1 | % ================================================================ |
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2 | % Chapter Ñ Appendix B : Diffusive Operators |
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3 | % ================================================================ |
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4 | \chapter{Appendix B : Diffusive Operators} |
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5 | \label{Apdx_B} |
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6 | \minitoc |
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7 | |
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8 | |
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9 | \newpage |
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10 | $\ $\newline % force a new ligne |
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11 | |
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12 | % ================================================================ |
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13 | % Horizontal/Vertical 2nd Order Tracer Diffusive Operators |
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14 | % ================================================================ |
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15 | \section{Horizontal/Vertical 2nd Order Tracer Diffusive Operators} |
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16 | \label{Apdx_B_1} |
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17 | |
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18 | |
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19 | In the $z$-coordinate, the horizontal/vertical second order tracer diffusion operator |
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20 | is given by: |
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21 | \begin{eqnarray} \label{Apdx_B1} |
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22 | &D^T = \frac{1}{e_1 \, e_2} \left[ |
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23 | \left. \frac{\partial}{\partial i} \left( \frac{e_2}{e_1}A^{lT} \;\left. \frac{\partial T}{\partial i} \right|_z \right) \right|_z \right. |
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24 | \left. |
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25 | + \left. \frac{\partial}{\partial j} \left( \frac{e_1}{e_2}A^{lT} \;\left. \frac{\partial T}{\partial j} \right|_z \right) \right|_z \right] |
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26 | + \frac{\partial }{\partial z}\left( {A^{vT} \;\frac{\partial T}{\partial z}} \right) |
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27 | \end{eqnarray} |
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28 | |
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29 | In the $s$-coordinate, we defined the slopes of $s$-surfaces, $\sigma_1$ and |
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30 | $\sigma_2$ by \eqref{Apdx_A_s_slope} and the vertical/horizontal ratio of diffusion |
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31 | coefficient by $\epsilon = A^{vT} / A^{lT}$. The diffusion operator is given by: |
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32 | |
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33 | \begin{equation} \label{Apdx_B2} |
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34 | D^T = \left. \nabla \right|_s \cdot |
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35 | \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T \right] \\ |
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36 | \;\;\text{where} \;\Re =\left( {{\begin{array}{*{20}c} |
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37 | 1 \hfill & 0 \hfill & {-\sigma _1 } \hfill \\ |
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38 | 0 \hfill & 1 \hfill & {-\sigma _2 } \hfill \\ |
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39 | {-\sigma _1 } \hfill & {-\sigma _2 } \hfill & {\varepsilon +\sigma _1 |
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40 | ^2+\sigma _2 ^2} \hfill \\ |
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41 | \end{array} }} \right) |
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42 | \end{equation} |
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43 | or in expanded form: |
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44 | \begin{subequations} |
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45 | \begin{align*} {\begin{array}{*{20}l} |
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46 | D^T=& \frac{1}{e_1\,e_2\,e_3 }\;\left[ {\ \ \ \ e_2\,e_3\,A^{lT} \;\left. |
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47 | {\frac{\partial }{\partial i}\left( {\frac{1}{e_1}\;\left. {\frac{\partial T}{\partial i}} \right|_s -\frac{\sigma _1 }{e_3 }\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right. \\ |
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48 | &\qquad \quad \ \ \ +e_1\,e_3\,A^{lT} \;\left. {\frac{\partial }{\partial j}\left( {\frac{1}{e_2 }\;\left. {\frac{\partial T}{\partial j}} \right|_s -\frac{\sigma _2 }{e_3 }\;\frac{\partial T}{\partial s}} \right)} \right|_s \\ |
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49 | &\qquad \quad \ \ \ + e_1\,e_2\,A^{lT} \;\frac{\partial }{\partial s}\left( {-\frac{\sigma _1 }{e_1 }\;\left. {\frac{\partial T}{\partial i}} \right|_s -\frac{\sigma _2 }{e_2 }\;\left. {\frac{\partial T}{\partial j}} \right|_s } \right. |
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50 | \left. {\left. {+\left( {\varepsilon +\sigma _1^2+\sigma _2 ^2} \right)\;\frac{1}{e_3 }\;\frac{\partial T}{\partial s}} \right)\;\;} \right] |
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51 | \end{array} } |
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52 | \end{align*} |
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53 | \end{subequations} |
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54 | |
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55 | Equation \eqref{Apdx_B2} is obtained from \eqref{Apdx_B1} without any |
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56 | additional assumption. Indeed, for the special case $k=z$ and thus $e_3 =1$, |
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57 | we introduce an arbitrary vertical coordinate $s = s (i,j,z)$ as in Appendix~\ref{Apdx_A} |
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58 | and use \eqref{Apdx_A_s_slope} and \eqref{Apdx_A_s_chain_rule}. |
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59 | Since no cross horizontal derivative $\partial _i \partial _j $ appears in |
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60 | \eqref{Apdx_B1}, the ($i$,$z$) and ($j$,$z$) planes are independent. |
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61 | The derivation can then be demonstrated for the ($i$,$z$)~$\to$~($j$,$s$) |
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62 | transformation without any loss of generality: |
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63 | |
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64 | \begin{subequations} |
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65 | \begin{align*} {\begin{array}{*{20}l} |
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66 | 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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67 | +\frac{\partial }{\partial z}\left( {A^{vT}\;\frac{\partial T}{\partial z}} \right) \\ |
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68 | \\ |
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69 | \allowdisplaybreaks |
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70 | &=\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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71 | -\frac{e_1\,\sigma _1 }{e_3 }\frac{\partial T}{\partial s}} \right)} \right)} \right|_s } \right. \\ |
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72 | & \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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73 | \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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74 | \\ |
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75 | \allowdisplaybreaks |
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76 | &=\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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77 | & \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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78 | & \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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79 | \end{array} } \\ |
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80 | {\begin{array}{*{20}l} |
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81 | % |
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82 | \allowdisplaybreaks |
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83 | \intertext{Noting that $\frac{1}{e_1} \left. \frac{\partial e_3 }{\partial i} \right|_s = \frac{\partial \sigma _1 }{\partial s}$, it becomes:} |
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84 | % |
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85 | & =\frac{1}{e_1\,e_2\,e_3 }\left[ {\left. {\;\;\;\frac{\partial }{\partial i}\left( {\frac{e_2\,e_3 }{e_1}\,A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)} \right|_s \left. -\, {e_3 \frac{\partial }{\partial i}\left( {\frac{e_2 \,\sigma _1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right. \\ |
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86 | & \qquad \qquad \quad -e_2 A^{lT}\;\frac{\partial \sigma _1 }{\partial s}\left. {\frac{\partial T}{\partial i}} \right|_s -e_1 \,\sigma_1 \frac{\partial }{\partial s}\left( {\frac{e_2 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right) \\ |
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87 | & \qquad \qquad \quad\shoveright{ \left. { +e_1 \,\sigma _1 \frac{\partial }{\partial s}\left( {\frac{e_2 \,\sigma _1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right)+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 }{e_3 }A^{vT}\;\frac{\partial T}{\partial z}} \right)\;\;\;} \right] }\\ |
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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{\partial }{\partial i}\left( {e_2 \,\sigma _1 A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right. \\ |
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90 | & \qquad \qquad \quad \left. {+\frac{e_2 \,\sigma _1 }{e_3}A^{lT}\;\frac{\partial T}{\partial s} \;\frac{\partial e_3 }{\partial i}} \right|_s -e_2 A^{lT}\;\frac{\partial \sigma _1 }{\partial s}\left. {\frac{\partial T}{\partial i}} \right|_s \\ |
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91 | & \qquad \qquad \quad-e_2 \,\sigma _1 \frac{\partial}{\partial s}\left( {A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 \,\sigma _1 ^2}{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right) \\ |
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92 | & \qquad \qquad \quad\shoveright{ \left. {-\frac{\partial \left( {e_1 \,e_2 \,\sigma _1 } \right)}{\partial s} \left( {\frac{\sigma _1 }{e_3}A^{lT}\;\frac{\partial T}{\partial s}} \right) + \frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 }{e_3 }A^{vT}\;\frac{\partial T}{\partial s}} \right)\;\;\;} \right]} \\ |
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93 | % |
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94 | \allowdisplaybreaks |
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95 | \intertext{using the same remark as just above, it becomes:} |
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96 | % |
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97 | &= \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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98 | & \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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99 | & \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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100 | & \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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101 | % |
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102 | \allowdisplaybreaks |
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103 | \intertext{Since the horizontal scale factors do not depend on the vertical coordinate, |
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104 | the last term of the first line and the first term of the last line cancel, while |
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105 | the second line reduces to a single vertical derivative, so it becomes:} |
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106 | % |
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107 | & =\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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108 | & \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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109 | % |
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110 | \allowdisplaybreaks |
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111 | \intertext{in other words, the horizontal Laplacian operator in the ($i$,$s$) plane takes the following form :} |
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112 | \end{array} } \\ |
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113 | % |
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114 | D^T = {\frac{1}{e_1\,e_2\,e_3}} |
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115 | \left( {{\begin{array}{*{30}c} |
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116 | {\left. {\frac{\partial \left( {e_2 e_3 \bullet } \right)}{\partial i}} \right|_s } \hfill \\ |
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117 | {\frac{\partial \left( {e_1 e_2 \bullet } \right)}{\partial s}} \hfill \\ |
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118 | \end{array}}}\right) |
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119 | \cdot \left[ {A^{lT} |
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120 | \left( {{\begin{array}{*{30}c} |
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121 | {1} \hfill & {-\sigma_1 } \hfill \\ |
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122 | {-\sigma_1} \hfill & {\varepsilon_1^2} \hfill \\ |
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123 | \end{array} }} \right) |
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124 | \cdot |
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125 | \left( {{\begin{array}{*{30}c} |
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126 | {\frac{1}{e_1 }\;\left. {\frac{\partial \bullet }{\partial i}} \right|_s } \hfill \\ |
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127 | {\frac{1}{e_3 }\;\frac{\partial \bullet }{\partial s}} \hfill \\ |
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128 | \end{array}}} \right) \left( T \right)} \right] |
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129 | \end{align*} |
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130 | \end{subequations} |
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131 | |
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132 | |
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133 | % ================================================================ |
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134 | % Isopycnal/Vertical 2nd Order Tracer Diffusive Operators |
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135 | % ================================================================ |
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136 | \section{Iso/diapycnal 2nd Order Tracer Diffusive Operators} |
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137 | \label{Apdx_B_2} |
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138 | |
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139 | |
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140 | The iso/diapycnal diffusive tensor $\textbf {A}_{\textbf I}$ expressed in the ($i$,$j$,$k$) |
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141 | curvilinear coordinate system in which the equations of the ocean circulation model are |
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142 | formulated, takes the following form \citep{Redi_JPO82}: |
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143 | |
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144 | \begin{equation*} |
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145 | \textbf {A}_{\textbf I} = \frac{A^{lT}}{\left( {1+a_1 ^2+a_2 ^2} \right)} |
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146 | \left[ {{\begin{array}{*{20}c} |
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147 | {1+a_1 ^2} \hfill & {-a_1 a_2 } \hfill & {-a_1 } \hfill \\ |
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148 | {-a_1 a_2 } \hfill & {1+a_2 ^2} \hfill & {-a_2 } \hfill \\ |
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149 | {-a_1 } \hfill & {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\ |
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150 | \end{array} }} \right] |
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151 | \end{equation*} |
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152 | where ($a_1$, $a_2$) are the isopycnal slopes in ($\textbf{i}$, $\textbf{j}$) directions: |
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153 | \begin{equation*} |
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154 | 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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155 | \qquad , \qquad |
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156 | a_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}} |
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157 | \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1} |
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158 | \end{equation*} |
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159 | |
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160 | In practice, the isopycnal slopes are generally less than $10^{-2}$ in the ocean, so |
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161 | $\textbf {A}_{\textbf I}$ can be simplified appreciably \citep{Cox1987}: |
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162 | \begin{equation*} |
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163 | {\textbf{A}_{\textbf{I}}} \approx A^{lT} |
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164 | \left[ {{\begin{array}{*{20}c} |
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165 | 1 \hfill & 0 \hfill & {-a_1 } \hfill \\ |
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166 | 0 \hfill & 1 \hfill & {-a_2 } \hfill \\ |
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167 | {-a_1 } \hfill & {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\ |
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168 | \end{array} }} \right] |
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169 | \end{equation*} |
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170 | |
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171 | The resulting isopycnal operator conserves the quantity and dissipates its square. |
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172 | The demonstration of the first property is trivial as \eqref{Apdx_B2} is the divergence |
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173 | of fluxes. Let us demonstrate the second one: |
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174 | \begin{equation*} |
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175 | \iiint\limits_D T\;\nabla .\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv |
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176 | = -\iiint\limits_D \nabla T\;.\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv |
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177 | \end{equation*} |
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178 | since |
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179 | \begin{subequations} |
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180 | \begin{align*} {\begin{array}{*{20}l} |
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181 | \nabla T\;.\left( {{\rm {\bf A}}_{\rm {\bf I}} \nabla T} |
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182 | \right)&=A^{lT}\left[ {\left( {\frac{\partial T}{\partial i}} \right)^2-2a_1 |
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183 | \frac{\partial T}{\partial i}\frac{\partial T}{\partial k}+\left( |
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184 | {\frac{\partial T}{\partial j}} \right)^2} \right. \\ |
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185 | &\qquad \qquad \qquad |
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186 | { \left. -\,{2a_2 \frac{\partial T}{\partial j}\frac{\partial T}{\partial k}+\left( {a_1 ^2+a_2 ^2} \right)\left( {\frac{\partial T}{\partial k}} \right)^2} \right]} \\ |
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187 | &=A_h \left[ {\left( {\frac{\partial T}{\partial i}-a_1 \frac{\partial T}{\partial k}} \right)^2+\left( {\frac{\partial T}{\partial j}-a_2 \frac{\partial T}{\partial k}} \right)^2} \right] \\ |
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188 | & \geq 0 |
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189 | \end{array} } |
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190 | \end{align*} |
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191 | \end{subequations} |
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192 | the property becomes obvious. |
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193 | |
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194 | The resulting diffusion operator in $z$-coordinate has the following form : |
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195 | \begin{multline*} \label{Apdx_B_ldfiso} |
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196 | D^T=\frac{1}{e_1 e_2 }\left\{ |
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197 | {\;\frac{\partial }{\partial i}\left[ {A_h \left( {\frac{e_2}{e_1}\frac{\partial T}{\partial i}-a_1 \frac{e_2}{e_3}\frac{\partial T}{\partial k}} \right)} \right]} |
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198 | {+\frac{\partial}{\partial j}\left[ {A_h \left( {\frac{e_1}{e_2}\frac{\partial T}{\partial j}-a_2 \frac{e_1}{e_3}\frac{\partial T}{\partial k}} \right)} \right]\;} \right\} \\ |
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199 | \shoveright{+\frac{1}{e_3 }\frac{\partial }{\partial k}\left[ {A_h \left( {-\frac{a_1 }{e_1 }\frac{\partial T}{\partial i}-\frac{a_2 }{e_2 }\frac{\partial T}{\partial j}+\frac{\left( {a_1 ^2+a_2 ^2} \right)}{e_3 }\frac{\partial T}{\partial k}} \right)} \right]} \\ |
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200 | \end{multline*} |
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201 | |
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202 | It has to be emphasised that the simplification introduced, leads to a decoupling |
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203 | between ($i$,$z$) and ($j$,$z$) planes. The operator has therefore the same |
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204 | expression as \eqref{Apdx_B3}, the diffusion operator obtained for geopotential |
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205 | diffusion in the $s$-coordinate. |
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206 | |
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207 | % ================================================================ |
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208 | % Lateral/Vertical Momentum Diffusive Operators |
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209 | % ================================================================ |
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210 | \section{Lateral/Vertical Momentum Diffusive Operators} |
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211 | \label{Apdx_B_3} |
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212 | |
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213 | The second order momentum diffusion operator (Laplacian) in the $z$-coordinate |
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214 | is found by applying \eqref{Eq_PE_lap_vector}, the expression for the Laplacian |
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215 | of a vector, to the horizontal velocity vector : |
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216 | \begin{align*} |
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217 | \Delta {\textbf{U}}_h |
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218 | &=\nabla \left( {\nabla \cdot {\textbf{U}}_h } \right)- |
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219 | \nabla \times \left( {\nabla \times {\textbf{U}}_h } \right) \\ |
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220 | \\ |
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221 | &=\left( {{\begin{array}{*{20}c} |
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222 | {\frac{1}{e_1 }\frac{\partial \chi }{\partial i}} \hfill \\ |
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223 | {\frac{1}{e_2 }\frac{\partial \chi }{\partial j}} \hfill \\ |
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224 | {\frac{1}{e_3 }\frac{\partial \chi }{\partial k}} \hfill \\ |
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225 | \end{array} }} \right)-\left( {{\begin{array}{*{20}c} |
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226 | {\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}-\frac{1}{e_3 |
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227 | }\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial |
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228 | u}{\partial k}} \right)} \hfill \\ |
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229 | {\frac{1}{e_3 }\frac{\partial }{\partial k}\left( {-\frac{1}{e_3 |
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230 | }\frac{\partial v}{\partial k}} \right)-\frac{1}{e_1 }\frac{\partial \zeta |
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231 | }{\partial i}} \hfill \\ |
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232 | {\frac{1}{e_1 e_2 }\left[ {\frac{\partial }{\partial i}\left( {\frac{e_2 |
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233 | }{e_3 }\frac{\partial u}{\partial k}} \right)-\frac{\partial }{\partial |
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234 | j}\left( {-\frac{e_1 }{e_3 }\frac{\partial v}{\partial k}} \right)} \right]} |
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235 | \hfill \\ |
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236 | \end{array} }} \right) |
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237 | \\ |
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238 | \\ |
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239 | &=\left( {{\begin{array}{*{20}c} |
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240 | {\frac{1}{e_1 }\frac{\partial \chi }{\partial i}-\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}} \\ |
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241 | {\frac{1}{e_2 }\frac{\partial \chi }{\partial j}+\frac{1}{e_1 }\frac{\partial \zeta }{\partial i}} \\ |
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242 | 0 \\ |
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243 | \end{array} }} \right) |
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244 | +\frac{1}{e_3 } |
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245 | \left( {{\begin{array}{*{20}c} |
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246 | {\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial u}{\partial k}} \right)} \\ |
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247 | {\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial v}{\partial k}} \right)} \\ |
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248 | {\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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249 | \end{array} }} \right) |
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250 | \end{align*} |
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251 | Using \eqref{Eq_PE_div}, the definition of the horizontal divergence, the third |
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252 | componant of the second vector is obviously zero and thus : |
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253 | \begin{equation*} |
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254 | \Delta {\textbf{U}}_h = \nabla _h \left( \chi \right) - \nabla _h \times \left( \zeta \right) + \frac {1}{e_3 } \frac {\partial }{\partial k} \left( {\frac {1}{e_3 } \frac{\partial {\textbf{ U}}_h }{\partial k}} \right) |
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255 | \end{equation*} |
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256 | |
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257 | Note that this operator ensures a full separation between the vorticity and horizontal |
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258 | divergence fields (see Appendix~\ref{Apdx_C}). It is only equal to a Laplacian |
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259 | applied to each component in Cartesian coordinates, not on the sphere. |
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260 | |
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261 | The horizontal/vertical second order (Laplacian type) operator used to diffuse |
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262 | horizontal momentum in the $z$-coordinate therefore takes the following form : |
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263 | \begin{equation} \label{Apdx_B_Lap_U} |
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264 | {\textbf{D}}^{\textbf{U}} = |
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265 | \nabla _h \left( {A^{lm}\;\chi } \right) |
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266 | - \nabla _h \times \left( {A^{lm}\;\zeta \;{\textbf{k}}} \right) |
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267 | + \frac{1}{e_3 }\frac{\partial }{\partial k}\left( {\frac{A^{vm}\;}{e_3 } |
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268 | \frac{\partial {\rm {\bf U}}_h }{\partial k}} \right) \\ |
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269 | \end{equation} |
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270 | that is, in expanded form: |
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271 | \begin{align*} |
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272 | D^{\textbf{U}}_u |
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273 | & = \frac{1}{e_1} \frac{\partial \left( {A^{lm}\chi } \right)}{\partial i} |
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274 | -\frac{1}{e_2} \frac{\partial \left( {A^{lm}\zeta } \right)}{\partial j} |
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275 | +\frac{1}{e_3} \frac{\partial u}{\partial k} \\ |
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276 | D^{\textbf{U}}_v |
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277 | & = \frac{1}{e_2 }\frac{\partial \left( {A^{lm}\chi } \right)}{\partial j} |
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278 | +\frac{1}{e_1 }\frac{\partial \left( {A^{lm}\zeta } \right)}{\partial i} |
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279 | +\frac{1}{e_3} \frac{\partial v}{\partial k} |
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280 | \end{align*} |
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281 | |
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282 | Note Bene: introducing a rotation in \eqref{Apdx_B_Lap_U} does not lead to a |
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283 | useful expression for the iso/diapycnal Laplacian operator in the $z$-coordinate. |
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284 | Similarly, we did not found an expression of practical use for the geopotential |
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285 | horizontal/vertical Laplacian operator in the $s$-coordinate. Generally, |
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286 | \eqref{Apdx_B_Lap_U} is used in both $z$- and $s$-coordinate systems, that is |
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287 | a Laplacian diffusion is applied on momentum along the coordinate directions. |
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