Changeset 10406 for NEMO/trunk/doc/latex/NEMO/subfiles/annex_C.tex
- Timestamp:
- 2018-12-18T11:25:09+01:00 (5 years ago)
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- NEMO/trunk/doc/latex
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- 4 edited
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NEMO/trunk/doc/latex/NEMO/subfiles/annex_C.tex
r10354 r10406 25 25 26 26 fluxes at the faces of a $T$-box: 27 \ begin{equation*}27 \[ 28 28 U = e_{2u}\,e_{3u}\; u \qquad V = e_{1v}\,e_{3v}\; v \qquad W = e_{1w}\,e_{2w}\; \omega \\ 29 \ end{equation*}29 \] 30 30 31 31 volume of cells at $u$-, $v$-, and $T$-points: 32 \ begin{equation*}32 \[ 33 33 b_u = e_{1u}\,e_{2u}\,e_{3u} \qquad b_v = e_{1v}\,e_{2v}\,e_{3v} \qquad b_t = e_{1t}\,e_{2t}\,e_{3t} \\ 34 \ end{equation*}34 \] 35 35 36 36 partial derivative notation: $\partial_\bullet = \frac{\partial}{\partial \bullet}$ … … 47 47 48 48 Continuity equation with the above notation: 49 \ begin{equation*}49 \[ 50 50 \frac{1}{e_{3t}} \partial_t (e_{3t})+ \frac{1}{b_t} \biggl\{ \delta_i [U] + \delta_j [V] + \delta_k [W] \biggr\} = 0 51 \ end{equation*}51 \] 52 52 53 53 A quantity, $Q$ is conserved when its domain averaged time change is zero, that is when: 54 \ begin{equation*}54 \[ 55 55 \partial_t \left( \int_D{ Q\;dv } \right) =0 56 \ end{equation*}56 \] 57 57 Noting that the coordinate system used .... blah blah 58 \ begin{equation*}58 \[ 59 59 \partial_t \left( \int_D {Q\;dv} \right) = \int_D { \partial_t \left( e_3 \, Q \right) e_1e_2\;di\,dj\,dk } 60 60 = \int_D { \frac{1}{e_3} \partial_t \left( e_3 \, Q \right) dv } =0 61 \ end{equation*}61 \] 62 62 equation of evolution of $Q$ written as 63 63 the time evolution of the vertical content of $Q$ like for tracers, or momentum in flux form, … … 162 162 $\ $\newline % force a new ligne 163 163 The prognostic ocean dynamics equation can be summarized as follows: 164 \ begin{equation*}164 \[ 165 165 \text{NXT} = \dbinom {\text{VOR} + \text{KEG} + \text {ZAD} } 166 166 {\text{COR} + \text{ADV} } 167 167 + \text{HPG} + \text{SPG} + \text{LDF} + \text{ZDF} 168 \ end{equation*}168 \] 169 169 $\ $\newline % force a new ligne 170 170 … … 365 365 366 366 For the ENE scheme, the two components of the vorticity term are given by: 367 \ begin{equation*}367 \[ 368 368 - e_3 \, q \;{\textbf{k}}\times {\textbf {U}}_h \equiv 369 369 \left( {{ \begin{array} {*{20}c} … … 373 373 \overline {\, q \ \overline {\left( e_{2u}\,e_{3u}\,u \right)}^{\,j+1/2}} ^{\,i} \hfill \\ 374 374 \end{array}} } \right) 375 \ end{equation*}375 \] 376 376 377 377 This formulation does not conserve the enstrophy but it does conserve the total kinetic energy. … … 471 471 472 472 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~: 473 \ begin{equation*}473 \[ 474 474 \int_D \textbf{U}_h \cdot \frac{1}{e_3 } \omega \partial_k \textbf{U}_h \;dv 475 475 = - \int_D \textbf{U}_h \cdot \nabla_h \left( \frac{1}{2}\;{\textbf{U}_h}^2 \right)\;dv 476 476 + \frac{1}{2} \int_D { \frac{{\textbf{U}_h}^2}{e_3} \partial_t ( e_3) \;dv } \\ 477 \ end{equation*}477 \] 478 478 Indeed, using successively \autoref{eq:DOM_di_adj} ($i.e.$ the skew symmetry property of the $\delta$ operator) 479 479 and the continuity equation, then \autoref{eq:DOM_di_adj} again, … … 544 544 For example KE can also be discretized as $1/2\,({\overline u^{\,i}}^2 + {\overline v^{\,j}}^2)$. 545 545 This leads to the following expression for the vertical advection: 546 \ begin{equation*}546 \[ 547 547 \frac{1} {e_3 }\; \omega\; \partial_k \textbf{U}_h 548 548 \equiv \left( {{\begin{array} {*{20}c} … … 552 552 \left[ \overline v^{\,j+1/2} \right]}}^{\,j+1/2,k} \hfill \\ 553 553 \end{array}} } \right) 554 \ end{equation*}554 \] 555 555 a formulation that requires an additional horizontal mean in contrast with the one used in NEMO. 556 556 Nine velocity points have to be used instead of 3. … … 588 588 the change of KE due to the work of pressure forces is balanced by 589 589 the change of potential energy due to buoyancy forces: 590 \ begin{equation*}590 \[ 591 591 - \int_D \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv 592 592 = - \int_D \nabla \cdot \left( \rho \,\textbf {U} \right) \,g\,z \;dv 593 593 + \int_D g\, \rho \; \partial_t (z) \;dv 594 \ end{equation*}594 \] 595 595 596 596 This property can be satisfied in a discrete sense for both $z$- and $s$-coordinates. … … 771 771 This altered Coriolis parameter is discretised at an f-point. 772 772 It is given by: 773 \ begin{equation*}773 \[ 774 774 f+\frac{1} {e_1 e_2 } \left( v \frac{\partial e_2 } {\partial i} - u \frac{\partial e_1 } {\partial j}\right)\; 775 775 \equiv \; 776 776 f+\frac{1} {e_{1f}\,e_{2f}} \left( \overline v^{\,i+1/2} \delta_{i+1/2} \left[ e_{2u} \right] 777 777 -\overline u^{\,j+1/2} \delta_{j+1/2} \left[ e_{1u} \right] \right) 778 \ end{equation*}778 \] 779 779 780 780 Either the ENE or EEN scheme is then applied to obtain the vorticity term in flux form. … … 842 842 \end{flalign*} 843 843 Applying similar manipulation applied to the second term of the scalar product leads to: 844 \ begin{equation*}844 \[ 845 845 - \int_D \textbf{U}_h \cdot \left( {{\begin{array} {*{20}c} 846 846 \nabla \cdot \left( \textbf{U}\,u \right) \hfill \\ … … 848 848 \equiv + \sum\limits_{i,j,k} \frac{1}{2} \left( \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right) 849 849 \biggl\{ \left( \frac{1}{e_{3t}} \frac{\partial e_{3t}}{\partial t} \right) \; b_t \biggr\} 850 \ end{equation*}850 \] 851 851 which is the discrete form of $ \frac{1}{2} \int_D u \cdot \nabla \cdot \left( \textbf{U}\,u \right) \; dv $. 852 852 \autoref{eq:C_ADV_KE_flux} is thus satisfied. … … 1032 1032 1033 1033 conservation of a tracer, $T$: 1034 \ begin{equation*}1034 \[ 1035 1035 \frac{\partial }{\partial t} \left( \int_D {T\;dv} \right) 1036 1036 = \int_D { \frac{1}{e_3}\frac{\partial \left( e_3 \, T \right)}{\partial t} \;dv }=0 1037 \ end{equation*}1037 \] 1038 1038 1039 1039 conservation of its variance: … … 1156 1156 The lateral momentum diffusion term dissipates the horizontal kinetic energy: 1157 1157 %\begin{flalign*} 1158 \ begin{equation*}1158 \[ 1159 1159 \begin{split} 1160 1160 \int_D \textbf{U}_h \cdot … … 1198 1198 \quad \leq 0 \\ 1199 1199 \end{split} 1200 \ end{equation*}1200 \] 1201 1201 1202 1202 % -------------------------------------------------------------------------------------------------------------
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