1 | \documentclass[NEMO_book]{subfiles} |
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2 | \begin{document} |
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3 | % ================================================================ |
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4 | % Chapter 1 ——— Model Basics |
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5 | % ================================================================ |
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6 | % ================================================================ |
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7 | % Curvilinear z*- s*-coordinate System |
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8 | % ================================================================ |
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9 | \chapter{ essai z* s*} |
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10 | \section{Curvilinear \textit{z*}- or \textit{s*} coordinate System} |
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11 | |
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12 | % ------------------------------------------------------------------------------------------------------------- |
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13 | % ???? |
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14 | % ------------------------------------------------------------------------------------------------------------- |
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15 | |
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16 | \colorbox{yellow}{ to be updated } |
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17 | |
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18 | In that case, the free surface equation is nonlinear, and the variations of |
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19 | volume are fully taken into account. These coordinates systems is presented in |
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20 | a report \citep{Levier2007} available on the \NEMO web site. |
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21 | |
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22 | \colorbox{yellow}{ end of to be updated} |
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23 | \newline |
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24 | |
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25 | % from MOM4p1 documentation |
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26 | |
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27 | To overcome problems with vanishing surface and/or bottom cells, we consider the |
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28 | zstar coordinate |
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29 | \begin{equation} \label{PE_} |
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30 | z^\star = H \left( \frac{z-\eta}{H+\eta} \right) |
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31 | \end{equation} |
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32 | |
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33 | This coordinate is closely related to the "eta" coordinate used in many atmospheric |
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34 | models (see Black (1994) for a review of eta coordinate atmospheric models). It |
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35 | was originally used in ocean models by Stacey et al. (1995) for studies of tides |
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36 | next to shelves, and it has been recently promoted by Adcroft and Campin (2004) |
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37 | for global climate modelling. |
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38 | |
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39 | 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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40 | undulations of the bottom topography versus undulations in the surface height, it |
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41 | 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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42 | Additionally, since $z^\star$ when $\eta = 0$, no flow is spontaneously generated in an |
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43 | 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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44 | 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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45 | 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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46 | the same techniques as in $z$-models (see Chapters 13-16 of Griffies (2004) for a |
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47 | discussion of neutral physics in $z$-models, as well as Section \S\ref{LDF_slp} |
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48 | in this document for treatment in \NEMO). |
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49 | |
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50 | The range over which $z^\star$ varies is time independent $-H \leq z^\star \leq 0$. Hence, all |
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51 | cells remain nonvanishing, so long as the surface height maintains $\eta > ?H$. This |
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52 | is a minor constraint relative to that encountered on the surface height when using |
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53 | $s = z$ or $s = z - \eta$. |
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54 | |
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55 | Because $z^\star$ has a time independent range, all grid cells have static increments |
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56 | ds, and the sum of the ver tical increments yields the time independent ocean |
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57 | depth %�k ds = H. |
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58 | The $z^\star$ coordinate is therefore invisible to undulations of the |
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59 | free surface, since it moves along with the free surface. This proper ty means that |
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60 | no spurious ver tical transpor t is induced across surfaces of constant $z^\star$ by the |
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61 | motion of external gravity waves. Such spurious transpor t can be a problem in |
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62 | z-models, especially those with tidal forcing. Quite generally, the time independent |
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63 | range for the $z^\star$ coordinate is a very convenient proper ty that allows for a nearly |
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64 | arbitrary ver tical resolution even in the presence of large amplitude fluctuations of |
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65 | the surface height, again so long as $\eta > -H$. |
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66 | |
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67 | |
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68 | |
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69 | %%% |
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70 | % essai update time splitting... |
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71 | %%% |
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72 | |
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73 | |
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74 | % ================================================================ |
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75 | % Surface Pressure Gradient and Sea Surface Height |
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76 | % ================================================================ |
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77 | \section{Surface pressure gradient and Sea Surface Heigth (\protect\mdl{dynspg})} |
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78 | \label{DYN_hpg_spg} |
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79 | %-----------------------------------------nam_dynspg---------------------------------------------------- |
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80 | \namdisplay{nam_dynspg} |
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81 | %------------------------------------------------------------------------------------------------------------ |
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82 | Options are defined through the \ngn{nam\_dynspg} namelist variables. |
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83 | 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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84 | |
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85 | %------------------------------------------------------------- |
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86 | % Explicit |
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87 | %------------------------------------------------------------- |
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88 | \subsubsection{Explicit (\protect\key{dynspg\_exp})} |
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89 | \label{DYN_spg_exp} |
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90 | |
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91 | 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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92 | \begin{equation} \label{Eq_dynspg_ssh} |
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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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96 | \end{equation} |
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97 | |
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98 | 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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99 | |
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100 | The surface pressure gradient, also evaluated using a leap-frog scheme, is then simply given by : |
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101 | \begin{equation} \label{Eq_dynspg_exp} |
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102 | \left\{ \begin{aligned} |
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103 | - \frac{1} {e_{1u}} \; \delta _{i+1/2} \left[ \,\eta\, \right] \\ |
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104 | \\ |
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105 | - \frac{1} {e_{2v}} \; \delta _{j+1/2} \left[ \,\eta\, \right] |
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106 | \end{aligned} \right. |
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107 | \end{equation} |
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108 | |
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109 | Consistent with the linearization, a $\left. \rho \right|_{k=1} / \rho _o$ factor is omitted in (\ref{Eq_dynspg_exp}). |
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110 | |
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111 | %------------------------------------------------------------- |
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112 | % Split-explicit time-stepping |
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113 | %------------------------------------------------------------- |
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114 | \subsubsection{Split-explicit time-stepping (\protect\key{dynspg\_ts})} |
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115 | \label{DYN_spg_ts} |
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116 | %--------------------------------------------namdom---------------------------------------------------- |
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117 | \namdisplay{namdom} |
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118 | %-------------------------------------------------------------------------------------------------------------- |
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119 | 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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120 | in the \ngn{namdom} namelist. |
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121 | (Figure III.3). |
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122 | |
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123 | %> > > > > > > > > > > > > > > > > > > > > > > > > > > > |
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124 | \begin{figure}[!t] \begin{center} |
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125 | \includegraphics[width=0.90\textwidth]{Fig_DYN_dynspg_ts} |
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126 | \caption{ \protect\label{Fig_DYN_dynspg_ts} |
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127 | Schematic of the split-explicit time stepping scheme for the barotropic and baroclinic modes, |
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128 | after \citet{Griffies2004}. Time increases to the right. Baroclinic time steps are denoted by |
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129 | $t-\Delta t$, $t, t+\Delta t$, and $t+2\Delta t$. The curved line represents a leap-frog time step, |
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130 | and the smaller barotropic time steps $N \Delta t=2\Delta t$ are denoted by the zig-zag line. |
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131 | The vertically integrated forcing \textbf{M}(t) computed at baroclinic time step t represents |
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132 | the interaction between the barotropic and baroclinic motions. While keeping the total depth, |
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133 | tracer, and freshwater forcing fields fixed, a leap-frog integration carries the surface height |
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134 | and vertically integrated velocity from t to $t+2 \Delta t$ using N barotropic time steps of length |
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135 | $\Delta t$. Time averaging the barotropic fields over the N+1 time steps (endpoints included) |
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136 | centers the vertically integrated velocity at the baroclinic timestep $t+\Delta t$. |
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137 | A baroclinic leap-frog time step carries the surface height to $t+\Delta t$ using the convergence |
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138 | of the time averaged vertically integrated velocity taken from baroclinic time step t. } |
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139 | \end{center} |
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140 | \end{figure} |
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141 | %> > > > > > > > > > > > > > > > > > > > > > > > > > > > |
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142 | |
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143 | 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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144 | |
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145 | %from griffies book: ..... copy past ! |
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146 | |
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147 | \textbf{title: Time stepping the barotropic system } |
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148 | |
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149 | Assume knowledge of the full velocity and tracer fields at baroclinic time $\tau$. Hence, |
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150 | we can update the surface height and vertically integrated velocity with a leap-frog |
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151 | scheme using the small barotropic time step $\Delta t$. We have |
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152 | |
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153 | \begin{equation} \label{DYN_spg_ts_eta} |
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154 | \eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1}) |
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155 | = 2 \Delta t \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right] |
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156 | \end{equation} |
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157 | \begin{multline} \label{DYN_spg_ts_u} |
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158 | \textbf{U}^{(b)}(\tau,t_{n+1}) - \textbf{U}^{(b)}(\tau,t_{n-1}) \\ |
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159 | = 2\Delta t \left[ - f \textbf{k} \times \textbf{U}^{(b)}(\tau,t_{n}) |
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160 | - H(\tau) \nabla p_s^{(b)}(\tau,t_{n}) +\textbf{M}(\tau) \right] |
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161 | \end{multline} |
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162 | \ |
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163 | |
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164 | 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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165 | 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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166 | that sets the barotropic time steps via |
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167 | \begin{equation} \label{DYN_spg_ts_t} |
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168 | t_n=\tau+n\Delta t |
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169 | \end{equation} |
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170 | with $n$ an integer. The density scaled surface pressure is evaluated via |
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171 | \begin{equation} \label{DYN_spg_ts_ps} |
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172 | p_s^{(b)}(\tau,t_{n}) = \begin{cases} |
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173 | g \;\eta_s^{(b)}(\tau,t_{n}) \;\rho(\tau)_{k=1}) / \rho_o & \text{non-linear case} \\ |
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174 | g \;\eta_s^{(b)}(\tau,t_{n}) & \text{linear case} |
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175 | \end{cases} |
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176 | \end{equation} |
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177 | To get started, we assume the following initial conditions |
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178 | \begin{equation} \label{DYN_spg_ts_eta} |
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179 | \begin{split} |
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180 | \eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)} |
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181 | \\ |
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182 | \eta^{(b)}(\tau,t_{n=1}) &= \eta^{(b)}(\tau,t_{n=0}) + \Delta t \ \text{RHS}_{n=0} |
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183 | \end{split} |
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184 | \end{equation} |
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185 | with |
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186 | \begin{equation} \label{DYN_spg_ts_etaF} |
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187 | \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\Delta t,t_{n}) |
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188 | \end{equation} |
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189 | the time averaged surface height taken from the previous barotropic cycle. Likewise, |
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190 | \begin{equation} \label{DYN_spg_ts_u} |
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191 | \textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)} \\ |
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192 | \\ |
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193 | \textbf{U}(\tau,t_{n=1}) = \textbf{U}^{(b)}(\tau,t_{n=0}) + \Delta t \ \text{RHS}_{n=0} |
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194 | \end{equation} |
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195 | with |
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196 | \begin{equation} \label{DYN_spg_ts_u} |
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197 | \overline{\textbf{U}^{(b)}(\tau)} |
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198 | = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\Delta t,t_{n}) |
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199 | \end{equation} |
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200 | 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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201 | |
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202 | 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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203 | \begin{equation} \label{DYN_spg_ts_u} |
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204 | \textbf{U}(\tau+\Delta t) = \overline{\textbf{U}^{(b)}(\tau+\Delta t)} |
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205 | = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n}) |
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206 | \end{equation} |
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207 | 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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208 | |
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209 | \begin{equation} \label{DYN_spg_ts_ssh} |
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210 | \eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\Delta t \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right] |
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211 | \end{equation} |
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212 | |
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213 | 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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214 | |
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215 | In general, some form of time filter is needed to maintain integrity of the surface |
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216 | height field due to the leap-frog splitting mode in equation \ref{DYN_spg_ts_ssh}. We |
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217 | have tried various forms of such filtering, with the following method discussed in |
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218 | Griffies et al. (2001) chosen due to its stability and reasonably good maintenance of |
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219 | tracer conservation properties (see Section ??) |
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220 | |
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221 | \begin{equation} \label{DYN_spg_ts_sshf} |
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222 | \eta^{F}(\tau-\Delta) = \overline{\eta^{(b)}(\tau)} |
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223 | \end{equation} |
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224 | Another approach tried was |
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225 | |
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226 | \begin{equation} \label{DYN_spg_ts_sshf2} |
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227 | \eta^{F}(\tau-\Delta) = \eta(\tau) |
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228 | + (\alpha/2) \left[\overline{\eta^{(b)}}(\tau+\Delta t) |
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229 | + \overline{\eta^{(b)}}(\tau-\Delta t) -2 \;\eta(\tau) \right] |
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230 | \end{equation} |
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231 | |
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232 | which is useful since it isolates all the time filtering aspects into the term multiplied |
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233 | by $\alpha$. This isolation allows for an easy check that tracer conservation is exact when |
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234 | 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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235 | |
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236 | |
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237 | |
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238 | |
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239 | |
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240 | %------------------------------------------------------------- |
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241 | % Filtered formulation |
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242 | %------------------------------------------------------------- |
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243 | \subsubsection{Filtered formulation (\protect\key{dynspg\_flt})} |
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244 | \label{DYN_spg_flt} |
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245 | |
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246 | 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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247 | 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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248 | |
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249 | \colorbox{red}{\np{rnu}=1 to be suppressed from namelist !} |
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250 | |
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251 | %------------------------------------------------------------- |
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252 | % Non-linear free surface formulation |
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253 | %------------------------------------------------------------- |
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254 | \subsection{Non-linear free surface formulation (\protect\key{vvl})} |
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255 | \label{DYN_spg_vvl} |
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256 | |
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257 | 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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258 | |
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259 | |
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260 | \end{document} |
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