Changeset 11435 for NEMO/trunk/doc/latex/NEMO/subfiles/annex_C.tex
- Timestamp:
- 2019-08-14T14:45:08+02:00 (5 years ago)
- File:
-
- 1 edited
-
NEMO/trunk/doc/latex/NEMO/subfiles/annex_C.tex (modified) (33 diffs)
Legend:
- Unmodified
- Added
- Removed
-
NEMO/trunk/doc/latex/NEMO/subfiles/annex_C.tex
r10442 r11435 8 8 \label{apdx:C} 9 9 10 \ minitoc10 \chaptertoc 11 11 12 12 %%% Appendix put in gmcomment as it has not been updated for \zstar and s coordinate … … 39 39 $dv=e_1\,e_2\,e_3 \,di\,dj\,dk$ is the volume element, with only $e_3$ that depends on time. 40 40 $D$ and $S$ are the ocean domain volume and surface, respectively. 41 No wetting/drying is allow (\ie $\frac{\partial S}{\partial t} = 0$).41 No wetting/drying is allow (\ie\ $\frac{\partial S}{\partial t} = 0$). 42 42 Let $k_s$ and $k_b$ be the ocean surface and bottom, resp. 43 (\ie $s(k_s) = \eta$ and $s(k_b)=-H$, where $H$ is the bottom depth).43 (\ie\ $s(k_s) = \eta$ and $s(k_b)=-H$, where $H$ is the bottom depth). 44 44 \begin{flalign*} 45 45 z(k) = \eta - \int\limits_{\tilde{k}=k}^{\tilde{k}=k_s} e_3(\tilde{k}) \;d\tilde{k} … … 99 99 \label{sec:C.1} 100 100 101 The discretization of pimitive equation in $s$-coordinate (\ie time and space varying vertical coordinate)102 must be chosen so that the discrete equation of the model satisfy integral constrains on energy and enstrophy. 101 The discretization of pimitive equation in $s$-coordinate (\ie\ time and space varying vertical coordinate) 102 must be chosen so that the discrete equation of the model satisfy integral constrains on energy and enstrophy. 103 103 104 104 Let us first establish those constraint in the continuous world. 105 The total energy (\ie kinetic plus potential energies) is conserved:105 The total energy (\ie\ kinetic plus potential energies) is conserved: 106 106 \begin{flalign} 107 107 \label{eq:Tot_Energy} … … 109 109 \end{flalign} 110 110 under the following assumptions: no dissipation, no forcing (wind, buoyancy flux, atmospheric pressure variations), 111 mass conservation, and closed domain. 111 mass conservation, and closed domain. 112 112 113 113 This equation can be transformed to obtain several sub-equalities. … … 211 211 \end{subequations} 212 212 213 \autoref{eq:E_tot_pg_3} is the balance between the conversion KE to PE and PE to KE. 213 \autoref{eq:E_tot_pg_3} is the balance between the conversion KE to PE and PE to KE. 214 214 Indeed the left hand side of \autoref{eq:E_tot_pg_3} can be transformed as follows: 215 215 \begin{flalign*} … … 224 224 &=+ \int\limits_D g\, \rho \; w \; dv &&&\\ 225 225 \end{flalign*} 226 where the last equality is obtained by noting that the brackets is exactly the expression of $w$, 227 the vertical velocity referenced to the fixe $z$-coordinate system (see \autoref{apdx:A_w_s}). 228 226 where the last equality is obtained by noting that the brackets is exactly the expression of $w$, 227 the vertical velocity referenced to the fixe $z$-coordinate system (see \autoref{apdx:A_w_s}). 228 229 229 The left hand side of \autoref{eq:E_tot_pg_3} can be transformed as follows: 230 230 \begin{flalign*} … … 367 367 % ------------------------------------------------------------------------------------------------------------- 368 368 \subsubsection{Vorticity term with ENE scheme (\protect\np{ln\_dynvor\_ene}\forcode{ = .true.})} 369 \label{subsec:C_vorENE} 369 \label{subsec:C_vorENE} 370 370 371 371 For the ENE scheme, the two components of the vorticity term are given by: … … 399 399 - \overline{ U }^{\,j+1/2}\; \overline{ V }^{\,i+1/2} \biggr\} \quad \equiv 0 400 400 \end{array} 401 } 401 } 402 402 \end{flalign*} 403 403 In other words, the domain averaged kinetic energy does not change due to the vorticity term. … … 407 407 % ------------------------------------------------------------------------------------------------------------- 408 408 \subsubsection{Vorticity term with EEN scheme (\protect\np{ln\_dynvor\_een}\forcode{ = .true.})} 409 \label{subsec:C_vorEEN_vect} 410 411 With the EEN scheme, the vorticity terms are represented as: 409 \label{subsec:C_vorEEN_vect} 410 411 With the EEN scheme, the vorticity terms are represented as: 412 412 \begin{equation} 413 413 \tag{\ref{eq:dynvor_een}} … … 420 420 \end{aligned} 421 421 } \right. 422 \end{equation} 422 \end{equation} 423 423 where the indices $i_p$ and $j_p$ take the following value: $i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$, 424 and the vorticity triads, ${^i_j}\mathbb{Q}^{i_p}_{j_p}$, defined at $T$-point, are given by: 424 and the vorticity triads, ${^i_j}\mathbb{Q}^{i_p}_{j_p}$, defined at $T$-point, are given by: 425 425 \begin{equation} 426 426 \tag{\ref{eq:Q_triads}} … … 479 479 % ------------------------------------------------------------------------------------------------------------- 480 480 \subsubsection{Gradient of kinetic energy / Vertical advection} 481 \label{subsec:C_zad} 481 \label{subsec:C_zad} 482 482 483 483 The change of Kinetic Energy (KE) due to the vertical advection is exactly balanced by the change of KE due to the horizontal gradient of KE~: … … 487 487 + \frac{1}{2} \int_D { \frac{{\textbf{U}_h}^2}{e_3} \partial_t ( e_3) \;dv } 488 488 \] 489 Indeed, using successively \autoref{eq:DOM_di_adj} (\ie the skew symmetry property of the $\delta$ operator)489 Indeed, using successively \autoref{eq:DOM_di_adj} (\ie\ the skew symmetry property of the $\delta$ operator) 490 490 and the continuity equation, then \autoref{eq:DOM_di_adj} again, 491 491 then the commutativity of operators $\overline {\,\cdot \,}$ and $\delta$, and finally \autoref{eq:DOM_mi_adj} 492 (\ie the symmetry property of the $\overline {\,\cdot \,}$ operator)492 (\ie\ the symmetry property of the $\overline {\,\cdot \,}$ operator) 493 493 applied in the horizontal and vertical directions, it becomes: 494 494 \begin{flalign*} … … 566 566 } } \right) 567 567 \] 568 a formulation that requires an additional horizontal mean in contrast with the one used in NEMO.568 a formulation that requires an additional horizontal mean in contrast with the one used in \NEMO. 569 569 Nine velocity points have to be used instead of 3. 570 570 This is the reason why it has not been chosen. … … 595 595 A pressure gradient has no contribution to the evolution of the vorticity as the curl of a gradient is zero. 596 596 In the $z$-coordinate, this property is satisfied locally on a C-grid with 2nd order finite differences 597 (property \autoref{eq:DOM_curl_grad}). 597 (property \autoref{eq:DOM_curl_grad}). 598 598 } 599 599 600 600 When the equation of state is linear 601 (\ie when an advection-diffusion equation for density can be derived from those of temperature and salinity)601 (\ie\ when an advection-diffusion equation for density can be derived from those of temperature and salinity) 602 602 the change of KE due to the work of pressure forces is balanced by 603 the change of potential energy due to buoyancy forces: 603 the change of potential energy due to buoyancy forces: 604 604 \[ 605 605 - \int_D \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv … … 621 621 % 622 622 \allowdisplaybreaks 623 \intertext{Using successively \autoref{eq:DOM_di_adj}, \ie the skew symmetry property of623 \intertext{Using successively \autoref{eq:DOM_di_adj}, \ie\ the skew symmetry property of 624 624 the $\delta$ operator, \autoref{eq:wzv}, the continuity equation, \autoref{eq:dynhpg_sco}, 625 625 the hydrostatic equation in the $s$-coordinate, and $\delta_{k+1/2} \left[ z_t \right] \equiv e_{3w} $, … … 771 771 % ------------------------------------------------------------------------------------------------------------- 772 772 \subsubsection{Coriolis plus ``metric'' term} 773 \label{subsec:C.3.3} 773 \label{subsec:C.3.3} 774 774 775 775 In flux from the vorticity term reduces to a Coriolis term in which … … 792 792 % ------------------------------------------------------------------------------------------------------------- 793 793 \subsubsection{Flux form advection} 794 \label{subsec:C.3.4} 794 \label{subsec:C.3.4} 795 795 796 796 The flux form operator of the momentum advection is evaluated using … … 811 811 812 812 Let us first consider the first term of the scalar product 813 (\ie just the the terms associated with the i-component of the advection):813 (\ie\ just the the terms associated with the i-component of the advection): 814 814 \begin{flalign*} 815 815 & - \int_D u \cdot \nabla \cdot \left( \textbf{U}\,u \right) \; dv \\ … … 867 867 When the UBS scheme is used to evaluate the flux form momentum advection, 868 868 the discrete operator does not contribute to the global budget of linear momentum (flux form). 869 The horizontal kinetic energy is not conserved, but forced to decay (\ie the scheme is diffusive).869 The horizontal kinetic energy is not conserved, but forced to decay (\ie\ the scheme is diffusive). 870 870 871 871 % ================================================================ … … 879 879 % ------------------------------------------------------------------------------------------------------------- 880 880 \subsubsection{Vorticity term with ENS scheme (\protect\np{ln\_dynvor\_ens}\forcode{ = .true.})} 881 \label{subsec:C_vorENS} 881 \label{subsec:C_vorENS} 882 882 883 883 In the ENS scheme, the vorticity term is descretized as follows: … … 890 890 \end{aligned} 891 891 \right. 892 \end{equation} 892 \end{equation} 893 893 894 894 The scheme does not allow but the conservation of the total kinetic energy but the conservation of $q^2$, 895 the potential enstrophy for a horizontally non-divergent flow (\ie when $\chi$=$0$).895 the potential enstrophy for a horizontally non-divergent flow (\ie\ when $\chi$=$0$). 896 896 Indeed, using the symmetry or skew symmetry properties of the operators 897 897 ( \autoref{eq:DOM_mi_adj} and \autoref{eq:DOM_di_adj}), … … 942 942 } 943 943 \end{flalign*} 944 The later equality is obtain only when the flow is horizontally non-divergent, \ie $\chi$=$0$.944 The later equality is obtain only when the flow is horizontally non-divergent, \ie\ $\chi$=$0$. 945 945 946 946 % ------------------------------------------------------------------------------------------------------------- … … 948 948 % ------------------------------------------------------------------------------------------------------------- 949 949 \subsubsection{Vorticity Term with EEN scheme (\protect\np{ln\_dynvor\_een}\forcode{ = .true.})} 950 \label{subsec:C_vorEEN} 951 952 With the EEN scheme, the vorticity terms are represented as: 950 \label{subsec:C_vorEEN} 951 952 With the EEN scheme, the vorticity terms are represented as: 953 953 \begin{equation} 954 954 \tag{\ref{eq:dynvor_een}} … … 961 961 \end{aligned} 962 962 } \right. 963 \end{equation} 964 where the indices $i_p$ and $k_p$ take the following values: 963 \end{equation} 964 where the indices $i_p$ and $k_p$ take the following values: 965 965 $i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$, 966 and the vorticity triads, ${^i_j}\mathbb{Q}^{i_p}_{j_p}$, defined at $T$-point, are given by: 966 and the vorticity triads, ${^i_j}\mathbb{Q}^{i_p}_{j_p}$, defined at $T$-point, are given by: 967 967 \begin{equation} 968 968 \tag{\ref{eq:Q_triads}} … … 971 971 \end{equation} 972 972 973 This formulation does conserve the potential enstrophy for a horizontally non-divergent flow (\ie $\chi=0$).973 This formulation does conserve the potential enstrophy for a horizontally non-divergent flow (\ie\ $\chi=0$). 974 974 975 975 Let consider one of the vorticity triad, for example ${^{i}_j}\mathbb{Q}^{+1/2}_{+1/2} $, … … 1023 1023 \label{sec:C.5} 1024 1024 1025 All the numerical schemes used in NEMOare written such that the tracer content is conserved by1025 All the numerical schemes used in \NEMO\ are written such that the tracer content is conserved by 1026 1026 the internal dynamics and physics (equations in flux form). 1027 1027 For advection, 1028 only the CEN2 scheme (\ie $2^{nd}$ order finite different scheme) conserves the global variance of tracer.1028 only the CEN2 scheme (\ie\ $2^{nd}$ order finite different scheme) conserves the global variance of tracer. 1029 1029 Nevertheless the other schemes ensure that the global variance decreases 1030 (\ie they are at least slightly diffusive).1030 (\ie\ they are at least slightly diffusive). 1031 1031 For diffusion, all the schemes ensure the decrease of the total tracer variance, except the iso-neutral operator. 1032 1032 There is generally no strict conservation of mass, 1033 1033 as the equation of state is non linear with respect to $T$ and $S$. 1034 In practice, the mass is conserved to a very high accuracy. 1034 In practice, the mass is conserved to a very high accuracy. 1035 1035 % ------------------------------------------------------------------------------------------------------------- 1036 1036 % Advection Term … … 1056 1056 Indeed, let $T$ be the tracer and its $\tau_u$, $\tau_v$, and $\tau_w$ interpolated values at velocity point 1057 1057 (whatever the interpolation is), 1058 the conservation of the tracer content due to the advection tendency is obtained as follows: 1058 the conservation of the tracer content due to the advection tendency is obtained as follows: 1059 1059 \begin{flalign*} 1060 1060 &\int_D { \frac{1}{e_3}\frac{\partial \left( e_3 \, T \right)}{\partial t} \;dv } = - \int_D \nabla \cdot \left( T \textbf{U} \right)\;dv &&&\\ … … 1072 1072 1073 1073 The conservation of the variance of tracer due to the advection tendency can be achieved only with the CEN2 scheme, 1074 \ie when $\tau_u= \overline T^{\,i+1/2}$, $\tau_v= \overline T^{\,j+1/2}$, and $\tau_w= \overline T^{\,k+1/2}$.1074 \ie\ when $\tau_u= \overline T^{\,i+1/2}$, $\tau_v= \overline T^{\,j+1/2}$, and $\tau_w= \overline T^{\,k+1/2}$. 1075 1075 It can be demonstarted as follows: 1076 1076 \begin{flalign*} … … 1108 1108 the conservation of potential vorticity and the horizontal divergence, 1109 1109 and the dissipation of the square of these quantities 1110 (\ie enstrophy and the variance of the horizontal divergence) as well as1110 (\ie\ enstrophy and the variance of the horizontal divergence) as well as 1111 1111 the dissipation of the horizontal kinetic energy. 1112 1112 In particular, when the eddy coefficients are horizontally uniform, 1113 1113 it ensures a complete separation of vorticity and horizontal divergence fields, 1114 1114 so that diffusion (dissipation) of vorticity (enstrophy) does not generate horizontal divergence 1115 (variance of the horizontal divergence) and \textit{vice versa}. 1115 (variance of the horizontal divergence) and \textit{vice versa}. 1116 1116 1117 1117 These properties of the horizontal diffusion operator are a direct consequence of 1118 1118 properties \autoref{eq:DOM_curl_grad} and \autoref{eq:DOM_div_curl}. 1119 1119 When the vertical curl of the horizontal diffusion of momentum (discrete sense) is taken, 1120 the term associated with the horizontal gradient of the divergence is locally zero. 1120 the term associated with the horizontal gradient of the divergence is locally zero. 1121 1121 1122 1122 % ------------------------------------------------------------------------------------------------------------- … … 1237 1237 1238 1238 When the horizontal divergence of the horizontal diffusion of momentum (discrete sense) is taken, 1239 the term associated with the vertical curl of the vorticity is zero locally, due to \autoref{eq:DOM_div_curl}. 1239 the term associated with the vertical curl of the vorticity is zero locally, due to \autoref{eq:DOM_div_curl}. 1240 1240 The resulting term conserves the $\chi$ and dissipates $\chi^2$ when the eddy coefficients are horizontally uniform. 1241 1241 \begin{flalign*} … … 1396 1396 \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv = 0 &&& 1397 1397 \end{flalign*} 1398 and the square of the horizontal divergence decreases (\ie the horizontal divergence is dissipated) if1398 and the square of the horizontal divergence decreases (\ie\ the horizontal divergence is dissipated) if 1399 1399 the vertical diffusion coefficient is uniform over the whole domain: 1400 1400 … … 1463 1463 the heat and salt contents are conserved (equations in flux form). 1464 1464 Since a flux form is used to compute the temperature and salinity, 1465 the quadratic form of these quantities (\ie their variance) globally tends to diminish.1466 As for the advection term, there is conservation of mass only if the Equation Of Seawater is linear. 1465 the quadratic form of these quantities (\ie\ their variance) globally tends to diminish. 1466 As for the advection term, there is conservation of mass only if the Equation Of Seawater is linear. 1467 1467 1468 1468 % ------------------------------------------------------------------------------------------------------------- … … 1497 1497 \end{flalign*} 1498 1498 1499 In fact, this property simply results from the flux form of the operator. 1499 In fact, this property simply results from the flux form of the operator. 1500 1500 1501 1501 % -------------------------------------------------------------------------------------------------------------
Note: See TracChangeset
for help on using the changeset viewer.
