[10414] | 1 | \documentclass[../main/NEMO_manual]{subfiles} |
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| 2 | |
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[6997] | 3 | \begin{document} |
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[10442] | 4 | \chapter{ essai \zstar \sstar} |
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[11597] | 5 | %% ================================================================================================= |
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[10442] | 6 | \section{Curvilinear \zstar- or \sstar coordinate system} |
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[707] | 7 | |
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| 8 | \colorbox{yellow}{ to be updated } |
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| 9 | |
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[10354] | 10 | In that case, the free surface equation is nonlinear, and the variations of volume are fully taken into account. |
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[11435] | 11 | These coordinates systems is presented in a report \citep{levier.treguier.ea_rpt07} available on the \NEMO\ web site. |
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[707] | 12 | |
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| 13 | \colorbox{yellow}{ end of to be updated} |
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| 14 | |
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| 15 | % from MOM4p1 documentation |
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| 16 | |
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[11435] | 17 | To overcome problems with vanishing surface and/or bottom cells, we consider the zstar coordinate |
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[10414] | 18 | \[ |
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[11544] | 19 | % \label{eq:MBZ_PE_} |
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[10414] | 20 | z^\star = H \left( \frac{z-\eta}{H+\eta} \right) |
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| 21 | \] |
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[707] | 22 | |
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[10354] | 23 | This coordinate is closely related to the "eta" coordinate used in many atmospheric models |
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| 24 | (see Black (1994) for a review of eta coordinate atmospheric models). |
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| 25 | It was originally used in ocean models by Stacey et al. (1995) for studies of tides next to shelves, |
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| 26 | and it has been recently promoted by Adcroft and Campin (2004) for global climate modelling. |
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[707] | 27 | |
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[10354] | 28 | The surfaces of constant $z^\star$ are quasi-horizontal. |
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| 29 | Indeed, the $z^\star$ coordinate reduces to $z$ when $\eta$ is zero. |
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| 30 | In general, when noting the large differences between undulations of the bottom topography versus undulations in |
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| 31 | the surface height, it is clear that surfaces constant $z^\star$ are very similar to the depth surfaces. |
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| 32 | These properties greatly reduce difficulties of computing the horizontal pressure gradient relative to |
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[11543] | 33 | terrain following sigma models discussed in \autoref{subsec:MB_sco}. |
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[10354] | 34 | Additionally, since $z^\star$ when $\eta = 0$, no flow is spontaneously generated in |
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| 35 | an unforced ocean starting from rest, regardless the bottom topography. |
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| 36 | This behaviour is in contrast to the case with "s"-models, where pressure gradient errors in the presence of |
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| 37 | nontrivial topographic variations can generate nontrivial spontaneous flow from a resting state, |
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| 38 | depending on the sophistication of the pressure gradient solver. |
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| 39 | The quasi-horizontal nature of the coordinate surfaces also facilitates the implementation of |
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| 40 | neutral physics parameterizations in $z^\star$ models using the same techniques as in $z$-models |
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| 41 | (see Chapters 13-16 of Griffies (2004) for a discussion of neutral physics in $z$-models, |
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[11435] | 42 | as well as \autoref{sec:LDF_slp} in this document for treatment in \NEMO). |
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[707] | 43 | |
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[10354] | 44 | The range over which $z^\star$ varies is time independent $-H \leq z^\star \leq 0$. |
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| 45 | Hence, all cells remain nonvanishing, so long as the surface height maintains $\eta > ?H$. |
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[11435] | 46 | This is a minor constraint relative to that encountered on the surface height when using $s = z$ or $s = z - \eta$. |
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[707] | 47 | |
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[10354] | 48 | Because $z^\star$ has a time independent range, all grid cells have static increments ds, |
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[11435] | 49 | and the sum of the ver tical increments yields the time independent ocean depth %�k ds = H. |
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[10354] | 50 | The $z^\star$ coordinate is therefore invisible to undulations of the free surface, |
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| 51 | since it moves along with the free surface. |
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| 52 | This proper ty means that no spurious ver tical transpor t is induced across surfaces of |
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| 53 | constant $z^\star$ by the motion of external gravity waves. |
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| 54 | Such spurious transpor t can be a problem in z-models, especially those with tidal forcing. |
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| 55 | Quite generally, the time independent range for the $z^\star$ coordinate is a very convenient property that |
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| 56 | allows for a nearly arbitrary vertical resolution even in the presence of large amplitude fluctuations of |
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[11435] | 57 | the surface height, again so long as $\eta > -H$. |
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[707] | 58 | |
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| 59 | %%% |
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| 60 | % essai update time splitting... |
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| 61 | %%% |
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| 62 | |
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[11597] | 63 | %% ================================================================================================= |
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[11571] | 64 | \section[Surface pressure gradient and sea surface heigth (\textit{dynspg.F90})]{Surface pressure gradient and sea surface heigth (\protect\mdl{dynspg})} |
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[11544] | 65 | \label{sec:MBZ_dyn_hpg_spg} |
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[10146] | 66 | |
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[11435] | 67 | %\nlst{nam_dynspg} |
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[11577] | 68 | Options are defined through the \nam{_dynspg}{\_dynspg} namelist variables. |
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[11543] | 69 | The surface pressure gradient term is related to the representation of the free surface (\autoref{sec:MB_hor_pg}). |
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[10354] | 70 | The main distinction is between the fixed volume case (linear free surface or rigid lid) and |
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| 71 | the variable volume case (nonlinear free surface, \key{vvl} is active). |
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[11543] | 72 | In the linear free surface case (\autoref{subsec:MB_free_surface}) and rigid lid (\autoref{PE_rigid_lid}), |
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[10354] | 73 | the vertical scale factors $e_{3}$ are fixed in time, |
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[11543] | 74 | while in the nonlinear case (\autoref{subsec:MB_free_surface}) they are time-dependent. |
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[10354] | 75 | With both linear and nonlinear free surface, external gravity waves are allowed in the equations, |
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| 76 | which imposes a very small time step when an explicit time stepping is used. |
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| 77 | Two methods are proposed to allow a longer time step for the three-dimensional equations: |
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[11543] | 78 | the filtered free surface, which is a modification of the continuous equations %(see \autoref{eq:MB_flt?}), |
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[10354] | 79 | and the split-explicit free surface described below. |
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| 80 | The extra term introduced in the filtered method is calculated implicitly, |
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| 81 | so that the update of the next velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}. |
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[707] | 82 | |
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| 83 | % Explicit |
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[11597] | 84 | %% ================================================================================================= |
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[11571] | 85 | \subsubsection[Explicit (\texttt{\textbf{key\_dynspg\_exp}})]{Explicit (\protect\key{dynspg\_exp})} |
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[11544] | 86 | \label{subsec:MBZ_dyn_spg_exp} |
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[707] | 87 | |
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[10354] | 88 | In the explicit free surface formulation, the model time step is chosen small enough to |
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| 89 | describe the external gravity waves (typically a few ten seconds). |
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| 90 | The sea surface height is given by: |
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[10414] | 91 | \begin{equation} |
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[11544] | 92 | \label{eq:MBZ_dynspg_ssh} |
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[10414] | 93 | \frac{\partial \eta }{\partial t}\equiv \frac{\text{EMP}}{\rho_w }+\frac{1}{e_{1T} |
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| 94 | e_{2T} }\sum\limits_k {\left( {\delta_i \left[ {e_{2u} e_{3u} u} |
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| 95 | \right]+\delta_j \left[ {e_{1v} e_{3v} v} \right]} \right)} |
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[707] | 96 | \end{equation} |
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| 97 | |
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[10354] | 98 | where EMP is the surface freshwater budget (evaporation minus precipitation, and minus river runoffs |
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| 99 | (if the later are introduced as a surface freshwater flux, see \autoref{chap:SBC}) expressed in $Kg.m^{-2}.s^{-1}$, |
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[10406] | 100 | and $\rho_w =1,000\,Kg.m^{-3}$ is the volumic mass of pure water. |
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[10354] | 101 | The sea-surface height is evaluated using a leapfrog scheme in combination with an Asselin time filter, |
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[11543] | 102 | (\ie\ the velocity appearing in (\autoref{eq:DYN_spg_ssh}) is centred in time (\textit{now} velocity). |
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[707] | 103 | |
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[10354] | 104 | The surface pressure gradient, also evaluated using a leap-frog scheme, is then simply given by: |
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[10414] | 105 | \begin{equation} |
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[11544] | 106 | \label{eq:MBZ_dynspg_exp} |
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[10414] | 107 | \left\{ |
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| 108 | \begin{aligned} |
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| 109 | - \frac{1} {e_{1u}} \; \delta_{i+1/2} \left[ \,\eta\, \right] \\ \\ |
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| 110 | - \frac{1} {e_{2v}} \; \delta_{j+1/2} \left[ \,\eta\, \right] |
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| 111 | \end{aligned} |
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| 112 | \right. |
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[11435] | 113 | \end{equation} |
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[707] | 114 | |
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[10406] | 115 | Consistent with the linearization, a $\left. \rho \right|_{k=1} / \rho_o$ factor is omitted in |
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[11543] | 116 | (\autoref{eq:DYN_spg_exp}). |
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[707] | 117 | |
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| 118 | % Split-explicit time-stepping |
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[11597] | 119 | %% ================================================================================================= |
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[11571] | 120 | \subsubsection[Split-explicit time-stepping (\texttt{\textbf{key\_dynspg\_ts}})]{Split-explicit time-stepping (\protect\key{dynspg\_ts})} |
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[11544] | 121 | \label{subsec:MBZ_dyn_spg_ts} |
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[10146] | 122 | |
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[11123] | 123 | The split-explicit free surface formulation used in OPA follows the one proposed by \citet{Griffies2004?}. |
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[10354] | 124 | The general idea is to solve the free surface equation with a small time step, |
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| 125 | while the three dimensional prognostic variables are solved with a longer time step that |
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[11578] | 126 | is a multiple of \np{rdtbt}{rdtbt} in the \nam{dom}{dom} namelist (Figure III.3). |
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[707] | 127 | |
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| 128 | %> > > > > > > > > > > > > > > > > > > > > > > > > > > > |
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[10414] | 129 | \begin{figure}[!t] |
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[11558] | 130 | \centering |
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[11561] | 131 | \includegraphics[width=0.66\textwidth]{Fig_DYN_dynspg_ts} |
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[11558] | 132 | \caption[Schematic of the split-explicit time stepping scheme for |
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| 133 | the barotropic and baroclinic modes, after \citet{Griffies2004?}]{ |
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| 134 | Schematic of the split-explicit time stepping scheme for the barotropic and baroclinic modes, |
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| 135 | after \citet{Griffies2004?}. |
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| 136 | Time increases to the right. |
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| 137 | Baroclinic time steps are denoted by $t-\Delta t$, $t, t+\Delta t$, and $t+2\Delta t$. |
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| 138 | The curved line represents a leap-frog time step, |
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| 139 | and the smaller barotropic time steps $N \Delta t=2\Delta t$ are denoted by the zig-zag line. |
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| 140 | The vertically integrated forcing \textbf{M}(t) computed at |
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| 141 | baroclinic time step t represents the interaction between the barotropic and baroclinic motions. |
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| 142 | While keeping the total depth, tracer, and freshwater forcing fields fixed, |
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| 143 | a leap-frog integration carries the surface height and vertically integrated velocity from |
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| 144 | t to $t+2 \Delta t$ using N barotropic time steps of length $\Delta t$. |
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| 145 | Time averaging the barotropic fields over the N+1 time steps (endpoints included) |
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| 146 | centers the vertically integrated velocity at the baroclinic timestep $t+\Delta t$. |
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| 147 | A baroclinic leap-frog time step carries the surface height to $t+\Delta t$ using |
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| 148 | the convergence of the time averaged vertically integrated velocity taken from |
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| 149 | baroclinic time step t.} |
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| 150 | \label{fig:MBZ_dyn_dynspg_ts} |
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[707] | 151 | \end{figure} |
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| 152 | %> > > > > > > > > > > > > > > > > > > > > > > > > > > > |
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| 153 | |
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[10354] | 154 | The split-explicit formulation has a damping effect on external gravity waves, |
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[11123] | 155 | which is weaker than the filtered free surface but still significant as shown by \citet{levier.treguier.ea_rpt07} in |
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[11435] | 156 | the case of an analytical barotropic Kelvin wave. |
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[707] | 157 | |
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| 158 | %from griffies book: ..... copy past ! |
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| 159 | |
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| 160 | \textbf{title: Time stepping the barotropic system } |
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| 161 | |
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[10354] | 162 | Assume knowledge of the full velocity and tracer fields at baroclinic time $\tau$. |
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| 163 | Hence, we can update the surface height and vertically integrated velocity with a leap-frog scheme using |
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| 164 | the small barotropic time step $\Delta t$. |
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| 165 | We have |
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[10414] | 166 | \[ |
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[11544] | 167 | % \label{eq:MBZ_dyn_spg_ts_eta} |
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[10414] | 168 | \eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1}) |
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[11435] | 169 | = 2 \Delta t \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right] |
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[10414] | 170 | \] |
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| 171 | \begin{multline*} |
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[11544] | 172 | % \label{eq:MBZ_dyn_spg_ts_u} |
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[10414] | 173 | \textbf{U}^{(b)}(\tau,t_{n+1}) - \textbf{U}^{(b)}(\tau,t_{n-1}) \\ |
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| 174 | = 2\Delta t \left[ - f \textbf{k} \times \textbf{U}^{(b)}(\tau,t_{n}) |
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| 175 | - H(\tau) \nabla p_s^{(b)}(\tau,t_{n}) +\textbf{M}(\tau) \right] |
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| 176 | \end{multline*} |
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[707] | 177 | \ |
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| 178 | |
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[10354] | 179 | In these equations, araised (b) denotes values of surface height and |
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| 180 | vertically integrated velocity updated with the barotropic time steps. |
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| 181 | The $\tau$ time label on $\eta^{(b)}$ and $U^{(b)}$ denotes the baroclinic time at which |
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| 182 | the vertically integrated forcing $\textbf{M}(\tau)$ |
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| 183 | (note that this forcing includes the surface freshwater forcing), the tracer fields, |
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| 184 | the freshwater flux $\text{EMP}_w(\tau)$, and total depth of the ocean $H(\tau)$ are held for |
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| 185 | the duration of the barotropic time stepping over a single cycle. |
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[11435] | 186 | This is also the time that sets the barotropic time steps via |
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[10414] | 187 | \[ |
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[11544] | 188 | % \label{eq:MBZ_dyn_spg_ts_t} |
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[11435] | 189 | t_n=\tau+n\Delta t |
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[10414] | 190 | \] |
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[10354] | 191 | with $n$ an integer. |
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[11435] | 192 | The density scaled surface pressure is evaluated via |
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[10414] | 193 | \[ |
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[11544] | 194 | % \label{eq:MBZ_dyn_spg_ts_ps} |
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[10414] | 195 | p_s^{(b)}(\tau,t_{n}) = |
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| 196 | \begin{cases} |
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| 197 | g \;\eta_s^{(b)}(\tau,t_{n}) \;\rho(\tau)_{k=1}) / \rho_o & \text{non-linear case} \\ |
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| 198 | g \;\eta_s^{(b)}(\tau,t_{n}) & \text{linear case} |
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| 199 | \end{cases} |
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| 200 | \] |
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[11435] | 201 | To get started, we assume the following initial conditions |
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[10414] | 202 | \[ |
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[11544] | 203 | % \label{eq:MBZ_dyn_spg_ts_eta} |
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[10414] | 204 | \begin{split} |
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| 205 | \eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)} \\ |
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| 206 | \eta^{(b)}(\tau,t_{n=1}) &= \eta^{(b)}(\tau,t_{n=0}) + \Delta t \ \text{RHS}_{n=0} |
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| 207 | \end{split} |
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| 208 | \] |
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[11435] | 209 | with |
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[10414] | 210 | \[ |
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[11544] | 211 | % \label{eq:MBZ_dyn_spg_ts_etaF} |
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[10414] | 212 | \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\Delta t,t_{n}) |
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| 213 | \] |
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[10354] | 214 | the time averaged surface height taken from the previous barotropic cycle. |
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| 215 | Likewise, |
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[10414] | 216 | \[ |
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[11544] | 217 | % \label{eq:MBZ_dyn_spg_ts_u} |
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[10414] | 218 | \textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)} \\ \\ |
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| 219 | \textbf{U}(\tau,t_{n=1}) = \textbf{U}^{(b)}(\tau,t_{n=0}) + \Delta t \ \text{RHS}_{n=0} |
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| 220 | \] |
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[11435] | 221 | with |
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[10414] | 222 | \[ |
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[11544] | 223 | % \label{eq:MBZ_dyn_spg_ts_u} |
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[10414] | 224 | \overline{\textbf{U}^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\Delta t,t_{n}) |
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| 225 | \] |
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[10354] | 226 | the time averaged vertically integrated transport. |
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[11435] | 227 | Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration. |
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[707] | 228 | |
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[10354] | 229 | Upon reaching $t_{n=N} = \tau + 2\Delta \tau$ , the vertically integrated velocity is time averaged to |
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[11435] | 230 | produce the updated vertically integrated velocity at baroclinic time $\tau + \Delta \tau$ |
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[10414] | 231 | \[ |
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[11544] | 232 | % \label{eq:MBZ_dyn_spg_ts_u} |
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[10414] | 233 | \textbf{U}(\tau+\Delta t) = \overline{\textbf{U}^{(b)}(\tau+\Delta t)} |
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| 234 | = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n}) |
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| 235 | \] |
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[10354] | 236 | The surface height on the new baroclinic time step is then determined via |
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[11435] | 237 | a baroclinic leap-frog using the following form |
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[10414] | 238 | \begin{equation} |
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[11544] | 239 | \label{eq:MBZ_dyn_spg_ts_ssh} |
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[10414] | 240 | \eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\Delta t \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right] |
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[707] | 241 | \end{equation} |
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| 242 | |
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[10354] | 243 | The use of this "big-leap-frog" scheme for the surface height ensures compatibility between |
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| 244 | the mass/volume budgets and the tracer budgets. |
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[11435] | 245 | More discussion of this point is provided in Chapter 10 (see in particular Section 10.2). |
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| 246 | |
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[10354] | 247 | In general, some form of time filter is needed to maintain integrity of the surface height field due to |
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[11544] | 248 | the leap-frog splitting mode in equation \autoref{eq:MBZ_dyn_spg_ts_ssh}. |
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[10354] | 249 | We have tried various forms of such filtering, |
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| 250 | with the following method discussed in Griffies et al. (2001) chosen due to its stability and |
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[11435] | 251 | reasonably good maintenance of tracer conservation properties (see ??) |
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[707] | 252 | |
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[10414] | 253 | \begin{equation} |
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[11544] | 254 | \label{eq:MBZ_dyn_spg_ts_sshf} |
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[10414] | 255 | \eta^{F}(\tau-\Delta) = \overline{\eta^{(b)}(\tau)} |
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[707] | 256 | \end{equation} |
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[11435] | 257 | Another approach tried was |
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[707] | 258 | |
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[10414] | 259 | \[ |
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[11544] | 260 | % \label{eq:MBZ_dyn_spg_ts_sshf2} |
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[10414] | 261 | \eta^{F}(\tau-\Delta) = \eta(\tau) |
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| 262 | + (\alpha/2) \left[\overline{\eta^{(b)}}(\tau+\Delta t) |
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| 263 | + \overline{\eta^{(b)}}(\tau-\Delta t) -2 \;\eta(\tau) \right] |
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| 264 | \] |
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[707] | 265 | |
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[10354] | 266 | which is useful since it isolates all the time filtering aspects into the term multiplied by $\alpha$. |
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| 267 | This isolation allows for an easy check that tracer conservation is exact when eliminating tracer and |
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| 268 | surface height time filtering (see ?? for more complete discussion). |
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[11544] | 269 | However, in the general case with a non-zero $\alpha$, the filter \autoref{eq:MBZ_dyn_spg_ts_sshf} was found to |
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[11435] | 270 | be more conservative, and so is recommended. |
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[707] | 271 | |
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[11435] | 272 | % Filtered formulation |
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[11597] | 273 | %% ================================================================================================= |
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[11571] | 274 | \subsubsection[Filtered formulation (\texttt{\textbf{key\_dynspg\_flt}})]{Filtered formulation (\protect\key{dynspg\_flt})} |
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[11544] | 275 | \label{subsec:MBZ_dyn_spg_flt} |
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[707] | 276 | |
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[11123] | 277 | The filtered formulation follows the \citet{Roullet2000?} implementation. |
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[10354] | 278 | The extra term introduced in the equations (see {\S}I.2.2) is solved implicitly. |
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| 279 | The elliptic solvers available in the code are documented in \autoref{chap:MISC}. |
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[11578] | 280 | The amplitude of the extra term is given by the namelist variable \np{rnu}{rnu}. |
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[11123] | 281 | The default value is 1, as recommended by \citet{Roullet2000?} |
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[707] | 282 | |
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[11582] | 283 | \colorbox{red}{\np[=1]{rnu}{rnu} to be suppressed from namelist !} |
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[707] | 284 | |
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[11435] | 285 | % Non-linear free surface formulation |
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[11597] | 286 | %% ================================================================================================= |
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[11571] | 287 | \subsection[Non-linear free surface formulation (\texttt{\textbf{key\_vvl}})]{Non-linear free surface formulation (\protect\key{vvl})} |
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[11544] | 288 | \label{subsec:MBZ_dyn_spg_vvl} |
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[707] | 289 | |
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[10354] | 290 | In the non-linear free surface formulation, the variations of volume are fully taken into account. |
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[11435] | 291 | This option is presented in a report \citep{levier.treguier.ea_rpt07} available on the \NEMO\ web site. |
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[10354] | 292 | The three time-stepping methods (explicit, split-explicit and filtered) are the same as in |
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[11543] | 293 | \autoref{?:DYN_spg_linear?} except that the ocean depth is now time-dependent. |
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[10354] | 294 | In particular, this means that in filtered case, the matrix to be inverted has to be recomputed at each time-step. |
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[707] | 295 | |
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[11584] | 296 | \onlyinsubfile{\input{../../global/epilogue}} |
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[707] | 297 | |
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[6997] | 298 | \end{document} |
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