Changeset 11123 for NEMO/trunk/doc/latex/NEMO/subfiles/chap_DYN.tex
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NEMO/trunk/doc/latex/NEMO/subfiles/chap_DYN.tex
r10499 r11123 127 127 Replacing $T$ by the number $1$ in the tracer equation and summing over the water column must lead to 128 128 the sea surface height equation otherwise tracer content will not be conserved 129 \citep{ Griffies_al_MWR01, Leclair_Madec_OM09}.129 \citep{griffies.pacanowski.ea_MWR01, leclair.madec_OM09}. 130 130 131 131 The vertical velocity is computed by an upward integration of the horizontal divergence starting at the bottom, … … 287 287 Nevertheless, this technique strongly distort the phase and group velocity of Rossby waves....} 288 288 289 A very nice solution to the problem of double averaging was proposed by \citet{ Arakawa_Hsu_MWR90}.289 A very nice solution to the problem of double averaging was proposed by \citet{arakawa.hsu_MWR90}. 290 290 The idea is to get rid of the double averaging by considering triad combinations of vorticity. 291 291 It is noteworthy that this solution is conceptually quite similar to the one proposed by 292 \citep{ Griffies_al_JPO98} for the discretization of the iso-neutral diffusion operator (see \autoref{apdx:C}).293 294 The \citet{ Arakawa_Hsu_MWR90} vorticity advection scheme for a single layer is modified295 for spherical coordinates as described by \citet{ Arakawa_Lamb_MWR81} to obtain the EEN scheme.292 \citep{griffies.gnanadesikan.ea_JPO98} for the discretization of the iso-neutral diffusion operator (see \autoref{apdx:C}). 293 294 The \citet{arakawa.hsu_MWR90} vorticity advection scheme for a single layer is modified 295 for spherical coordinates as described by \citet{arakawa.lamb_MWR81} to obtain the EEN scheme. 296 296 First consider the discrete expression of the potential vorticity, $q$, defined at an $f$-point: 297 297 \[ … … 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 (\ie 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.le-sommer.ea_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 … … 356 356 (\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 the noise in the vertical velocity field \citep{ Le_Sommer_al_OM09}.358 the noise in the vertical velocity field \citep{le-sommer.penduff.ea_OM09}. 359 359 Furthermore, used in combination with a partial steps representation of bottom topography, 360 360 it improves the interaction between current and topography, 361 leading to a larger topostrophy of the flow \citep{ Barnier_al_OD06, Penduff_al_OS07}.361 leading to a larger topostrophy of the flow \citep{barnier.madec.ea_OD06, penduff.le-sommer.ea_OS07}. 362 362 363 363 %-------------------------------------------------------------------------------------------------------------- … … 403 403 When \np{ln\_dynzad\_zts}\forcode{ = .true.}, 404 404 a split-explicit time stepping with 5 sub-timesteps is used on the vertical advection term. 405 This option can be useful when the value of the timestep is limited by vertical advection \citep{ Lemarie_OM2015}.405 This option can be useful when the value of the timestep is limited by vertical advection \citep{lemarie.debreu.ea_OM15}. 406 406 Note that in this case, 407 407 a similar split-explicit time stepping should be used on vertical advection of tracer to ensure a better stability, … … 475 475 a $2^{nd}$ order centered finite difference scheme, CEN2, 476 476 or a $3^{rd}$ order upstream biased scheme, UBS. 477 The latter is described in \citet{ Shchepetkin_McWilliams_OM05}.477 The latter is described in \citet{shchepetkin.mcwilliams_OM05}. 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 … … 523 523 where $u"_{i+1/2} =\delta_{i+1/2} \left[ {\delta_i \left[ u \right]} \right]$. 524 524 This results in a dissipatively dominant (\ie hyper-diffusive) truncation error 525 \citep{ Shchepetkin_McWilliams_OM05}.526 The overall performance of the advection scheme is similar to that reported in \citet{ Farrow1995}.525 \citep{shchepetkin.mcwilliams_OM05}. 526 The overall performance of the advection scheme is similar to that reported in \citet{farrow.stevens_JPO95}. 527 527 It is a relatively good compromise between accuracy and smoothness. 528 528 It is not a \emph{positive} scheme, meaning that false extrema are permitted. … … 542 542 while the second term, which is the diffusion part of the scheme, 543 543 is evaluated using the \textit{before} velocity (forward in time). 544 This is discussed by \citet{ Webb_al_JAOT98} in the context of the Quick advection scheme.544 This is discussed by \citet{webb.de-cuevas.ea_JAOT98} in the context of the Quick advection scheme. 545 545 546 546 Note that the UBS and QUICK (Quadratic Upstream Interpolation for Convective Kinematics) schemes only differ by 547 547 one coefficient. 548 Replacing $1/6$ by $1/8$ in (\autoref{eq:dynadv_ubs}) leads to the QUICK advection scheme \citep{ Webb_al_JAOT98}.548 Replacing $1/6$ by $1/8$ in (\autoref{eq:dynadv_ubs}) leads to the QUICK advection scheme \citep{webb.de-cuevas.ea_JAOT98}. 549 549 This option is not available through a namelist parameter, since the $1/6$ coefficient is hard coded. 550 550 Nevertheless it is quite easy to make the substitution in the \mdl{dynadv\_ubs} module and obtain a QUICK scheme. … … 652 652 653 653 Pressure gradient formulations in an $s$-coordinate have been the subject of a vast number of papers 654 (\eg, \citet{ Song1998, Shchepetkin_McWilliams_OM05}).654 (\eg, \citet{song_MWR98, 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. 657 657 658 $\bullet$ Traditional coding (see for example \citet{ Madec_al_JPO96}: (\np{ln\_dynhpg\_sco}\forcode{ = .true.})658 $\bullet$ Traditional coding (see for example \citet{madec.delecluse.ea_JPO96}: (\np{ln\_dynhpg\_sco}\forcode{ = .true.}) 659 659 \begin{equation} 660 660 \label{eq:dynhpg_sco} … … 679 679 $\bullet$ Pressure Jacobian scheme (prj) (a research paper in preparation) (\np{ln\_dynhpg\_prj}\forcode{ = .true.}) 680 680 681 $\bullet$ Density Jacobian with cubic polynomial scheme (DJC) \citep{ Shchepetkin_McWilliams_OM05}681 $\bullet$ Density Jacobian with cubic polynomial scheme (DJC) \citep{shchepetkin.mcwilliams_OM05} 682 682 (\np{ln\_dynhpg\_djc}\forcode{ = .true.}) (currently disabled; under development) 683 683 684 684 Note that expression \autoref{eq:dynhpg_sco} is commonly used when the variable volume formulation is activated 685 685 (\key{vvl}) because in that case, even with a flat bottom, 686 the coordinate surfaces are not horizontal but follow the free surface \citep{ Levier2007}.686 the coordinate surfaces are not horizontal but follow the free surface \citep{levier.treguier.ea_rpt07}. 687 687 The pressure jacobian scheme (\np{ln\_dynhpg\_prj}\forcode{ = .true.}) is available as 688 688 an improved option to \np{ln\_dynhpg\_sco}\forcode{ = .true.} when \key{vvl} is active. … … 704 704 corresponds to the water replaced by the ice shelf. 705 705 This top pressure is constant over time. 706 A detailed description of this method is described in \citet{ Losch2008}.\\706 A detailed description of this method is described in \citet{losch_JGR08}.\\ 707 707 708 708 The pressure gradient due to ocean load is computed using the expression \autoref{eq:dynhpg_sco} described in … … 722 722 the physical phenomenon that controls the time-step is internal gravity waves (IGWs). 723 723 A semi-implicit scheme for doubling the stability limit associated with IGWs can be used 724 \citep{ Brown_Campana_MWR78, Maltrud1998}.724 \citep{brown.campana_MWR78, maltrud.smith.ea_JGR98}. 725 725 It involves the evaluation of the hydrostatic pressure gradient as 726 726 an average over the three time levels $t-\rdt$, $t$, and $t+\rdt$ … … 790 790 which imposes a very small time step when an explicit time stepping is used. 791 791 Two methods are proposed to allow a longer time step for the three-dimensional equations: 792 the filtered free surface, which is a modification of the continuous equations (see \autoref{eq:PE_flt }),792 the filtered free surface, which is a modification of the continuous equations (see \autoref{eq:PE_flt?}), 793 793 and the split-explicit free surface described below. 794 794 The extra term introduced in the filtered method is calculated implicitly, … … 845 845 846 846 The split-explicit free surface formulation used in \NEMO (\key{dynspg\_ts} defined), 847 also called the time-splitting formulation, follows the one proposed by \citet{ Shchepetkin_McWilliams_OM05}.847 also called the time-splitting formulation, follows the one proposed by \citet{shchepetkin.mcwilliams_OM05}. 848 848 The general idea is to solve the free surface equation and the associated barotropic velocity equations with 849 849 a smaller time step than $\rdt$, the time step used for the three dimensional prognostic variables … … 876 876 (see section \autoref{sec:ZDF_bfr}), explicitly accounted for at each barotropic iteration. 877 877 Temporal discretization of the system above follows a three-time step Generalized Forward Backward algorithm 878 detailed in \citet{ Shchepetkin_McWilliams_OM05}.878 detailed in \citet{shchepetkin.mcwilliams_OM05}. 879 879 AB3-AM4 coefficients used in \NEMO follow the second-order accurate, 880 "multi-purpose" stability compromise as defined in \citet{ Shchepetkin_McWilliams_Bk08}880 "multi-purpose" stability compromise as defined in \citet{shchepetkin.mcwilliams_ibk09} 881 881 (see their figure 12, lower left). 882 882 … … 936 936 and time splitting not compatible. 937 937 Advective barotropic velocities are obtained by using a secondary set of filtering weights, 938 uniquely defined from the filter coefficients used for the time averaging (\citet{ Shchepetkin_McWilliams_OM05}).938 uniquely defined from the filter coefficients used for the time averaging (\citet{shchepetkin.mcwilliams_OM05}). 939 939 Consistency between the time averaged continuity equation and the time stepping of tracers is here the key to 940 940 obtain exact conservation. … … 953 953 external gravity waves in idealized or weakly non-linear cases. 954 954 Although the damping is lower than for the filtered free surface, 955 it is still significant as shown by \citet{ Levier2007} in the case of an analytical barotropic Kelvin wave.955 it is still significant as shown by \citet{levier.treguier.ea_rpt07} in the case of an analytical barotropic Kelvin wave. 956 956 957 957 %>>>>>=============== … … 1051 1051 the leap-frog splitting mode in equation \autoref{eq:DYN_spg_ts_ssh}. 1052 1052 We have tried various forms of such filtering, 1053 with the following method discussed in \cite{ Griffies_al_MWR01} chosen due to1053 with the following method discussed in \cite{griffies.pacanowski.ea_MWR01} chosen due to 1054 1054 its stability and reasonably good maintenance of tracer conservation properties (see ??). 1055 1055 … … 1084 1084 \label{subsec:DYN_spg_fltp} 1085 1085 1086 The filtered formulation follows the \citet{ Roullet_Madec_JGR00} implementation.1086 The filtered formulation follows the \citet{roullet.madec_JGR00} implementation. 1087 1087 The extra term introduced in the equations (see \autoref{subsec:PE_free_surface}) is solved implicitly. 1088 1088 The elliptic solvers available in the code are documented in \autoref{chap:MISC}. … … 1326 1326 There are two main options for wetting and drying code (wd): 1327 1327 (a) an iterative limiter (il) and (b) a directional limiter (dl). 1328 The directional limiter is based on the scheme developed by \cite{ WarnerEtal13} for RO1328 The directional limiter is based on the scheme developed by \cite{warner.defne.ea_CG13} for RO 1329 1329 MS 1330 which was in turn based on ideas developed for POM by \cite{ Oey06}. The iterative1330 which was in turn based on ideas developed for POM by \cite{oey_OM06}. The iterative 1331 1331 limiter is a new scheme. The iterative limiter is activated by setting $\mathrm{ln\_wd\_il} = \mathrm{.true.}$ 1332 1332 and $\mathrm{ln\_wd\_dl} = \mathrm{.false.}$. The directional limiter is activated … … 1400 1400 1401 1401 1402 \cite{ WarnerEtal13} state that in their scheme the velocity masks at the cell faces for the baroclinic1402 \cite{warner.defne.ea_CG13} state that in their scheme the velocity masks at the cell faces for the baroclinic 1403 1403 timesteps are set to 0 or 1 depending on whether the average of the masks over the barotropic sub-steps is respectively less than 1404 1404 or greater than 0.5. That scheme does not conserve tracers in integrations started from constant tracer
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