Changeset 11544 for NEMO/trunk/doc/latex/NEMO/subfiles/apdx_algos.tex
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NEMO/trunk/doc/latex/NEMO/subfiles/apdx_algos.tex
r11543 r11544 18 18 % ------------------------------------------------------------------------------------------------------------- 19 19 \section{Upstream Biased Scheme (UBS) (\protect\np{ln\_traadv\_ubs}\forcode{ = .true.})} 20 \label{sec: TRA_adv_ubs}20 \label{sec:ALGOS_tra_adv_ubs} 21 21 22 22 The UBS advection scheme is an upstream biased third order scheme based on … … 25 25 For example, in the $i$-direction: 26 26 \begin{equation} 27 \label{eq: tra_adv_ubs2}27 \label{eq:ALGOS_tra_adv_ubs2} 28 28 \tau_u^{ubs} = \left\{ 29 29 \begin{aligned} … … 35 35 or equivalently, the advective flux is 36 36 \begin{equation} 37 \label{eq: tra_adv_ubs2}37 \label{eq:ALGOS_tra_adv_ubs2} 38 38 U_{i+1/2} \ \tau_u^{ubs} 39 39 =U_{i+1/2} \ \overline{ T_i - \frac{1}{6}\,\tau"_i }^{\,i+1/2} … … 85 85 NB 3: It is straight forward to rewrite \autoref{eq:TRA_adv_ubs} as follows: 86 86 \begin{equation} 87 \label{eq: tra_adv_ubs2}87 \label{eq:ALGOS_tra_adv_ubs2} 88 88 \tau_u^{ubs} = \left\{ 89 89 \begin{aligned} … … 95 95 or equivalently 96 96 \begin{equation} 97 \label{eq: tra_adv_ubs2}97 \label{eq:ALGOS_tra_adv_ubs2} 98 98 \begin{split} 99 99 e_{2u} e_{3u}\,u_{i+1/2} \ \tau_u^{ubs} … … 112 112 laplacian diffusion: 113 113 \begin{equation} 114 \label{eq: tra_ldf_lap}114 \label{eq:ALGOS_tra_ldf_lap} 115 115 \begin{split} 116 116 D_T^{lT} =\frac{1}{e_{1T} \; e_{2T}\; e_{3T} } &\left[ {\quad \delta_i … … 125 125 bilaplacian: 126 126 \begin{equation} 127 \label{eq: tra_ldf_lap}127 \label{eq:ALGOS_tra_ldf_lap} 128 128 \begin{split} 129 129 D_T^{lT} =&-\frac{1}{e_{1T} \; e_{2T}\; e_{3T}} \\ … … 138 138 it comes: 139 139 \begin{equation} 140 \label{eq: tra_ldf_lap}140 \label{eq:ALGOS_tra_ldf_lap} 141 141 \begin{split} 142 142 D_T^{lT} =&-\frac{1}{12}\,\frac{1}{e_{1T} \; e_{2T}\; e_{3T}} \\ … … 149 149 if the velocity is uniform (\ie\ $|u|=cst$) then the diffusive flux is 150 150 \begin{equation} 151 \label{eq: tra_ldf_lap}151 \label{eq:ALGOS_tra_ldf_lap} 152 152 \begin{split} 153 153 F_u^{lT} = - \frac{1}{12} … … 161 161 162 162 \begin{equation} 163 \label{eq: tra_adv_ubs2}163 \label{eq:ALGOS_tra_adv_ubs2} 164 164 \begin{split} 165 165 F_u^{lT} &= - \frac{1}{2} e_{2u} e_{3u}\,|u|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i] … … 171 171 sol 1 coefficient at T-point ( add $e_{1u}$ and $e_{1T}$ on both side of first $\delta$): 172 172 \begin{equation} 173 \label{eq: tra_adv_ubs2}173 \label{eq:ALGOS_tra_adv_ubs2} 174 174 \begin{split} 175 175 F_u^{lT} &= - \frac{1}{12} \frac{e_{2u} e_{3u}}{e_{1u}}\;\delta_{i+1/2}\left[ \frac{e_{1T}^3\,|u|}{e_{1T}e_{2T}\,e_{3T}}\,\delta_i \left[ \frac{e_{2u} e_{3u} }{e_{1u} } \delta_{i+1/2}[\tau] \right] \right] … … 180 180 sol 2 coefficient at u-point: split $|u|$ into $\sqrt{|u|}$ and $e_{1T}$ into $\sqrt{e_{1u}}$ 181 181 \begin{equation} 182 \label{eq: tra_adv_ubs2}182 \label{eq:ALGOS_tra_adv_ubs2} 183 183 \begin{split} 184 184 F_u^{lT} &= - \frac{1}{12} {e_{1u}}^1 \sqrt{e_{1u}|u|} \frac{e_{2u} e_{3u}}{e_{1u}}\;\delta_{i+1/2}\left[ \frac{1}{e_{2T}\,e_{3T}}\,\delta_i \left[ \sqrt{e_{1u}|u|} \frac{e_{2u} e_{3u} }{e_{1u} } \delta_{i+1/2}[\tau] \right] \right] \\ … … 192 192 % ------------------------------------------------------------------------------------------------------------- 193 193 \section{Leapfrog energetic} 194 \label{sec: LF}194 \label{sec:ALGOS_LF} 195 195 196 196 We adopt the following semi-discrete notation for time derivative. … … 198 198 the time derivation and averaging operators at the mid time step are: 199 199 \[ 200 % \label{eq: dt_mt}200 % \label{eq:ALGOS_dt_mt} 201 201 \begin{split} 202 202 \delta_{t+\rdt/2} [q] &= \ \ \, q^{t+\rdt} - q^{t} \\ … … 210 210 The Leap-frog time stepping given by \autoref{eq:DOM_nxt} can be defined as: 211 211 \[ 212 % \label{eq: LF}212 % \label{eq:ALGOS_LF} 213 213 \frac{\partial q}{\partial t} 214 214 \equiv \frac{1}{\rdt} \overline{ \delta_{t+\rdt/2}[q]}^{\,t} … … 220 220 As such it respects the quadratic invariant in integral forms, \ie\ the following continuous property, 221 221 \[ 222 % \label{eq: Energy}222 % \label{eq:ALGOS_Energy} 223 223 \int_{t_0}^{t_1} {q\, \frac{\partial q}{\partial t} \;dt} 224 224 =\int_{t_0}^{t_1} {\frac{1}{2}\, \frac{\partial q^2}{\partial t} \;dt} … … 278 278 For example in the (\textbf{i},\textbf{k}) plane, the four triads are defined at the $(i,k)$ $T$-point as follows: 279 279 \begin{equation} 280 \label{eq: Gf_triads}280 \label{eq:ALGOS_Gf_triads} 281 281 _i^k \mathbb{T}_{i_p}^{k_p} (T) 282 282 = \frac{1}{4} \ {b_u}_{\,i+i_p}^{\,k} \ A_i^k \left( … … 291 291 and $_i^k \mathbb{R}_{i_p}^{k_p}$ is the slope associated with each triad: 292 292 \begin{equation} 293 \label{eq: Gf_slopes}293 \label{eq:ALGOS_Gf_slopes} 294 294 _i^k \mathbb{R}_{i_p}^{k_p} 295 295 =\frac{ {e_{3w}}_{\,i}^{\,k+k_p}} { {e_{1u}}_{\,i+i_p}^{\,k}} \ \frac … … 297 297 {\left(\alpha / \beta \right)_i^k \ \delta_{k+k_p}[T^i ] - \delta_{k+k_p}[S^i ] } 298 298 \end{equation} 299 Note that in \autoref{eq: Gf_slopes} we use the ratio $\alpha / \beta$ instead of299 Note that in \autoref{eq:ALGOS_Gf_slopes} we use the ratio $\alpha / \beta$ instead of 300 300 multiplying the temperature derivative by $\alpha$ and the salinity derivative by $\beta$. 301 301 This is more efficient as the ratio $\alpha / \beta$ can to be evaluated directly. 302 302 303 Note that in \autoref{eq: Gf_triads}, we chose to use ${b_u}_{\,i+i_p}^{\,k}$ instead of ${b_{uw}}_{\,i+i_p}^{\,k+k_p}$.303 Note that in \autoref{eq:ALGOS_Gf_triads}, we chose to use ${b_u}_{\,i+i_p}^{\,k}$ instead of ${b_{uw}}_{\,i+i_p}^{\,k+k_p}$. 304 304 This choice has been motivated by the decrease of tracer variance and 305 the presence of partial cell at the ocean bottom (see \autoref{ apdx:Gf_operator}).305 the presence of partial cell at the ocean bottom (see \autoref{subsec:ALGOS_Gf_operator}). 306 306 307 307 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 310 310 \includegraphics[width=\textwidth]{Fig_ISO_triad} 311 311 \caption{ 312 \protect\label{fig: ISO_triad}312 \protect\label{fig:ALGOS_ISO_triad} 313 313 Triads used in the Griffies's like iso-neutral diffision scheme for 314 314 $u$-component (upper panel) and $w$-component (lower panel). … … 321 321 They take the following expression: 322 322 \begin{flalign*} 323 % \label{eq: Gf_fluxes}323 % \label{eq:ALGOS_Gf_fluxes} 324 324 \begin{split} 325 325 {_i^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) … … 334 334 the sum of the fluxes that cross the $u$- and $w$-face (\autoref{fig:TRIADS_ISO_triad}): 335 335 \begin{flalign} 336 \label{eq: iso_flux}336 \label{eq:ALGOS_iso_flux} 337 337 \textbf{F}_{iso}(T) 338 338 &\equiv \sum_{\substack{i_p,\,k_p}} … … 364 364 the divergence of the sum of all the four triad fluxes: 365 365 \begin{equation} 366 \label{eq: Gf_operator}366 \label{eq:ALGOS_Gf_operator} 367 367 D_l^T = \frac{1}{b_T} \sum_{\substack{i_p,\,k_p}} \left\{ 368 368 \delta_{i} \left[{_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \right] … … 377 377 the limit of flat iso-neutral direction: 378 378 \[ 379 % \label{eq: Gf_property1a}379 % \label{eq:ALGOS_Gf_property1a} 380 380 D_l^T = \frac{1}{b_T} \ \delta_{i} 381 381 \left[ \frac{e_{2u}\,e_{3u}}{e_{1u}} \; \overline{A}^{\,i} \; \delta_{i+1/2}[T] \right] … … 401 401 The iso-neutral flux of locally referenced potential density is zero, \ie 402 402 \begin{align*} 403 % \label{eq: Gf_property2}403 % \label{eq:ALGOS_Gf_property2} 404 404 \begin{matrix} 405 405 &{_i^k {\mathbb{F}_u}_{i_p}^{k_p} (\rho)} … … 411 411 \end{matrix} 412 412 \end{align*} 413 This result is trivially obtained using the \autoref{eq: Gf_triads} applied to $T$ and $S$ and414 the definition of the triads' slopes \autoref{eq: Gf_slopes}.413 This result is trivially obtained using the \autoref{eq:ALGOS_Gf_triads} applied to $T$ and $S$ and 414 the definition of the triads' slopes \autoref{eq:ALGOS_Gf_slopes}. 415 415 416 416 \item[$\bullet$ conservation of tracer] 417 417 The iso-neutral diffusion term conserve the total tracer content, \ie 418 418 \[ 419 % \label{eq: Gf_property1}419 % \label{eq:ALGOS_Gf_property1} 420 420 \sum_{i,j,k} \left\{ D_l^T \ b_T \right\} = 0 421 421 \] … … 425 425 The iso-neutral diffusion term does not increase the total tracer variance, \ie 426 426 \[ 427 % \label{eq: Gf_property1}427 % \label{eq:ALGOS_Gf_property1} 428 428 \sum_{i,j,k} \left\{ T \ D_l^T \ b_T \right\} \leq 0 429 429 \] 430 The property is demonstrated in the \autoref{ apdx:Gf_operator}.430 The property is demonstrated in the \autoref{subsec:ALGOS_Gf_operator}. 431 431 It is a key property for a diffusion term. 432 432 It means that the operator is also a dissipation term, … … 438 438 The iso-neutral diffusion operator is self-adjoint, \ie 439 439 \[ 440 % \label{eq: Gf_property1}440 % \label{eq:ALGOS_Gf_property1} 441 441 \sum_{i,j,k} \left\{ S \ D_l^T \ b_T \right\} = \sum_{i,j,k} \left\{ D_l^S \ T \ b_T \right\} 442 442 \] … … 444 444 We just have to apply the same routine. 445 445 This properties can be demonstrated quite easily in a similar way the "non increase of tracer variance" property 446 has been proved (see \autoref{apdx: Gf_operator}).446 has been proved (see \autoref{apdx:ALGOS_Gf_operator}). 447 447 \end{description} 448 448 … … 462 462 The eddy induced velocity is given by: 463 463 \begin{equation} 464 \label{eq: eiv_v}464 \label{eq:ALGOS_eiv_v} 465 465 \begin{split} 466 466 u^* & = - \frac{1}{e_2\,e_{3}} \;\partial_k \left( e_2 \, A_e \; r_i \right) … … 487 487 % give here the expression using the triads. It is different from the one given in \autoref{eq:LDF_eiv} 488 488 % see just below a copy of this equation: 489 %\begin{equation} \label{eq: ldfeiv}489 %\begin{equation} \label{eq:ALGOS_ldfeiv} 490 490 %\begin{split} 491 491 % u^* & = \frac{1}{e_{2u}e_{3u}}\; \delta_k \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right]\\ … … 495 495 %\end{equation} 496 496 \[ 497 % \label{eq: eiv_vd}497 % \label{eq:ALGOS_eiv_vd} 498 498 \textbf{F}_{eiv}^T \equiv \left( 499 499 \begin{aligned} … … 540 540 we end up with the skew form of the eddy induced advective fluxes: 541 541 \begin{equation} 542 \label{eq: eiv_skew_continuous}542 \label{eq:ALGOS_eiv_skew_continuous} 543 543 \textbf{F}_{eiv}^T = 544 544 \begin{pmatrix} … … 548 548 \end{equation} 549 549 The tendency associated with eddy induced velocity is then simply the divergence of 550 the \autoref{eq: eiv_skew_continuous} fluxes.550 the \autoref{eq:ALGOS_eiv_skew_continuous} fluxes. 551 551 It naturally conserves the tracer content, as it is expressed in flux form and, 552 552 as the advective form, it preserves the tracer variance. 553 Another interesting property of \autoref{eq: eiv_skew_continuous} form is that when $A=A_e$,553 Another interesting property of \autoref{eq:ALGOS_eiv_skew_continuous} form is that when $A=A_e$, 554 554 a simplification occurs in the sum of the iso-neutral diffusion and eddy induced velocity terms: 555 555 \begin{flalign*} 556 % \label{eq: eiv_skew+eiv_continuous}556 % \label{eq:ALGOS_eiv_skew+eiv_continuous} 557 557 \textbf{F}_{iso}^T + \textbf{F}_{eiv}^T &= 558 558 \begin{pmatrix} … … 576 576 This property has been used to reduce the computational time \citep{griffies_JPO98}, 577 577 but it is not of practical use as usually $A \neq A_e$. 578 Nevertheless this property can be used to choose a discret form of \autoref{eq: eiv_skew_continuous} which579 is consistent with the iso-neutral operator \autoref{eq: Gf_operator}.580 Using the slopes \autoref{eq: Gf_slopes} and defining $A_e$ at $T$-point(\ie\ as $A$,578 Nevertheless this property can be used to choose a discret form of \autoref{eq:ALGOS_eiv_skew_continuous} which 579 is consistent with the iso-neutral operator \autoref{eq:ALGOS_Gf_operator}. 580 Using the slopes \autoref{eq:ALGOS_Gf_slopes} and defining $A_e$ at $T$-point(\ie\ as $A$, 581 581 the eddy diffusivity coefficient), the resulting discret form is given by: 582 582 \begin{equation} 583 \label{eq: eiv_skew}583 \label{eq:ALGOS_eiv_skew} 584 584 \textbf{F}_{eiv}^T \equiv \frac{1}{4} \left( 585 585 \begin{aligned} … … 593 593 \right) 594 594 \end{equation} 595 Note that \autoref{eq: eiv_skew} is valid in $z$-coordinate with or without partial cells.595 Note that \autoref{eq:ALGOS_eiv_skew} is valid in $z$-coordinate with or without partial cells. 596 596 In $z^*$ or $s$-coordinate, the slope between the level and the geopotential surfaces must be added to 597 597 $\mathbb{R}$ for the discret form to be exact. … … 599 599 Such a choice of discretisation is consistent with the iso-neutral operator as 600 600 it uses the same definition for the slopes. 601 It also ensures the conservation of the tracer variance (see Appendix \autoref{apdx:eiv_skew}),601 It also ensures the conservation of the tracer variance (see \autoref{subsec:ALGOS_eiv_skew}), 602 602 \ie\ it does not include a diffusive component but is a "pure" advection term. 603 603 … … 607 607 % ================================================================ 608 608 \subsection{Discrete invariants of the iso-neutral diffrusion} 609 \label{subsec: Gf_operator}609 \label{subsec:ALGOS_Gf_operator} 610 610 611 611 Demonstration of the decrease of the tracer variance in the (\textbf{i},\textbf{j}) plane. … … 702 702 \intertext{ 703 703 Then outing in factor the triad in each of the four terms of the summation and 704 substituting the triads by their expression given in \autoref{eq: Gf_triads}.704 substituting the triads by their expression given in \autoref{eq:ALGOS_Gf_triads}. 705 705 It becomes: 706 706 } … … 774 774 % ================================================================ 775 775 \subsection{Discrete invariants of the skew flux formulation} 776 \label{subsec: eiv_skew}776 \label{subsec:ALGOS_eiv_skew} 777 777 778 778 Demonstration for the conservation of the tracer variance in the (\textbf{i},\textbf{j}) plane. … … 784 784 \int_D \nabla \cdot \textbf{F}_{eiv}(T) \; T \;dv \equiv 0 785 785 \end{align*} 786 The discrete form of its left hand side is obtained using \autoref{eq: eiv_skew}786 The discrete form of its left hand side is obtained using \autoref{eq:ALGOS_eiv_skew} 787 787 \begin{align*} 788 788 \sum\limits_{i,k} \sum_{\substack{i_p,\,k_p}} \Biggl\{ \;\;
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