion perpendicular to this plane. |
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| 212 | However, |
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| 213 | the approximate `weak-slope' tensor \autoref{apdx:B4a} represents strong diffusion along the isoneutral surface, |
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| 214 | with weak \emph{vertical} diffusion -- the principal axes of the tensor are no longer orthogonal. |
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| 215 | This simplification also decouples the ($i$,$z$) and ($j$,$z$) planes of the tensor. |
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| 216 | The weak-slope operator therefore takes the same form, \autoref{apdx:B4}, as \autoref{apdx:B2}, |
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| 217 | the diffusion operator for geopotential diffusion written in non-orthogonal $i,j,s$-coordinates. |
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| 218 | Written out explicitly, |
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[3294] | 219 | |
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[10414] | 220 | \begin{multline} |
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| 221 | \label{apdx:B_ldfiso} |
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| 222 | D^T=\frac{1}{e_1 e_2 }\left\{ |
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| 223 | {\;\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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| 224 | {+\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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| 225 | \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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[3294] | 226 | \end{multline} |
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| 227 | |
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[9407] | 228 | The isopycnal diffusion operator \autoref{apdx:B4}, |
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[10354] | 229 | \autoref{apdx:B_ldfiso} conserves tracer quantity and dissipates its square. |
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| 230 | The demonstration of the first property is trivial as \autoref{apdx:B4} is the divergence of fluxes. |
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| 231 | Let us demonstrate the second one: |
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[10406] | 232 | \[ |
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[10414] | 233 | \iiint\limits_D T\;\nabla .\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv |
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| 234 | = -\iiint\limits_D \nabla T\;.\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv, |
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[10406] | 235 | \] |
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[3294] | 236 | and since |
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[10414] | 237 | \begin{align*} |
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| 238 | { |
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| 239 | \begin{array}{*{20}l} |
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[11151] | 240 | \nabla T\;.\left( {{\mathrm {\mathbf A}}_{\mathrm {\mathbf I}} \nabla T} |
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[10414] | 241 | \right)&=A^{lT}\left[ {\left( {\frac{\partial T}{\partial i}} \right)^2-2a_1 |
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| 242 | \frac{\partial T}{\partial i}\frac{\partial T}{\partial k}+\left( |
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| 243 | {\frac{\partial T}{\partial j}} \right)^2} \right. \\ |
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| 244 | &\qquad \qquad \qquad |
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| 245 | { \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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| 246 | &=A_h \left[ {\left( {\frac{\partial T}{\partial i}-a_1 \frac{\partial |
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| 247 | T}{\partial k}} \right)^2+\left( {\frac{\partial T}{\partial |
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| 248 | j}-a_2 \frac{\partial T}{\partial k}} \right)^2} |
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| 249 | +\varepsilon \left(\frac{\partial T}{\partial k}\right) ^2\right] \\ |
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| 250 | & \geq 0 |
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| 251 | \end{array} |
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| 252 | } |
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[817] | 253 | \end{align*} |
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[10414] | 254 | %\addtocounter{equation}{-1} |
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[10354] | 255 | the property becomes obvious. |
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[707] | 256 | |
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[3294] | 257 | \subsubsection*{In generalized vertical coordinates} |
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[707] | 258 | |
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[10354] | 259 | Because the weak-slope operator \autoref{apdx:B4}, |
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| 260 | \autoref{apdx:B_ldfiso} is decoupled in the ($i$,$z$) and ($j$,$z$) planes, |
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| 261 | it may be transformed into generalized $s$-coordinates in the same way as |
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| 262 | \autoref{sec:B_1} was transformed into \autoref{sec:B_2}. |
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| 263 | The resulting operator then takes the simple form |
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[707] | 264 | |
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[10414] | 265 | \begin{equation} |
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| 266 | \label{apdx:B_ldfiso_s} |
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| 267 | D^T = \left. \nabla \right|_s \cdot |
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| 268 | \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T \right] \\ |
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| 269 | \;\;\text{where} \;\Re =\left( {{ |
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| 270 | \begin{array}{*{20}c} |
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| 271 | 1 \hfill & 0 \hfill & {-r _1 } \hfill \\ |
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| 272 | 0 \hfill & 1 \hfill & {-r _2 } \hfill \\ |
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| 273 | {-r _1 } \hfill & {-r _2 } \hfill & {\varepsilon +r _1 |
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| 274 | ^2+r _2 ^2} \hfill \\ |
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| 275 | \end{array} |
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| 276 | }} \right), |
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[3294] | 277 | \end{equation} |
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| 278 | |
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[10354] | 279 | where ($r_1$, $r_2$) are the isopycnal slopes in ($\textbf{i}$, $\textbf{j}$) directions, |
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| 280 | relative to $s$-coordinate surfaces: |
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[10406] | 281 | \[ |
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[10414] | 282 | r_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1} |
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| 283 | \qquad , \qquad |
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| 284 | r_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}} |
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| 285 | \right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1}. |
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[10406] | 286 | \] |
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[3294] | 287 | |
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[11331] | 288 | To prove \autoref{apdx:B_ldfiso_s} by direct re-expression of \autoref{apdx:B_ldfiso} is straightforward, but laborious. |
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[10354] | 289 | An easier way is first to note (by reversing the derivation of \autoref{sec:B_2} from \autoref{sec:B_1} ) that |
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| 290 | the weak-slope operator may be \emph{exactly} reexpressed in non-orthogonal $i,j,\rho$-coordinates as |
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[3294] | 291 | |
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[10414] | 292 | \begin{equation} |
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| 293 | \label{apdx:B5} |
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| 294 | D^T = \left. \nabla \right|_\rho \cdot |
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| 295 | \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_\rho T \right] \\ |
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| 296 | \;\;\text{where} \;\Re =\left( {{ |
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| 297 | \begin{array}{*{20}c} |
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| 298 | 1 \hfill & 0 \hfill &0 \hfill \\ |
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| 299 | 0 \hfill & 1 \hfill & 0 \hfill \\ |
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| 300 | 0 \hfill & 0 \hfill & \varepsilon \hfill \\ |
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| 301 | \end{array} |
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| 302 | }} \right). |
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[3294] | 303 | \end{equation} |
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[10354] | 304 | Then direct transformation from $i,j,\rho$-coordinates to $i,j,s$-coordinates gives |
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| 305 | \autoref{apdx:B_ldfiso_s} immediately. |
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[3294] | 306 | |
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[10354] | 307 | Note that the weak-slope approximation is only made in transforming from |
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| 308 | the (rotated,orthogonal) isoneutral axes to the non-orthogonal $i,j,\rho$-coordinates. |
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| 309 | The further transformation into $i,j,s$-coordinates is exact, whatever the steepness of the $s$-surfaces, |
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[11331] | 310 | in the same way as the transformation of horizontal/vertical Laplacian diffusion in $z$-coordinates in |
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[9407] | 311 | \autoref{sec:B_1} onto $s$-coordinates is exact, however steep the $s$-surfaces. |
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[3294] | 312 | |
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| 313 | |
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[707] | 314 | % ================================================================ |
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| 315 | % Lateral/Vertical Momentum Diffusive Operators |
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| 316 | % ================================================================ |
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[9393] | 317 | \section{Lateral/Vertical momentum diffusive operators} |
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[9407] | 318 | \label{sec:B_3} |
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[707] | 319 | |
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[11331] | 320 | The second order momentum diffusion operator (Laplacian) in $z$-coordinates is found by |
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[10354] | 321 | applying \autoref{eq:PE_lap_vector}, the expression for the Laplacian of a vector, |
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| 322 | to the horizontal velocity vector: |
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[817] | 323 | \begin{align*} |
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[10414] | 324 | \Delta {\textbf{U}}_h |
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| 325 | &=\nabla \left( {\nabla \cdot {\textbf{U}}_h } \right)- |
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| 326 | \nabla \times \left( {\nabla \times {\textbf{U}}_h } \right) \\ \\ |
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| 327 | &=\left( {{ |
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| 328 | \begin{array}{*{20}c} |
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| 329 | {\frac{1}{e_1 }\frac{\partial \chi }{\partial i}} \hfill \\ |
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| 330 | {\frac{1}{e_2 }\frac{\partial \chi }{\partial j}} \hfill \\ |
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| 331 | {\frac{1}{e_3 }\frac{\partial \chi }{\partial k}} \hfill \\ |
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| 332 | \end{array} |
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| 333 | }} \right) |
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| 334 | -\left( {{ |
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| 335 | \begin{array}{*{20}c} |
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| 336 | {\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}-\frac{1}{e_3 |
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| 337 | }\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial |
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| 338 | u}{\partial k}} \right)} \hfill \\ |
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| 339 | {\frac{1}{e_3 }\frac{\partial }{\partial k}\left( {-\frac{1}{e_3 |
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| 340 | }\frac{\partial v}{\partial k}} \right)-\frac{1}{e_1 }\frac{\partial \zeta |
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| 341 | }{\partial i}} \hfill \\ |
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| 342 | {\frac{1}{e_1 e_2 }\left[ {\frac{\partial }{\partial i}\left( {\frac{e_2 |
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| 343 | }{e_3 }\frac{\partial u}{\partial k}} \right)-\frac{\partial }{\partial |
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| 344 | j}\left( {-\frac{e_1 }{e_3 }\frac{\partial v}{\partial k}} \right)} \right]} |
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| 345 | \hfill \\ |
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| 346 | \end{array} |
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| 347 | }} \right) \\ \\ |
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| 348 | &=\left( {{ |
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| 349 | \begin{array}{*{20}c} |
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| 350 | {\frac{1}{e_1 }\frac{\partial \chi }{\partial i}-\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}} \\ |
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| 351 | {\frac{1}{e_2 }\frac{\partial \chi }{\partial j}+\frac{1}{e_1 }\frac{\partial \zeta }{\partial i}} \\ |
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| 352 | 0 \\ |
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| 353 | \end{array} |
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| 354 | }} \right) |
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| 355 | +\frac{1}{e_3 } |
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| 356 | \left( {{ |
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| 357 | \begin{array}{*{20}c} |
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| 358 | {\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial u}{\partial k}} \right)} \\ |
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| 359 | {\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial v}{\partial k}} \right)} \\ |
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| 360 | {\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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| 361 | \end{array} |
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| 362 | }} \right) |
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[817] | 363 | \end{align*} |
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[10354] | 364 | Using \autoref{eq:PE_div}, the definition of the horizontal divergence, |
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[11331] | 365 | the third component of the second vector is obviously zero and thus : |
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[10406] | 366 | \[ |
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[10414] | 367 | \Delta {\textbf{U}}_h = \nabla _h \left( \chi \right) - \nabla _h \times \left( \zeta \right) + \frac {1}{e_3 } \frac {\partial }{\partial k} \left( {\frac {1}{e_3 } \frac{\partial {\textbf{ U}}_h }{\partial k}} \right) |
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[10406] | 368 | \] |
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[707] | 369 | |
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[10354] | 370 | Note that this operator ensures a full separation between |
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| 371 | the vorticity and horizontal divergence fields (see \autoref{apdx:C}). |
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| 372 | It is only equal to a Laplacian applied to each component in Cartesian coordinates, not on the sphere. |
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[707] | 373 | |
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[10354] | 374 | The horizontal/vertical second order (Laplacian type) operator used to diffuse horizontal momentum in |
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| 375 | the $z$-coordinate therefore takes the following form: |
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[10414] | 376 | \begin{equation} |
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| 377 | \label{apdx:B_Lap_U} |
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| 378 | { |
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| 379 | \textbf{D}}^{\textbf{U}} = |
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| 380 | \nabla _h \left( {A^{lm}\;\chi } \right) |
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| 381 | - \nabla _h \times \left( {A^{lm}\;\zeta \;{\textbf{k}}} \right) |
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| 382 | + \frac{1}{e_3 }\frac{\partial }{\partial k}\left( {\frac{A^{vm}\;}{e_3 } |
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[11151] | 383 | \frac{\partial {\mathrm {\mathbf U}}_h }{\partial k}} \right) \\ |
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[817] | 384 | \end{equation} |
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[1223] | 385 | that is, in expanded form: |
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[817] | 386 | \begin{align*} |
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[10414] | 387 | D^{\textbf{U}}_u |
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| 388 | & = \frac{1}{e_1} \frac{\partial \left( {A^{lm}\chi } \right)}{\partial i} |
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| 389 | -\frac{1}{e_2} \frac{\partial \left( {A^{lm}\zeta } \right)}{\partial j} |
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[11331] | 390 | +\frac{1}{e_3} \frac{\partial}{\partial k} \left( \frac{A^{vm}}{e_3} \frac{\partial u}{\partial k} \right) \\ |
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[10414] | 391 | D^{\textbf{U}}_v |
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| 392 | & = \frac{1}{e_2 }\frac{\partial \left( {A^{lm}\chi } \right)}{\partial j} |
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| 393 | +\frac{1}{e_1 }\frac{\partial \left( {A^{lm}\zeta } \right)}{\partial i} |
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[11331] | 394 | +\frac{1}{e_3} \frac{\partial}{\partial k} \left( \frac{A^{vm}}{e_3} \frac{\partial v}{\partial k} \right) |
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[817] | 395 | \end{align*} |
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[707] | 396 | |
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[10354] | 397 | Note Bene: introducing a rotation in \autoref{apdx:B_Lap_U} does not lead to |
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| 398 | a useful expression for the iso/diapycnal Laplacian operator in the $z$-coordinate. |
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| 399 | Similarly, we did not found an expression of practical use for |
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| 400 | the geopotential horizontal/vertical Laplacian operator in the $s$-coordinate. |
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| 401 | Generally, \autoref{apdx:B_Lap_U} is used in both $z$- and $s$-coordinate systems, |
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| 402 | that is a Laplacian diffusion is applied on momentum along the coordinate directions. |
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[10414] | 403 | |
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| 404 | \biblio |
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| 405 | |
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[10442] | 406 | \pindex |
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| 407 | |
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[6997] | 408 | \end{document} |
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