[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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[11543] | 7 | \chapter{Diffusive Operators} |
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| 8 | \label{apdx:DIFFOPERS} |
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[10414] | 9 | |
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[11435] | 10 | \chaptertoc |
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[707] | 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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[11543] | 18 | \label{sec:DIFFOPERS_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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[11543] | 24 | \label{eq:DIFFOPERS_1} |
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[10414] | 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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[11543] | 34 | In $s$-coordinates, we defined the slopes of $s$-surfaces, $\sigma_1$ and $\sigma_2$ by \autoref{eq:SCOORD_s_slope} and |
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[10354] | 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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[11543] | 39 | \label{eq:DIFFOPERS_2} |
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[10414] | 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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[11335] | 55 | D^T= \frac{1}{e_1\,e_2\,e_3 } & \left\{ \quad \quad \frac{\partial }{\partial i} \left. \left[ e_2\,e_3 \, A^{lT} |
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[11543] | 56 | \left( \ \frac{1}{e_1}\; \left. \frac{\partial T}{\partial i} \right|_s |
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[11335] | 57 | -\frac{\sigma_1 }{e_3 } \; \frac{\partial T}{\partial s} \right) \right] \right|_s \right. \\ |
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| 58 | & \quad \ + \ \left. \frac{\partial }{\partial j} \left. \left[ e_1\,e_3 \, A^{lT} |
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[11543] | 59 | \left( \ \frac{1}{e_2 }\; \left. \frac{\partial T}{\partial j} \right|_s |
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[11335] | 60 | -\frac{\sigma_2 }{e_3 } \; \frac{\partial T}{\partial s} \right) \right] \right|_s \right. \\ |
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[11543] | 61 | & \quad \ + \ \left. e_1\,e_2\, \frac{\partial }{\partial s} \left[ A^{lT} \; \left( |
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| 62 | -\frac{\sigma_1 }{e_1 } \; \left. \frac{\partial T}{\partial i} \right|_s |
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| 63 | -\frac{\sigma_2 }{e_2 } \; \left. \frac{\partial T}{\partial j} \right|_s |
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[11335] | 64 | +\left( \varepsilon +\sigma_1^2+\sigma_2 ^2 \right) \; \frac{1}{e_3 } \; \frac{\partial T}{\partial s} \right) \; \right] \; \right\} . |
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[10414] | 65 | \end{array} |
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| 66 | } |
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[2282] | 67 | \end{align*} |
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[707] | 68 | |
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[11543] | 69 | \autoref{eq:DIFFOPERS_2} is obtained from \autoref{eq:DIFFOPERS_1} without any additional assumption. |
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[10354] | 70 | Indeed, for the special case $k=z$ and thus $e_3 =1$, |
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[11543] | 71 | we introduce an arbitrary vertical coordinate $s = s (i,j,z)$ as in \autoref{apdx:SCOORD} and |
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[11558] | 72 | use \autoref{eq:SCOORD_s_slope} and \autoref{eq:SCOORD_s_chain_rule1}. |
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[11543] | 73 | Since no cross horizontal derivative $\partial _i \partial _j $ appears in \autoref{eq:DIFFOPERS_1}, |
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[10354] | 74 | the ($i$,$z$) and ($j$,$z$) planes are independent. |
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| 75 | The derivation can then be demonstrated for the ($i$,$z$)~$\to$~($j$,$s$) transformation without |
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| 76 | any loss of generality: |
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[707] | 77 | |
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[10414] | 78 | \begin{align*} |
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| 79 | { |
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| 80 | \begin{array}{*{20}l} |
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| 81 | 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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| 82 | +\frac{\partial }{\partial z}\left( {A^{vT}\;\frac{\partial T}{\partial z}} \right) \\ \\ |
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| 83 | % |
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| 84 | &=\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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| 85 | -\frac{e_1\,\sigma_1 }{e_3 }\frac{\partial T}{\partial s}} \right)} \right)} \right|_s } \right. \\ |
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| 86 | & \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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| 87 | \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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| 88 | % |
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| 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{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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| 90 | & \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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| 91 | & \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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| 92 | \end{array} |
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| 93 | } \\ |
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| 94 | % |
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| 95 | { |
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| 96 | \begin{array}{*{20}l} |
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[11335] | 97 | \intertext{Noting that $\frac{1}{e_1} \left. \frac{\partial e_3 }{\partial i} \right|_s = \frac{\partial \sigma_1 }{\partial s}$, this becomes:} |
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[10414] | 98 | % |
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[11335] | 99 | D^T & =\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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[10414] | 100 | & \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] | 101 | & \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] | 102 | \\ |
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| 103 | &=\frac{1}{e_1 \,e_2 \,e_3 } \left[ {\left. {\;\;\;\frac{\partial }{\partial i} \left( {\frac{e_2 \,e_3 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \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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| 104 | & \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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| 105 | & \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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[11335] | 106 | & \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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[10414] | 107 | \end{array} |
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| 108 | } \\ |
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| 109 | { |
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| 110 | \begin{array}{*{20}l} |
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| 111 | % |
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[11335] | 112 | \intertext{Using the same remark as just above, $D^T$ becomes:} |
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[10414] | 113 | % |
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[11335] | 114 | D^T &= \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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[10414] | 115 | & \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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| 116 | & \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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[11335] | 117 | & \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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[10414] | 118 | \end{array} |
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| 119 | } \\ |
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| 120 | { |
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| 121 | \begin{array}{*{20}l} |
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| 122 | % |
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| 123 | \intertext{Since the horizontal scale factors do not depend on the vertical coordinate, |
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[11331] | 124 | the two terms on the second line cancel, while |
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| 125 | the third line reduces to a single vertical derivative, so it becomes:} |
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[10414] | 126 | % |
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[11335] | 127 | D^T & =\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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[10414] | 128 | & \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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| 129 | % |
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[11331] | 130 | \intertext{In other words, the horizontal/vertical Laplacian operator in the ($i$,$s$) plane takes the following form:} |
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[10414] | 131 | \end{array} |
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[11335] | 132 | } \\ |
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[10414] | 133 | % |
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| 134 | {\frac{1}{e_1\,e_2\,e_3}} |
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| 135 | \left( {{ |
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| 136 | \begin{array}{*{30}c} |
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| 137 | {\left. {\frac{\partial \left( {e_2 e_3 \bullet } \right)}{\partial i}} \right|_s } \hfill \\ |
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| 138 | {\frac{\partial \left( {e_1 e_2 \bullet } \right)}{\partial s}} \hfill \\ |
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| 139 | \end{array}}} |
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| 140 | \right) |
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| 141 | \cdot \left[ {A^{lT} |
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| 142 | \left( {{ |
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| 143 | \begin{array}{*{30}c} |
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| 144 | {1} \hfill & {-\sigma_1 } \hfill \\ |
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| 145 | {-\sigma_1} \hfill & {\varepsilon + \sigma_1^2} \hfill \\ |
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| 146 | \end{array} |
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| 147 | }} \right) |
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| 148 | \cdot |
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| 149 | \left( {{ |
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| 150 | \begin{array}{*{30}c} |
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| 151 | {\frac{1}{e_1 }\;\left. {\frac{\partial \bullet }{\partial i}} \right|_s } \hfill \\ |
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| 152 | {\frac{1}{e_3 }\;\frac{\partial \bullet }{\partial s}} \hfill \\ |
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| 153 | \end{array} |
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| 154 | }} \right) \left( T \right)} \right] |
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[2282] | 155 | \end{align*} |
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[10414] | 156 | %\addtocounter{equation}{-2} |
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[707] | 157 | |
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| 158 | % ================================================================ |
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[817] | 159 | % Isopycnal/Vertical 2nd Order Tracer Diffusive Operators |
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[707] | 160 | % ================================================================ |
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[9393] | 161 | \section{Iso/Diapycnal $2^{nd}$ order tracer diffusive operators} |
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[11543] | 162 | \label{sec:DIFFOPERS_2} |
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[707] | 163 | |
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[3294] | 164 | \subsubsection*{In z-coordinates} |
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[707] | 165 | |
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[10354] | 166 | The iso/diapycnal diffusive tensor $\textbf {A}_{\textbf I}$ expressed in |
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| 167 | the ($i$,$j$,$k$) curvilinear coordinate system in which |
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| 168 | the equations of the ocean circulation model are formulated, |
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[11123] | 169 | takes the following form \citep{redi_JPO82}: |
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[707] | 170 | |
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[10414] | 171 | \begin{equation} |
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[11543] | 172 | \label{eq:DIFFOPERS_3} |
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[10414] | 173 | \textbf {A}_{\textbf I} = \frac{A^{lT}}{\left( {1+a_1 ^2+a_2 ^2} \right)} |
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| 174 | \left[ {{ |
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| 175 | \begin{array}{*{20}c} |
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[11331] | 176 | {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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| 177 | {-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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| 178 | {-a_1 (1-\varepsilon)} \hfill & {-a_2 (1-\varepsilon)} \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\ |
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[10414] | 179 | \end{array} |
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| 180 | }} \right] |
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[3294] | 181 | \end{equation} |
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[11351] | 182 | where ($a_1$, $a_2$) are $(-1) \times$ the isopycnal slopes in |
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| 183 | ($\textbf{i}$, $\textbf{j}$) directions, relative to geopotentials (or |
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| 184 | equivalently the slopes of the geopotential surfaces in the isopycnal |
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| 185 | coordinate framework): |
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[10406] | 186 | \[ |
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[10414] | 187 | a_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1} |
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| 188 | \qquad , \qquad |
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| 189 | a_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}} |
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| 190 | \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1} |
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[10406] | 191 | \] |
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[11331] | 192 | and, as before, $\epsilon = A^{vT} / A^{lT}$. |
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[817] | 193 | |
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[11331] | 194 | In practice, $\epsilon$ is small and isopycnal slopes are generally less than $10^{-2}$ in the ocean, |
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[11543] | 195 | 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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[11331] | 196 | 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] | 197 | \begin{subequations} |
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[11543] | 198 | \label{eq:DIFFOPERS_4} |
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[10414] | 199 | \begin{equation} |
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[11543] | 200 | \label{eq:DIFFOPERS_4a} |
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[10414] | 201 | {\textbf{A}_{\textbf{I}}} \approx A^{lT}\;\Re\;\text{where} \;\Re = |
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| 202 | \left[ {{ |
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| 203 | \begin{array}{*{20}c} |
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| 204 | 1 \hfill & 0 \hfill & {-a_1 } \hfill \\ |
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| 205 | 0 \hfill & 1 \hfill & {-a_2 } \hfill \\ |
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| 206 | {-a_1 } \hfill & {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\ |
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| 207 | \end{array} |
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| 208 | }} \right], |
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| 209 | \end{equation} |
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| 210 | and the iso/dianeutral diffusive operator in $z$-coordinates is then |
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| 211 | \begin{equation} |
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[11543] | 212 | \label{eq:DIFFOPERS_4b} |
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[10414] | 213 | D^T = \left. \nabla \right|_z \cdot |
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| 214 | \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_z T \right]. \\ |
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| 215 | \end{equation} |
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[3294] | 216 | \end{subequations} |
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[817] | 217 | |
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[11543] | 218 | Physically, the full tensor \autoref{eq:DIFFOPERS_3} represents strong isoneutral diffusion on a plane parallel to |
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[10354] | 219 | the isoneutral surface and weak dianeutral diffusion perpendicular to this plane. |
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| 220 | However, |
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[11543] | 221 | the approximate `weak-slope' tensor \autoref{eq:DIFFOPERS_4a} represents strong diffusion along the isoneutral surface, |
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[10354] | 222 | with weak \emph{vertical} diffusion -- the principal axes of the tensor are no longer orthogonal. |
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| 223 | This simplification also decouples the ($i$,$z$) and ($j$,$z$) planes of the tensor. |
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[11543] | 224 | The weak-slope operator therefore takes the same form, \autoref{eq:DIFFOPERS_4}, as \autoref{eq:DIFFOPERS_2}, |
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[10354] | 225 | the diffusion operator for geopotential diffusion written in non-orthogonal $i,j,s$-coordinates. |
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| 226 | Written out explicitly, |
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[3294] | 227 | |
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[10414] | 228 | \begin{multline} |
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[11543] | 229 | \label{eq:DIFFOPERS_ldfiso} |
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[10414] | 230 | D^T=\frac{1}{e_1 e_2 }\left\{ |
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| 231 | {\;\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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| 232 | {+\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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| 233 | \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] | 234 | \end{multline} |
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| 235 | |
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[11543] | 236 | The isopycnal diffusion operator \autoref{eq:DIFFOPERS_4}, |
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| 237 | \autoref{eq:DIFFOPERS_ldfiso} conserves tracer quantity and dissipates its square. |
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| 238 | As \autoref{eq:DIFFOPERS_4} is the divergence of a flux, the demonstration of the first property is trivial, providing that the flux normal to the boundary is zero |
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[11335] | 239 | (as it is when $A_h$ is zero at the boundary). Let us demonstrate the second one: |
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[10406] | 240 | \[ |
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[10414] | 241 | \iiint\limits_D T\;\nabla .\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv |
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| 242 | = -\iiint\limits_D \nabla T\;.\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv, |
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[10406] | 243 | \] |
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[3294] | 244 | and since |
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[10414] | 245 | \begin{align*} |
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| 246 | { |
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| 247 | \begin{array}{*{20}l} |
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[11151] | 248 | \nabla T\;.\left( {{\mathrm {\mathbf A}}_{\mathrm {\mathbf I}} \nabla T} |
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[10414] | 249 | \right)&=A^{lT}\left[ {\left( {\frac{\partial T}{\partial i}} \right)^2-2a_1 |
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| 250 | \frac{\partial T}{\partial i}\frac{\partial T}{\partial k}+\left( |
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| 251 | {\frac{\partial T}{\partial j}} \right)^2} \right. \\ |
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| 252 | &\qquad \qquad \qquad |
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| 253 | { \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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| 254 | &=A_h \left[ {\left( {\frac{\partial T}{\partial i}-a_1 \frac{\partial |
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| 255 | T}{\partial k}} \right)^2+\left( {\frac{\partial T}{\partial |
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| 256 | j}-a_2 \frac{\partial T}{\partial k}} \right)^2} |
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| 257 | +\varepsilon \left(\frac{\partial T}{\partial k}\right) ^2\right] \\ |
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[11543] | 258 | & \geq 0 . |
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[10414] | 259 | \end{array} |
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| 260 | } |
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[817] | 261 | \end{align*} |
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[10414] | 262 | %\addtocounter{equation}{-1} |
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[10354] | 263 | the property becomes obvious. |
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[707] | 264 | |
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[3294] | 265 | \subsubsection*{In generalized vertical coordinates} |
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[707] | 266 | |
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[11543] | 267 | Because the weak-slope operator \autoref{eq:DIFFOPERS_4}, |
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| 268 | \autoref{eq:DIFFOPERS_ldfiso} is decoupled in the ($i$,$z$) and ($j$,$z$) planes, |
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[10354] | 269 | it may be transformed into generalized $s$-coordinates in the same way as |
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[11543] | 270 | \autoref{sec:DIFFOPERS_1} was transformed into \autoref{sec:DIFFOPERS_2}. |
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[10354] | 271 | The resulting operator then takes the simple form |
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[707] | 272 | |
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[10414] | 273 | \begin{equation} |
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[11543] | 274 | \label{eq:DIFFOPERS_ldfiso_s} |
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[10414] | 275 | D^T = \left. \nabla \right|_s \cdot |
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| 276 | \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T \right] \\ |
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| 277 | \;\;\text{where} \;\Re =\left( {{ |
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| 278 | \begin{array}{*{20}c} |
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| 279 | 1 \hfill & 0 \hfill & {-r _1 } \hfill \\ |
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| 280 | 0 \hfill & 1 \hfill & {-r _2 } \hfill \\ |
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| 281 | {-r _1 } \hfill & {-r _2 } \hfill & {\varepsilon +r _1 |
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| 282 | ^2+r _2 ^2} \hfill \\ |
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| 283 | \end{array} |
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| 284 | }} \right), |
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[3294] | 285 | \end{equation} |
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| 286 | |
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[11351] | 287 | where ($r_1$, $r_2$) are $(-1)\times$ the isopycnal slopes in ($\textbf{i}$, $\textbf{j}$) directions, |
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| 288 | relative to $s$-coordinate surfaces (or equivalently the slopes of the |
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| 289 | $s$-coordinate surfaces in the isopycnal coordinate framework): |
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[10406] | 290 | \[ |
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[10414] | 291 | r_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1} |
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| 292 | \qquad , \qquad |
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| 293 | r_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}} |
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| 294 | \right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1}. |
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[10406] | 295 | \] |
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[3294] | 296 | |
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[11543] | 297 | To prove \autoref{eq:DIFFOPERS_ldfiso_s} by direct re-expression of \autoref{eq:DIFFOPERS_ldfiso} is straightforward, but laborious. |
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| 298 | An easier way is first to note (by reversing the derivation of \autoref{sec:DIFFOPERS_2} from \autoref{sec:DIFFOPERS_1} ) that |
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[10354] | 299 | the weak-slope operator may be \emph{exactly} reexpressed in non-orthogonal $i,j,\rho$-coordinates as |
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[3294] | 300 | |
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[10414] | 301 | \begin{equation} |
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[11543] | 302 | \label{eq:DIFFOPERS_5} |
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[10414] | 303 | D^T = \left. \nabla \right|_\rho \cdot |
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| 304 | \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_\rho T \right] \\ |
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| 305 | \;\;\text{where} \;\Re =\left( {{ |
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| 306 | \begin{array}{*{20}c} |
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| 307 | 1 \hfill & 0 \hfill &0 \hfill \\ |
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| 308 | 0 \hfill & 1 \hfill & 0 \hfill \\ |
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| 309 | 0 \hfill & 0 \hfill & \varepsilon \hfill \\ |
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| 310 | \end{array} |
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| 311 | }} \right). |
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[3294] | 312 | \end{equation} |
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[10354] | 313 | Then direct transformation from $i,j,\rho$-coordinates to $i,j,s$-coordinates gives |
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[11543] | 314 | \autoref{eq:DIFFOPERS_ldfiso_s} immediately. |
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[3294] | 315 | |
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[10354] | 316 | Note that the weak-slope approximation is only made in transforming from |
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| 317 | the (rotated,orthogonal) isoneutral axes to the non-orthogonal $i,j,\rho$-coordinates. |
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| 318 | The further transformation into $i,j,s$-coordinates is exact, whatever the steepness of the $s$-surfaces, |
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[11331] | 319 | in the same way as the transformation of horizontal/vertical Laplacian diffusion in $z$-coordinates in |
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[11543] | 320 | \autoref{sec:DIFFOPERS_1} onto $s$-coordinates is exact, however steep the $s$-surfaces. |
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[3294] | 321 | |
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| 322 | |
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[707] | 323 | % ================================================================ |
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| 324 | % Lateral/Vertical Momentum Diffusive Operators |
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| 325 | % ================================================================ |
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[9393] | 326 | \section{Lateral/Vertical momentum diffusive operators} |
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[11543] | 327 | \label{sec:DIFFOPERS_3} |
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[707] | 328 | |
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[11331] | 329 | The second order momentum diffusion operator (Laplacian) in $z$-coordinates is found by |
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[11543] | 330 | applying \autoref{eq:MB_lap_vector}, the expression for the Laplacian of a vector, |
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[10354] | 331 | to the horizontal velocity vector: |
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[817] | 332 | \begin{align*} |
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[10414] | 333 | \Delta {\textbf{U}}_h |
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| 334 | &=\nabla \left( {\nabla \cdot {\textbf{U}}_h } \right)- |
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| 335 | \nabla \times \left( {\nabla \times {\textbf{U}}_h } \right) \\ \\ |
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| 336 | &=\left( {{ |
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| 337 | \begin{array}{*{20}c} |
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| 338 | {\frac{1}{e_1 }\frac{\partial \chi }{\partial i}} \hfill \\ |
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| 339 | {\frac{1}{e_2 }\frac{\partial \chi }{\partial j}} \hfill \\ |
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| 340 | {\frac{1}{e_3 }\frac{\partial \chi }{\partial k}} \hfill \\ |
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| 341 | \end{array} |
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| 342 | }} \right) |
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| 343 | -\left( {{ |
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| 344 | \begin{array}{*{20}c} |
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| 345 | {\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}-\frac{1}{e_3 |
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| 346 | }\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial |
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| 347 | u}{\partial k}} \right)} \hfill \\ |
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| 348 | {\frac{1}{e_3 }\frac{\partial }{\partial k}\left( {-\frac{1}{e_3 |
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| 349 | }\frac{\partial v}{\partial k}} \right)-\frac{1}{e_1 }\frac{\partial \zeta |
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| 350 | }{\partial i}} \hfill \\ |
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| 351 | {\frac{1}{e_1 e_2 }\left[ {\frac{\partial }{\partial i}\left( {\frac{e_2 |
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| 352 | }{e_3 }\frac{\partial u}{\partial k}} \right)-\frac{\partial }{\partial |
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| 353 | j}\left( {-\frac{e_1 }{e_3 }\frac{\partial v}{\partial k}} \right)} \right]} |
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| 354 | \hfill \\ |
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| 355 | \end{array} |
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| 356 | }} \right) \\ \\ |
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| 357 | &=\left( {{ |
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| 358 | \begin{array}{*{20}c} |
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| 359 | {\frac{1}{e_1 }\frac{\partial \chi }{\partial i}-\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}} \\ |
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| 360 | {\frac{1}{e_2 }\frac{\partial \chi }{\partial j}+\frac{1}{e_1 }\frac{\partial \zeta }{\partial i}} \\ |
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| 361 | 0 \\ |
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| 362 | \end{array} |
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| 363 | }} \right) |
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| 364 | +\frac{1}{e_3 } |
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| 365 | \left( {{ |
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| 366 | \begin{array}{*{20}c} |
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| 367 | {\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial u}{\partial k}} \right)} \\ |
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| 368 | {\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial v}{\partial k}} \right)} \\ |
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| 369 | {\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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| 370 | \end{array} |
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| 371 | }} \right) |
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[817] | 372 | \end{align*} |
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[11543] | 373 | Using \autoref{eq:MB_div}, the definition of the horizontal divergence, |
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[11331] | 374 | the third component of the second vector is obviously zero and thus : |
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[10406] | 375 | \[ |
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[11543] | 376 | \Delta {\textbf{U}}_h = \nabla _h \left( \chi \right) - \nabla _h \times \left( \zeta \textbf{k} \right) + \frac {1}{e_3 } \frac {\partial }{\partial k} \left( {\frac {1}{e_3 } \frac{\partial {\textbf{ U}}_h }{\partial k}} \right) . |
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[10406] | 377 | \] |
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[707] | 378 | |
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[10354] | 379 | Note that this operator ensures a full separation between |
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[11543] | 380 | the vorticity and horizontal divergence fields (see \autoref{apdx:INVARIANTS}). |
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[10354] | 381 | It is only equal to a Laplacian applied to each component in Cartesian coordinates, not on the sphere. |
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[707] | 382 | |
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[10354] | 383 | The horizontal/vertical second order (Laplacian type) operator used to diffuse horizontal momentum in |
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| 384 | the $z$-coordinate therefore takes the following form: |
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[10414] | 385 | \begin{equation} |
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[11543] | 386 | \label{eq:DIFFOPERS_Lap_U} |
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[10414] | 387 | { |
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| 388 | \textbf{D}}^{\textbf{U}} = |
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| 389 | \nabla _h \left( {A^{lm}\;\chi } \right) |
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| 390 | - \nabla _h \times \left( {A^{lm}\;\zeta \;{\textbf{k}}} \right) |
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| 391 | + \frac{1}{e_3 }\frac{\partial }{\partial k}\left( {\frac{A^{vm}\;}{e_3 } |
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[11335] | 392 | \frac{\partial {\mathrm {\mathbf U}}_h }{\partial k}} \right) , \\ |
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[817] | 393 | \end{equation} |
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[1223] | 394 | that is, in expanded form: |
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[817] | 395 | \begin{align*} |
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[10414] | 396 | D^{\textbf{U}}_u |
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| 397 | & = \frac{1}{e_1} \frac{\partial \left( {A^{lm}\chi } \right)}{\partial i} |
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| 398 | -\frac{1}{e_2} \frac{\partial \left( {A^{lm}\zeta } \right)}{\partial j} |
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[11335] | 399 | +\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] | 400 | D^{\textbf{U}}_v |
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| 401 | & = \frac{1}{e_2 }\frac{\partial \left( {A^{lm}\chi } \right)}{\partial j} |
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| 402 | +\frac{1}{e_1 }\frac{\partial \left( {A^{lm}\zeta } \right)}{\partial i} |
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[11335] | 403 | +\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] | 404 | \end{align*} |
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[707] | 405 | |
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[11543] | 406 | Note Bene: introducing a rotation in \autoref{eq:DIFFOPERS_Lap_U} does not lead to |
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[10354] | 407 | a useful expression for the iso/diapycnal Laplacian operator in the $z$-coordinate. |
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| 408 | Similarly, we did not found an expression of practical use for |
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| 409 | the geopotential horizontal/vertical Laplacian operator in the $s$-coordinate. |
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[11543] | 410 | Generally, \autoref{eq:DIFFOPERS_Lap_U} is used in both $z$- and $s$-coordinate systems, |
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[10354] | 411 | that is a Laplacian diffusion is applied on momentum along the coordinate directions. |
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[10414] | 412 | |
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| 413 | \biblio |
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| 414 | |
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[10442] | 415 | \pindex |
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| 416 | |
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[6997] | 417 | \end{document} |
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