[707] | 1 | % ================================================================ |
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[4147] | 2 | % Chapter 1 � Model Basics |
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[707] | 3 | % ================================================================ |
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| 4 | % ================================================================ |
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| 5 | % Curvilinear z*- s*-coordinate System |
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| 6 | % ================================================================ |
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| 7 | \chapter{ essai z* s*} |
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| 8 | \section{Curvilinear \textit{z*}- or \textit{s*} coordinate System} |
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| 9 | |
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| 10 | % ------------------------------------------------------------------------------------------------------------- |
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| 11 | % ???? |
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| 12 | % ------------------------------------------------------------------------------------------------------------- |
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| 13 | |
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| 14 | \colorbox{yellow}{ to be updated } |
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| 15 | |
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| 16 | In that case, the free surface equation is nonlinear, and the variations of |
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[996] | 17 | volume are fully taken into account. These coordinates systems is presented in |
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| 18 | a report \citep{Levier2007} available on the \NEMO web site. |
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[707] | 19 | |
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| 20 | \colorbox{yellow}{ end of to be updated} |
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| 21 | \newline |
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| 22 | |
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| 23 | % from MOM4p1 documentation |
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| 24 | |
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| 25 | To overcome problems with vanishing surface and/or bottom cells, we consider the |
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| 26 | zstar coordinate |
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| 27 | \begin{equation} \label{PE_} |
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| 28 | z^\star = H \left( \frac{z-\eta}{H+\eta} \right) |
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| 29 | \end{equation} |
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| 30 | |
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| 31 | This coordinate is closely related to the "eta" coordinate used in many atmospheric |
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| 32 | models (see Black (1994) for a review of eta coordinate atmospheric models). It |
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| 33 | was originally used in ocean models by Stacey et al. (1995) for studies of tides |
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| 34 | next to shelves, and it has been recently promoted by Adcroft and Campin (2004) |
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| 35 | for global climate modelling. |
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| 36 | |
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| 37 | The surfaces of constant $z^\star$ are quasi-horizontal. Indeed, the $z^\star$ coordinate reduces to $z$ when $\eta$ is zero. In general, when noting the large differences between |
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| 38 | undulations of the bottom topography versus undulations in the surface height, it |
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| 39 | is clear that surfaces constant $z^\star$ are very similar to the depth surfaces. These properties greatly reduce difficulties of computing the horizontal pressure gradient relative to terrain following sigma models discussed in \S\ref{PE_sco}. |
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| 40 | Additionally, since $z^\star$ when $\eta = 0$, no flow is spontaneously generated in an |
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| 41 | unforced ocean starting from rest, regardless the bottom topography. This behaviour is in contrast to the case with "s"-models, where pressure gradient errors in |
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| 42 | the presence of nontrivial topographic variations can generate nontrivial spontaneous flow from a resting state, depending on the sophistication of the pressure |
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| 43 | gradient solver. The quasi-horizontal nature of the coordinate surfaces also facilitates the implementation of neutral physics parameterizations in $z^\star$ models using |
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| 44 | the same techniques as in $z$-models (see Chapters 13-16 of Griffies (2004) for a |
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| 45 | discussion of neutral physics in $z$-models, as well as Section \S\ref{LDF_slp} |
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| 46 | in this document for treatment in \NEMO). |
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| 47 | |
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| 48 | The range over which $z^\star$ varies is time independent $-H \leq z^\star \leq 0$. Hence, all |
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| 49 | cells remain nonvanishing, so long as the surface height maintains $\eta > ?H$. This |
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| 50 | is a minor constraint relative to that encountered on the surface height when using |
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| 51 | $s = z$ or $s = z - \eta$. |
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| 52 | |
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| 53 | Because $z^\star$ has a time independent range, all grid cells have static increments |
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| 54 | ds, and the sum of the ver tical increments yields the time independent ocean |
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[4147] | 55 | depth %�k ds = H. |
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[707] | 56 | The $z^\star$ coordinate is therefore invisible to undulations of the |
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| 57 | free surface, since it moves along with the free surface. This proper ty means that |
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| 58 | no spurious ver tical transpor t is induced across surfaces of constant $z^\star$ by the |
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| 59 | motion of external gravity waves. Such spurious transpor t can be a problem in |
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| 60 | z-models, especially those with tidal forcing. Quite generally, the time independent |
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| 61 | range for the $z^\star$ coordinate is a very convenient proper ty that allows for a nearly |
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| 62 | arbitrary ver tical resolution even in the presence of large amplitude fluctuations of |
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| 63 | the surface height, again so long as $\eta > -H$. |
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| 64 | |
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| 65 | |
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| 66 | |
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| 67 | %%% |
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| 68 | % essai update time splitting... |
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| 69 | %%% |
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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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| 75 | \section{Surface pressure gradient and Sea Surface Heigth (\mdl{dynspg})} |
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| 76 | \label{DYN_hpg_spg} |
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| 77 | %-----------------------------------------nam_dynspg---------------------------------------------------- |
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| 78 | \namdisplay{nam_dynspg} |
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| 79 | %------------------------------------------------------------------------------------------------------------ |
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[4147] | 80 | Options are defined through the \ngn{nam\_dynspg} namelist variables. |
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[707] | 81 | The surface pressure gradient term is related to the representation of the free surface (\S\ref{PE_hor_pg}). The main distinction is between the fixed volume case (linear free surface or rigid lid) and the variable volume case (nonlinear free surface, \key{vvl} is active). In the linear free surface case (\S\ref{PE_free_surface}) and rigid lid (\S\ref{PE_rigid_lid}), the vertical scale factors $e_{3}$ are fixed in time, while in the nonlinear case (\S\ref{PE_free_surface}) they are time-dependent. With both linear and nonlinear free surface, external gravity waves are allowed in the equations, which imposes a very small time step when an explicit time stepping is used. Two methods are proposed to allow a longer time step for the three-dimensional equations: the filtered free surface, which is a modification of the continuous equations (see \eqref{Eq_PE_flt}), and the split-explicit free surface described below. The extra term introduced in the filtered method is calculated implicitly, so that the update of the next velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}. |
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| 82 | |
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| 83 | %------------------------------------------------------------- |
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| 84 | % Explicit |
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| 85 | %------------------------------------------------------------- |
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| 86 | \subsubsection{Explicit (\key{dynspg\_exp})} |
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| 87 | \label{DYN_spg_exp} |
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| 88 | |
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| 89 | In the explicit free surface formulation, the model time step is chosen small enough to describe the external gravity waves (typically a few ten seconds). The sea surface height is given by : |
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| 90 | \begin{equation} \label{Eq_dynspg_ssh} |
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| 91 | \frac{\partial \eta }{\partial t}\equiv \frac{\text{EMP}}{\rho _w }+\frac{1}{e_{1T} |
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| 92 | e_{2T} }\sum\limits_k {\left( {\delta _i \left[ {e_{2u} e_{3u} u} |
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| 93 | \right]+\delta _j \left[ {e_{1v} e_{3v} v} \right]} \right)} |
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| 94 | \end{equation} |
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| 95 | |
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| 96 | where EMP is the surface freshwater budget (evaporation minus precipitation, and minus river runoffs (if the later are introduced as a surface freshwater flux, see \S\ref{SBC}) expressed in $Kg.m^{-2}.s^{-1}$, and $\rho _w =1,000\,Kg.m^{-3}$ is the volumic mass of pure water. The sea-surface height is evaluated using a leapfrog scheme in combination with an Asselin time filter, i.e. the velocity appearing in (\ref{Eq_dynspg_ssh}) is centred in time (\textit{now} velocity). |
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| 97 | |
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| 98 | The surface pressure gradient, also evaluated using a leap-frog scheme, is then simply given by : |
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| 99 | \begin{equation} \label{Eq_dynspg_exp} |
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| 100 | \left\{ \begin{aligned} |
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| 101 | - \frac{1} {e_{1u}} \; \delta _{i+1/2} \left[ \,\eta\, \right] \\ |
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| 102 | \\ |
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| 103 | - \frac{1} {e_{2v}} \; \delta _{j+1/2} \left[ \,\eta\, \right] |
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| 104 | \end{aligned} \right. |
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| 105 | \end{equation} |
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| 106 | |
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| 107 | Consistent with the linearization, a $\left. \rho \right|_{k=1} / \rho _o$ factor is omitted in (\ref{Eq_dynspg_exp}). |
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| 108 | |
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| 109 | %------------------------------------------------------------- |
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| 110 | % Split-explicit time-stepping |
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| 111 | %------------------------------------------------------------- |
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| 112 | \subsubsection{Split-explicit time-stepping (\key{dynspg\_ts})} |
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| 113 | \label{DYN_spg_ts} |
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| 114 | %--------------------------------------------namdom---------------------------------------------------- |
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| 115 | \namdisplay{namdom} |
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| 116 | %-------------------------------------------------------------------------------------------------------------- |
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[4147] | 117 | The split-explicit free surface formulation used in OPA follows the one proposed by \citet{Griffies2004}. The general idea is to solve the free surface equation with a small time step, while the three dimensional prognostic variables are solved with a longer time step that is a multiple of \np{rdtbt} |
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| 118 | in the \ngn{namdom} namelist. |
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| 119 | (Figure III.3). |
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[707] | 120 | |
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| 121 | %> > > > > > > > > > > > > > > > > > > > > > > > > > > > |
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[2376] | 122 | \begin{figure}[!t] \begin{center} |
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[707] | 123 | \includegraphics[width=0.90\textwidth]{./Figures/Fig_DYN_dynspg_ts.pdf} |
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[2376] | 124 | \caption{ \label{Fig_DYN_dynspg_ts} |
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| 125 | Schematic of the split-explicit time stepping scheme for the barotropic and baroclinic modes, |
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| 126 | after \citet{Griffies2004}. Time increases to the right. Baroclinic time steps are denoted by |
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| 127 | $t-\Delta t$, $t, t+\Delta t$, and $t+2\Delta t$. The curved line represents a leap-frog time step, |
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| 128 | and the smaller barotropic time steps $N \Delta t=2\Delta t$ are denoted by the zig-zag line. |
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| 129 | The vertically integrated forcing \textbf{M}(t) computed at baroclinic time step t represents |
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| 130 | the interaction between the barotropic and baroclinic motions. While keeping the total depth, |
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| 131 | tracer, and freshwater forcing fields fixed, a leap-frog integration carries the surface height |
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| 132 | and vertically integrated velocity from t to $t+2 \Delta t$ using N barotropic time steps of length |
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| 133 | $\Delta t$. Time averaging the barotropic fields over the N+1 time steps (endpoints included) |
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| 134 | centers the vertically integrated velocity at the baroclinic timestep $t+\Delta t$. |
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| 135 | A baroclinic leap-frog time step carries the surface height to $t+\Delta t$ using the convergence |
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| 136 | of the time averaged vertically integrated velocity taken from baroclinic time step t. } |
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[707] | 137 | \end{center} |
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| 138 | \end{figure} |
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| 139 | %> > > > > > > > > > > > > > > > > > > > > > > > > > > > |
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| 140 | |
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| 141 | The split-explicit formulation has a damping effect on external gravity waves, which is weaker than the filtered free surface but still significant as shown by \citet{Levier2007} in the case of an analytical barotropic Kelvin wave. |
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| 142 | |
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| 143 | %from griffies book: ..... copy past ! |
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| 144 | |
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| 145 | \textbf{title: Time stepping the barotropic system } |
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| 146 | |
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| 147 | Assume knowledge of the full velocity and tracer fields at baroclinic time $\tau$. Hence, |
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| 148 | we can update the surface height and vertically integrated velocity with a leap-frog |
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| 149 | scheme using the small barotropic time step $\Delta t$. We have |
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| 150 | |
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| 151 | \begin{equation} \label{DYN_spg_ts_eta} |
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| 152 | \eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1}) |
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| 153 | = 2 \Delta t \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right] |
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| 154 | \end{equation} |
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| 155 | \begin{multline} \label{DYN_spg_ts_u} |
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| 156 | \textbf{U}^{(b)}(\tau,t_{n+1}) - \textbf{U}^{(b)}(\tau,t_{n-1}) \\ |
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| 157 | = 2\Delta t \left[ - f \textbf{k} \times \textbf{U}^{(b)}(\tau,t_{n}) |
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| 158 | - H(\tau) \nabla p_s^{(b)}(\tau,t_{n}) +\textbf{M}(\tau) \right] |
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| 159 | \end{multline} |
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| 160 | \ |
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| 161 | |
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| 162 | In these equations, araised (b) denotes values of surface height and vertically integrated velocity updated with the barotropic time steps. The $\tau$ time label on $\eta^{(b)}$ |
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| 163 | and $U^{(b)}$ denotes the baroclinic time at which the vertically integrated forcing $\textbf{M}(\tau)$ (note that this forcing includes the surface freshwater forcing), the tracer fields, the freshwater flux $\text{EMP}_w(\tau)$, and total depth of the ocean $H(\tau)$ are held for the duration of the barotropic time stepping over a single cycle. This is also the time |
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| 164 | that sets the barotropic time steps via |
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| 165 | \begin{equation} \label{DYN_spg_ts_t} |
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| 166 | t_n=\tau+n\Delta t |
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| 167 | \end{equation} |
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| 168 | with $n$ an integer. The density scaled surface pressure is evaluated via |
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| 169 | \begin{equation} \label{DYN_spg_ts_ps} |
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| 170 | p_s^{(b)}(\tau,t_{n}) = \begin{cases} |
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| 171 | g \;\eta_s^{(b)}(\tau,t_{n}) \;\rho(\tau)_{k=1}) / \rho_o & \text{non-linear case} \\ |
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| 172 | g \;\eta_s^{(b)}(\tau,t_{n}) & \text{linear case} |
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| 173 | \end{cases} |
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| 174 | \end{equation} |
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| 175 | To get started, we assume the following initial conditions |
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| 176 | \begin{equation} \label{DYN_spg_ts_eta} |
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| 177 | \begin{split} |
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| 178 | \eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)} |
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| 179 | \\ |
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| 180 | \eta^{(b)}(\tau,t_{n=1}) &= \eta^{(b)}(\tau,t_{n=0}) + \Delta t \ \text{RHS}_{n=0} |
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| 181 | \end{split} |
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| 182 | \end{equation} |
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| 183 | with |
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| 184 | \begin{equation} \label{DYN_spg_ts_etaF} |
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| 185 | \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\Delta t,t_{n}) |
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| 186 | \end{equation} |
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| 187 | the time averaged surface height taken from the previous barotropic cycle. Likewise, |
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| 188 | \begin{equation} \label{DYN_spg_ts_u} |
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| 189 | \textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)} \\ |
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| 190 | \\ |
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| 191 | \textbf{U}(\tau,t_{n=1}) = \textbf{U}^{(b)}(\tau,t_{n=0}) + \Delta t \ \text{RHS}_{n=0} |
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| 192 | \end{equation} |
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| 193 | with |
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| 194 | \begin{equation} \label{DYN_spg_ts_u} |
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| 195 | \overline{\textbf{U}^{(b)}(\tau)} |
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| 196 | = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\Delta t,t_{n}) |
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| 197 | \end{equation} |
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| 198 | the time averaged vertically integrated transport. Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration. |
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| 199 | |
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| 200 | Upon reaching $t_{n=N} = \tau + 2\Delta \tau$ , the vertically integrated velocity is time averaged to produce the updated vertically integrated velocity at baroclinic time $\tau + \Delta \tau$ |
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| 201 | \begin{equation} \label{DYN_spg_ts_u} |
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| 202 | \textbf{U}(\tau+\Delta t) = \overline{\textbf{U}^{(b)}(\tau+\Delta t)} |
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| 203 | = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n}) |
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| 204 | \end{equation} |
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| 205 | The surface height on the new baroclinic time step is then determined via a baroclinic leap-frog using the following form |
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| 206 | |
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| 207 | \begin{equation} \label{DYN_spg_ts_ssh} |
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| 208 | \eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\Delta t \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right] |
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| 209 | \end{equation} |
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| 210 | |
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| 211 | The use of this "big-leap-frog" scheme for the surface height ensures compatibility between the mass/volume budgets and the tracer budgets. More discussion of this point is provided in Chapter 10 (see in particular Section 10.2). |
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| 212 | |
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| 213 | In general, some form of time filter is needed to maintain integrity of the surface |
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| 214 | height field due to the leap-frog splitting mode in equation \ref{DYN_spg_ts_ssh}. We |
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| 215 | have tried various forms of such filtering, with the following method discussed in |
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| 216 | Griffies et al. (2001) chosen due to its stability and reasonably good maintenance of |
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| 217 | tracer conservation properties (see Section ??) |
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| 218 | |
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| 219 | \begin{equation} \label{DYN_spg_ts_sshf} |
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| 220 | \eta^{F}(\tau-\Delta) = \overline{\eta^{(b)}(\tau)} |
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| 221 | \end{equation} |
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| 222 | Another approach tried was |
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| 223 | |
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| 224 | \begin{equation} \label{DYN_spg_ts_sshf2} |
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| 225 | \eta^{F}(\tau-\Delta) = \eta(\tau) |
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| 226 | + (\alpha/2) \left[\overline{\eta^{(b)}}(\tau+\Delta t) |
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| 227 | + \overline{\eta^{(b)}}(\tau-\Delta t) -2 \;\eta(\tau) \right] |
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| 228 | \end{equation} |
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| 229 | |
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| 230 | which is useful since it isolates all the time filtering aspects into the term multiplied |
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| 231 | by $\alpha$. This isolation allows for an easy check that tracer conservation is exact when |
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| 232 | eliminating tracer and surface height time filtering (see Section ?? for more complete discussion). However, in the general case with a non-zero $\alpha$, the filter \ref{DYN_spg_ts_sshf} was found to be more conservative, and so is recommended. |
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| 233 | |
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| 234 | |
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| 235 | |
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| 236 | |
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| 237 | |
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| 238 | %------------------------------------------------------------- |
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| 239 | % Filtered formulation |
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| 240 | %------------------------------------------------------------- |
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| 241 | \subsubsection{Filtered formulation (\key{dynspg\_flt})} |
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| 242 | \label{DYN_spg_flt} |
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| 243 | |
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| 244 | The filtered formulation follows the \citet{Roullet2000} implementation. The extra term introduced in the equations (see {\S}I.2.2) is solved implicitly. The elliptic solvers available in the code are |
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| 245 | documented in \S\ref{MISC}. The amplitude of the extra term is given by the namelist variable \np{rnu}. The default value is 1, as recommended by \citet{Roullet2000} |
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| 246 | |
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| 247 | \colorbox{red}{\np{rnu}=1 to be suppressed from namelist !} |
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| 248 | |
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| 249 | %------------------------------------------------------------- |
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| 250 | % Non-linear free surface formulation |
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| 251 | %------------------------------------------------------------- |
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| 252 | \subsection{Non-linear free surface formulation (\key{vvl})} |
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| 253 | \label{DYN_spg_vvl} |
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| 254 | |
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| 255 | In the non-linear free surface formulation, the variations of volume are fully taken into account. This option is presented in a report \citep{Levier2007} available on the NEMO web site. The three time-stepping methods (explicit, split-explicit and filtered) are the same as in \S\ref{DYN_spg_linear} except that the ocean depth is now time-dependent. 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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| 256 | |
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| 257 | |
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