Changeset 10442 for NEMO/trunk/doc/latex/NEMO/subfiles/chap_DYN.tex
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NEMO/trunk/doc/latex/NEMO/subfiles/chap_DYN.tex
r10414 r10442 68 68 \label{subsec:DYN_divcur} 69 69 70 The vorticity is defined at an $f$-point ( $i.e.$corner point) as follows:70 The vorticity is defined at an $f$-point (\ie corner point) as follows: 71 71 \begin{equation} 72 72 \label{eq:divcur_cur} … … 123 123 the tracer equation \autoref{eq:tra_nxt}: 124 124 a leapfrog scheme in combination with an Asselin time filter, 125 $i.e.$the velocity appearing in \autoref{eq:dynspg_ssh} is centred in time (\textit{now} velocity).125 \ie the velocity appearing in \autoref{eq:dynspg_ssh} is centred in time (\textit{now} velocity). 126 126 This is of paramount importance. 127 127 Replacing $T$ by the number $1$ in the tracer equation and summing over the water column must lead to … … 149 149 The upper boundary condition applies at a fixed level $z=0$. 150 150 The top vertical velocity is thus equal to the divergence of the barotropic transport 151 ( $i.e.$the first term in the right-hand-side of \autoref{eq:dynspg_ssh}).151 (\ie the first term in the right-hand-side of \autoref{eq:dynspg_ssh}). 152 152 153 153 Note also that whereas the vertical velocity has the same discrete expression in $z$- and $s$-coordinates, 154 154 its physical meaning is not the same: 155 155 in the second case, $w$ is the velocity normal to the $s$-surfaces. 156 Note also that the $k$-axis is re-orientated downwards in the \ textsc{fortran}code compared to156 Note also that the $k$-axis is re-orientated downwards in the \fortran code compared to 157 157 the indexing used in the semi-discrete equations such as \autoref{eq:wzv} 158 158 (see \autoref{subsec:DOM_Num_Index_vertical}). … … 174 174 Options are defined through the \ngn{namdyn\_adv} namelist variables Coriolis and 175 175 momentum advection terms are evaluated using a leapfrog scheme, 176 $i.e.$the velocity appearing in these expressions is centred in time (\textit{now} velocity).176 \ie the velocity appearing in these expressions is centred in time (\textit{now} velocity). 177 177 At the lateral boundaries either free slip, no slip or partial slip boundary conditions are applied following 178 178 \autoref{chap:LBC}. … … 208 208 In the enstrophy conserving case (ENS scheme), 209 209 the discrete formulation of the vorticity term provides a global conservation of the enstrophy 210 ($ [ (\zeta +f ) / e_{3f} ]^2 $ in $s$-coordinates) for a horizontally non-divergent flow ( $i.e.$$\chi$=$0$),210 ($ [ (\zeta +f ) / e_{3f} ]^2 $ in $s$-coordinates) for a horizontally non-divergent flow (\ie $\chi$=$0$), 211 211 but does not conserve the total kinetic energy. 212 212 It is given by: … … 278 278 the presence of grid point oscillation structures that will be invisible to the operator. 279 279 These structures are \textit{computational modes} that will be at least partly damped by 280 the momentum diffusion operator ( $i.e.$the subgrid-scale advection), but not by the resolved advection term.280 the momentum diffusion operator (\ie the subgrid-scale advection), but not by the resolved advection term. 281 281 The ENS and ENE schemes therefore do not contribute to dump any grid point noise in the horizontal velocity field. 282 282 Such noise would result in more noise in the vertical velocity field, an undesirable feature. … … 327 327 (with a systematic reduction of $e_{3f}$ when a model level intercept the bathymetry) 328 328 that tends to reinforce the topostrophy of the flow 329 ( $i.e.$the tendency of the flow to follow the isobaths) \citep{Penduff_al_OS07}.329 (\ie the tendency of the flow to follow the isobaths) \citep{Penduff_al_OS07}. 330 330 331 331 Next, the vorticity triads, $ {^i_j}\mathbb{Q}^{i_p}_{j_p}$ can be defined at a $T$-point as … … 354 354 This EEN scheme in fact combines the conservation properties of the ENS and ENE schemes. 355 355 It conserves both total energy and potential enstrophy in the limit of horizontally nondivergent flow 356 ( $i.e.$$\chi$=$0$) (see \autoref{subsec:C_vorEEN}).356 (\ie $\chi$=$0$) (see \autoref{subsec:C_vorEEN}). 357 357 Applied to a realistic ocean configuration, it has been shown that it leads to a significant reduction of 358 358 the noise in the vertical velocity field \citep{Le_Sommer_al_OM09}. … … 422 422 In the flux form (as in the vector invariant form), 423 423 the Coriolis and momentum advection terms are evaluated using a leapfrog scheme, 424 $i.e.$the velocity appearing in their expressions is centred in time (\textit{now} velocity).424 \ie the velocity appearing in their expressions is centred in time (\textit{now} velocity). 425 425 At the lateral boundaries either free slip, 426 426 no slip or partial slip boundary conditions are applied following \autoref{chap:LBC}. … … 446 446 compute the product of the Coriolis parameter and the vorticity. 447 447 However, the energy-conserving scheme (\autoref{eq:dynvor_een}) has exclusively been used to date. 448 This term is evaluated using a leapfrog scheme, $i.e.$the velocity is centred in time (\textit{now} velocity).448 This term is evaluated using a leapfrog scheme, \ie the velocity is centred in time (\textit{now} velocity). 449 449 450 450 %-------------------------------------------------------------------------------------------------------------- … … 478 478 The schemes are selected using the namelist logicals \np{ln\_dynadv\_cen2} and \np{ln\_dynadv\_ubs}. 479 479 In flux form, the schemes differ by the choice of a space and time interpolation to define the value of 480 $u$ and $v$ at the centre of each face of $u$- and $v$-cells, $i.e.$at the $T$-, $f$-,480 $u$ and $v$ at the centre of each face of $u$- and $v$-cells, \ie at the $T$-, $f$-, 481 481 and $uw$-points for $u$ and at the $f$-, $T$- and $vw$-points for $v$. 482 482 … … 498 498 \end{equation} 499 499 500 The scheme is non diffusive ( i.e. conserves the kinetic energy) but dispersive ($i.e.$it may create false extrema).500 The scheme is non diffusive (\ie conserves the kinetic energy) but dispersive (\ie it may create false extrema). 501 501 It is therefore notoriously noisy and must be used in conjunction with an explicit diffusion operator to 502 502 produce a sensible solution. … … 522 522 \end{equation} 523 523 where $u"_{i+1/2} =\delta_{i+1/2} \left[ {\delta_i \left[ u \right]} \right]$. 524 This results in a dissipatively dominant ( $i.e.$hyper-diffusive) truncation error524 This results in a dissipatively dominant (\ie hyper-diffusive) truncation error 525 525 \citep{Shchepetkin_McWilliams_OM05}. 526 526 The overall performance of the advection scheme is similar to that reported in \citet{Farrow1995}. … … 529 529 But the amplitudes of the false extrema are significantly reduced over those in the centred second order method. 530 530 As the scheme already includes a diffusion component, it can be used without explicit lateral diffusion on momentum 531 ( $i.e.$\np{ln\_dynldf\_lap}\forcode{ = }\np{ln\_dynldf\_bilap}\forcode{ = .false.}),531 (\ie \np{ln\_dynldf\_lap}\forcode{ = }\np{ln\_dynldf\_bilap}\forcode{ = .false.}), 532 532 and it is recommended to do so. 533 533 534 534 The UBS scheme is not used in all directions. 535 In the vertical, the centred $2^{nd}$ order evaluation of the advection is preferred, $i.e.$$u_{uw}^{ubs}$ and535 In the vertical, the centred $2^{nd}$ order evaluation of the advection is preferred, \ie $u_{uw}^{ubs}$ and 536 536 $u_{vw}^{ubs}$ in \autoref{eq:dynadv_cen2} are used. 537 537 UBS is diffusive and is associated with vertical mixing of momentum. \gmcomment{ gm pursue the … … 570 570 The key distinction between the different algorithms used for 571 571 the hydrostatic pressure gradient is the vertical coordinate used, 572 since HPG is a \emph{horizontal} pressure gradient, $i.e.$computed along geopotential surfaces.572 since HPG is a \emph{horizontal} pressure gradient, \ie computed along geopotential surfaces. 573 573 As a result, any tilt of the surface of the computational levels will require a specific treatment to 574 574 compute the hydrostatic pressure gradient. 575 575 576 576 The hydrostatic pressure gradient term is evaluated either using a leapfrog scheme, 577 $i.e.$the density appearing in its expression is centred in time (\emph{now} $\rho$),577 \ie the density appearing in its expression is centred in time (\emph{now} $\rho$), 578 578 or a semi-implcit scheme. 579 579 At the lateral boundaries either free slip, no slip or partial slip boundary conditions are applied. … … 652 652 653 653 Pressure gradient formulations in an $s$-coordinate have been the subject of a vast number of papers 654 ( $e.g.$, \citet{Song1998, Shchepetkin_McWilliams_OM05}).654 (\eg, \citet{Song1998, Shchepetkin_McWilliams_OM05}). 655 655 A number of different pressure gradient options are coded but the ROMS-like, 656 656 density Jacobian with cubic polynomial method is currently disabled whilst known bugs are under investigation. … … 704 704 $\bullet$ The main hypothesis to compute the ice shelf load is that the ice shelf is in an isostatic equilibrium. 705 705 The top pressure is computed integrating from surface to the base of the ice shelf a reference density profile 706 (prescribed as density of a water at 34.4 PSU and -1.9\deg C) and706 (prescribed as density of a water at 34.4 PSU and -1.9\deg{C}) and 707 707 corresponds to the water replaced by the ice shelf. 708 708 This top pressure is constant over time. … … 728 728 It involves the evaluation of the hydrostatic pressure gradient as 729 729 an average over the three time levels $t-\rdt$, $t$, and $t+\rdt$ 730 ( $i.e.$ \textit{before},\textit{now} and \textit{after} time-steps),730 (\ie \textit{before}, \textit{now} and \textit{after} time-steps), 731 731 rather than at the central time level $t$ only, as in the standard leapfrog scheme. 732 732 … … 820 820 the model time step is chosen to be small enough to resolve the external gravity waves 821 821 (typically a few tens of seconds). 822 The surface pressure gradient, evaluated using a leap-frog scheme ( $i.e.$centered in time),822 The surface pressure gradient, evaluated using a leap-frog scheme (\ie centered in time), 823 823 is thus simply given by : 824 824 \begin{equation} … … 832 832 \end{equation} 833 833 834 Note that in the non-linear free surface case ( $i.e.$\key{vvl} defined),834 Note that in the non-linear free surface case (\ie \key{vvl} defined), 835 835 the surface pressure gradient is already included in the momentum tendency through 836 836 the level thickness variation allowed in the computation of the hydrostatic pressure gradient. … … 948 948 (\np{ln\_bt\_av}\forcode{ = .false.}). 949 949 In that case, external mode equations are continuous in time, 950 $i.e.$they are not re-initialized when starting a new sub-stepping sequence.950 \ie they are not re-initialized when starting a new sub-stepping sequence. 951 951 This is the method used so far in the POM model, the stability being maintained by 952 952 refreshing at (almost) each barotropic time step advection and horizontal diffusion terms. … … 1124 1124 the description of the coefficients is found in the chapter on lateral physics (\autoref{chap:LDF}). 1125 1125 The lateral diffusion of momentum is evaluated using a forward scheme, 1126 $i.e.$the velocity appearing in its expression is the \textit{before} velocity in time,1126 \ie the velocity appearing in its expression is the \textit{before} velocity in time, 1127 1127 except for the pure vertical component that appears when a tensor of rotation is used. 1128 1128 This latter term is solved implicitly together with the vertical diffusion term (see \autoref{chap:STP}). … … 1140 1140 In finite difference methods, 1141 1141 the biharmonic operator is frequently the method of choice to achieve this scale selective dissipation since 1142 its damping time ( $i.e.$its spin down time) scale like $\lambda^{-4}$ for disturbances of wavelength $\lambda$1142 its damping time (\ie its spin down time) scale like $\lambda^{-4}$ for disturbances of wavelength $\lambda$ 1143 1143 (so that short waves damped more rapidelly than long ones), 1144 1144 whereas the Laplace operator damping time scales only like $\lambda^{-2}$. … … 1315 1315 1316 1316 (3) When \np{nn\_ice\_embd}\forcode{ = 2} and LIM or CICE is used 1317 ( $i.e.$when the sea-ice is embedded in the ocean),1317 (\ie when the sea-ice is embedded in the ocean), 1318 1318 the snow-ice mass is taken into account when computing the surface pressure gradient. 1319 1319 … … 1335 1335 Options are defined through the \ngn{namdom} namelist variables. 1336 1336 The general framework for dynamics time stepping is a leap-frog scheme, 1337 $i.e.$a three level centred time scheme associated with an Asselin time filter (cf. \autoref{chap:STP}).1337 \ie a three level centred time scheme associated with an Asselin time filter (cf. \autoref{chap:STP}). 1338 1338 The scheme is applied to the velocity, except when 1339 1339 using the flux form of momentum advection (cf. \autoref{sec:DYN_adv_cor_flux}) … … 1379 1379 \biblio 1380 1380 1381 \pindex 1382 1381 1383 \end{document}
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