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r11263 r11512 8 8 \label{chap:DYN} 9 9 10 \ minitoc10 \chaptertoc 11 11 12 12 Using the representation described in \autoref{chap:DOM}, … … 36 36 (surface wind stress calculation using bulk formulae, estimation of mixing coefficients) 37 37 that are carried out in modules SBC, LDF and ZDF and are described in 38 \autoref{chap:SBC}, \autoref{chap:LDF} and \autoref{chap:ZDF}, respectively. 38 \autoref{chap:SBC}, \autoref{chap:LDF} and \autoref{chap:ZDF}, respectively. 39 39 40 40 In the present chapter we also describe the diagnostic equations used to compute the horizontal divergence, 41 41 curl of the velocities (\emph{divcur} module) and the vertical velocity (\emph{wzvmod} module). 42 42 43 The different options available to the user are managed by namelist variables. 44 For term \textit{ttt} in the momentum equations, the logical namelist variables are \textit{ln\_dynttt\_xxx}, 43 The different options available to the user are managed by namelist variables. 44 For term \textit{ttt} in the momentum equations, the logical namelist variables are \textit{ln\_dynttt\_xxx}, 45 45 where \textit{xxx} is a 3 or 4 letter acronym corresponding to each optional scheme. 46 If a CPP key is used for this term its name is \key{ttt}.46 %If a CPP key is used for this term its name is \key{ttt}. 47 47 The corresponding code can be found in the \textit{dynttt\_xxx} module in the DYN directory, 48 48 and it is usually computed in the \textit{dyn\_ttt\_xxx} subroutine. 49 49 50 50 The user has the option of extracting and outputting each tendency term from the 3D momentum equations 51 (\ key{trddyn} defined), as described in \autoref{chap:MISC}.52 Furthermore, the tendency terms associated with the 2D barotropic vorticity balance (when \ key{trdvor} is defined)51 (\texttt{trddyn?} defined), as described in \autoref{chap:MISC}. 52 Furthermore, the tendency terms associated with the 2D barotropic vorticity balance (when \texttt{trdvor?} is defined) 53 53 can be derived from the 3D terms. 54 54 %%% 55 \gmcomment{STEVEN: not quite sure I've got the sense of the last sentence. does 55 \gmcomment{STEVEN: not quite sure I've got the sense of the last sentence. does 56 56 MISC correspond to "extracting tendency terms" or "vorticity balance"?} 57 57 … … 69 69 \label{subsec:DYN_divcur} 70 70 71 The vorticity is defined at an $f$-point (\ie corner point) as follows:71 The vorticity is defined at an $f$-point (\ie\ corner point) as follows: 72 72 \begin{equation} 73 73 \label{eq:divcur_cur} 74 74 \zeta =\frac{1}{e_{1f}\,e_{2f} }\left( {\;\delta_{i+1/2} \left[ {e_{2v}\;v} \right] 75 75 -\delta_{j+1/2} \left[ {e_{1u}\;u} \right]\;} \right) 76 \end{equation} 76 \end{equation} 77 77 78 78 The horizontal divergence is defined at a $T$-point. … … 97 97 ensure perfect restartability. 98 98 The vorticity and divergence at the \textit{now} time step are used for the computation of 99 the nonlinear advection and of the vertical velocity respectively. 99 the nonlinear advection and of the vertical velocity respectively. 100 100 101 101 %-------------------------------------------------------------------------------------------------------------- … … 117 117 \end{aligned} 118 118 \end{equation} 119 where \textit{emp} is the surface freshwater budget (evaporation minus precipitation), 119 where \textit{emp} is the surface freshwater budget (evaporation minus precipitation), 120 120 expressed in Kg/m$^2$/s (which is equal to mm/s), 121 121 and $\rho_w$=1,035~Kg/m$^3$ is the reference density of sea water (Boussinesq approximation). 122 122 If river runoff is expressed as a surface freshwater flux (see \autoref{chap:SBC}) then 123 \textit{emp} can be written as the evaporation minus precipitation, minus the river runoff. 123 \textit{emp} can be written as the evaporation minus precipitation, minus the river runoff. 124 124 The sea-surface height is evaluated using exactly the same time stepping scheme as 125 125 the tracer equation \autoref{eq:tra_nxt}: 126 126 a leapfrog scheme in combination with an Asselin time filter, 127 \ie the velocity appearing in \autoref{eq:dynspg_ssh} is centred in time (\textit{now} velocity).127 \ie\ the velocity appearing in \autoref{eq:dynspg_ssh} is centred in time (\textit{now} velocity). 128 128 This is of paramount importance. 129 129 Replacing $T$ by the number $1$ in the tracer equation and summing over the water column must lead to … … 144 144 \end{equation} 145 145 146 In the case of a non-linear free surface (\ key{vvl}), the top vertical velocity is $-\textit{emp}/\rho_w$,146 In the case of a non-linear free surface (\texttt{vvl?}), the top vertical velocity is $-\textit{emp}/\rho_w$, 147 147 as changes in the divergence of the barotropic transport are absorbed into the change of the level thicknesses, 148 148 re-orientated downward. … … 151 151 The upper boundary condition applies at a fixed level $z=0$. 152 152 The top vertical velocity is thus equal to the divergence of the barotropic transport 153 (\ie the first term in the right-hand-side of \autoref{eq:dynspg_ssh}).153 (\ie\ the first term in the right-hand-side of \autoref{eq:dynspg_ssh}). 154 154 155 155 Note also that whereas the vertical velocity has the same discrete expression in $z$- and $s$-coordinates, … … 158 158 Note also that the $k$-axis is re-orientated downwards in the \fortran code compared to 159 159 the indexing used in the semi-discrete equations such as \autoref{eq:wzv} 160 (see \autoref{subsec:DOM_Num_Index_vertical}). 160 (see \autoref{subsec:DOM_Num_Index_vertical}). 161 161 162 162 … … 168 168 %-----------------------------------------nam_dynadv---------------------------------------------------- 169 169 170 \nlst{namdyn_adv} 170 \nlst{namdyn_adv} 171 171 %------------------------------------------------------------------------------------------------------------- 172 172 173 173 The vector invariant form of the momentum equations is the one most often used in 174 applications of the \NEMO ocean model.174 applications of the \NEMO\ ocean model. 175 175 The flux form option (see next section) has been present since version $2$. 176 Options are defined through the \n gn{namdyn\_adv} namelist variables Coriolis and176 Options are defined through the \nam{dyn\_adv} namelist variables Coriolis and 177 177 momentum advection terms are evaluated using a leapfrog scheme, 178 \ie the velocity appearing in these expressions is centred in time (\textit{now} velocity).178 \ie\ the velocity appearing in these expressions is centred in time (\textit{now} velocity). 179 179 At the lateral boundaries either free slip, no slip or partial slip boundary conditions are applied following 180 180 \autoref{chap:LBC}. 181 181 182 182 % ------------------------------------------------------------------------------------------------------------- 183 % Vorticity term 183 % Vorticity term 184 184 % ------------------------------------------------------------------------------------------------------------- 185 185 \subsection[Vorticity term (\textit{dynvor.F90})] … … 188 188 %------------------------------------------nam_dynvor---------------------------------------------------- 189 189 190 \nlst{namdyn_vor} 190 \nlst{namdyn_vor} 191 191 %------------------------------------------------------------------------------------------------------------- 192 192 193 Options are defined through the \n gn{namdyn\_vor} namelist variables.194 Four discretisations of the vorticity term (\ np{ln\_dynvor\_xxx}\forcode{ = .true.}) are available:193 Options are defined through the \nam{dyn\_vor} namelist variables. 194 Four discretisations of the vorticity term (\texttt{ln\_dynvor\_xxx}\forcode{ = .true.}) are available: 195 195 conserving potential enstrophy of horizontally non-divergent flow (ENS scheme); 196 196 conserving horizontal kinetic energy (ENE scheme); … … 212 212 In the enstrophy conserving case (ENS scheme), 213 213 the discrete formulation of the vorticity term provides a global conservation of the enstrophy 214 ($ [ (\zeta +f ) / e_{3f} ]^2 $ in $s$-coordinates) for a horizontally non-divergent flow (\ie $\chi$=$0$),214 ($ [ (\zeta +f ) / e_{3f} ]^2 $ in $s$-coordinates) for a horizontally non-divergent flow (\ie\ $\chi$=$0$), 215 215 but does not conserve the total kinetic energy. 216 216 It is given by: … … 225 225 \end{aligned} 226 226 \right. 227 \end{equation} 227 \end{equation} 228 228 229 229 %------------------------------------------------------------- … … 246 246 \end{aligned} 247 247 \right. 248 \end{equation} 248 \end{equation} 249 249 250 250 %------------------------------------------------------------- … … 285 285 the presence of grid point oscillation structures that will be invisible to the operator. 286 286 These structures are \textit{computational modes} that will be at least partly damped by 287 the momentum diffusion operator (\ie the subgrid-scale advection), but not by the resolved advection term.287 the momentum diffusion operator (\ie\ the subgrid-scale advection), but not by the resolved advection term. 288 288 The ENS and ENE schemes therefore do not contribute to dump any grid point noise in the horizontal velocity field. 289 289 Such noise would result in more noise in the vertical velocity field, an undesirable feature. … … 291 291 $u$ and $v$ are located at different grid points, 292 292 a price worth paying to avoid a double averaging in the pressure gradient term as in the $B$-grid. 293 \gmcomment{ To circumvent this, Adcroft (ADD REF HERE) 293 \gmcomment{ To circumvent this, Adcroft (ADD REF HERE) 294 294 Nevertheless, this technique strongly distort the phase and group velocity of Rossby waves....} 295 295 … … 299 299 \citep{griffies.gnanadesikan.ea_JPO98} for the discretization of the iso-neutral diffusion operator (see \autoref{apdx:C}). 300 300 301 The \citet{arakawa.hsu_MWR90} vorticity advection scheme for a single layer is modified 302 for spherical coordinates as described by \citet{arakawa.lamb_MWR81} to obtain the EEN scheme. 303 First consider the discrete expression of the potential vorticity, $q$, defined at an $f$-point: 301 The \citet{arakawa.hsu_MWR90} vorticity advection scheme for a single layer is modified 302 for spherical coordinates as described by \citet{arakawa.lamb_MWR81} to obtain the EEN scheme. 303 First consider the discrete expression of the potential vorticity, $q$, defined at an $f$-point: 304 304 \[ 305 305 % \label{eq:pot_vor} … … 307 307 \] 308 308 where the relative vorticity is defined by (\autoref{eq:divcur_cur}), 309 the Coriolis parameter is given by $f=2 \,\Omega \;\sin \varphi _f $ and the layer thickness at $f$-points is: 309 the Coriolis parameter is given by $f=2 \,\Omega \;\sin \varphi _f $ and the layer thickness at $f$-points is: 310 310 \begin{equation} 311 311 \label{eq:een_e3f} … … 326 326 % >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 327 327 328 A key point in \autoref{eq:een_e3f} is how the averaging in the \textbf{i}- and \textbf{j}- directions is made. 328 A key point in \autoref{eq:een_e3f} is how the averaging in the \textbf{i}- and \textbf{j}- directions is made. 329 329 It uses the sum of masked t-point vertical scale factor divided either by the sum of the four t-point masks 330 330 (\np{nn\_een\_e3f}\forcode{ = 1}), or just by $4$ (\np{nn\_een\_e3f}\forcode{ = .true.}). … … 334 334 (with a systematic reduction of $e_{3f}$ when a model level intercept the bathymetry) 335 335 that tends to reinforce the topostrophy of the flow 336 (\ie the tendency of the flow to follow the isobaths) \citep{penduff.le-sommer.ea_OS07}.336 (\ie\ the tendency of the flow to follow the isobaths) \citep{penduff.le-sommer.ea_OS07}. 337 337 338 338 Next, the vorticity triads, $ {^i_j}\mathbb{Q}^{i_p}_{j_p}$ can be defined at a $T$-point as 339 339 the following triad combinations of the neighbouring potential vorticities defined at f-points 340 (\autoref{fig:DYN_een_triad}): 340 (\autoref{fig:DYN_een_triad}): 341 341 \begin{equation} 342 342 \label{eq:Q_triads} … … 344 344 = \frac{1}{12} \ \left( q^{i-i_p}_{j+j_p} + q^{i+j_p}_{j+i_p} + q^{i+i_p}_{j-j_p} \right) 345 345 \end{equation} 346 where the indices $i_p$ and $k_p$ take the values: $i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$. 347 348 Finally, the vorticity terms are represented as: 346 where the indices $i_p$ and $k_p$ take the values: $i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$. 347 348 Finally, the vorticity terms are represented as: 349 349 \begin{equation} 350 350 \label{eq:dynvor_een} … … 357 357 \end{aligned} 358 358 } \right. 359 \end{equation} 359 \end{equation} 360 360 361 361 This EEN scheme in fact combines the conservation properties of the ENS and ENE schemes. 362 362 It conserves both total energy and potential enstrophy in the limit of horizontally nondivergent flow 363 (\ie $\chi$=$0$) (see \autoref{subsec:C_vorEEN}).363 (\ie\ $\chi$=$0$) (see \autoref{subsec:C_vorEEN}). 364 364 Applied to a realistic ocean configuration, it has been shown that it leads to a significant reduction of 365 365 the noise in the vertical velocity field \citep{le-sommer.penduff.ea_OM09}. 366 366 Furthermore, used in combination with a partial steps representation of bottom topography, 367 367 it improves the interaction between current and topography, 368 leading to a larger topostrophy of the flow \citep{barnier.madec.ea_OD06, penduff.le-sommer.ea_OS07}. 368 leading to a larger topostrophy of the flow \citep{barnier.madec.ea_OD06, penduff.le-sommer.ea_OS07}. 369 369 370 370 %-------------------------------------------------------------------------------------------------------------- … … 412 412 When \np{ln\_dynzad\_zts}\forcode{ = .true.}, 413 413 a split-explicit time stepping with 5 sub-timesteps is used on the vertical advection term. 414 This option can be useful when the value of the timestep is limited by vertical advection \citep{lemarie.debreu.ea_OM15}. 414 This option can be useful when the value of the timestep is limited by vertical advection \citep{lemarie.debreu.ea_OM15}. 415 415 Note that in this case, 416 416 a similar split-explicit time stepping should be used on vertical advection of tracer to ensure a better stability, … … 425 425 %------------------------------------------nam_dynadv---------------------------------------------------- 426 426 427 \nlst{namdyn_adv} 427 \nlst{namdyn_adv} 428 428 %------------------------------------------------------------------------------------------------------------- 429 429 430 Options are defined through the \n gn{namdyn\_adv} namelist variables.430 Options are defined through the \nam{dyn\_adv} namelist variables. 431 431 In the flux form (as in the vector invariant form), 432 432 the Coriolis and momentum advection terms are evaluated using a leapfrog scheme, 433 \ie the velocity appearing in their expressions is centred in time (\textit{now} velocity).433 \ie\ the velocity appearing in their expressions is centred in time (\textit{now} velocity). 434 434 At the lateral boundaries either free slip, 435 435 no slip or partial slip boundary conditions are applied following \autoref{chap:LBC}. … … 445 445 In flux form, the vorticity term reduces to a Coriolis term in which the Coriolis parameter has been modified to account for the "metric" term. 446 446 This altered Coriolis parameter is thus discretised at $f$-points. 447 It is given by: 447 It is given by: 448 448 \begin{multline*} 449 449 % \label{eq:dyncor_metric} … … 451 451 \equiv f + \frac{1}{e_{1f} e_{2f} } \left( { \ \overline v ^{i+1/2}\delta_{i+1/2} \left[ {e_{2u} } \right] 452 452 - \overline u ^{j+1/2}\delta_{j+1/2} \left[ {e_{1u} } \right] } \ \right) 453 \end{multline*} 453 \end{multline*} 454 454 455 455 Any of the (\autoref{eq:dynvor_ens}), (\autoref{eq:dynvor_ene}) and (\autoref{eq:dynvor_een}) schemes can be used to 456 456 compute the product of the Coriolis parameter and the vorticity. 457 457 However, the energy-conserving scheme (\autoref{eq:dynvor_een}) has exclusively been used to date. 458 This term is evaluated using a leapfrog scheme, \ie the velocity is centred in time (\textit{now} velocity).458 This term is evaluated using a leapfrog scheme, \ie\ the velocity is centred in time (\textit{now} velocity). 459 459 460 460 %-------------------------------------------------------------------------------------------------------------- … … 487 487 or a $3^{rd}$ order upstream biased scheme, UBS. 488 488 The latter is described in \citet{shchepetkin.mcwilliams_OM05}. 489 The schemes are selected using the namelist logicals \np{ln\_dynadv\_cen2} and \np{ln\_dynadv\_ubs}. 489 The schemes are selected using the namelist logicals \np{ln\_dynadv\_cen2} and \np{ln\_dynadv\_ubs}. 490 490 In flux form, the schemes differ by the choice of a space and time interpolation to define the value of 491 $u$ and $v$ at the centre of each face of $u$- and $v$-cells, \ie at the $T$-, $f$-,492 and $uw$-points for $u$ and at the $f$-, $T$- and $vw$-points for $v$. 491 $u$ and $v$ at the centre of each face of $u$- and $v$-cells, \ie\ at the $T$-, $f$-, 492 and $uw$-points for $u$ and at the $f$-, $T$- and $vw$-points for $v$. 493 493 494 494 %------------------------------------------------------------- … … 508 508 \end{aligned} 509 509 \right. 510 \end{equation} 511 512 The scheme is non diffusive (\ie conserves the kinetic energy) but dispersive (\ieit may create false extrema).510 \end{equation} 511 512 The scheme is non diffusive (\ie\ conserves the kinetic energy) but dispersive (\ie\ it may create false extrema). 513 513 It is therefore notoriously noisy and must be used in conjunction with an explicit diffusion operator to 514 514 produce a sensible solution. … … 535 535 \end{equation} 536 536 where $u"_{i+1/2} =\delta_{i+1/2} \left[ {\delta_i \left[ u \right]} \right]$. 537 This results in a dissipatively dominant (\ie hyper-diffusive) truncation error537 This results in a dissipatively dominant (\ie\ hyper-diffusive) truncation error 538 538 \citep{shchepetkin.mcwilliams_OM05}. 539 539 The overall performance of the advection scheme is similar to that reported in \citet{farrow.stevens_JPO95}. … … 541 541 It is not a \emph{positive} scheme, meaning that false extrema are permitted. 542 542 But the amplitudes of the false extrema are significantly reduced over those in the centred second order method. 543 As the scheme already includes a diffusion component, it can be used without explicit lateral diffusion on momentum 544 (\ie \np{ln\_dynldf\_lap}\forcode{ = }\np{ln\_dynldf\_bilap}\forcode{ = .false.}),543 As the scheme already includes a diffusion component, it can be used without explicit lateral diffusion on momentum 544 (\ie\ \np{ln\_dynldf\_lap}\forcode{ = }\np{ln\_dynldf\_bilap}\forcode{ = .false.}), 545 545 and it is recommended to do so. 546 546 547 547 The UBS scheme is not used in all directions. 548 In the vertical, the centred $2^{nd}$ order evaluation of the advection is preferred, \ie $u_{uw}^{ubs}$ and548 In the vertical, the centred $2^{nd}$ order evaluation of the advection is preferred, \ie\ $u_{uw}^{ubs}$ and 549 549 $u_{vw}^{ubs}$ in \autoref{eq:dynadv_cen2} are used. 550 UBS is diffusive and is associated with vertical mixing of momentum. \gmcomment{ gm pursue the 550 UBS is diffusive and is associated with vertical mixing of momentum. \gmcomment{ gm pursue the 551 551 sentence:Since vertical mixing of momentum is a source term of the TKE equation... } 552 552 … … 578 578 %------------------------------------------nam_dynhpg--------------------------------------------------- 579 579 580 \nlst{namdyn_hpg} 580 \nlst{namdyn_hpg} 581 581 %------------------------------------------------------------------------------------------------------------- 582 582 583 Options are defined through the \n gn{namdyn\_hpg} namelist variables.583 Options are defined through the \nam{dyn\_hpg} namelist variables. 584 584 The key distinction between the different algorithms used for 585 585 the hydrostatic pressure gradient is the vertical coordinate used, 586 since HPG is a \emph{horizontal} pressure gradient, \ie computed along geopotential surfaces.586 since HPG is a \emph{horizontal} pressure gradient, \ie\ computed along geopotential surfaces. 587 587 As a result, any tilt of the surface of the computational levels will require a specific treatment to 588 588 compute the hydrostatic pressure gradient. 589 589 590 590 The hydrostatic pressure gradient term is evaluated either using a leapfrog scheme, 591 \ie the density appearing in its expression is centred in time (\emph{now} $\rho$),591 \ie\ the density appearing in its expression is centred in time (\emph{now} $\rho$), 592 592 or a semi-implcit scheme. 593 593 At the lateral boundaries either free slip, no slip or partial slip boundary conditions are applied. … … 616 616 \end{aligned} 617 617 \right. 618 \end{equation} 618 \end{equation} 619 619 620 620 for $1<k<km$ (interior layer) … … 631 631 \end{aligned} 632 632 \right. 633 \end{equation} 633 \end{equation} 634 634 635 635 Note that the $1/2$ factor in (\autoref{eq:dynhpg_zco_surf}) is adequate because of the definition of $e_{3w}$ as 636 636 the vertical derivative of the scale factor at the surface level ($z=0$). 637 Note also that in case of variable volume level (\ key{vvl} defined),637 Note also that in case of variable volume level (\texttt{vvl?} defined), 638 638 the surface pressure gradient is included in \autoref{eq:dynhpg_zco_surf} and 639 639 \autoref{eq:dynhpg_zco} through the space and time variations of the vertical scale factor $e_{3w}$. … … 649 649 Before taking horizontal gradients between these tracer points, 650 650 a linear interpolation is used to approximate the deeper tracer as if 651 it actually lived at the depth of the shallower tracer point. 651 it actually lived at the depth of the shallower tracer point. 652 652 653 653 Apart from this modification, … … 668 668 669 669 Pressure gradient formulations in an $s$-coordinate have been the subject of a vast number of papers 670 (\eg, \citet{song_MWR98, shchepetkin.mcwilliams_OM05}). 670 (\eg, \citet{song_MWR98, shchepetkin.mcwilliams_OM05}). 671 671 A number of different pressure gradient options are coded but the ROMS-like, 672 672 density Jacobian with cubic polynomial method is currently disabled whilst known bugs are under investigation. … … 683 683 \end{aligned} 684 684 \right. 685 \end{equation} 685 \end{equation} 686 686 687 687 Where the first term is the pressure gradient along coordinates, 688 688 computed as in \autoref{eq:dynhpg_zco_surf} - \autoref{eq:dynhpg_zco}, 689 and $z_T$ is the depth of the $T$-point evaluated from the sum of the vertical scale factors at the $w$-point 689 and $z_T$ is the depth of the $T$-point evaluated from the sum of the vertical scale factors at the $w$-point 690 690 ($e_{3w}$). 691 691 692 692 $\bullet$ Traditional coding with adaptation for ice shelf cavities (\np{ln\_dynhpg\_isf}\forcode{ = .true.}). 693 693 This scheme need the activation of ice shelf cavities (\np{ln\_isfcav}\forcode{ = .true.}). … … 695 695 $\bullet$ Pressure Jacobian scheme (prj) (a research paper in preparation) (\np{ln\_dynhpg\_prj}\forcode{ = .true.}) 696 696 697 $\bullet$ Density Jacobian with cubic polynomial scheme (DJC) \citep{shchepetkin.mcwilliams_OM05} 697 $\bullet$ Density Jacobian with cubic polynomial scheme (DJC) \citep{shchepetkin.mcwilliams_OM05} 698 698 (\np{ln\_dynhpg\_djc}\forcode{ = .true.}) (currently disabled; under development) 699 699 700 700 Note that expression \autoref{eq:dynhpg_sco} is commonly used when the variable volume formulation is activated 701 (\ key{vvl}) because in that case, even with a flat bottom,701 (\texttt{vvl?}) because in that case, even with a flat bottom, 702 702 the coordinate surfaces are not horizontal but follow the free surface \citep{levier.treguier.ea_rpt07}. 703 703 The pressure jacobian scheme (\np{ln\_dynhpg\_prj}\forcode{ = .true.}) is available as 704 an improved option to \np{ln\_dynhpg\_sco}\forcode{ = .true.} when \ key{vvl} is active.704 an improved option to \np{ln\_dynhpg\_sco}\forcode{ = .true.} when \texttt{vvl?} is active. 705 705 The pressure Jacobian scheme uses a constrained cubic spline to 706 706 reconstruct the density profile across the water column. … … 723 723 724 724 The pressure gradient due to ocean load is computed using the expression \autoref{eq:dynhpg_sco} described in 725 \autoref{subsec:DYN_hpg_sco}. 725 \autoref{subsec:DYN_hpg_sco}. 726 726 727 727 %-------------------------------------------------------------------------------------------------------------- … … 742 742 It involves the evaluation of the hydrostatic pressure gradient as 743 743 an average over the three time levels $t-\rdt$, $t$, and $t+\rdt$ 744 (\ie \textit{before}, \textit{now} and \textit{after} time-steps),745 rather than at the central time level $t$ only, as in the standard leapfrog scheme. 744 (\ie\ \textit{before}, \textit{now} and \textit{after} time-steps), 745 rather than at the central time level $t$ only, as in the standard leapfrog scheme. 746 746 747 747 $\bullet$ leapfrog scheme (\np{ln\_dynhpg\_imp}\forcode{ = .true.}): … … 795 795 %-----------------------------------------nam_dynspg---------------------------------------------------- 796 796 797 \nlst{namdyn_spg} 797 \nlst{namdyn_spg} 798 798 %------------------------------------------------------------------------------------------------------------ 799 799 800 Options are defined through the \n gn{namdyn\_spg} namelist variables.800 Options are defined through the \nam{dyn\_spg} namelist variables. 801 801 The surface pressure gradient term is related to the representation of the free surface (\autoref{sec:PE_hor_pg}). 802 802 The main distinction is between the fixed volume case (linear free surface) and 803 the variable volume case (nonlinear free surface, \ key{vvl} is defined).803 the variable volume case (nonlinear free surface, \texttt{vvl?} is defined). 804 804 In the linear free surface case (\autoref{subsec:PE_free_surface}) 805 805 the vertical scale factors $e_{3}$ are fixed in time, 806 806 while they are time-dependent in the nonlinear case (\autoref{subsec:PE_free_surface}). 807 With both linear and nonlinear free surface, external gravity waves are allowed in the equations, 807 With both linear and nonlinear free surface, external gravity waves are allowed in the equations, 808 808 which imposes a very small time step when an explicit time stepping is used. 809 Two methods are proposed to allow a longer time step for the three-dimensional equations: 810 the filtered free surface, which is a modification of the continuous equations (see \autoref{eq:PE_flt?}), 809 Two methods are proposed to allow a longer time step for the three-dimensional equations: 810 the filtered free surface, which is a modification of the continuous equations (see \autoref{eq:PE_flt?}), 811 811 and the split-explicit free surface described below. 812 The extra term introduced in the filtered method is calculated implicitly, 812 The extra term introduced in the filtered method is calculated implicitly, 813 813 so that the update of the next velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}. 814 814 … … 819 819 an explicit formulation which requires a small time step; 820 820 a filtered free surface formulation which allows a larger time step by 821 adding a filtering term into the momentum equation; 821 adding a filtering term into the momentum equation; 822 822 and a split-explicit free surface formulation, described below, which also allows a larger time step. 823 823 … … 829 829 % Explicit free surface formulation 830 830 %-------------------------------------------------------------------------------------------------------------- 831 \subsection[Explicit free surface (\texttt{ \textbf{key\_dynspg\_exp}})]832 {Explicit free surface (\protect\ key{dynspg\_exp})}831 \subsection[Explicit free surface (\texttt{ln\_dynspg\_exp}\forcode{ = .true.})] 832 {Explicit free surface (\protect\np{ln\_dynspg\_exp}\forcode{ = .true.})} 833 833 \label{subsec:DYN_spg_exp} 834 834 835 In the explicit free surface formulation (\ key{dynspg\_exp} defined),835 In the explicit free surface formulation (\np{ln\_dynspg\_exp} set to true), 836 836 the model time step is chosen to be small enough to resolve the external gravity waves 837 837 (typically a few tens of seconds). 838 The surface pressure gradient, evaluated using a leap-frog scheme (\ie centered in time),838 The surface pressure gradient, evaluated using a leap-frog scheme (\ie\ centered in time), 839 839 is thus simply given by : 840 840 \begin{equation} … … 846 846 \end{aligned} 847 847 \right. 848 \end{equation} 849 850 Note that in the non-linear free surface case (\ie \key{vvl} defined),848 \end{equation} 849 850 Note that in the non-linear free surface case (\ie\ \texttt{vvl?} defined), 851 851 the surface pressure gradient is already included in the momentum tendency through 852 852 the level thickness variation allowed in the computation of the hydrostatic pressure gradient. … … 856 856 % Split-explict free surface formulation 857 857 %-------------------------------------------------------------------------------------------------------------- 858 \subsection[Split-explicit free surface (\texttt{ \textbf{key\_dynspg\_ts}})]859 {Split-explicit free surface (\protect\ key{dynspg\_ts})}858 \subsection[Split-explicit free surface (\texttt{ln\_dynspg\_ts}\forcode{ = .true.})] 859 {Split-explicit free surface (\protect\np{ln\_dynspg\_ts}\forcode{ = .true.})} 860 860 \label{subsec:DYN_spg_ts} 861 861 %------------------------------------------namsplit----------------------------------------------------------- … … 864 864 %------------------------------------------------------------------------------------------------------------- 865 865 866 The split-explicit free surface formulation used in \NEMO (\key{dynspg\_ts} defined),866 The split-explicit free surface formulation used in \NEMO\ (\np{ln\_dynspg\_ts} set to true), 867 867 also called the time-splitting formulation, follows the one proposed by \citet{shchepetkin.mcwilliams_OM05}. 868 868 The general idea is to solve the free surface equation and the associated barotropic velocity equations with … … 897 897 Temporal discretization of the system above follows a three-time step Generalized Forward Backward algorithm 898 898 detailed in \citet{shchepetkin.mcwilliams_OM05}. 899 AB3-AM4 coefficients used in \NEMO follow the second-order accurate,899 AB3-AM4 coefficients used in \NEMO\ follow the second-order accurate, 900 900 "multi-purpose" stability compromise as defined in \citet{shchepetkin.mcwilliams_ibk09} 901 (see their figure 12, lower left). 901 (see their figure 12, lower left). 902 902 903 903 %> > > > > > > > > > > > > > > > > > > > > > > > > > > > … … 935 935 provide time filtered quantities. 936 936 These are used for the subsequent initialization of the barotropic mode in the following baroclinic step. 937 Since external mode equations written at baroclinic time steps finally follow a forward time stepping scheme, 937 Since external mode equations written at baroclinic time steps finally follow a forward time stepping scheme, 938 938 asselin filtering is not applied to barotropic quantities.\\ 939 939 Alternatively, one can choose to integrate barotropic equations starting from \textit{before} time step … … 963 963 964 964 One can eventually choose to feedback instantaneous values by not using any time filter 965 (\np{ln\_bt\_av}\forcode{ = .false.}). 965 (\np{ln\_bt\_av}\forcode{ = .false.}). 966 966 In that case, external mode equations are continuous in time, 967 \ie they are not re-initialized when starting a new sub-stepping sequence.967 \ie\ they are not re-initialized when starting a new sub-stepping sequence. 968 968 This is the method used so far in the POM model, the stability being maintained by 969 969 refreshing at (almost) each barotropic time step advection and horizontal diffusion terms. 970 Since the latter terms have not been added in \NEMO for computational efficiency,970 Since the latter terms have not been added in \NEMO\ for computational efficiency, 971 971 removing time filtering is not recommended except for debugging purposes. 972 972 This may be used for instance to appreciate the damping effect of the standard formulation on … … 976 976 977 977 %>>>>>=============== 978 \gmcomment{ %%% copy from griffies Book 978 \gmcomment{ %%% copy from griffies Book 979 979 980 980 \textbf{title: Time stepping the barotropic system } … … 983 983 Hence, we can update the surface height and vertically integrated velocity with a leap-frog scheme using 984 984 the small barotropic time step $\rdt$. 985 We have 985 We have 986 986 987 987 \[ … … 1006 1006 and total depth of the ocean $H(\tau)$ are held for the duration of the barotropic time stepping over 1007 1007 a single cycle. 1008 This is also the time that sets the barotropic time steps via 1008 This is also the time that sets the barotropic time steps via 1009 1009 \[ 1010 1010 % \label{eq:DYN_spg_ts_t} … … 1012 1012 \] 1013 1013 with $n$ an integer. 1014 The density scaled surface pressure is evaluated via 1014 The density scaled surface pressure is evaluated via 1015 1015 \[ 1016 1016 % \label{eq:DYN_spg_ts_ps} … … 1021 1021 \end{cases} 1022 1022 \] 1023 To get started, we assume the following initial conditions 1023 To get started, we assume the following initial conditions 1024 1024 \[ 1025 1025 % \label{eq:DYN_spg_ts_eta} … … 1029 1029 \end{split} 1030 1030 \] 1031 with 1031 with 1032 1032 \[ 1033 1033 % \label{eq:DYN_spg_ts_etaF} … … 1035 1035 \] 1036 1036 the time averaged surface height taken from the previous barotropic cycle. 1037 Likewise, 1037 Likewise, 1038 1038 \[ 1039 1039 % \label{eq:DYN_spg_ts_u} … … 1041 1041 \textbf{U}(\tau,t_{n=1}) = \textbf{U}^{(b)}(\tau,t_{n=0}) + \rdt \ \text{RHS}_{n=0} 1042 1042 \] 1043 with 1043 with 1044 1044 \[ 1045 1045 % \label{eq:DYN_spg_ts_u} … … 1047 1047 \] 1048 1048 the time averaged vertically integrated transport. 1049 Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration. 1049 Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration. 1050 1050 1051 1051 Upon reaching $t_{n=N} = \tau + 2\rdt \tau$ , 1052 1052 the vertically integrated velocity is time averaged to produce the updated vertically integrated velocity at 1053 baroclinic time $\tau + \rdt \tau$ 1053 baroclinic time $\tau + \rdt \tau$ 1054 1054 \[ 1055 1055 % \label{eq:DYN_spg_ts_u} … … 1057 1057 \] 1058 1058 The surface height on the new baroclinic time step is then determined via a baroclinic leap-frog using 1059 the following form 1059 the following form 1060 1060 1061 1061 \begin{equation} 1062 1062 \label{eq:DYN_spg_ts_ssh} 1063 \eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\rdt \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right] 1063 \eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\rdt \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right] 1064 1064 \end{equation} 1065 1065 1066 1066 The use of this "big-leap-frog" scheme for the surface height ensures compatibility between 1067 1067 the mass/volume budgets and the tracer budgets. 1068 More discussion of this point is provided in Chapter 10 (see in particular Section 10.2). 1069 1068 More discussion of this point is provided in Chapter 10 (see in particular Section 10.2). 1069 1070 1070 In general, some form of time filter is needed to maintain integrity of the surface height field due to 1071 1071 the leap-frog splitting mode in equation \autoref{eq:DYN_spg_ts_ssh}. … … 1078 1078 \eta^{F}(\tau-\Delta) = \overline{\eta^{(b)}(\tau)} 1079 1079 \end{equation} 1080 Another approach tried was 1080 Another approach tried was 1081 1081 1082 1082 \[ … … 1091 1091 eliminating tracer and surface height time filtering (see ?? for more complete discussion). 1092 1092 However, in the general case with a non-zero $\alpha$, 1093 the filter \autoref{eq:DYN_spg_ts_sshf} was found to be more conservative, and so is recommended. 1093 the filter \autoref{eq:DYN_spg_ts_sshf} was found to be more conservative, and so is recommended. 1094 1094 1095 1095 } %%end gm comment (copy of griffies book) … … 1101 1101 % Filtered free surface formulation 1102 1102 %-------------------------------------------------------------------------------------------------------------- 1103 \subsection[Filtered free surface (\texttt{ \textbf{key\_dynspg\_flt}})]1104 {Filtered free surface (\protect\ key{dynspg\_flt})}1103 \subsection[Filtered free surface (\texttt{dynspg\_flt?})] 1104 {Filtered free surface (\protect\texttt{dynspg\_flt?})} 1105 1105 \label{subsec:DYN_spg_fltp} 1106 1106 1107 The filtered formulation follows the \citet{roullet.madec_JGR00} implementation. 1108 The extra term introduced in the equations (see \autoref{subsec:PE_free_surface}) is solved implicitly. 1107 The filtered formulation follows the \citet{roullet.madec_JGR00} implementation. 1108 The extra term introduced in the equations (see \autoref{subsec:PE_free_surface}) is solved implicitly. 1109 1109 The elliptic solvers available in the code are documented in \autoref{chap:MISC}. 1110 1110 1111 1111 %% gm %%======>>>> given here the discrete eqs provided to the solver 1112 \gmcomment{ %%% copy from chap-model basics 1112 \gmcomment{ %%% copy from chap-model basics 1113 1113 \[ 1114 1114 % \label{eq:spg_flt} … … 1123 1123 } %end gmcomment 1124 1124 1125 Note that in the linear free surface formulation (\ key{vvl} not defined),1125 Note that in the linear free surface formulation (\texttt{vvl?} not defined), 1126 1126 the ocean depth is time-independent and so is the matrix to be inverted. 1127 It is computed once and for all and applies to all ocean time steps. 1127 It is computed once and for all and applies to all ocean time steps. 1128 1128 1129 1129 % ================================================================ … … 1135 1135 %------------------------------------------nam_dynldf---------------------------------------------------- 1136 1136 1137 \nlst{namdyn_ldf} 1137 \nlst{namdyn_ldf} 1138 1138 %------------------------------------------------------------------------------------------------------------- 1139 1139 1140 Options are defined through the \n gn{namdyn\_ldf} namelist variables.1140 Options are defined through the \nam{dyn\_ldf} namelist variables. 1141 1141 The options available for lateral diffusion are to use either laplacian (rotated or not) or biharmonic operators. 1142 1142 The coefficients may be constant or spatially variable; 1143 1143 the description of the coefficients is found in the chapter on lateral physics (\autoref{chap:LDF}). 1144 1144 The lateral diffusion of momentum is evaluated using a forward scheme, 1145 \ie the velocity appearing in its expression is the \textit{before} velocity in time,1145 \ie\ the velocity appearing in its expression is the \textit{before} velocity in time, 1146 1146 except for the pure vertical component that appears when a tensor of rotation is used. 1147 1147 This latter term is solved implicitly together with the vertical diffusion term (see \autoref{chap:STP}). … … 1159 1159 In finite difference methods, 1160 1160 the biharmonic operator is frequently the method of choice to achieve this scale selective dissipation since 1161 its damping time (\ie its spin down time) scale like $\lambda^{-4}$ for disturbances of wavelength $\lambda$1161 its damping time (\ie\ its spin down time) scale like $\lambda^{-4}$ for disturbances of wavelength $\lambda$ 1162 1162 (so that short waves damped more rapidelly than long ones), 1163 1163 whereas the Laplace operator damping time scales only like $\lambda^{-2}$. … … 1169 1169 \label{subsec:DYN_ldf_lap} 1170 1170 1171 For lateral iso-level diffusion, the discrete operator is: 1171 For lateral iso-level diffusion, the discrete operator is: 1172 1172 \begin{equation} 1173 1173 \label{eq:dynldf_lap} … … 1175 1175 \begin{aligned} 1176 1176 D_u^{l{\mathrm {\mathbf U}}} =\frac{1}{e_{1u} }\delta_{i+1/2} \left[ {A_T^{lm} 1177 \;\chi } \right]-\frac{1}{e_{2u} {\kern 1pt}e_{3u} }\delta_j \left[ 1177 \;\chi } \right]-\frac{1}{e_{2u} {\kern 1pt}e_{3u} }\delta_j \left[ 1178 1178 {A_f^{lm} \;e_{3f} \zeta } \right] \\ \\ 1179 1179 D_v^{l{\mathrm {\mathbf U}}} =\frac{1}{e_{2v} }\delta_{j+1/2} \left[ {A_T^{lm} 1180 \;\chi } \right]+\frac{1}{e_{1v} {\kern 1pt}e_{3v} }\delta_i \left[ 1180 \;\chi } \right]+\frac{1}{e_{1v} {\kern 1pt}e_{3v} }\delta_i \left[ 1181 1181 {A_f^{lm} \;e_{3f} \zeta } \right] 1182 1182 \end{aligned} 1183 1183 \right. 1184 \end{equation} 1184 \end{equation} 1185 1185 1186 1186 As explained in \autoref{subsec:PE_ldf}, 1187 1187 this formulation (as the gradient of a divergence and curl of the vorticity) preserves symmetry and 1188 ensures a complete separation between the vorticity and divergence parts of the momentum diffusion. 1188 ensures a complete separation between the vorticity and divergence parts of the momentum diffusion. 1189 1189 1190 1190 %-------------------------------------------------------------------------------------------------------------- … … 1230 1230 -e_{2f} \; r_{1f} \,\overline{\overline {\delta_{k+1/2}[v]}}^{\,i+1/2,\,k}} 1231 1231 \right)} \right]} \right. \\ 1232 & \qquad +\ \delta_j \left[ {A_T^{lm} \left( {\frac{e_{1t}\,e_{3t} }{e_{2t} 1232 & \qquad +\ \delta_j \left[ {A_T^{lm} \left( {\frac{e_{1t}\,e_{3t} }{e_{2t} 1233 1233 }\,\delta_{j} [v] - e_{1t}\, r_{2t} 1234 1234 \,\overline{\overline {\delta_{k+1/2} [v]}} ^{\,j,\,k}} 1235 1235 \right)} \right] \\ 1236 & \qquad +\ \delta_k \left[ {A_{vw}^{lm} \left( {-e_{2v} \, r_{1vw} \,\overline{\overline 1236 & \qquad +\ \delta_k \left[ {A_{vw}^{lm} \left( {-e_{2v} \, r_{1vw} \,\overline{\overline 1237 1237 {\delta_{i+1/2} [v]}}^{\,i+1/2,\,k+1/2} }\right.} \right. \\ 1238 1238 & \ \qquad \qquad \qquad \quad\ 1239 1239 - e_{1v} \, r_{2vw} \,\overline{\overline {\delta_{j+1/2} [v]}} ^{\,j+1/2,\,k+1/2} \\ 1240 & \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\ 1240 & \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\ 1241 1241 +\frac{e_{1v}\, e_{2v} }{e_{3vw} }\,\left( {r_{1vw}^2+r_{2vw}^2} 1242 \right)\,\delta_{k+1/2} [v]} \right)} \right]\;\;\;} \right\} 1242 \right)\,\delta_{k+1/2} [v]} \right)} \right]\;\;\;} \right\} 1243 1243 \end{split} 1244 1244 \end{equation} 1245 1245 where $r_1$ and $r_2$ are the slopes between the surface along which the diffusion operator acts and 1246 the surface of computation ($z$- or $s$-surfaces). 1246 the surface of computation ($z$- or $s$-surfaces). 1247 1247 The way these slopes are evaluated is given in the lateral physics chapter (\autoref{chap:LDF}). 1248 1248 … … 1270 1270 %----------------------------------------------namzdf------------------------------------------------------ 1271 1271 1272 \nlst{namzdf} 1272 \nlst{namzdf} 1273 1273 %------------------------------------------------------------------------------------------------------------- 1274 1274 1275 Options are defined through the \n gn{namzdf} namelist variables.1275 Options are defined through the \nam{zdf} namelist variables. 1276 1276 The large vertical diffusion coefficient found in the surface mixed layer together with high vertical resolution implies that in the case of explicit time stepping there would be too restrictive a constraint on the time step. 1277 1277 Two time stepping schemes can be used for the vertical diffusion term: … … 1280 1280 $(b)$ a backward (or implicit) time differencing scheme (\np{ln\_zdfexp}\forcode{ = .false.}) 1281 1281 (see \autoref{chap:STP}). 1282 Note that namelist variables \np{ln\_zdfexp} and \np{nn\_zdfexp} apply to both tracers and dynamics. 1282 Note that namelist variables \np{ln\_zdfexp} and \np{nn\_zdfexp} apply to both tracers and dynamics. 1283 1283 1284 1284 The formulation of the vertical subgrid scale physics is the same whatever the vertical coordinate is. … … 1309 1309 where $\left( \tau_u ,\tau_v \right)$ are the two components of the wind stress vector in 1310 1310 the (\textbf{i},\textbf{j}) coordinate system. 1311 The high mixing coefficients in the surface mixed layer ensure that the surface wind stress is distributed in 1311 The high mixing coefficients in the surface mixed layer ensure that the surface wind stress is distributed in 1312 1312 the vertical over the mixed layer depth. 1313 1313 If the vertical mixing coefficient is small (when no mixed layer scheme is used) … … 1326 1326 Besides the surface and bottom stresses (see the above section) 1327 1327 which are introduced as boundary conditions on the vertical mixing, 1328 three other forcings may enter the dynamical equations by affecting the surface pressure gradient. 1328 three other forcings may enter the dynamical equations by affecting the surface pressure gradient. 1329 1329 1330 1330 (1) When \np{ln\_apr\_dyn}\forcode{ = .true.} (see \autoref{sec:SBC_apr}), … … 1335 1335 1336 1336 (3) When \np{nn\_ice\_embd}\forcode{ = 2} and LIM or CICE is used 1337 (\ie when the sea-ice is embedded in the ocean),1337 (\ie\ when the sea-ice is embedded in the ocean), 1338 1338 the snow-ice mass is taken into account when computing the surface pressure gradient. 1339 1339 … … 1343 1343 1344 1344 % ================================================================ 1345 % Wetting and drying 1345 % Wetting and drying 1346 1346 % ================================================================ 1347 1347 \section{Wetting and drying } … … 1359 1359 1360 1360 The following terminology is used. The depth of the topography (positive downwards) 1361 at each $(i,j)$ point is the quantity stored in array $\mathrm{ht\_wd}$ in the NEMOcode.1361 at each $(i,j)$ point is the quantity stored in array $\mathrm{ht\_wd}$ in the \NEMO\ code. 1362 1362 The height of the free surface (positive upwards) is denoted by $ \mathrm{ssh}$. Given the sign 1363 1363 conventions used, the water depth, $h$, is the height of the free surface plus the depth of the … … 1367 1367 covered by water. They require the topography specified with a model 1368 1368 configuration to have negative depths at points where the land is higher than the 1369 topography's reference sea-level. The vertical grid in NEMOis normally computed relative to an1369 topography's reference sea-level. The vertical grid in \NEMO\ is normally computed relative to an 1370 1370 initial state with zero sea surface height elevation. 1371 1371 The user can choose to compute the vertical grid and heights in the model relative to … … 1386 1386 All these configurations have used pure sigma coordinates. It is expected that 1387 1387 the wetting and drying code will work in domains with more general s-coordinates provided 1388 the coordinates are pure sigma in the region where wetting and drying actually occurs. 1388 the coordinates are pure sigma in the region where wetting and drying actually occurs. 1389 1389 1390 1390 The next sub-section descrbies the directional limiter and the following sub-section the iterative limiter. … … 1399 1399 \label{subsec:DYN_wd_directional_limiter} 1400 1400 The principal idea of the directional limiter is that 1401 water should not be allowed to flow out of a dry tracer cell (i.e. one whose water depth is less than rn\_wdmin1).1401 water should not be allowed to flow out of a dry tracer cell (i.e. one whose water depth is less than \np{rn\_wdmin1}). 1402 1402 1403 1403 All the changes associated with this option are made to the barotropic solver for the non-linear … … 1409 1409 1410 1410 The flux across each $u$-face of a tracer cell is multiplied by a factor zuwdmask (an array which depends on ji and jj). 1411 If the user sets ln\_wd\_dl\_ramp = .False.then zuwdmask is 1 when the1412 flux is from a cell with water depth greater than rn\_wdmin1and 0 otherwise. If the user sets1413 ln\_wd\_dl\_ramp = .True.the flux across the face is ramped down as the water depth decreases1414 from 2 * rn\_wdmin1 to rn\_wdmin1. The use of this ramp reduced grid-scale noise in idealised test cases.1411 If the user sets \np{ln\_wd\_dl\_ramp}\forcode{ = .false.} then zuwdmask is 1 when the 1412 flux is from a cell with water depth greater than \np{rn\_wdmin1} and 0 otherwise. If the user sets 1413 \np{ln\_wd\_dl\_ramp}\forcode{ = .true.} the flux across the face is ramped down as the water depth decreases 1414 from 2 * \np{rn\_wdmin1} to \np{rn\_wdmin1}. The use of this ramp reduced grid-scale noise in idealised test cases. 1415 1415 1416 1416 At the point where the flux across a $u$-face is multiplied by zuwdmask , we have chosen … … 1428 1428 fields (tracers independent of $x$, $y$ and $z$). Our scheme conserves constant tracers because 1429 1429 the velocities used at the tracer cell faces on the baroclinic timesteps are carefully calculated by dynspg\_ts 1430 to equal their mean value during the barotropic steps. If the user sets ln\_wd\_dl\_bc = .True., the1431 baroclinic velocities are also multiplied by a suitably weighted average of zuwdmask. 1430 to equal their mean value during the barotropic steps. If the user sets \np{ln\_wd\_dl\_bc}\forcode{ = .true.}, the 1431 baroclinic velocities are also multiplied by a suitably weighted average of zuwdmask. 1432 1432 1433 1433 %----------------------------------------------------------------------------------------- … … 1455 1455 1456 1456 \begin{align} \label{dyn_wd_continuity_2} 1457 \frac{e_1 e_2}{\Delta t} ( h_{i,j}(t_{n+1}) - h_{i,j}(t_e) ) 1457 \frac{e_1 e_2}{\Delta t} ( h_{i,j}(t_{n+1}) - h_{i,j}(t_e) ) 1458 1458 &= - ( \mathrm{flxu}_{i+1,j} - \mathrm{flxu}_{i,j} + \mathrm{flxv}_{i,j+1} - \mathrm{flxv}_{i,j} ) \\ 1459 1459 &= \mathrm{zzflx}_{i,j} . … … 1471 1471 1472 1472 \begin{equation} \label{dyn_wd_zzflx_p_n_1} 1473 \mathrm{zzflx}_{i,j} = \mathrm{zzflxp}_{i,j} + \mathrm{zzflxn}_{i,j} . 1473 \mathrm{zzflx}_{i,j} = \mathrm{zzflxp}_{i,j} + \mathrm{zzflxn}_{i,j} . 1474 1474 \end{equation} 1475 1475 … … 1495 1495 1496 1496 \begin{equation} \label{dyn_wd_zzflx_initial} 1497 \mathrm{zzflxp^{(0)}}_{i,j} = \mathrm{zzflxp}_{i,j} , \quad \mathrm{zzflxn^{(0)}}_{i,j} = \mathrm{zzflxn}_{i,j} . 1497 \mathrm{zzflxp^{(0)}}_{i,j} = \mathrm{zzflxp}_{i,j} , \quad \mathrm{zzflxn^{(0)}}_{i,j} = \mathrm{zzflxn}_{i,j} . 1498 1498 \end{equation} 1499 1499 … … 1525 1525 \begin{split} 1526 1526 \mathrm{zcoef}^{(m+1)}_{i,j} = \Big[ (h_{i,j}(t_e) & - \mathrm{rn\_wdmin1} - \mathrm{rn\_wdmin2}) \frac{e_1 e_2}{\Delta t} \phantom{]} \\ 1527 \phantom{[} & - \mathrm{zzflxn}^{(m)}_{i,j} \Big] \frac{1}{ \mathrm{zzflxp}^{(0)}_{i,j} } 1527 \phantom{[} & - \mathrm{zzflxn}^{(m)}_{i,j} \Big] \frac{1}{ \mathrm{zzflxp}^{(0)}_{i,j} } 1528 1528 \end{split} 1529 1529 \end{equation} … … 1635 1635 1636 1636 % ================================================================ 1637 % Time evolution term 1637 % Time evolution term 1638 1638 % ================================================================ 1639 1639 \section[Time evolution term (\textit{dynnxt.F90})] … … 1643 1643 %----------------------------------------------namdom---------------------------------------------------- 1644 1644 1645 \nlst{namdom} 1645 \nlst{namdom} 1646 1646 %------------------------------------------------------------------------------------------------------------- 1647 1647 1648 Options are defined through the \n gn{namdom} namelist variables.1648 Options are defined through the \nam{dom} namelist variables. 1649 1649 The general framework for dynamics time stepping is a leap-frog scheme, 1650 \ie a three level centred time scheme associated with an Asselin time filter (cf. \autoref{chap:STP}).1650 \ie\ a three level centred time scheme associated with an Asselin time filter (cf. \autoref{chap:STP}). 1651 1651 The scheme is applied to the velocity, except when 1652 1652 using the flux form of momentum advection (cf. \autoref{sec:DYN_adv_cor_flux}) 1653 in the variable volume case (\ key{vvl} defined),1654 where it has to be applied to the thickness weighted velocity (see \autoref{sec:A_momentum}) 1653 in the variable volume case (\texttt{vvl?} defined), 1654 where it has to be applied to the thickness weighted velocity (see \autoref{sec:A_momentum}) 1655 1655 1656 1656 $\bullet$ vector invariant form or linear free surface 1657 (\np{ln\_dynhpg\_vec}\forcode{ = .true.} ; \ key{vvl} not defined):1657 (\np{ln\_dynhpg\_vec}\forcode{ = .true.} ; \texttt{vvl?} not defined): 1658 1658 \[ 1659 1659 % \label{eq:dynnxt_vec} … … 1667 1667 1668 1668 $\bullet$ flux form and nonlinear free surface 1669 (\np{ln\_dynhpg\_vec}\forcode{ = .false.} ; \ key{vvl} defined):1669 (\np{ln\_dynhpg\_vec}\forcode{ = .false.} ; \texttt{vvl?} defined): 1670 1670 \[ 1671 1671 % \label{eq:dynnxt_flux}
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