1 | \documentclass[NEMO_book]{subfiles} |
---|
2 | \begin{document} |
---|
3 | |
---|
4 | % ================================================================ |
---|
5 | % Chapter 2 ——— Time Domain (step.F90) |
---|
6 | % ================================================================ |
---|
7 | \chapter{Time Domain (STP) } |
---|
8 | \label{STP} |
---|
9 | \minitoc |
---|
10 | |
---|
11 | % Missing things: |
---|
12 | % - daymod: definition of the time domain (nit000, nitend andd the calendar) |
---|
13 | |
---|
14 | |
---|
15 | \gmcomment{STEVEN :maybe a picture of the directory structure in the introduction |
---|
16 | which could be referred to here, would help ==> to be added} |
---|
17 | %%%% |
---|
18 | |
---|
19 | |
---|
20 | \newpage |
---|
21 | $\ $\newline % force a new ligne |
---|
22 | |
---|
23 | |
---|
24 | Having defined the continuous equations in Chap.~\ref{PE}, we need now to choose |
---|
25 | a time discretization, a key feature of an ocean model as it exerts a strong influence |
---|
26 | on the structure of the computer code ($i.e.$ on its flowchart). |
---|
27 | In the present chapter, we provide a general description of the \NEMO |
---|
28 | time stepping strategy and the consequences for the order in which the equations are |
---|
29 | solved. |
---|
30 | |
---|
31 | $\ $\newline % force a new ligne |
---|
32 | % ================================================================ |
---|
33 | % Time Discretisation |
---|
34 | % ================================================================ |
---|
35 | \section{Time stepping environment} |
---|
36 | \label{STP_environment} |
---|
37 | |
---|
38 | The time stepping used in \NEMO is a three level scheme that can be |
---|
39 | represented as follows: |
---|
40 | \begin{equation} \label{Eq_STP} |
---|
41 | x^{t+\rdt} = x^{t-\rdt} + 2 \, \rdt \ \text{RHS}_x^{t-\rdt,\,t,\,t+\rdt} |
---|
42 | \end{equation} |
---|
43 | where $x$ stands for $u$, $v$, $T$ or $S$; RHS is the Right-Hand-Side of the |
---|
44 | corresponding time evolution equation; $\rdt$ is the time step; and the |
---|
45 | superscripts indicate the time at which a quantity is evaluated. Each term of the |
---|
46 | RHS is evaluated at a specific time step depending on the physics with which |
---|
47 | it is associated. |
---|
48 | |
---|
49 | The choice of the time step used for this evaluation is discussed below as |
---|
50 | well as the implications for starting or restarting a model |
---|
51 | simulation. Note that the time stepping calculation is generally performed in a single |
---|
52 | operation. With such a complex and nonlinear system of equations it would be |
---|
53 | dangerous to let a prognostic variable evolve in time for each term separately. |
---|
54 | |
---|
55 | The three level scheme requires three arrays for each prognostic variable. |
---|
56 | For each variable $x$ there is $x_b$ (before), $x_n$ (now) and $x_a$. The third array, |
---|
57 | although referred to as $x_a$ (after) in the code, is usually not the variable at |
---|
58 | the after time step; but rather it is used to store the time derivative (RHS in |
---|
59 | \eqref{Eq_STP}) prior to time-stepping the equation. Generally, the time |
---|
60 | stepping is performed once at each time step in the \mdl{tranxt} and \mdl{dynnxt} |
---|
61 | modules, except when using implicit vertical diffusion or calculating sea surface height |
---|
62 | in which case time-splitting options are used. |
---|
63 | |
---|
64 | % ------------------------------------------------------------------------------------------------------------- |
---|
65 | % Non-Diffusive Part---Leapfrog Scheme |
---|
66 | % ------------------------------------------------------------------------------------------------------------- |
---|
67 | \section{Non-Diffusive Part --- Leapfrog Scheme} |
---|
68 | \label{STP_leap_frog} |
---|
69 | |
---|
70 | The time stepping used for processes other than diffusion is the well-known leapfrog |
---|
71 | scheme \citep{Mesinger_Arakawa_Bk76}. This scheme is widely used for advection |
---|
72 | processes in low-viscosity fluids. It is a time centred scheme, $i.e.$ |
---|
73 | the RHS in \eqref{Eq_STP} is evaluated at time step $t$, the now time step. |
---|
74 | It may be used for momentum and tracer advection, |
---|
75 | pressure gradient, and Coriolis terms, but not for diffusion terms. |
---|
76 | It is an efficient method that achieves |
---|
77 | second-order accuracy with just one right hand side evaluation per time step. |
---|
78 | Moreover, it does not artificially damp linear oscillatory motion nor does it produce |
---|
79 | instability by amplifying the oscillations. These advantages are somewhat diminished |
---|
80 | by the large phase-speed error of the leapfrog scheme, and the unsuitability |
---|
81 | of leapfrog differencing for the representation of diffusion and Rayleigh |
---|
82 | damping processes. However, the scheme allows the coexistence of a numerical |
---|
83 | and a physical mode due to its leading third order dispersive error. In other words a |
---|
84 | divergence of odd and even time steps may occur. To prevent it, the leapfrog scheme |
---|
85 | is often used in association with a Robert-Asselin time filter (hereafter the LF-RA scheme). |
---|
86 | This filter, first designed by \citet{Robert_JMSJ66} and more comprehensively studied |
---|
87 | by \citet{Asselin_MWR72}, is a kind of laplacian diffusion in time that mixes odd and |
---|
88 | even time steps: |
---|
89 | \begin{equation} \label{Eq_STP_asselin} |
---|
90 | x_F^t = x^t + \gamma \, \left[ x_F^{t-\rdt} - 2 x^t + x^{t+\rdt} \right] |
---|
91 | \end{equation} |
---|
92 | where the subscript $F$ denotes filtered values and $\gamma$ is the Asselin |
---|
93 | coefficient. $\gamma$ is initialized as \np{rn\_atfp} (namelist parameter). |
---|
94 | Its default value is \np{rn\_atfp}=$10^{-3}$ (see \S~\ref{STP_mLF}), |
---|
95 | causing only a weak dissipation of high frequency motions (\citep{Farge1987}). |
---|
96 | The addition of a time filter degrades the accuracy of the |
---|
97 | calculation from second to first order. However, the second order truncation |
---|
98 | error is proportional to $\gamma$, which is small compared to 1. Therefore, |
---|
99 | the LF-RA is a quasi second order accurate scheme. The LF-RA scheme |
---|
100 | is preferred to other time differencing schemes such as |
---|
101 | predictor corrector or trapezoidal schemes, because the user has an explicit |
---|
102 | and simple control of the magnitude of the time diffusion of the scheme. |
---|
103 | When used with the 2nd order space centred discretisation of the |
---|
104 | advection terms in the momentum and tracer equations, LF-RA avoids implicit |
---|
105 | numerical diffusion: diffusion is set explicitly by the user through the Robert-Asselin |
---|
106 | filter parameter and the viscosity and diffusion coefficients. |
---|
107 | |
---|
108 | % ------------------------------------------------------------------------------------------------------------- |
---|
109 | % Diffusive Part---Forward or Backward Scheme |
---|
110 | % ------------------------------------------------------------------------------------------------------------- |
---|
111 | \section{Diffusive Part --- Forward or Backward Scheme} |
---|
112 | \label{STP_forward_imp} |
---|
113 | |
---|
114 | The leapfrog differencing scheme is unsuitable for the representation of |
---|
115 | diffusion and damping processes. For a tendancy $D_x$, representing a |
---|
116 | diffusion term or a restoring term to a tracer climatology |
---|
117 | (when present, see \S~\ref{TRA_dmp}), a forward time differencing scheme |
---|
118 | is used : |
---|
119 | \begin{equation} \label{Eq_STP_euler} |
---|
120 | x^{t+\rdt} = x^{t-\rdt} + 2 \, \rdt \ {D_x}^{t-\rdt} |
---|
121 | \end{equation} |
---|
122 | |
---|
123 | This is diffusive in time and conditionally stable. The |
---|
124 | conditions for stability of second and fourth order horizontal diffusion schemes are \citep{Griffies_Bk04}: |
---|
125 | \begin{equation} \label{Eq_STP_euler_stability} |
---|
126 | A^h < \left\{ |
---|
127 | \begin{aligned} |
---|
128 | &\frac{e^2}{ 8 \; \rdt } &&\quad \text{laplacian diffusion} \\ |
---|
129 | &\frac{e^4}{64 \; \rdt } &&\quad \text{bilaplacian diffusion} |
---|
130 | \end{aligned} |
---|
131 | \right. |
---|
132 | \end{equation} |
---|
133 | where $e$ is the smallest grid size in the two horizontal directions and $A^h$ is |
---|
134 | the mixing coefficient. The linear constraint \eqref{Eq_STP_euler_stability} |
---|
135 | is a necessary condition, but not sufficient. If it is not satisfied, even mildly, |
---|
136 | then the model soon becomes wildly unstable. The instability can be removed |
---|
137 | by either reducing the length of the time steps or reducing the mixing coefficient. |
---|
138 | |
---|
139 | For the vertical diffusion terms, a forward time differencing scheme can be |
---|
140 | used, but usually the numerical stability condition imposes a strong |
---|
141 | constraint on the time step. Two solutions are available in \NEMO to overcome |
---|
142 | the stability constraint: $(a)$ a forward time differencing scheme using a |
---|
143 | time splitting technique (\np{ln\_zdfexp} = true) or $(b)$ a backward (or implicit) |
---|
144 | time differencing scheme (\np{ln\_zdfexp} = false). In $(a)$, the master |
---|
145 | time step $\Delta $t is cut into $N$ fractional time steps so that the |
---|
146 | stability criterion is reduced by a factor of $N$. The computation is performed as |
---|
147 | follows: |
---|
148 | \begin{equation} \label{Eq_STP_ts} |
---|
149 | \begin{split} |
---|
150 | & x_\ast ^{t-\rdt} = x^{t-\rdt} \\ |
---|
151 | & x_\ast ^{t-\rdt+L\frac{2\rdt}{N}}=x_\ast ^{t-\rdt+\left( {L-1} |
---|
152 | \right)\frac{2\rdt}{N}}+\frac{2\rdt}{N}\;\text{DF}^{t-\rdt+\left( {L-1} \right)\frac{2\rdt}{N}} |
---|
153 | \quad \text{for $L=1$ to $N$} \\ |
---|
154 | & x^{t+\rdt} = x_\ast^{t+\rdt} |
---|
155 | \end{split} |
---|
156 | \end{equation} |
---|
157 | with DF a vertical diffusion term. The number of fractional time steps, $N$, is given |
---|
158 | by setting \np{nn\_zdfexp}, (namelist parameter). The scheme $(b)$ is unconditionally |
---|
159 | stable but diffusive. It can be written as follows: |
---|
160 | \begin{equation} \label{Eq_STP_imp} |
---|
161 | x^{t+\rdt} = x^{t-\rdt} + 2 \, \rdt \ \text{RHS}_x^{t+\rdt} |
---|
162 | \end{equation} |
---|
163 | |
---|
164 | %%gm |
---|
165 | %%gm UPDATE the next paragraphs with time varying thickness ... |
---|
166 | %%gm |
---|
167 | |
---|
168 | This scheme is rather time consuming since it requires a matrix inversion, |
---|
169 | but it becomes attractive since a value of 3 or more is needed for N in |
---|
170 | the forward time differencing scheme. For example, the finite difference |
---|
171 | approximation of the temperature equation is: |
---|
172 | \begin{equation} \label{Eq_STP_imp_zdf} |
---|
173 | \frac{T(k)^{t+1}-T(k)^{t-1}}{2\;\rdt}\equiv \text{RHS}+\frac{1}{e_{3t} }\delta |
---|
174 | _k \left[ {\frac{A_w^{vT} }{e_{3w} }\delta _{k+1/2} \left[ {T^{t+1}} \right]} |
---|
175 | \right] |
---|
176 | \end{equation} |
---|
177 | where RHS is the right hand side of the equation except for the vertical diffusion term. |
---|
178 | We rewrite \eqref{Eq_STP_imp} as: |
---|
179 | \begin{equation} \label{Eq_STP_imp_mat} |
---|
180 | -c(k+1)\;T^{t+1}(k+1) + d(k)\;T^{t+1}(k) - \;c(k)\;T^{t+1}(k-1) \equiv b(k) |
---|
181 | \end{equation} |
---|
182 | where |
---|
183 | \begin{align*} |
---|
184 | c(k) &= A_w^{vT} (k) \, / \, e_{3w} (k) \\ |
---|
185 | d(k) &= e_{3t} (k) \, / \, (2\rdt) + c_k + c_{k+1} \\ |
---|
186 | b(k) &= e_{3t} (k) \; \left( T^{t-1}(k) \, / \, (2\rdt) + \text{RHS} \right) |
---|
187 | \end{align*} |
---|
188 | |
---|
189 | \eqref{Eq_STP_imp_mat} is a linear system of equations with an associated |
---|
190 | matrix which is tridiagonal. Moreover, $c(k)$ and $d(k)$ are positive and the diagonal |
---|
191 | term is greater than the sum of the two extra-diagonal terms, therefore a special |
---|
192 | adaptation of the Gauss elimination procedure is used to find the solution |
---|
193 | (see for example \citet{Richtmyer1967}). |
---|
194 | |
---|
195 | |
---|
196 | |
---|
197 | % ------------------------------------------------------------------------------------------------------------- |
---|
198 | % Surface Pressure gradient |
---|
199 | % ------------------------------------------------------------------------------------------------------------- |
---|
200 | \section{Surface Pressure Gradient} |
---|
201 | \label{STP_spg_ts} |
---|
202 | |
---|
203 | ===>>>> TO BE written.... :-) |
---|
204 | |
---|
205 | %\gmcomment{ |
---|
206 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
207 | \begin{figure}[!t] \begin{center} |
---|
208 | \includegraphics[width=0.7\textwidth]{Fig_TimeStepping_flowchart} |
---|
209 | \caption{ \label{Fig_TimeStep_flowchart} |
---|
210 | Sketch of the leapfrog time stepping sequence in \NEMO from \citet{Leclair_Madec_OM09}. |
---|
211 | The use of a semi-implicit computation of the hydrostatic pressure gradient requires |
---|
212 | the tracer equation to be stepped forward prior to the momentum equation. |
---|
213 | The need for knowledge of the vertical scale factor (here denoted as $h$) |
---|
214 | requires the sea surface height and the continuity equation to be stepped forward |
---|
215 | prior to the computation of the tracer equation. |
---|
216 | Note that the method for the evaluation of the surface pressure gradient $\nabla p_s$ is not presented here |
---|
217 | (see \S~\ref{DYN_spg}). } |
---|
218 | \end{center} \end{figure} |
---|
219 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
220 | %} |
---|
221 | |
---|
222 | % ------------------------------------------------------------------------------------------------------------- |
---|
223 | % The Modified Leapfrog -- Asselin Filter scheme |
---|
224 | % ------------------------------------------------------------------------------------------------------------- |
---|
225 | \section{The Modified Leapfrog -- Asselin Filter scheme} |
---|
226 | \label{STP_mLF} |
---|
227 | |
---|
228 | Significant changes have been introduced by \cite{Leclair_Madec_OM09} in the |
---|
229 | LF-RA scheme in order to ensure tracer conservation and to allow the use of |
---|
230 | a much smaller value of the Asselin filter parameter. The modifications affect |
---|
231 | both the forcing and filtering treatments in the LF-RA scheme. |
---|
232 | |
---|
233 | In a classical LF-RA environment, the forcing term is centred in time, $i.e.$ |
---|
234 | it is time-stepped over a $2\rdt$ period: $x^t = x^t + 2\rdt Q^t $ where $Q$ |
---|
235 | is the forcing applied to $x$, and the time filter is given by \eqref{Eq_STP_asselin} |
---|
236 | so that $Q$ is redistributed over several time step. |
---|
237 | In the modified LF-RA environment, these two formulations have been replaced by: |
---|
238 | \begin{align} |
---|
239 | x^{t+\rdt} &= x^{t-\rdt} + \rdt \left( Q^{t-\rdt/2} + Q^{t+\rdt/2} \right) \label{Eq_STP_forcing} \\ |
---|
240 | % |
---|
241 | x_F^t &= x^t + \gamma \, \left[ x_F^{t-\rdt} - 2 x^t + x^{t+\rdt} \right] |
---|
242 | - \gamma\,\rdt \, \left[ Q^{t+\rdt/2} - Q^{t-\rdt/2} \right] \label{Eq_STP_RA} |
---|
243 | \end{align} |
---|
244 | The change in the forcing formulation given by \eqref{Eq_STP_forcing} |
---|
245 | (see Fig.\ref{Fig_MLF_forcing}) has a significant effect: the forcing term no |
---|
246 | longer excites the divergence of odd and even time steps \citep{Leclair_Madec_OM09}. |
---|
247 | % forcing seen by the model.... |
---|
248 | This property improves the LF-RA scheme in two respects. |
---|
249 | First, the LF-RA can now ensure the local and global conservation of tracers. |
---|
250 | Indeed, time filtering is no longer required on the forcing part. The influence of |
---|
251 | the Asselin filter on the forcing is be removed by adding a new term in the filter |
---|
252 | (last term in \eqref{Eq_STP_RA} compared to \eqref{Eq_STP_asselin}). Since |
---|
253 | the filtering of the forcing was the source of non-conservation in the classical |
---|
254 | LF-RA scheme, the modified formulation becomes conservative \citep{Leclair_Madec_OM09}. |
---|
255 | Second, the LF-RA becomes a truly quasi-second order scheme. Indeed, |
---|
256 | \eqref{Eq_STP_forcing} used in combination with a careful treatment of static |
---|
257 | instability (\S\ref{ZDF_evd}) and of the TKE physics (\S\ref{ZDF_tke_ene}), |
---|
258 | the two other main sources of time step divergence, allows a reduction by |
---|
259 | two orders of magnitude of the Asselin filter parameter. |
---|
260 | |
---|
261 | Note that the forcing is now provided at the middle of a time step: $Q^{t+\rdt/2}$ |
---|
262 | is the forcing applied over the $[t,t+\rdt]$ time interval. This and the change |
---|
263 | in the time filter, \eqref{Eq_STP_RA}, allows an exact evaluation of the |
---|
264 | contribution due to the forcing term between any two time steps, |
---|
265 | even if separated by only $\rdt$ since the time filter is no longer applied to the |
---|
266 | forcing term. |
---|
267 | |
---|
268 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
269 | \begin{figure}[!t] \begin{center} |
---|
270 | \includegraphics[width=0.90\textwidth]{Fig_MLF_forcing} |
---|
271 | \caption{ \label{Fig_MLF_forcing} |
---|
272 | Illustration of forcing integration methods. |
---|
273 | (top) ''Traditional'' formulation : the forcing is defined at the same time as the variable |
---|
274 | to which it is applied (integer value of the time step index) and it is applied over a $2\rdt$ period. |
---|
275 | (bottom) modified formulation : the forcing is defined in the middle of the time (integer and a half |
---|
276 | value of the time step index) and the mean of two successive forcing values ($n-1/2$, $n+1/2$). |
---|
277 | is applied over a $2\rdt$ period.} |
---|
278 | \end{center} \end{figure} |
---|
279 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
280 | |
---|
281 | % ------------------------------------------------------------------------------------------------------------- |
---|
282 | % Start/Restart strategy |
---|
283 | % ------------------------------------------------------------------------------------------------------------- |
---|
284 | \section{Start/Restart strategy} |
---|
285 | \label{STP_rst} |
---|
286 | %--------------------------------------------namrun------------------------------------------- |
---|
287 | \namdisplay{namrun} |
---|
288 | %-------------------------------------------------------------------------------------------------------------- |
---|
289 | |
---|
290 | The first time step of this three level scheme when starting from initial conditions |
---|
291 | is a forward step (Euler time integration): |
---|
292 | \begin{equation} \label{Eq_DOM_euler} |
---|
293 | x^1 = x^0 + \rdt \ \text{RHS}^0 |
---|
294 | \end{equation} |
---|
295 | This is done simply by keeping the leapfrog environment ($i.e.$ the \eqref{Eq_STP} |
---|
296 | three level time stepping) but setting all $x^0$ (\textit{before}) and $x^{1}$ (\textit{now}) fields |
---|
297 | equal at the first time step and using half the value of $\rdt$. |
---|
298 | |
---|
299 | It is also possible to restart from a previous computation, by using a |
---|
300 | restart file. The restart strategy is designed to ensure perfect |
---|
301 | restartability of the code: the user should obtain the same results to |
---|
302 | machine precision either by running the model for $2N$ time steps in one go, |
---|
303 | or by performing two consecutive experiments of $N$ steps with a restart. |
---|
304 | This requires saving two time levels and many auxiliary data in the restart |
---|
305 | files in machine precision. |
---|
306 | |
---|
307 | Note that when a semi-implicit scheme is used to evaluate the hydrostatic pressure |
---|
308 | gradient (see \S\ref{DYN_hpg_imp}), an extra three-dimensional field has to be |
---|
309 | added to the restart file to ensure an exact restartability. This is done optionally |
---|
310 | via the \np{nn\_dynhpg\_rst} namelist parameter, so that the size of the |
---|
311 | restart file can be reduced when restartability is not a key issue (operational |
---|
312 | oceanography or in ensemble simulations for seasonal forecasting). |
---|
313 | |
---|
314 | Note the size of the time step used, $\rdt$, is also saved in the restart file. |
---|
315 | When restarting, if the the time step has been changed, a restart using an Euler time |
---|
316 | stepping scheme is imposed. |
---|
317 | Options are defined through the \ngn{namrun} namelist variables. |
---|
318 | %%% |
---|
319 | \gmcomment{ |
---|
320 | add here how to force the restart to contain only one time step for operational purposes |
---|
321 | |
---|
322 | add also the idea of writing several restart for seasonal forecast : how is it done ? |
---|
323 | |
---|
324 | verify that all namelist pararmeters are truly described |
---|
325 | |
---|
326 | a word on the check of restart ..... |
---|
327 | } |
---|
328 | %%% |
---|
329 | |
---|
330 | \gmcomment{ % add a subsection here |
---|
331 | |
---|
332 | %------------------------------------------------------------------------------------------------------------- |
---|
333 | % Time Domain |
---|
334 | % ------------------------------------------------------------------------------------------------------------- |
---|
335 | \subsection{Time domain} |
---|
336 | \label{STP_time} |
---|
337 | %--------------------------------------------namrun------------------------------------------- |
---|
338 | \namdisplay{namdom} |
---|
339 | %-------------------------------------------------------------------------------------------------------------- |
---|
340 | |
---|
341 | Options are defined through the \ngn{namdom} namelist variables. |
---|
342 | \colorbox{yellow}{add here a few word on nit000 and nitend} |
---|
343 | |
---|
344 | \colorbox{yellow}{Write documentation on the calendar and the key variable adatrj} |
---|
345 | |
---|
346 | add a description of daymod, and the model calandar (leap-year and co) |
---|
347 | |
---|
348 | } %% end add |
---|
349 | |
---|
350 | |
---|
351 | |
---|
352 | %% |
---|
353 | \gmcomment{ % add implicit in vvl case and Crant-Nicholson scheme |
---|
354 | |
---|
355 | Implicit time stepping in case of variable volume thickness. |
---|
356 | |
---|
357 | Tracer case (NB for momentum in vector invariant form take care!) |
---|
358 | |
---|
359 | \begin{flalign*} |
---|
360 | &\frac{\left( e_{3t}\,T \right)_k^{t+1}-\left( e_{3t}\,T \right)_k^{t-1}}{2\rdt} |
---|
361 | \equiv \text{RHS}+ \delta _k \left[ {\frac{A_w^{vt} }{e_{3w}^{t+1} }\delta _{k+1/2} \left[ {T^{t+1}} \right]} |
---|
362 | \right] \\ |
---|
363 | &\left( e_{3t}\,T \right)_k^{t+1}-\left( e_{3t}\,T \right)_k^{t-1} |
---|
364 | \equiv {2\rdt} \ \text{RHS}+ {2\rdt} \ \delta _k \left[ {\frac{A_w^{vt} }{e_{3w}^{t+1} }\delta _{k+1/2} \left[ {T^{t+1}} \right]} |
---|
365 | \right] \\ |
---|
366 | &\left( e_{3t}\,T \right)_k^{t+1}-\left( e_{3t}\,T \right)_k^{t-1} |
---|
367 | \equiv 2\rdt \ \text{RHS} |
---|
368 | + 2\rdt \ \left\{ \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k+1/2} [ T_{k+1}^{t+1} - T_k ^{t+1} ] |
---|
369 | - \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k-1/2} [ T_k ^{t+1} - T_{k-1}^{t+1} ] \right\} \\ |
---|
370 | &\\ |
---|
371 | &\left( e_{3t}\,T \right)_k^{t+1} |
---|
372 | - {2\rdt} \ \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k+1/2} T_{k+1}^{t+1} |
---|
373 | + {2\rdt} \ \left\{ \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k+1/2} |
---|
374 | + \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k-1/2} \right\} T_{k }^{t+1} |
---|
375 | - {2\rdt} \ \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k-1/2} T_{k-1}^{t+1} \\ |
---|
376 | &\equiv \left( e_{3t}\,T \right)_k^{t-1} + {2\rdt} \ \text{RHS} \\ |
---|
377 | % |
---|
378 | \end{flalign*} |
---|
379 | |
---|
380 | \begin{flalign*} |
---|
381 | \allowdisplaybreaks |
---|
382 | \intertext{ Tracer case } |
---|
383 | % |
---|
384 | & \qquad \qquad \quad - {2\rdt} \ \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k+1/2} |
---|
385 | \qquad \qquad \qquad \qquad T_{k+1}^{t+1} \\ |
---|
386 | &+ {2\rdt} \ \biggl\{ (e_{3t})_{k }^{t+1} \bigg. + \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k+1/2} |
---|
387 | + \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k-1/2} \bigg. \biggr\} \ \ \ T_{k }^{t+1} &&\\ |
---|
388 | & \qquad \qquad \qquad \qquad \qquad \quad \ \ - {2\rdt} \ \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k-1/2} \quad \ \ T_{k-1}^{t+1} |
---|
389 | \ \equiv \ \left( e_{3t}\,T \right)_k^{t-1} + {2\rdt} \ \text{RHS} \\ |
---|
390 | % |
---|
391 | \end{flalign*} |
---|
392 | \begin{flalign*} |
---|
393 | \allowdisplaybreaks |
---|
394 | \intertext{ Tracer content case } |
---|
395 | % |
---|
396 | & - {2\rdt} \ & \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} &\\ |
---|
397 | & + {2\rdt} \ \left[ 1 \right.+ & \frac{(A_w^{vt})_{k+1/2}} {(e_{3w})_{k+1/2}^{t+1}\;(e_{3t})_k^{t+1}} |
---|
398 | + & \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} &\\ |
---|
399 | & - {2\rdt} \ & \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} |
---|
400 | \equiv \left( e_{3t}\,T \right)_k^{t-1} + {2\rdt} \ \text{RHS} & |
---|
401 | \end{flalign*} |
---|
402 | |
---|
403 | %% |
---|
404 | } |
---|
405 | %% |
---|
406 | \end{document} |
---|