- Timestamp:
- 2010-04-12T16:59:59+02:00 (14 years ago)
- File:
-
- 1 edited
Legend:
- Unmodified
- Added
- Removed
-
branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Chap_DOM.tex
r1224 r1831 31 31 as well as the DOM (DOMain) directory. 32 32 33 $\ $\newline % force a new ligne 34 33 35 % ================================================================ 34 36 % Fundamentals of the Discretisation … … 46 48 \begin{figure}[!tb] \label{Fig_cell} \begin{center} 47 49 \includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_cell.pdf} 48 \caption{Arrangement of variables. $ T$ indicates scalar points where temperature,50 \caption{Arrangement of variables. $t$ indicates scalar points where temperature, 49 51 salinity, density, pressure and horizontal divergence are defined. ($u$,$v$,$w$) 50 52 indicates vector points, and $f$ indicates vorticity points where both relative and … … 57 59 Special attention has been given to the homogeneity of the solution in the three 58 60 space directions. The arrangement of variables is the same in all directions. 59 It consists of cells centred on scalar points ($ T$, $S$, $p$, $\rho$) with vector61 It consists of cells centred on scalar points ($t$, $S$, $p$, $\rho$) with vector 60 62 points $(u, v, w)$ defined in the centre of each face of the cells (Fig. \ref{Fig_cell}). 61 63 This is the generalisation to three dimensions of the well-known ``C'' grid in … … 120 122 Similar operators are defined with respect to $i+1/2$, $j$, $j+1/2$, $k$, and 121 123 $k+1/2$. Following \eqref{Eq_PE_grad} and \eqref{Eq_PE_lap}, the gradient of a 122 variable $q$ defined at a $ T$-point has its three components defined at $u$-, $v$-123 and $w$-points while its Laplacien is defined at $ T$-point. These operators have124 variable $q$ defined at a $t$-point has its three components defined at $u$-, $v$- 125 and $w$-points while its Laplacien is defined at $t$-point. These operators have 124 126 the following discrete forms in the curvilinear $s$-coordinate system: 125 127 \begin{equation} \label{Eq_DOM_grad} 126 \nabla q\equiv \frac{1}{e_{1u} } \delta _{i+1/2} \left[ q \right]\;\,{\rm {\bf i}}127 + \frac{1}{e_{2v} }\delta _{j+1/2} \left[ q \right]\;\,{\rm {\bf j}}128 + \frac{1}{e_{3w} }\delta _{k+1/2} \left[ q \right]\;\,{\rm {\bf k}}128 \nabla q\equiv \frac{1}{e_{1u} } \delta _{i+1/2 } [q] \;\,{\rm {\bf i}} 129 + \frac{1}{e_{2v} } \delta _{j+1/2 } [q] \;\,{\rm {\bf j}} 130 + \frac{1}{e_{3w}} \delta _{k+1/2} [q] \;\,{\rm {\bf k}} 129 131 \end{equation} 130 132 \begin{multline} \label{Eq_DOM_lap} 131 \Delta q\equiv \frac{1}{e_{1T} {\kern 1pt}e_{2T} {\kern 1pt}e_{3T} }\;\left( 132 {\delta _i \left[ {\frac{e_{2u} e_{3u} }{e_{1u} }\;\delta _{i+1/2} 133 \left[ q \right]} \right] 134 +\delta _j \left[ {\frac{e_{1v} e_{3v} }{e_{2v} 135 }\;\delta _{j+1/2} \left[ q \right]} \right]\;} \right) \\ 136 +\frac{1}{e_{3T} }\delta _k \left[ {\frac{1}{e_{3w} }\;\delta _{k+1/2} 137 \left[ q \right]} \right] 133 \Delta q\equiv \frac{1}{e_{1t}\,e_{2t}\,e_{3t} } 134 \;\left( \delta_i \left[ \frac{e_{2u}\,e_{3u}} {e_{1u}} \;\delta_{i+1/2} [q] \right] 135 + \delta_j \left[ \frac{e_{1v}\,e_{3v}} {e_{2v}} \;\delta_{j+1/2} [q] \right] \; \right) \\ 136 +\frac{1}{e_{3t}} \delta_k \left[ \frac{1}{e_{3w} } \;\delta_{k+1/2} [q] \right] 138 137 \end{multline} 139 138 140 Following \eqref{Eq_PE_curl} and \eqref{Eq_PE_div}, a vector ${\rm {\bf A}}=\left( a_1,a_2,a_3\right)$ defined at vector points $(u,v,w)$ has its three curl 141 components defined at $vw$-, $uw$, and $f$-points, and its divergence defined 142 at $T$-points: 143 \begin{equation} \label{Eq_DOM_curl} 144 \begin{split} 145 \nabla \times {\rm {\bf A}}\equiv \frac{1}{e_{2v} {\kern 1pt}e_{3vw} 146 }{\kern 1pt}\,\;\left( {\delta _{j+1/2} \left[ {e_{3w} a_3 } \right]-\delta 147 _{k+1/2} \left[ {e_{2v} a_2 } \right]} \right) &\;\;{\rm {\bf i}} \\ 148 +\frac{1}{e_{2u} {\kern 1pt}e_{3uw} }\;\left( {\delta _{k+1/2} \left[ {e_{1u} a_1 } 149 \right]-\delta _{i+1/2} \left[ {e_{3w} a_3 } \right]} \right) &\;\;{\rm{\bf j}} \\ 150 +\frac{e_{3f} }{e_{1f} {\kern 1pt}e_{2f} }\,{\kern 1pt}\;\left( {\delta 151 _{i+1/2} \left[ {e_{2v} a_2 } \right]-\delta _{j+12} \left[ {e_{1u} a_1 } \right]} 152 \right) &\;\;{\rm {\bf k}} 153 \end{split} 154 \end{equation} 139 Following \eqref{Eq_PE_curl} and \eqref{Eq_PE_div}, a vector ${\rm {\bf A}}=\left( a_1,a_2,a_3\right)$ 140 defined at vector points $(u,v,w)$ has its three curl components defined at $vw$-, $uw$, 141 and $f$-points, and its divergence defined at $t$-points: 142 \begin{eqnarray} \label{Eq_DOM_curl} 143 \nabla \times {\rm {\bf A}}\equiv & 144 \frac{1}{e_{2v}\,e_{3vw} } \ \left( \delta_{j +1/2} \left[e_{3w}\,a_3 \right] -\delta_{k+1/2} \left[e_{2v} \,a_2 \right] \right) &\ \rm{\bf i} \\ 145 +& \frac{1}{e_{2u}\,e_{3uw}} \ \left( \delta_{k+1/2} \left[e_{1u}\,a_1 \right] -\delta_{i +1/2} \left[e_{3w}\,a_3 \right] \right) &\ \rm{\bf j} \\ 146 +& \frac{1}{e_{1f} \,e_{2f} } \ \left( \delta_{i +1/2} \left[e_{2v}\,a_2 \right] -\delta_{j +1/2} \left[e_{1u}\,a_1 \right] \right) &\ \rm{\bf k} 147 \end{eqnarray} 155 148 \begin{equation} \label{Eq_DOM_div} 156 \nabla \cdot {\rm {\bf A}}=\frac{1}{e_{1T} e_{2T} e_{3T} }\left( {\delta 157 _i \left[ {e_{2u} e_{3u} a_1 } \right]+\delta _j \left[ {e_{1v} e_{3v} a_2 } 158 \right]} \right)+\frac{1}{e_{3T} }\delta _k \left[ {a_3 } \right] 149 \nabla \cdot \rm{\bf A}=\frac{1}{e_{1t}\,e_{2t}\,e_{3t}}\left( \delta_i \left[e_{2u}\,e_{3u}\,a_1 \right] 150 +\delta_j \left[e_{1v}\,e_{3v}\,a_2 \right] \right)+\frac{1}{e_{3t} }\delta_k \left[a_3 \right] 159 151 \end{equation} 160 152 … … 164 156 horizontal location of a grid point. For example \eqref{Eq_DOM_div} reduces to: 165 157 \begin{equation*} 166 \nabla \cdot {\rm {\bf A}}=\frac{1}{e_{1T} e_{2T} }\left( {\delta167 _i \left[ {e_{2u} a_1 } \right]+\delta _j \left[ {e_{1v} a_2 } 168 \right]} \right)+\frac{1}{e_{3T} }\delta _k \left[ {a_3 }\right]158 \nabla \cdot \rm{\bf A}=\frac{1}{e_{1t}\,e_{2t}} \left( \delta_i \left[e_{2u}\,a_1 \right] 159 +\delta_j \left[e_{1v}\, a_2 \right] \right) 160 +\frac{1}{e_{3t}} \delta_k \left[ a_3 \right] 169 161 \end{equation*} 170 162 … … 172 164 for a quantity $q$ which is a masked field (i.e. equal to zero inside solid area): 173 165 \begin{equation} \label{DOM_bar} 174 \bar q = \frac{1}{H}\int_{k^b}^{k^o} {q\;e_{3q} \,dk}166 \bar q = \frac{1}{H} \int_{k^b}^{k^o} {q\;e_{3q} \,dk} 175 167 \equiv \frac{1}{H_q }\sum\limits_k {q\;e_{3q} } 176 168 \end{equation} … … 189 181 190 182 It is straightforward to demonstrate that these properties are verified locally in 191 discrete form as soon as the scalar $q$ is taken at $ T$-points and the vector183 discrete form as soon as the scalar $q$ is taken at $t$-points and the vector 192 184 \textbf{A} has its components defined at vector points $(u,v,w)$. 193 185 … … 230 222 The array representation used in the \textsc{Fortran} code requires an integer 231 223 indexing while the analytical definition of the mesh (see \S\ref{DOM_cell}) is 232 associated with the use of integer values for $ T$-points and both integer and224 associated with the use of integer values for $t$-points and both integer and 233 225 integer and a half values for all the other points. Therefore a specific integer 234 indexing must be defined for points other than $ T$-points ($i.e.$ velocity and226 indexing must be defined for points other than $t$-points ($i.e.$ velocity and 235 227 vorticity grid-points). Furthermore, the direction of the vertical indexing has 236 228 been changed so that the surface level is at $k=1$. … … 242 234 \label{DOM_Num_Index_hor} 243 235 244 The indexing in the horizontal plane has been chosen as shown in Fig.\ref{Fig_index_hor}. For an increasing $i$ index ($j$ index), the $T$-point245 and the eastward $u$-point (northward $v$-point) have the same index246 ( see the dashed area in Fig.\ref{Fig_index_hor}). A $T$-point and its247 nearest northeast $f$-point have the same $i$-and $j$-indices.236 The indexing in the horizontal plane has been chosen as shown in Fig.\ref{Fig_index_hor}. 237 For an increasing $i$ index ($j$ index), the $t$-point and the eastward $u$-point 238 (northward $v$-point) have the same index (see the dashed area in Fig.\ref{Fig_index_hor}). 239 A $t$-point and its nearest northeast $f$-point have the same $i$-and $j$-indices. 248 240 249 241 % ----------------------------------- … … 257 249 to the indexing used in the semi-discrete equations and given in \S\ref{DOM_cell}. 258 250 The sea surface corresponds to the $w$-level $k=1$ which is the same index 259 as $ T$-level just below (Fig.\ref{Fig_index_vert}). The last $w$-level ($k=jpk$)251 as $t$-level just below (Fig.\ref{Fig_index_vert}). The last $w$-level ($k=jpk$) 260 252 either corresponds to the ocean floor or is inside the bathymetry while the last 261 $ T$-level is always inside the bathymetry (Fig.\ref{Fig_index_vert}). Note that262 for an increasing $k$ index, a $w$-point and the $ T$-point just below have the253 $t$-level is always inside the bathymetry (Fig.\ref{Fig_index_vert}). Note that 254 for an increasing $k$ index, a $w$-point and the $t$-point just below have the 263 255 same $k$ index, in opposition to what is done in the horizontal plane where 264 it is the $ T$-point and the nearest velocity points in the direction of the horizontal256 it is the $t$-point and the nearest velocity points in the direction of the horizontal 265 257 axis that have the same $i$ or $j$ index (compare the dashed area in 266 258 Fig.\ref{Fig_index_hor} and \ref{Fig_index_vert}). Since the scale factors are … … 309 301 \S\ref{LBC_mpp}). 310 302 303 304 $\ $\newline % force a new ligne 305 311 306 % ================================================================ 312 307 % Domain: Horizontal Grid (mesh) … … 341 336 factors in the horizontal plane as follows: 342 337 \begin{flalign*} 343 \lambda_ T &\equiv \text{glamt}= \lambda(i) & \varphi_T&\equiv \text{gphit} = \varphi(j)\\338 \lambda_t &\equiv \text{glamt}= \lambda(i) & \varphi_t &\equiv \text{gphit} = \varphi(j)\\ 344 339 \lambda_u &\equiv \text{glamu}= \lambda(i+1/2)& \varphi_u &\equiv \text{gphiu}= \varphi(j)\\ 345 340 \lambda_v &\equiv \text{glamv}= \lambda(i) & \varphi_v &\equiv \text{gphiv} = \varphi(j+1/2)\\ … … 347 342 \end{flalign*} 348 343 \begin{flalign*} 349 e_{1 T} &\equiv \text{e1t} = r_a |\lambda'(i) \; \cos\varphi(j) |&350 e_{2 T} &\equiv \text{e2t} = r_a |\varphi'(j)| \\344 e_{1t} &\equiv \text{e1t} = r_a |\lambda'(i) \; \cos\varphi(j) |& 345 e_{2t} &\equiv \text{e2t} = r_a |\varphi'(j)| \\ 351 346 e_{1u} &\equiv \text{e1t} = r_a |\lambda'(i+1/2) \; \cos\varphi(j) |& 352 347 e_{2u} &\equiv \text{e2t} = r_a |\varphi'(j)|\\ … … 359 354 considered and $r_a$ is the earth radius (defined in \mdl{phycst} along with 360 355 all universal constants). Note that the horizontal position of and scale factors 361 at $w$-points are exactly equal to those of $ T$-points, thus no specific arrays356 at $w$-points are exactly equal to those of $t$-points, thus no specific arrays 362 357 are defined at $w$-points. 363 358 364 359 Note that the definition of the scale factors ($i.e.$ as the analytical first derivative 365 360 of the transformation that gives $(\lambda,\varphi,z)$ as a function of $(i,j,k)$) is 366 specific to the \NEMO model \citep{Marti 1992}. As an example, $e_{1T}$ is defined367 locally at a $ T$-point, whereas many other models on a C grid choose to define361 specific to the \NEMO model \citep{Marti_al_JGR92}. As an example, $e_{1t}$ is defined 362 locally at a $t$-point, whereas many other models on a C grid choose to define 368 363 such a scale factor as the distance between the $U$-points on each side of the 369 $ T$-point. Relying on an analytical transformation has two advantages: firstly, there364 $t$-point. Relying on an analytical transformation has two advantages: firstly, there 370 365 is no ambiguity in the scale factors appearing in the discrete equations, since they 371 366 are first introduced in the continuous equations; secondly, analytical transformations … … 380 375 in the vertical, and (b) analytically derived grid-point position and scale factors. For 381 376 both grids here, the same $w$-point depth has been chosen but in (a) the 382 $ T$-points are set half way between $w$-points while in (b) they are defined from377 $t$-points are set half way between $w$-points while in (b) they are defined from 383 378 an analytical function: $z(k)=5\,(i-1/2)^3 - 45\,(i-1/2)^2 + 140\,(i-1/2) - 150$. 384 379 Note the resulting difference between the value of the grid-size $\Delta_k$ and … … 422 417 and \pp{ppgphi0}). Note that for the Mercator grid the user need only provide 423 418 an approximate starting latitude: the real latitude will be recalculated analytically, 424 in order to ensure that the equator corresponds to line passing through $ T$-419 in order to ensure that the equator corresponds to line passing through $t$- 425 420 and $u$-points. 426 421 … … 431 426 and given by the parameters \pp{ppe1\_m} and \pp{ppe2\_m} respectively. 432 427 The zonal grid coordinate (\textit{glam} arrays) is in kilometers, starting at zero 433 with the first $ T$-point. The meridional coordinate (gphi. arrays) is in kilometers,434 and the second $ T$-point corresponds to coordinate $gphit=0$. The input428 with the first $t$-point. The meridional coordinate (gphi. arrays) is in kilometers, 429 and the second $t$-point corresponds to coordinate $gphit=0$. The input 435 430 parameter \pp{ppglam0} is ignored. \pp{ppgphi0} is used to set the reference 436 431 latitude for computation of the Coriolis parameter. In the case of the beta plane, … … 460 455 the output grid written when $\np{nmsh} \not=0$ is no more equal to the input grid. 461 456 457 $\ $\newline % force a new ligne 458 462 459 % ================================================================ 463 460 % Domain: Vertical Grid (domzgr) … … 467 464 \label{DOM_zgr} 468 465 %-----------------------------------------nam_zgr & namdom------------------------------------------- 469 \namdisplay{nam _zgr}466 \namdisplay{namzgr} 470 467 \namdisplay{namdom} 471 468 %------------------------------------------------------------------------------------------------------------- … … 593 590 594 591 The reference coordinate transformation $z_0 (k)$ defines the arrays $gdept_0$ 595 and $gdepw_0$ for $ T$- and $w$-points, respectively. As indicated on592 and $gdepw_0$ for $t$- and $w$-points, respectively. As indicated on 596 593 Fig.\ref{Fig_index_vert} \jp{jpk} is the number of $w$-levels. $gdepw_0(1)$ is the 597 ocean surface. There are at most \jp{jpk}-1 $ T$-points inside the ocean, the598 additional $ T$-point at $jk=jpk$ is below the sea floor and is not used.599 The vertical location of $w$- and $ T$-levels is defined from the analytic expression594 ocean surface. There are at most \jp{jpk}-1 $t$-points inside the ocean, the 595 additional $t$-point at $jk=jpk$ is below the sea floor and is not used. 596 The vertical location of $w$- and $t$-levels is defined from the analytic expression 600 597 of the depth $z_0(k)$ whose analytical derivative with respect to $k$ provides the 601 598 vertical scale factors. The user must provide the analytical expression of both … … 663 660 \begin{center} \begin{tabular}{c||r|r|r|r} 664 661 \hline 665 \textbf{LEVEL}& \textbf{ GDEPT}& \textbf{GDEPW}& \textbf{E3T }& \textbf{E3W} \\ \hline662 \textbf{LEVEL}& \textbf{gdept}& \textbf{gdepw}& \textbf{e3t }& \textbf{e3w } \\ \hline 666 663 1 & \textbf{ 5.00} & 0.00 & \textbf{ 10.00} & 10.00 \\ \hline 667 664 2 & \textbf{15.00} & 10.00 & \textbf{ 10.00} & 10.00 \\ \hline … … 728 725 Two variables in the namdom namelist are used to define the partial step 729 726 vertical grid. The mimimum water thickness (in meters) allowed for a cell 730 partially filled with bathymetry at level jk is the minimum of \np{ e3zpsmin}731 (thickness in meters, usually $20~m$) or $e_{3t}(jk)*\np{ e3zps\_rat}$ (a fraction,727 partially filled with bathymetry at level jk is the minimum of \np{rn\_e3zps\_min} 728 (thickness in meters, usually $20~m$) or $e_{3t}(jk)*\np{rn\_e3zps\_rat}$ (a fraction, 732 729 usually 10\%, of the default thickness $e_{3t}(jk)$). 733 730 … … 738 735 % ------------------------------------------------------------------------------------------------------------- 739 736 \subsection [$s$-coordinate (\np{ln\_sco})] 740 737 {$s$-coordinate (\np{ln\_sco}=true)} 741 738 \label{DOM_sco} 742 739 %------------------------------------------nam_zgr_sco--------------------------------------------------- 743 \namdisplay{nam _zgr_sco}740 \namdisplay{namzgr_sco} 744 741 %-------------------------------------------------------------------------------------------------------------- 745 742 In $s$-coordinate (\key{sco} is defined), the depth and thickness of the model … … 752 749 \end{split} 753 750 \end{equation} 754 where $h$ is the depth of the last $w$-level ($z_0(k)$) defined at the $ T$-point751 where $h$ is the depth of the last $w$-level ($z_0(k)$) defined at the $t$-point 755 752 location in the horizontal and $z_0(k)$ is a function which varies from $0$ at the sea 756 753 surface to $1$ at the ocean bottom. The depth field $h$ is not necessary the ocean … … 760 757 sharp bathymetric gradients. 761 758 762 A new flexible stretching function, modified from \citet{Song 1994} is provided as an example:759 A new flexible stretching function, modified from \citet{Song_Haidvogel_JCP94} is provided as an example: 763 760 \begin{equation} \label{DOM_sco_function} 764 761 \begin{split} … … 801 798 Whatever the vertical coordinate used, the model offers the possibility of 802 799 representing the bottom topography with steps that follow the face of the 803 model cells (step like topography) \citep{Madec 1996}. The distribution of800 model cells (step like topography) \citep{Madec_al_JPO96}. The distribution of 804 801 the steps in the horizontal is defined in a 2D integer array, mbathy, which 805 802 gives the number of ocean levels ($i.e.$ those that are not masked) at each 806 $ T$-point. mbathy is computed from the meter bathymetry using the definiton of807 gdept as the number of $ T$-points which gdept $\leq$ bathy. Note that in version803 $t$-point. mbathy is computed from the meter bathymetry using the definiton of 804 gdept as the number of $t$-points which gdept $\leq$ bathy. Note that in version 808 805 NEMO v2.3, the user still has to provide the "level" bathymetry in a NetCDF 809 806 file when using the full step option (\np{ln\_zco}), rather than the bathymetry … … 817 814 domain (\np{ln\_dynspg\_rl}=.true. and \key{island} is defined), the \textit{mbathy} 818 815 array must be provided and takes values from $-N$ to \jp{jpk}-1. It provides the 819 following information: $mbathy(i,j) = -n, \ n \in \left] 0,N \right]$, $ T$-points are820 land points on the $n^{th}$ island ; $mbathy(i,j) =0$, $ T$-points are land points821 on the main land (continent) ; $mbathy(i,j) =k$, the first $k$ $ T$- and $w$-points816 following information: $mbathy(i,j) = -n, \ n \in \left] 0,N \right]$, $t$-points are 817 land points on the $n^{th}$ island ; $mbathy(i,j) =0$, $t$-points are land points 818 on the main land (continent) ; $mbathy(i,j) =k$, the first $k$ $t$- and $w$-points 822 819 are ocean points, the others are points below the ocean floor. 823 820 … … 849 846 %%% 850 847 848 $\ $\newline % force a new ligne 849 851 850 % ================================================================ 852 851 % Time Discretisation … … 860 859 x^{t+\Delta t} = x^{t-\Delta t} + 2 \, \Delta t \ \text{RHS}_x^{t-\Delta t,t,t+\Delta t} 861 860 \end{equation} 862 where $x$ stands for $u$, $v$, $ T$ or $S$; RHS is the Right-Hand-Side of the861 where $x$ stands for $u$, $v$, $t$ or $S$; RHS is the Right-Hand-Side of the 863 862 corresponding time evolution equation; $\Delta t$ is the time step; and the 864 863 superscripts indicate the time at which a quantity is evaluated. Each term of the … … 995 994 996 995 This is diffusive in time and conditionally stable. For example, the 997 conditions for stability of second and fourth order horizontal diffusion schemes are \citep{Griffies 2004}:996 conditions for stability of second and fourth order horizontal diffusion schemes are \citep{Griffies_Bk04}: 998 997 \begin{equation} \label{Eq_DOM_nxt_euler_stability} 999 998 A^h < \left\{ … … 1040 1039 approximation of the temperature equation is: 1041 1040 \begin{equation} \label{Eq_DOM_nxt_imp_zdf} 1042 \frac{T(k)^{t+1}-T(k)^{t-1}}{2\;\Delta t}\equiv \text{RHS}+\frac{1}{e_{3 T} }\delta1041 \frac{T(k)^{t+1}-T(k)^{t-1}}{2\;\Delta t}\equiv \text{RHS}+\frac{1}{e_{3t} }\delta 1043 1042 _k \left[ {\frac{A_w^{vT} }{e_{3w} }\delta _{k+1/2} \left[ {T^{t+1}} \right]} 1044 1043 \right] … … 1109 1108 } %% end add 1110 1109 1110 1111 1112 1113 1114 Implicit time stepping in case of variable volume thickness. 1115 1116 Tracer case (NB for momentum in vector invariant form take care!) 1117 1118 \begin{flalign*} 1119 &\frac{\left( e_{3t}\,T \right)_k^{t+1}-\left( e_{3t}\,T \right)_k^{t-1}}{2\Delta t} 1120 \equiv \text{RHS}+ \delta _k \left[ {\frac{A_w^{vt} }{e_{3w}^{t+1} }\delta _{k+1/2} \left[ {T^{t+1}} \right]} 1121 \right] \\ 1122 &\left( e_{3t}\,T \right)_k^{t+1}-\left( e_{3t}\,T \right)_k^{t-1} 1123 \equiv {2\Delta t} \ \text{RHS}+ {2\Delta t} \ \delta _k \left[ {\frac{A_w^{vt} }{e_{3w}^{t+1} }\delta _{k+1/2} \left[ {T^{t+1}} \right]} 1124 \right] \\ 1125 &\left( e_{3t}\,T \right)_k^{t+1}-\left( e_{3t}\,T \right)_k^{t-1} 1126 \equiv 2\Delta t \ \text{RHS} 1127 + 2\Delta t \ \left\{ \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k+1/2} [ T_{k+1}^{t+1} - T_k ^{t+1} ] 1128 - \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k-1/2} [ T_k ^{t+1} - T_{k-1}^{t+1} ] \right\} \\ 1129 &\\ 1130 &\left( e_{3t}\,T \right)_k^{t+1} 1131 - {2\Delta t} \ \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k+1/2} T_{k+1}^{t+1} 1132 + {2\Delta t} \ \left\{ \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k+1/2} 1133 + \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k-1/2} \right\} T_{k }^{t+1} 1134 - {2\Delta t} \ \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k-1/2} T_{k-1}^{t+1} \\ 1135 &\equiv \left( e_{3t}\,T \right)_k^{t-1} + {2\Delta t} \ \text{RHS} \\ 1136 % 1137 \end{flalign*} 1138 1139 \begin{flalign*} 1140 \allowdisplaybreaks 1141 \intertext{ Tracer case } 1142 % 1143 & \qquad \qquad \quad - {2\Delta t} \ \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k+1/2} 1144 \qquad \qquad \qquad \qquad T_{k+1}^{t+1} \\ 1145 &+ {2\Delta t} \ \biggl\{ (e_{3t})_{k }^{t+1} \bigg. + \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k+1/2} 1146 + \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k-1/2} \bigg. \biggr\} \ \ \ T_{k }^{t+1} &&\\ 1147 & \qquad \qquad \qquad \qquad \qquad \quad \ \ - {2\Delta t} \ \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k-1/2} \quad \ \ T_{k-1}^{t+1} 1148 \ \equiv \ \left( e_{3t}\,T \right)_k^{t-1} + {2\Delta t} \ \text{RHS} \\ 1149 % 1150 \end{flalign*} 1151 \begin{flalign*} 1152 \allowdisplaybreaks 1153 \intertext{ Tracer content case } 1154 % 1155 & - {2\Delta t} \ & \frac{(A_w^{vt})_{k+1/2}} {(e_{3w})_{k+1/2}^{t+1}\;(e_{3t})_{k+1}^{t+1}} && \ \left( e_{3t}\,T \right)_{k+1}^{t+1} &\\ 1156 & + {2\Delta t} \ \left[ 1 \right.+ & \frac{(A_w^{vt})_{k+1/2}} {(e_{3w})_{k+1/2}^{t+1}\;(e_{3t})_k^{t+1}} 1157 + & \frac{(A_w^{vt})_{k -1/2}} {(e_{3w})_{k -1/2}^{t+1}\;(e_{3t})_k^{t+1}} \left. \right] & \left( e_{3t}\,T \right)_{k }^{t+1} &\\ 1158 & - {2\Delta t} \ & \frac{(A_w^{vt})_{k -1/2}} {(e_{3w})_{k -1/2}^{t+1}\;(e_{3t})_{k-1}^{t+1}} &\ \left( e_{3t}\,T \right)_{k-1}^{t+1} 1159 \equiv \left( e_{3t}\,T \right)_k^{t-1} + {2\Delta t} \ \text{RHS} & 1160 \end{flalign*} 1161
Note: See TracChangeset
for help on using the changeset viewer.
