[10414] | 1 | \documentclass[../main/NEMO_manual]{subfiles} |
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[6997] | 3 | \begin{document} |
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[707] | 4 | % ================================================================ |
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[10414] | 5 | % Chapter 1 Model Basics |
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[707] | 6 | % ================================================================ |
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| 7 | % ================================================================ |
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[10442] | 8 | % Curvilinear \zstar- \sstar-coordinate System |
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[707] | 9 | % ================================================================ |
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[10442] | 10 | \chapter{ essai \zstar \sstar} |
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| 11 | \section{Curvilinear \zstar- or \sstar coordinate system} |
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[707] | 12 | |
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| 13 | % ------------------------------------------------------------------------------------------------------------- |
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| 14 | % ???? |
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| 15 | % ------------------------------------------------------------------------------------------------------------- |
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| 16 | |
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| 17 | \colorbox{yellow}{ to be updated } |
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| 18 | |
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[10354] | 19 | In that case, the free surface equation is nonlinear, and the variations of volume are fully taken into account. |
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[11123] | 20 | These coordinates systems is presented in a report \citep{levier.treguier.ea_rpt07} available on the \NEMO web site. |
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[707] | 21 | |
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| 22 | \colorbox{yellow}{ end of to be updated} |
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| 23 | |
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| 24 | % from MOM4p1 documentation |
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| 25 | |
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[10354] | 26 | To overcome problems with vanishing surface and/or bottom cells, we consider the zstar coordinate |
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[10414] | 27 | \[ |
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| 28 | % \label{eq:PE_} |
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| 29 | z^\star = H \left( \frac{z-\eta}{H+\eta} \right) |
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| 30 | \] |
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[707] | 31 | |
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[10354] | 32 | This coordinate is closely related to the "eta" coordinate used in many atmospheric models |
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| 33 | (see Black (1994) for a review of eta coordinate atmospheric models). |
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| 34 | It was originally used in ocean models by Stacey et al. (1995) for studies of tides next to shelves, |
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| 35 | and it has been recently promoted by Adcroft and Campin (2004) for global climate modelling. |
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[707] | 36 | |
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[10354] | 37 | The surfaces of constant $z^\star$ are quasi-horizontal. |
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| 38 | Indeed, the $z^\star$ coordinate reduces to $z$ when $\eta$ is zero. |
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| 39 | In general, when noting the large differences between undulations of the bottom topography versus undulations in |
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| 40 | the surface height, it is clear that surfaces constant $z^\star$ are very similar to the depth surfaces. |
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| 41 | These properties greatly reduce difficulties of computing the horizontal pressure gradient relative to |
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| 42 | terrain following sigma models discussed in \autoref{subsec:PE_sco}. |
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| 43 | Additionally, since $z^\star$ when $\eta = 0$, no flow is spontaneously generated in |
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| 44 | an unforced ocean starting from rest, regardless the bottom topography. |
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| 45 | This behaviour is in contrast to the case with "s"-models, where pressure gradient errors in the presence of |
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| 46 | nontrivial topographic variations can generate nontrivial spontaneous flow from a resting state, |
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| 47 | depending on the sophistication of the pressure gradient solver. |
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| 48 | The quasi-horizontal nature of the coordinate surfaces also facilitates the implementation of |
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| 49 | neutral physics parameterizations in $z^\star$ models using the same techniques as in $z$-models |
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| 50 | (see Chapters 13-16 of Griffies (2004) for a discussion of neutral physics in $z$-models, |
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| 51 | as well as \autoref{sec:LDF_slp} in this document for treatment in \NEMO). |
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[707] | 52 | |
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[10354] | 53 | The range over which $z^\star$ varies is time independent $-H \leq z^\star \leq 0$. |
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| 54 | Hence, all cells remain nonvanishing, so long as the surface height maintains $\eta > ?H$. |
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| 55 | 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] | 56 | |
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[10354] | 57 | Because $z^\star$ has a time independent range, all grid cells have static increments ds, |
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| 58 | and the sum of the ver tical increments yields the time independent ocean depth %�k ds = H. |
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| 59 | The $z^\star$ coordinate is therefore invisible to undulations of the free surface, |
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| 60 | since it moves along with the free surface. |
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| 61 | This proper ty means that no spurious ver tical transpor t is induced across surfaces of |
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| 62 | constant $z^\star$ by the motion of external gravity waves. |
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| 63 | Such spurious transpor t can be a problem in z-models, especially those with tidal forcing. |
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| 64 | Quite generally, the time independent range for the $z^\star$ coordinate is a very convenient property that |
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| 65 | allows for a nearly arbitrary vertical resolution even in the presence of large amplitude fluctuations of |
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[707] | 66 | the surface height, again so long as $\eta > -H$. |
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| 67 | |
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| 68 | %%% |
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| 69 | % essai update time splitting... |
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| 70 | %%% |
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| 71 | |
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| 72 | % ================================================================ |
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| 73 | % Surface Pressure Gradient and Sea Surface Height |
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| 74 | % ================================================================ |
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[9393] | 75 | \section{Surface pressure gradient and sea surface heigth (\protect\mdl{dynspg})} |
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[9407] | 76 | \label{sec:DYN_hpg_spg} |
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[707] | 77 | %-----------------------------------------nam_dynspg---------------------------------------------------- |
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[10146] | 78 | |
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[10414] | 79 | %\nlst{nam_dynspg} |
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[707] | 80 | %------------------------------------------------------------------------------------------------------------ |
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[10354] | 81 | Options are defined through the \ngn{nam\_dynspg} namelist variables. |
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| 82 | The surface pressure gradient term is related to the representation of the free surface (\autoref{sec:PE_hor_pg}). |
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| 83 | The main distinction is between the fixed volume case (linear free surface or rigid lid) and |
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| 84 | the variable volume case (nonlinear free surface, \key{vvl} is active). |
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| 85 | In the linear free surface case (\autoref{subsec:PE_free_surface}) and rigid lid (\autoref{PE_rigid_lid}), |
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| 86 | the vertical scale factors $e_{3}$ are fixed in time, |
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| 87 | while in the nonlinear case (\autoref{subsec:PE_free_surface}) they are time-dependent. |
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| 88 | With both linear and nonlinear free surface, external gravity waves are allowed in the equations, |
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| 89 | which imposes a very small time step when an explicit time stepping is used. |
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| 90 | Two methods are proposed to allow a longer time step for the three-dimensional equations: |
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[11123] | 91 | the filtered free surface, which is a modification of the continuous equations %(see \autoref{eq:PE_flt?}), |
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[10354] | 92 | and the split-explicit free surface described below. |
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| 93 | The extra term introduced in the filtered method is calculated implicitly, |
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| 94 | 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] | 95 | |
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| 96 | %------------------------------------------------------------- |
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| 97 | % Explicit |
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| 98 | %------------------------------------------------------------- |
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[9363] | 99 | \subsubsection{Explicit (\protect\key{dynspg\_exp})} |
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[9407] | 100 | \label{subsec:DYN_spg_exp} |
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[707] | 101 | |
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[10354] | 102 | In the explicit free surface formulation, the model time step is chosen small enough to |
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| 103 | describe the external gravity waves (typically a few ten seconds). |
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| 104 | The sea surface height is given by: |
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[10414] | 105 | \begin{equation} |
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| 106 | \label{eq:dynspg_ssh} |
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| 107 | \frac{\partial \eta }{\partial t}\equiv \frac{\text{EMP}}{\rho_w }+\frac{1}{e_{1T} |
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| 108 | e_{2T} }\sum\limits_k {\left( {\delta_i \left[ {e_{2u} e_{3u} u} |
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| 109 | \right]+\delta_j \left[ {e_{1v} e_{3v} v} \right]} \right)} |
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[707] | 110 | \end{equation} |
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| 111 | |
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[10354] | 112 | where EMP is the surface freshwater budget (evaporation minus precipitation, and minus river runoffs |
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| 113 | (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] | 114 | and $\rho_w =1,000\,Kg.m^{-3}$ is the volumic mass of pure water. |
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[10354] | 115 | The sea-surface height is evaluated using a leapfrog scheme in combination with an Asselin time filter, |
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[10442] | 116 | (\ie the velocity appearing in (\autoref{eq:dynspg_ssh}) is centred in time (\textit{now} velocity). |
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[707] | 117 | |
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[10354] | 118 | The surface pressure gradient, also evaluated using a leap-frog scheme, is then simply given by: |
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[10414] | 119 | \begin{equation} |
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| 120 | \label{eq:dynspg_exp} |
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| 121 | \left\{ |
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| 122 | \begin{aligned} |
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| 123 | - \frac{1} {e_{1u}} \; \delta_{i+1/2} \left[ \,\eta\, \right] \\ \\ |
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| 124 | - \frac{1} {e_{2v}} \; \delta_{j+1/2} \left[ \,\eta\, \right] |
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| 125 | \end{aligned} |
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| 126 | \right. |
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[707] | 127 | \end{equation} |
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| 128 | |
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[10406] | 129 | Consistent with the linearization, a $\left. \rho \right|_{k=1} / \rho_o$ factor is omitted in |
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[10354] | 130 | (\autoref{eq:dynspg_exp}). |
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[707] | 131 | |
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| 132 | %------------------------------------------------------------- |
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| 133 | % Split-explicit time-stepping |
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| 134 | %------------------------------------------------------------- |
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[9363] | 135 | \subsubsection{Split-explicit time-stepping (\protect\key{dynspg\_ts})} |
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[9407] | 136 | \label{subsec:DYN_spg_ts} |
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[707] | 137 | %--------------------------------------------namdom---------------------------------------------------- |
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[10146] | 138 | |
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| 139 | \nlst{namdom} |
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[707] | 140 | %-------------------------------------------------------------------------------------------------------------- |
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[11123] | 141 | The split-explicit free surface formulation used in OPA follows the one proposed by \citet{Griffies2004?}. |
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[10354] | 142 | The general idea is to solve the free surface equation with a small time step, |
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| 143 | while the three dimensional prognostic variables are solved with a longer time step that |
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| 144 | is a multiple of \np{rdtbt} in the \ngn{namdom} namelist (Figure III.3). |
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[707] | 145 | |
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| 146 | %> > > > > > > > > > > > > > > > > > > > > > > > > > > > |
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[10414] | 147 | \begin{figure}[!t] |
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| 148 | \begin{center} |
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| 149 | \includegraphics[width=0.90\textwidth]{Fig_DYN_dynspg_ts} |
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| 150 | \caption{ |
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| 151 | \protect\label{fig:DYN_dynspg_ts} |
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| 152 | Schematic of the split-explicit time stepping scheme for the barotropic and baroclinic modes, |
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[11123] | 153 | after \citet{Griffies2004?}. |
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[10414] | 154 | Time increases to the right. |
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| 155 | Baroclinic time steps are denoted by $t-\Delta t$, $t, t+\Delta t$, and $t+2\Delta t$. |
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| 156 | The curved line represents a leap-frog time step, |
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| 157 | and the smaller barotropic time steps $N \Delta t=2\Delta t$ are denoted by the zig-zag line. |
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| 158 | The vertically integrated forcing \textbf{M}(t) computed at |
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| 159 | baroclinic time step t represents the interaction between the barotropic and baroclinic motions. |
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| 160 | While keeping the total depth, tracer, and freshwater forcing fields fixed, |
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| 161 | a leap-frog integration carries the surface height and vertically integrated velocity from |
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| 162 | t to $t+2 \Delta t$ using N barotropic time steps of length $\Delta t$. |
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| 163 | Time averaging the barotropic fields over the N+1 time steps (endpoints included) |
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| 164 | centers the vertically integrated velocity at the baroclinic timestep $t+\Delta t$. |
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| 165 | A baroclinic leap-frog time step carries the surface height to $t+\Delta t$ using the convergence of |
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| 166 | the time averaged vertically integrated velocity taken from baroclinic time step t. |
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| 167 | } |
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| 168 | \end{center} |
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[707] | 169 | \end{figure} |
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| 170 | %> > > > > > > > > > > > > > > > > > > > > > > > > > > > |
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| 171 | |
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[10354] | 172 | The split-explicit formulation has a damping effect on external gravity waves, |
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[11123] | 173 | which is weaker than the filtered free surface but still significant as shown by \citet{levier.treguier.ea_rpt07} in |
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[10354] | 174 | the case of an analytical barotropic Kelvin wave. |
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[707] | 175 | |
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| 176 | %from griffies book: ..... copy past ! |
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| 177 | |
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| 178 | \textbf{title: Time stepping the barotropic system } |
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| 179 | |
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[10354] | 180 | Assume knowledge of the full velocity and tracer fields at baroclinic time $\tau$. |
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| 181 | Hence, we can update the surface height and vertically integrated velocity with a leap-frog scheme using |
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| 182 | the small barotropic time step $\Delta t$. |
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| 183 | We have |
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[10414] | 184 | \[ |
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| 185 | % \label{eq:DYN_spg_ts_eta} |
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| 186 | \eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1}) |
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| 187 | = 2 \Delta t \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right] |
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| 188 | \] |
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| 189 | \begin{multline*} |
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| 190 | % \label{eq:DYN_spg_ts_u} |
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| 191 | \textbf{U}^{(b)}(\tau,t_{n+1}) - \textbf{U}^{(b)}(\tau,t_{n-1}) \\ |
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| 192 | = 2\Delta t \left[ - f \textbf{k} \times \textbf{U}^{(b)}(\tau,t_{n}) |
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| 193 | - H(\tau) \nabla p_s^{(b)}(\tau,t_{n}) +\textbf{M}(\tau) \right] |
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| 194 | \end{multline*} |
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[707] | 195 | \ |
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| 196 | |
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[10354] | 197 | In these equations, araised (b) denotes values of surface height and |
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| 198 | vertically integrated velocity updated with the barotropic time steps. |
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| 199 | The $\tau$ time label on $\eta^{(b)}$ and $U^{(b)}$ denotes the baroclinic time at which |
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| 200 | the vertically integrated forcing $\textbf{M}(\tau)$ |
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| 201 | (note that this forcing includes the surface freshwater forcing), the tracer fields, |
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| 202 | the freshwater flux $\text{EMP}_w(\tau)$, and total depth of the ocean $H(\tau)$ are held for |
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| 203 | the duration of the barotropic time stepping over a single cycle. |
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| 204 | This is also the time that sets the barotropic time steps via |
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[10414] | 205 | \[ |
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| 206 | % \label{eq:DYN_spg_ts_t} |
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| 207 | t_n=\tau+n\Delta t |
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| 208 | \] |
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[10354] | 209 | with $n$ an integer. |
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| 210 | The density scaled surface pressure is evaluated via |
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[10414] | 211 | \[ |
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| 212 | % \label{eq:DYN_spg_ts_ps} |
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| 213 | p_s^{(b)}(\tau,t_{n}) = |
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| 214 | \begin{cases} |
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| 215 | g \;\eta_s^{(b)}(\tau,t_{n}) \;\rho(\tau)_{k=1}) / \rho_o & \text{non-linear case} \\ |
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| 216 | g \;\eta_s^{(b)}(\tau,t_{n}) & \text{linear case} |
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| 217 | \end{cases} |
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| 218 | \] |
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[707] | 219 | To get started, we assume the following initial conditions |
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[10414] | 220 | \[ |
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| 221 | % \label{eq:DYN_spg_ts_eta} |
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| 222 | \begin{split} |
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| 223 | \eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)} \\ |
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| 224 | \eta^{(b)}(\tau,t_{n=1}) &= \eta^{(b)}(\tau,t_{n=0}) + \Delta t \ \text{RHS}_{n=0} |
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| 225 | \end{split} |
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| 226 | \] |
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[707] | 227 | with |
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[10414] | 228 | \[ |
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| 229 | % \label{eq:DYN_spg_ts_etaF} |
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| 230 | \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\Delta t,t_{n}) |
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| 231 | \] |
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[10354] | 232 | the time averaged surface height taken from the previous barotropic cycle. |
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| 233 | Likewise, |
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[10414] | 234 | \[ |
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| 235 | % \label{eq:DYN_spg_ts_u} |
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| 236 | \textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)} \\ \\ |
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| 237 | \textbf{U}(\tau,t_{n=1}) = \textbf{U}^{(b)}(\tau,t_{n=0}) + \Delta t \ \text{RHS}_{n=0} |
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| 238 | \] |
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[707] | 239 | with |
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[10414] | 240 | \[ |
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| 241 | % \label{eq:DYN_spg_ts_u} |
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| 242 | \overline{\textbf{U}^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\Delta t,t_{n}) |
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| 243 | \] |
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[10354] | 244 | the time averaged vertically integrated transport. |
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| 245 | Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration. |
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[707] | 246 | |
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[10354] | 247 | Upon reaching $t_{n=N} = \tau + 2\Delta \tau$ , the vertically integrated velocity is time averaged to |
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| 248 | produce the updated vertically integrated velocity at baroclinic time $\tau + \Delta \tau$ |
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[10414] | 249 | \[ |
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| 250 | % \label{eq:DYN_spg_ts_u} |
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| 251 | \textbf{U}(\tau+\Delta t) = \overline{\textbf{U}^{(b)}(\tau+\Delta t)} |
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| 252 | = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n}) |
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| 253 | \] |
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[10354] | 254 | The surface height on the new baroclinic time step is then determined via |
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| 255 | a baroclinic leap-frog using the following form |
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[10414] | 256 | \begin{equation} |
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| 257 | \label{eq:DYN_spg_ts_ssh} |
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| 258 | \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] | 259 | \end{equation} |
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| 260 | |
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[10354] | 261 | The use of this "big-leap-frog" scheme for the surface height ensures compatibility between |
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| 262 | the mass/volume budgets and the tracer budgets. |
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| 263 | More discussion of this point is provided in Chapter 10 (see in particular Section 10.2). |
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[707] | 264 | |
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[10354] | 265 | In general, some form of time filter is needed to maintain integrity of the surface height field due to |
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| 266 | the leap-frog splitting mode in equation \autoref{eq:DYN_spg_ts_ssh}. |
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| 267 | We have tried various forms of such filtering, |
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| 268 | with the following method discussed in Griffies et al. (2001) chosen due to its stability and |
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| 269 | reasonably good maintenance of tracer conservation properties (see ??) |
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[707] | 270 | |
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[10414] | 271 | \begin{equation} |
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| 272 | \label{eq:DYN_spg_ts_sshf} |
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| 273 | \eta^{F}(\tau-\Delta) = \overline{\eta^{(b)}(\tau)} |
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[707] | 274 | \end{equation} |
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| 275 | Another approach tried was |
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| 276 | |
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[10414] | 277 | \[ |
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| 278 | % \label{eq:DYN_spg_ts_sshf2} |
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| 279 | \eta^{F}(\tau-\Delta) = \eta(\tau) |
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| 280 | + (\alpha/2) \left[\overline{\eta^{(b)}}(\tau+\Delta t) |
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| 281 | + \overline{\eta^{(b)}}(\tau-\Delta t) -2 \;\eta(\tau) \right] |
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| 282 | \] |
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[707] | 283 | |
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[10354] | 284 | which is useful since it isolates all the time filtering aspects into the term multiplied by $\alpha$. |
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| 285 | This isolation allows for an easy check that tracer conservation is exact when eliminating tracer and |
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| 286 | surface height time filtering (see ?? for more complete discussion). |
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| 287 | However, in the general case with a non-zero $\alpha$, the filter \autoref{eq:DYN_spg_ts_sshf} was found to |
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| 288 | be more conservative, and so is recommended. |
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[707] | 289 | |
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| 290 | %------------------------------------------------------------- |
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| 291 | % Filtered formulation |
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| 292 | %------------------------------------------------------------- |
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[9363] | 293 | \subsubsection{Filtered formulation (\protect\key{dynspg\_flt})} |
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[9407] | 294 | \label{subsec:DYN_spg_flt} |
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[707] | 295 | |
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[11123] | 296 | The filtered formulation follows the \citet{Roullet2000?} implementation. |
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[10354] | 297 | The extra term introduced in the equations (see {\S}I.2.2) is solved implicitly. |
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| 298 | The elliptic solvers available in the code are documented in \autoref{chap:MISC}. |
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| 299 | The amplitude of the extra term is given by the namelist variable \np{rnu}. |
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[11123] | 300 | The default value is 1, as recommended by \citet{Roullet2000?} |
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[707] | 301 | |
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[9393] | 302 | \colorbox{red}{\np{rnu}\forcode{ = 1} to be suppressed from namelist !} |
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[707] | 303 | |
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| 304 | %------------------------------------------------------------- |
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| 305 | % Non-linear free surface formulation |
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| 306 | %------------------------------------------------------------- |
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[9363] | 307 | \subsection{Non-linear free surface formulation (\protect\key{vvl})} |
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[9407] | 308 | \label{subsec:DYN_spg_vvl} |
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[707] | 309 | |
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[10354] | 310 | In the non-linear free surface formulation, the variations of volume are fully taken into account. |
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[11123] | 311 | This option is presented in a report \citep{levier.treguier.ea_rpt07} available on the NEMO web site. |
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[10354] | 312 | The three time-stepping methods (explicit, split-explicit and filtered) are the same as in |
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| 313 | \autoref{DYN_spg_linear} except that the ocean depth is now time-dependent. |
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| 314 | 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] | 315 | |
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[10414] | 316 | \biblio |
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[707] | 317 | |
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[10442] | 318 | \pindex |
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| 319 | |
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[6997] | 320 | \end{document} |
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