1 | \documentclass[../main/NEMO_manual]{subfiles} |
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2 | |
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3 | \begin{document} |
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4 | |
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5 | \chapter{Diffusive Operators} |
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6 | \label{apdx:DIFFOPERS} |
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7 | |
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8 | \thispagestyle{plain} |
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9 | |
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10 | \chaptertoc |
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11 | |
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12 | \paragraph{Changes record} ~\\ |
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13 | |
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14 | {\footnotesize |
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15 | \begin{tabularx}{\textwidth}{l||X|X} |
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16 | Release & Author(s) & Modifications \\ |
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17 | \hline |
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18 | {\em 4.0} & {\em ...} & {\em ...} \\ |
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19 | {\em 3.6} & {\em ...} & {\em ...} \\ |
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20 | {\em 3.4} & {\em ...} & {\em ...} \\ |
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21 | {\em <=3.4} & {\em ...} & {\em ...} |
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22 | \end{tabularx} |
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23 | } |
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24 | |
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25 | \clearpage |
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26 | |
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27 | %% ================================================================================================= |
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28 | \section{Horizontal/Vertical $2^{nd}$ order tracer diffusive operators} |
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29 | \label{sec:DIFFOPERS_1} |
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30 | |
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31 | %% ================================================================================================= |
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32 | \subsubsection*{In z-coordinates} |
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33 | |
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34 | In $z$-coordinates, the horizontal/vertical second order tracer diffusion operator is given by: |
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35 | \begin{align} |
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36 | \label{eq:DIFFOPERS_1} |
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37 | &D^T = \frac{1}{e_1 \, e_2} \left[ |
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38 | \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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39 | \left. |
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40 | + \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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41 | + \frac{\partial }{\partial z}\left( {A^{vT} \;\frac{\partial T}{\partial z}} \right) |
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42 | \end{align} |
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43 | |
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44 | %% ================================================================================================= |
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45 | \subsubsection*{In generalized vertical coordinates} |
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46 | |
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47 | 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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48 | the vertical/horizontal ratio of diffusion coefficient by $\epsilon = A^{vT} / A^{lT}$. |
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49 | The diffusion operator is given by: |
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50 | |
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51 | \begin{equation} |
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52 | \label{eq:DIFFOPERS_2} |
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53 | D^T = \left. \nabla \right|_s \cdot |
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54 | \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T \right] \\ |
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55 | \;\;\text{where} \;\Re =\left( {{ |
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56 | \begin{array}{*{20}c} |
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57 | 1 \hfill & 0 \hfill & {-\sigma_1 } \hfill \\ |
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58 | 0 \hfill & 1 \hfill & {-\sigma_2 } \hfill \\ |
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59 | {-\sigma_1 } \hfill & {-\sigma_2 } \hfill & {\varepsilon +\sigma_1 |
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60 | ^2+\sigma_2 ^2} \hfill \\ |
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61 | \end{array} |
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62 | }} \right) |
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63 | \end{equation} |
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64 | or in expanded form: |
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65 | \begin{align*} |
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66 | { |
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67 | \begin{array}{*{20}l} |
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68 | 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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69 | \left( \ \frac{1}{e_1}\; \left. \frac{\partial T}{\partial i} \right|_s |
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70 | -\frac{\sigma_1 }{e_3 } \; \frac{\partial T}{\partial s} \right) \right] \right|_s \right. \\ |
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71 | & \quad \ + \ \left. \frac{\partial }{\partial j} \left. \left[ e_1\,e_3 \, A^{lT} |
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72 | \left( \ \frac{1}{e_2 }\; \left. \frac{\partial T}{\partial j} \right|_s |
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73 | -\frac{\sigma_2 }{e_3 } \; \frac{\partial T}{\partial s} \right) \right] \right|_s \right. \\ |
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74 | & \quad \ + \ \left. e_1\,e_2\, \frac{\partial }{\partial s} \left[ A^{lT} \; \left( |
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75 | -\frac{\sigma_1 }{e_1 } \; \left. \frac{\partial T}{\partial i} \right|_s |
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76 | -\frac{\sigma_2 }{e_2 } \; \left. \frac{\partial T}{\partial j} \right|_s |
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77 | +\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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78 | \end{array} |
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79 | } |
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80 | \end{align*} |
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81 | |
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82 | \autoref{eq:DIFFOPERS_2} is obtained from \autoref{eq:DIFFOPERS_1} without any additional assumption. |
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83 | Indeed, for the special case $k=z$ and thus $e_3 =1$, |
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84 | we introduce an arbitrary vertical coordinate $s = s (i,j,z)$ as in \autoref{apdx:SCOORD} and |
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85 | use \autoref{eq:SCOORD_s_slope} and \autoref{eq:SCOORD_s_chain_rule1}. |
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86 | Since no cross horizontal derivative $\partial _i \partial _j $ appears in \autoref{eq:DIFFOPERS_1}, |
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87 | the ($i$,$z$) and ($j$,$z$) planes are independent. |
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88 | The derivation can then be demonstrated for the ($i$,$z$)~$\to$~($j$,$s$) transformation without |
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89 | any loss of generality: |
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90 | |
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91 | \begin{align*} |
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92 | { |
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93 | \begin{array}{*{20}l} |
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94 | 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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95 | +\frac{\partial }{\partial z}\left( {A^{vT}\;\frac{\partial T}{\partial z}} \right) \\ \\ |
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96 | % |
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97 | &=\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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98 | -\frac{e_1\,\sigma_1 }{e_3 }\frac{\partial T}{\partial s}} \right)} \right)} \right|_s } \right. \\ |
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99 | & \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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100 | \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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101 | % |
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102 | &=\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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103 | & \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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104 | & \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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105 | \end{array} |
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106 | } \\ |
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107 | % |
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108 | { |
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109 | \begin{array}{*{20}l} |
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110 | \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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111 | % |
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112 | 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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113 | & \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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114 | & \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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115 | \\ |
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116 | &=\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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117 | & \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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118 | & \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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119 | & \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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120 | \end{array} |
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121 | } \\ |
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122 | { |
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123 | \begin{array}{*{20}l} |
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124 | % |
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125 | \intertext{Using the same remark as just above, $D^T$ becomes:} |
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126 | % |
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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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128 | & \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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129 | & \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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130 | & \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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131 | \end{array} |
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132 | } \\ |
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133 | { |
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134 | \begin{array}{*{20}l} |
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135 | % |
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136 | \intertext{Since the horizontal scale factors do not depend on the vertical coordinate, |
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137 | the two terms on the second line cancel, while |
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138 | the third line reduces to a single vertical derivative, so it becomes:} |
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139 | % |
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140 | 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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141 | & \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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142 | % |
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143 | \intertext{In other words, the horizontal/vertical Laplacian operator in the ($i$,$s$) plane takes the following form:} |
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144 | \end{array} |
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145 | } \\ |
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146 | % |
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147 | {\frac{1}{e_1\,e_2\,e_3}} |
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148 | \left( {{ |
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149 | \begin{array}{*{30}c} |
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150 | {\left. {\frac{\partial \left( {e_2 e_3 \bullet } \right)}{\partial i}} \right|_s } \hfill \\ |
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151 | {\frac{\partial \left( {e_1 e_2 \bullet } \right)}{\partial s}} \hfill \\ |
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152 | \end{array}}} |
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153 | \right) |
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154 | \cdot \left[ {A^{lT} |
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155 | \left( {{ |
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156 | \begin{array}{*{30}c} |
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157 | {1} \hfill & {-\sigma_1 } \hfill \\ |
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158 | {-\sigma_1} \hfill & {\varepsilon + \sigma_1^2} \hfill \\ |
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159 | \end{array} |
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160 | }} \right) |
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161 | \cdot |
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162 | \left( {{ |
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163 | \begin{array}{*{30}c} |
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164 | {\frac{1}{e_1 }\;\left. {\frac{\partial \bullet }{\partial i}} \right|_s } \hfill \\ |
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165 | {\frac{1}{e_3 }\;\frac{\partial \bullet }{\partial s}} \hfill \\ |
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166 | \end{array} |
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167 | }} \right) \left( T \right)} \right] |
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168 | \end{align*} |
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169 | %\addtocounter{equation}{-2} |
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170 | |
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171 | %% ================================================================================================= |
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172 | \section{Iso/Diapycnal $2^{nd}$ order tracer diffusive operators} |
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173 | \label{sec:DIFFOPERS_2} |
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174 | |
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175 | %% ================================================================================================= |
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176 | \subsubsection*{In z-coordinates} |
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177 | |
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178 | The iso/diapycnal diffusive tensor $\textbf {A}_{\textbf I}$ expressed in |
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179 | the ($i$,$j$,$k$) curvilinear coordinate system in which |
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180 | the equations of the ocean circulation model are formulated, |
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181 | takes the following form \citep{redi_JPO82}: |
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182 | |
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183 | \begin{equation} |
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184 | \label{eq:DIFFOPERS_3} |
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185 | \textbf {A}_{\textbf I} = \frac{A^{lT}}{\left( {1+a_1 ^2+a_2 ^2} \right)} |
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186 | \left[ {{ |
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187 | \begin{array}{*{20}c} |
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188 | {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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189 | {-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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190 | {-a_1 (1-\varepsilon)} \hfill & {-a_2 (1-\varepsilon)} \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\ |
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191 | \end{array} |
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192 | }} \right] |
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193 | \end{equation} |
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194 | where ($a_1$, $a_2$) are $(-1) \times$ the isopycnal slopes in |
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195 | ($\textbf{i}$, $\textbf{j}$) directions, relative to geopotentials (or |
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196 | equivalently the slopes of the geopotential surfaces in the isopycnal |
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197 | coordinate framework): |
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198 | \[ |
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199 | a_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1} |
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200 | \qquad , \qquad |
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201 | a_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}} |
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202 | \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1} |
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203 | \] |
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204 | and, as before, $\epsilon = A^{vT} / A^{lT}$. |
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205 | |
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206 | In practice, $\epsilon$ is small and isopycnal slopes are generally less than $10^{-2}$ in the ocean, |
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207 | 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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208 | 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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209 | \begin{subequations} |
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210 | \label{eq:DIFFOPERS_4} |
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211 | \begin{equation} |
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212 | \label{eq:DIFFOPERS_4a} |
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213 | {\textbf{A}_{\textbf{I}}} \approx A^{lT}\;\Re\;\text{where} \;\Re = |
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214 | \left[ {{ |
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215 | \begin{array}{*{20}c} |
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216 | 1 \hfill & 0 \hfill & {-a_1 } \hfill \\ |
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217 | 0 \hfill & 1 \hfill & {-a_2 } \hfill \\ |
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218 | {-a_1 } \hfill & {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\ |
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219 | \end{array} |
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220 | }} \right], |
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221 | \end{equation} |
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222 | and the iso/dianeutral diffusive operator in $z$-coordinates is then |
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223 | \begin{equation} |
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224 | \label{eq:DIFFOPERS_4b} |
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225 | D^T = \left. \nabla \right|_z \cdot |
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226 | \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_z T \right]. \\ |
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227 | \end{equation} |
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228 | \end{subequations} |
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229 | |
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230 | Physically, the full tensor \autoref{eq:DIFFOPERS_3} represents strong isoneutral diffusion on a plane parallel to |
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231 | the isoneutral surface and weak dianeutral diffusion perpendicular to this plane. |
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232 | However, |
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233 | the approximate `weak-slope' tensor \autoref{eq:DIFFOPERS_4a} represents strong diffusion along the isoneutral surface, |
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234 | with weak \emph{vertical} diffusion -- the principal axes of the tensor are no longer orthogonal. |
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235 | This simplification also decouples the ($i$,$z$) and ($j$,$z$) planes of the tensor. |
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236 | The weak-slope operator therefore takes the same form, \autoref{eq:DIFFOPERS_4}, as \autoref{eq:DIFFOPERS_2}, |
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237 | the diffusion operator for geopotential diffusion written in non-orthogonal $i,j,s$-coordinates. |
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238 | Written out explicitly, |
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239 | |
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240 | \begin{multline} |
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241 | \label{eq:DIFFOPERS_ldfiso} |
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242 | D^T=\frac{1}{e_1 e_2 }\left\{ |
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243 | {\;\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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244 | {+\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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245 | \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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246 | \end{multline} |
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247 | |
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248 | The isopycnal diffusion operator \autoref{eq:DIFFOPERS_4}, |
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249 | \autoref{eq:DIFFOPERS_ldfiso} conserves tracer quantity and dissipates its square. |
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250 | 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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251 | (as it is when $A_h$ is zero at the boundary). Let us demonstrate the second one: |
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252 | \[ |
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253 | \iiint\limits_D T\;\nabla .\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv |
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254 | = -\iiint\limits_D \nabla T\;.\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv, |
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255 | \] |
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256 | and since |
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257 | \begin{align*} |
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258 | { |
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259 | \begin{array}{*{20}l} |
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260 | \nabla T\;.\left( {{\mathrm {\mathbf A}}_{\mathrm {\mathbf I}} \nabla T} |
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261 | \right)&=A^{lT}\left[ {\left( {\frac{\partial T}{\partial i}} \right)^2-2a_1 |
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262 | \frac{\partial T}{\partial i}\frac{\partial T}{\partial k}+\left( |
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263 | {\frac{\partial T}{\partial j}} \right)^2} \right. \\ |
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264 | &\qquad \qquad \qquad |
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265 | { \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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266 | &=A_h \left[ {\left( {\frac{\partial T}{\partial i}-a_1 \frac{\partial |
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267 | T}{\partial k}} \right)^2+\left( {\frac{\partial T}{\partial |
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268 | j}-a_2 \frac{\partial T}{\partial k}} \right)^2} |
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269 | +\varepsilon \left(\frac{\partial T}{\partial k}\right) ^2\right] \\ |
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270 | & \geq 0 . |
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271 | \end{array} |
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272 | } |
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273 | \end{align*} |
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274 | %\addtocounter{equation}{-1} |
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275 | the property becomes obvious. |
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276 | |
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277 | %% ================================================================================================= |
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278 | \subsubsection*{In generalized vertical coordinates} |
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279 | |
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280 | Because the weak-slope operator \autoref{eq:DIFFOPERS_4}, |
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281 | \autoref{eq:DIFFOPERS_ldfiso} is decoupled in the ($i$,$z$) and ($j$,$z$) planes, |
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282 | it may be transformed into generalized $s$-coordinates in the same way as |
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283 | \autoref{sec:DIFFOPERS_1} was transformed into \autoref{sec:DIFFOPERS_2}. |
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284 | The resulting operator then takes the simple form |
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285 | |
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286 | \begin{equation} |
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287 | \label{eq:DIFFOPERS_ldfiso_s} |
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288 | D^T = \left. \nabla \right|_s \cdot |
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289 | \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T \right] \\ |
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290 | \;\;\text{where} \;\Re =\left( {{ |
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291 | \begin{array}{*{20}c} |
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292 | 1 \hfill & 0 \hfill & {-r _1 } \hfill \\ |
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293 | 0 \hfill & 1 \hfill & {-r _2 } \hfill \\ |
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294 | {-r _1 } \hfill & {-r _2 } \hfill & {\varepsilon +r _1 |
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295 | ^2+r _2 ^2} \hfill \\ |
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296 | \end{array} |
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297 | }} \right), |
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298 | \end{equation} |
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299 | |
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300 | where ($r_1$, $r_2$) are $(-1)\times$ the isopycnal slopes in ($\textbf{i}$, $\textbf{j}$) directions, |
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301 | relative to $s$-coordinate surfaces (or equivalently the slopes of the |
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302 | $s$-coordinate surfaces in the isopycnal coordinate framework): |
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303 | \[ |
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304 | r_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1} |
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305 | \qquad , \qquad |
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306 | r_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}} |
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307 | \right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1}. |
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308 | \] |
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309 | |
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310 | To prove \autoref{eq:DIFFOPERS_ldfiso_s} by direct re-expression of \autoref{eq:DIFFOPERS_ldfiso} is straightforward, but laborious. |
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311 | 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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312 | the weak-slope operator may be \emph{exactly} reexpressed in non-orthogonal $i,j,\rho$-coordinates as |
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313 | |
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314 | \begin{equation} |
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315 | \label{eq:DIFFOPERS_5} |
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316 | D^T = \left. \nabla \right|_\rho \cdot |
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317 | \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_\rho T \right] \\ |
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318 | \;\;\text{where} \;\Re =\left( {{ |
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319 | \begin{array}{*{20}c} |
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320 | 1 \hfill & 0 \hfill &0 \hfill \\ |
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321 | 0 \hfill & 1 \hfill & 0 \hfill \\ |
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322 | 0 \hfill & 0 \hfill & \varepsilon \hfill \\ |
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323 | \end{array} |
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324 | }} \right). |
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325 | \end{equation} |
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326 | Then direct transformation from $i,j,\rho$-coordinates to $i,j,s$-coordinates gives |
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327 | \autoref{eq:DIFFOPERS_ldfiso_s} immediately. |
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328 | |
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329 | Note that the weak-slope approximation is only made in transforming from |
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330 | the (rotated,orthogonal) isoneutral axes to the non-orthogonal $i,j,\rho$-coordinates. |
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331 | The further transformation into $i,j,s$-coordinates is exact, whatever the steepness of the $s$-surfaces, |
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332 | in the same way as the transformation of horizontal/vertical Laplacian diffusion in $z$-coordinates in |
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333 | \autoref{sec:DIFFOPERS_1} onto $s$-coordinates is exact, however steep the $s$-surfaces. |
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334 | |
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335 | %% ================================================================================================= |
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336 | \section{Lateral/Vertical momentum diffusive operators} |
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337 | \label{sec:DIFFOPERS_3} |
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338 | |
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339 | The second order momentum diffusion operator (Laplacian) in $z$-coordinates is found by |
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340 | applying \autoref{eq:MB_lap_vector}, the expression for the Laplacian of a vector, |
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341 | to the horizontal velocity vector: |
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342 | \begin{align*} |
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343 | \Delta {\textbf{U}}_h |
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344 | &=\nabla \left( {\nabla \cdot {\textbf{U}}_h } \right)- |
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345 | \nabla \times \left( {\nabla \times {\textbf{U}}_h } \right) \\ \\ |
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346 | &=\left( {{ |
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347 | \begin{array}{*{20}c} |
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348 | {\frac{1}{e_1 }\frac{\partial \chi }{\partial i}} \hfill \\ |
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349 | {\frac{1}{e_2 }\frac{\partial \chi }{\partial j}} \hfill \\ |
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350 | {\frac{1}{e_3 }\frac{\partial \chi }{\partial k}} \hfill \\ |
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351 | \end{array} |
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352 | }} \right) |
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353 | -\left( {{ |
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354 | \begin{array}{*{20}c} |
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355 | {\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}-\frac{1}{e_3 |
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356 | }\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial |
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357 | u}{\partial k}} \right)} \hfill \\ |
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358 | {\frac{1}{e_3 }\frac{\partial }{\partial k}\left( {-\frac{1}{e_3 |
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359 | }\frac{\partial v}{\partial k}} \right)-\frac{1}{e_1 }\frac{\partial \zeta |
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360 | }{\partial i}} \hfill \\ |
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361 | {\frac{1}{e_1 e_2 }\left[ {\frac{\partial }{\partial i}\left( {\frac{e_2 |
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362 | }{e_3 }\frac{\partial u}{\partial k}} \right)-\frac{\partial }{\partial |
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363 | j}\left( {-\frac{e_1 }{e_3 }\frac{\partial v}{\partial k}} \right)} \right]} |
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364 | \hfill \\ |
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365 | \end{array} |
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366 | }} \right) \\ \\ |
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367 | &=\left( {{ |
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368 | \begin{array}{*{20}c} |
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369 | {\frac{1}{e_1 }\frac{\partial \chi }{\partial i}-\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}} \\ |
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370 | {\frac{1}{e_2 }\frac{\partial \chi }{\partial j}+\frac{1}{e_1 }\frac{\partial \zeta }{\partial i}} \\ |
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371 | 0 \\ |
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372 | \end{array} |
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373 | }} \right) |
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374 | +\frac{1}{e_3 } |
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375 | \left( {{ |
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376 | \begin{array}{*{20}c} |
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377 | {\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial u}{\partial k}} \right)} \\ |
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378 | {\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial v}{\partial k}} \right)} \\ |
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379 | {\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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380 | \end{array} |
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381 | }} \right) |
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382 | \end{align*} |
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383 | Using \autoref{eq:MB_div}, the definition of the horizontal divergence, |
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384 | the third component of the second vector is obviously zero and thus : |
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385 | \[ |
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386 | \Delta {\textbf{U}}_h = \nabla _h \left( \chi \right) - \nabla _h \times \left( \zeta \textbf{k} \right) + \frac {1}{e_3 } \frac {\partial }{\partial k} \left( {\frac {1}{e_3 } \frac{\partial {\textbf{ U}}_h }{\partial k}} \right) . |
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387 | \] |
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388 | |
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389 | Note that this operator ensures a full separation between |
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390 | the vorticity and horizontal divergence fields (see \autoref{apdx:INVARIANTS}). |
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391 | It is only equal to a Laplacian applied to each component in Cartesian coordinates, not on the sphere. |
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392 | |
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393 | The horizontal/vertical second order (Laplacian type) operator used to diffuse horizontal momentum in |
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394 | the $z$-coordinate therefore takes the following form: |
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395 | \begin{equation} |
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396 | \label{eq:DIFFOPERS_Lap_U} |
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397 | { |
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398 | \textbf{D}}^{\textbf{U}} = |
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399 | \nabla _h \left( {A^{lm}\;\chi } \right) |
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400 | - \nabla _h \times \left( {A^{lm}\;\zeta \;{\textbf{k}}} \right) |
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401 | + \frac{1}{e_3 }\frac{\partial }{\partial k}\left( {\frac{A^{vm}\;}{e_3 } |
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402 | \frac{\partial {\mathrm {\mathbf U}}_h }{\partial k}} \right) , \\ |
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403 | \end{equation} |
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404 | that is, in expanded form: |
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405 | \begin{align*} |
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406 | D^{\textbf{U}}_u |
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407 | & = \frac{1}{e_1} \frac{\partial \left( {A^{lm}\chi } \right)}{\partial i} |
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408 | -\frac{1}{e_2} \frac{\partial \left( {A^{lm}\zeta } \right)}{\partial j} |
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409 | +\frac{1}{e_3} \frac{\partial }{\partial k} \left( \frac{A^{vm}}{e_3} \frac{\partial u}{\partial k} \right) , \\ |
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410 | D^{\textbf{U}}_v |
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411 | & = \frac{1}{e_2 }\frac{\partial \left( {A^{lm}\chi } \right)}{\partial j} |
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412 | +\frac{1}{e_1 }\frac{\partial \left( {A^{lm}\zeta } \right)}{\partial i} |
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413 | +\frac{1}{e_3} \frac{\partial }{\partial k} \left( \frac{A^{vm}}{e_3} \frac{\partial v}{\partial k} \right) . |
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414 | \end{align*} |
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415 | |
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416 | Note Bene: introducing a rotation in \autoref{eq:DIFFOPERS_Lap_U} does not lead to |
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417 | a useful expression for the iso/diapycnal Laplacian operator in the $z$-coordinate. |
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418 | Similarly, we did not found an expression of practical use for |
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419 | the geopotential horizontal/vertical Laplacian operator in the $s$-coordinate. |
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420 | Generally, \autoref{eq:DIFFOPERS_Lap_U} is used in both $z$- and $s$-coordinate systems, |
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421 | that is a Laplacian diffusion is applied on momentum along the coordinate directions. |
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422 | |
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423 | \subinc{\input{../../global/epilogue}} |
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424 | |
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425 | \end{document} |
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