1 | \documentclass[../main/NEMO_manual]{subfiles} |
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2 | |
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3 | \begin{document} |
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4 | |
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5 | % ================================================================ |
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6 | % Chapter 2 ——— Time Domain (step.F90) |
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7 | % ================================================================ |
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8 | \chapter{Time Domain (STP)} |
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9 | \label{chap:STP} |
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10 | \minitoc |
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11 | |
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12 | % Missing things: |
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13 | % - daymod: definition of the time domain (nit000, nitend andd the calendar) |
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14 | |
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15 | \gmcomment{STEVEN :maybe a picture of the directory structure in the introduction which could be referred to here, |
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16 | would help ==> to be added} |
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17 | %%%% |
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18 | |
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19 | \newpage |
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20 | |
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21 | Having defined the continuous equations in \autoref{chap:PE}, we need now to choose a time discretization, |
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22 | a key feature of an ocean model as it exerts a strong influence on the structure of the computer code |
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23 | (\ie on its flowchart). |
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24 | In the present chapter, we provide a general description of the \NEMO time stepping strategy and |
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25 | the consequences for the order in which the equations are solved. |
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26 | |
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27 | % ================================================================ |
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28 | % Time Discretisation |
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29 | % ================================================================ |
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30 | \section{Time stepping environment} |
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31 | \label{sec:STP_environment} |
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32 | |
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33 | The time stepping used in \NEMO is a three level scheme that can be represented as follows: |
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34 | \begin{equation} |
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35 | \label{eq:STP} |
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36 | x^{t + \rdt} = x^{t - \rdt} + 2 \, \rdt \ \text{RHS}_x^{t - \rdt, \, t, \, t + \rdt} |
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37 | \end{equation} |
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38 | where $x$ stands for $u$, $v$, $T$ or $S$; |
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39 | RHS is the Right-Hand-Side of the corresponding time evolution equation; |
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40 | $\rdt$ is the time step; |
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41 | and the superscripts indicate the time at which a quantity is evaluated. |
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42 | Each term of the RHS is evaluated at a specific time step depending on the physics with which it is associated. |
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43 | |
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44 | The choice of the time step used for this evaluation is discussed below as well as |
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45 | the implications for starting or restarting a model simulation. |
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46 | Note that the time stepping calculation is generally performed in a single operation. |
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47 | With such a complex and nonlinear system of equations it would be dangerous to let a prognostic variable evolve in |
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48 | time for each term separately. |
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49 | |
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50 | The three level scheme requires three arrays for each prognostic variable. |
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51 | For each variable $x$ there is $x_b$ (before), $x_n$ (now) and $x_a$. |
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52 | The third array, although referred to as $x_a$ (after) in the code, |
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53 | is usually not the variable at the after time step; |
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54 | but rather it is used to store the time derivative (RHS in \autoref{eq:STP}) prior to time-stepping the equation. |
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55 | Generally, the time stepping is performed once at each time step in the \mdl{tranxt} and \mdl{dynnxt} modules, |
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56 | except when using implicit vertical diffusion or calculating sea surface height in which |
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57 | case time-splitting options are used. |
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58 | |
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59 | % ------------------------------------------------------------------------------------------------------------- |
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60 | % Non-Diffusive Part---Leapfrog Scheme |
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61 | % ------------------------------------------------------------------------------------------------------------- |
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62 | \section{Non-diffusive part --- Leapfrog scheme} |
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63 | \label{sec:STP_leap_frog} |
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64 | |
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65 | The time stepping used for processes other than diffusion is the well-known leapfrog scheme |
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66 | \citep{Mesinger_Arakawa_Bk76}. |
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67 | This scheme is widely used for advection processes in low-viscosity fluids. |
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68 | It is a time centred scheme, \ie the RHS in \autoref{eq:STP} is evaluated at time step $t$, the now time step. |
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69 | It may be used for momentum and tracer advection, pressure gradient, and Coriolis terms, |
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70 | but not for diffusion terms. |
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71 | It is an efficient method that achieves second-order accuracy with |
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72 | just one right hand side evaluation per time step. |
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73 | Moreover, it does not artificially damp linear oscillatory motion nor does it produce instability by |
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74 | amplifying the oscillations. |
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75 | These advantages are somewhat diminished by the large phase-speed error of the leapfrog scheme, |
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76 | and the unsuitability of leapfrog differencing for the representation of diffusion and Rayleigh damping processes. |
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77 | However, the scheme allows the coexistence of a numerical and a physical mode due to |
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78 | its leading third order dispersive error. |
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79 | In other words a divergence of odd and even time steps may occur. |
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80 | To prevent it, the leapfrog scheme is often used in association with a Robert-Asselin time filter |
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81 | (hereafter the LF-RA scheme). |
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82 | This filter, first designed by \citet{Robert_JMSJ66} and more comprehensively studied by \citet{Asselin_MWR72}, |
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83 | is a kind of laplacian diffusion in time that mixes odd and even time steps: |
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84 | \begin{equation} |
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85 | \label{eq:STP_asselin} |
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86 | x_F^t = x^t + \gamma \, \lt[ x_F^{t - \rdt} - 2 x^t + x^{t + \rdt} \rt] |
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87 | \end{equation} |
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88 | where the subscript $F$ denotes filtered values and $\gamma$ is the Asselin coefficient. |
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89 | $\gamma$ is initialized as \np{rn\_atfp} (namelist parameter). |
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90 | Its default value is \np{rn\_atfp}~\forcode{= 10.e-3} (see \autoref{sec:STP_mLF}), |
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91 | causing only a weak dissipation of high frequency motions (\citep{Farge1987}). |
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92 | The addition of a time filter degrades the accuracy of the calculation from second to first order. |
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93 | However, the second order truncation error is proportional to $\gamma$, which is small compared to 1. |
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94 | Therefore, the LF-RA is a quasi second order accurate scheme. |
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95 | The LF-RA scheme is preferred to other time differencing schemes such as predictor corrector or trapezoidal schemes, |
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96 | because the user has an explicit and simple control of the magnitude of the time diffusion of the scheme. |
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97 | When used with the 2nd order space centred discretisation of the advection terms in |
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98 | the momentum and tracer equations, LF-RA avoids implicit numerical diffusion: |
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99 | diffusion is set explicitly by the user through the Robert-Asselin |
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100 | filter parameter and the viscosity and diffusion coefficients. |
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101 | |
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102 | % ------------------------------------------------------------------------------------------------------------- |
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103 | % Diffusive Part---Forward or Backward Scheme |
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104 | % ------------------------------------------------------------------------------------------------------------- |
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105 | \section{Diffusive part --- Forward or backward scheme} |
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106 | \label{sec:STP_forward_imp} |
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107 | |
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108 | The leapfrog differencing scheme is unsuitable for the representation of diffusion and damping processes. |
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109 | For a tendancy $D_x$, representing a diffusion term or a restoring term to a tracer climatology |
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110 | (when present, see \autoref{sec:TRA_dmp}), a forward time differencing scheme is used : |
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111 | \[ |
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112 | %\label{eq:STP_euler} |
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113 | x^{t + \rdt} = x^{t - \rdt} + 2 \, \rdt \ D_x^{t - \rdt} |
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114 | \] |
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115 | |
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116 | This is diffusive in time and conditionally stable. |
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117 | The conditions for stability of second and fourth order horizontal diffusion schemes are \citep{Griffies_Bk04}: |
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118 | \begin{equation} |
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119 | \label{eq:STP_euler_stability} |
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120 | A^h < |
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121 | \begin{cases} |
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122 | \frac{e^2}{ 8 \, \rdt} & \text{laplacian diffusion} \\ |
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123 | \frac{e^4}{64 \, \rdt} & \text{bilaplacian diffusion} |
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124 | \end{cases} |
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125 | \end{equation} |
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126 | where $e$ is the smallest grid size in the two horizontal directions and $A^h$ is the mixing coefficient. |
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127 | The linear constraint \autoref{eq:STP_euler_stability} is a necessary condition, but not sufficient. |
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128 | If it is not satisfied, even mildly, then the model soon becomes wildly unstable. |
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129 | The instability can be removed by either reducing the length of the time steps or reducing the mixing coefficient. |
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130 | |
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131 | For the vertical diffusion terms, a forward time differencing scheme can be used, |
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132 | but usually the numerical stability condition imposes a strong constraint on the time step. |
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133 | Two solutions are available in \NEMO to overcome the stability constraint: |
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134 | $(a)$ a forward time differencing scheme using a time splitting technique (\np{ln\_zdfexp}~\forcode{= .true.}) or |
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135 | $(b)$ a backward (or implicit) time differencing scheme (\np{ln\_zdfexp}~\forcode{= .false.}). |
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136 | In $(a)$, the master time step $\Delta$t is cut into $N$ fractional time steps so that |
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137 | the stability criterion is reduced by a factor of $N$. |
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138 | The computation is performed as follows: |
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139 | \begin{alignat*}{2} |
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140 | % \label{eq:STP_ts} |
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141 | &x_\ast^{t - \rdt} &= &x^{t - \rdt} \\ |
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142 | &x_\ast^{t - \rdt + L \frac{2 \rdt}{N}} &= &x_\ast ^{t - \rdt + (L - 1) \frac{2 \rdt}{N}} |
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143 | + \frac{2 \rdt}{N} \; DF^{t - \rdt + (L - 1) \frac{2 \rdt}{N}} |
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144 | \quad \text{for $L = 1$ to $N$} \\ |
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145 | &x^{t + \rdt} &= &x_\ast^{t + \rdt} |
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146 | \end{alignat*} |
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147 | with DF a vertical diffusion term. |
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148 | The number of fractional time steps, $N$, is given by setting \np{nn\_zdfexp}, (namelist parameter). |
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149 | The scheme $(b)$ is unconditionally stable but diffusive. It can be written as follows: |
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150 | \begin{equation} |
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151 | \label{eq:STP_imp} |
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152 | x^{t + \rdt} = x^{t - \rdt} + 2 \, \rdt \ \text{RHS}_x^{t + \rdt} |
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153 | \end{equation} |
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154 | |
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155 | %%gm |
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156 | %%gm UPDATE the next paragraphs with time varying thickness ... |
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157 | %%gm |
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158 | |
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159 | This scheme is rather time consuming since it requires a matrix inversion, |
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160 | but it becomes attractive since a value of 3 or more is needed for N in the forward time differencing scheme. |
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161 | For example, the finite difference approximation of the temperature equation is: |
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162 | \[ |
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163 | % \label{eq:STP_imp_zdf} |
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164 | \frac{T(k)^{t + 1} - T(k)^{t - 1}}{2 \; \rdt} |
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165 | \equiv |
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166 | \text{RHS} + \frac{1}{e_{3t}} \delta_k \lt[ \frac{A_w^{vT}}{e_{3w} } \delta_{k + 1/2} \lt[ T^{t + 1} \rt] \rt] |
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167 | \] |
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168 | where RHS is the right hand side of the equation except for the vertical diffusion term. |
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169 | We rewrite \autoref{eq:STP_imp} as: |
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170 | \begin{equation} |
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171 | \label{eq:STP_imp_mat} |
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172 | -c(k + 1) \; T^{t + 1}(k + 1) + d(k) \; T^{t + 1}(k) - \; c(k) \; T^{t + 1}(k - 1) \equiv b(k) |
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173 | \end{equation} |
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174 | where |
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175 | \begin{align*} |
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176 | c(k) &= A_w^{vT} (k) \, / \, e_{3w} (k) \\ |
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177 | d(k) &= e_{3t} (k) \, / \, (2 \rdt) + c_k + c_{k + 1} \\ |
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178 | b(k) &= e_{3t} (k) \; \lt( T^{t - 1}(k) \, / \, (2 \rdt) + \text{RHS} \rt) |
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179 | \end{align*} |
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180 | |
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181 | \autoref{eq:STP_imp_mat} is a linear system of equations with an associated matrix which is tridiagonal. |
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182 | Moreover, |
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183 | $c(k)$ and $d(k)$ are positive and the diagonal term is greater than the sum of the two extra-diagonal terms, |
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184 | therefore a special adaptation of the Gauss elimination procedure is used to find the solution |
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185 | (see for example \citet{Richtmyer1967}). |
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186 | |
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187 | % ------------------------------------------------------------------------------------------------------------- |
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188 | % Surface Pressure gradient |
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189 | % ------------------------------------------------------------------------------------------------------------- |
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190 | \section{Surface pressure gradient} |
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191 | \label{sec:STP_spg_ts} |
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192 | |
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193 | ===>>>> TO BE written.... :-) |
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194 | |
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195 | %\gmcomment{ |
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196 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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197 | \begin{figure}[!t] |
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198 | \begin{center} |
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199 | \includegraphics[]{Fig_TimeStepping_flowchart} |
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200 | \caption{ |
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201 | \protect\label{fig:TimeStep_flowchart} |
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202 | Sketch of the leapfrog time stepping sequence in \NEMO from \citet{Leclair_Madec_OM09}. |
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203 | The use of a semi -implicit computation of the hydrostatic pressure gradient requires the tracer equation to |
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204 | be stepped forward prior to the momentum equation. |
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205 | The need for knowledge of the vertical scale factor (here denoted as $h$) requires the sea surface height and |
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206 | the continuity equation to be stepped forward prior to the computation of the tracer equation. |
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207 | Note that the method for the evaluation of the surface pressure gradient $\nabla p_s$ is not presented here |
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208 | (see \autoref{sec:DYN_spg}). |
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209 | } |
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210 | \end{center} |
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211 | \end{figure} |
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212 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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213 | %} |
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214 | |
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215 | % ------------------------------------------------------------------------------------------------------------- |
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216 | % The Modified Leapfrog -- Asselin Filter scheme |
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217 | % ------------------------------------------------------------------------------------------------------------- |
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218 | \section{Modified Leapfrog -- Asselin filter scheme} |
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219 | \label{sec:STP_mLF} |
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220 | |
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221 | Significant changes have been introduced by \cite{Leclair_Madec_OM09} in the LF-RA scheme in order to |
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222 | ensure tracer conservation and to allow the use of a much smaller value of the Asselin filter parameter. |
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223 | The modifications affect both the forcing and filtering treatments in the LF-RA scheme. |
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224 | |
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225 | In a classical LF-RA environment, the forcing term is centred in time, |
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226 | \ie it is time-stepped over a $2 \rdt$ period: |
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227 | $x^t = x^t + 2 \rdt Q^t$ where $Q$ is the forcing applied to $x$, |
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228 | and the time filter is given by \autoref{eq:STP_asselin} so that $Q$ is redistributed over several time step. |
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229 | In the modified LF-RA environment, these two formulations have been replaced by: |
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230 | \begin{gather} |
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231 | \label{eq:STP_forcing} |
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232 | x^{t + \rdt} = x^{t - \rdt} + \rdt \lt( Q^{t - \rdt / 2} + Q^{t + \rdt / 2} \rt) \\ |
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233 | \label{eq:STP_RA} |
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234 | x_F^t = x^t + \gamma \, \lt( x_F^{t - \rdt} - 2 x^t + x^{t + \rdt} \rt) |
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235 | - \gamma \, \rdt \, \lt( Q^{t + \rdt / 2} - Q^{t - \rdt / 2} \rt) |
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236 | \end{gather} |
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237 | The change in the forcing formulation given by \autoref{eq:STP_forcing} (see \autoref{fig:MLF_forcing}) |
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238 | has a significant effect: |
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239 | the forcing term no longer excites the divergence of odd and even time steps \citep{Leclair_Madec_OM09}. |
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240 | % forcing seen by the model.... |
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241 | This property improves the LF-RA scheme in two respects. |
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242 | First, the LF-RA can now ensure the local and global conservation of tracers. |
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243 | Indeed, time filtering is no longer required on the forcing part. |
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244 | The influence of the Asselin filter on the forcing is be removed by adding a new term in the filter |
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245 | (last term in \autoref{eq:STP_RA} compared to \autoref{eq:STP_asselin}). |
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246 | Since the filtering of the forcing was the source of non-conservation in the classical LF-RA scheme, |
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247 | the modified formulation becomes conservative \citep{Leclair_Madec_OM09}. |
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248 | Second, the LF-RA becomes a truly quasi -second order scheme. |
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249 | Indeed, \autoref{eq:STP_forcing} used in combination with a careful treatment of static instability |
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250 | (\autoref{subsec:ZDF_evd}) and of the TKE physics (\autoref{subsec:ZDF_tke_ene}), |
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251 | the two other main sources of time step divergence, |
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252 | allows a reduction by two orders of magnitude of the Asselin filter parameter. |
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253 | |
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254 | Note that the forcing is now provided at the middle of a time step: |
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255 | $Q^{t + \rdt / 2}$ is the forcing applied over the $[t,t + \rdt]$ time interval. |
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256 | This and the change in the time filter, \autoref{eq:STP_RA}, |
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257 | allows an exact evaluation of the contribution due to the forcing term between any two time steps, |
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258 | even if separated by only $\rdt$ since the time filter is no longer applied to the forcing term. |
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259 | |
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260 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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261 | \begin{figure}[!t] |
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262 | \begin{center} |
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263 | \includegraphics[]{Fig_MLF_forcing} |
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264 | \caption{ |
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265 | \protect\label{fig:MLF_forcing} |
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266 | Illustration of forcing integration methods. |
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267 | (top) ''Traditional'' formulation: |
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268 | the forcing is defined at the same time as the variable to which it is applied |
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269 | (integer value of the time step index) and it is applied over a $2 \rdt$ period. |
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270 | (bottom) modified formulation: |
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271 | the forcing is defined in the middle of the time (integer and a half value of the time step index) and |
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272 | the mean of two successive forcing values ($n - 1 / 2$, $n + 1 / 2$) is applied over a $2 \rdt$ period. |
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273 | } |
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274 | \end{center} |
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275 | \end{figure} |
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276 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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277 | |
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278 | % ------------------------------------------------------------------------------------------------------------- |
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279 | % Start/Restart strategy |
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280 | % ------------------------------------------------------------------------------------------------------------- |
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281 | \section{Start/Restart strategy} |
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282 | \label{sec:STP_rst} |
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283 | |
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284 | %--------------------------------------------namrun------------------------------------------- |
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285 | \nlst{namrun} |
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286 | %-------------------------------------------------------------------------------------------------------------- |
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287 | |
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288 | The first time step of this three level scheme when starting from initial conditions is a forward step |
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289 | (Euler time integration): |
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290 | \[ |
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291 | % \label{eq:DOM_euler} |
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292 | x^1 = x^0 + \rdt \ \text{RHS}^0 |
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293 | \] |
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294 | This is done simply by keeping the leapfrog environment (\ie the \autoref{eq:STP} three level time stepping) but |
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295 | setting all $x^0$ (\textit{before}) and $x^1$ (\textit{now}) fields equal at the first time step and |
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296 | using half the value of $\rdt$. |
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297 | |
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298 | It is also possible to restart from a previous computation, by using a restart file. |
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299 | The restart strategy is designed to ensure perfect restartability of the code: |
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300 | the user should obtain the same results to machine precision either by |
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301 | running the model for $2N$ time steps in one go, |
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302 | or by performing two consecutive experiments of $N$ steps with a restart. |
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303 | This requires saving two time levels and many auxiliary data in the restart files in machine precision. |
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304 | |
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305 | Note that when a semi -implicit scheme is used to evaluate the hydrostatic pressure gradient |
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306 | (see \autoref{subsec:DYN_hpg_imp}), an extra three-dimensional field has to |
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307 | be added to the restart file to ensure an exact restartability. |
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308 | This is done optionally via the \np{nn\_dynhpg\_rst} namelist parameter, |
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309 | so that the size of the restart file can be reduced when restartability is not a key issue |
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310 | (operational oceanography or in ensemble simulations for seasonal forecasting). |
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311 | |
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312 | Note the size of the time step used, $\rdt$, is also saved in the restart file. |
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313 | When restarting, if the the time step has been changed, a restart using an Euler time stepping scheme is imposed. |
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314 | Options are defined through the \ngn{namrun} namelist variables. |
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315 | %%% |
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316 | \gmcomment{ |
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317 | add here how to force the restart to contain only one time step for operational purposes |
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318 | |
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319 | add also the idea of writing several restart for seasonal forecast : how is it done ? |
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320 | |
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321 | verify that all namelist pararmeters are truly described |
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322 | |
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323 | a word on the check of restart ..... |
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324 | } |
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325 | %%% |
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326 | |
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327 | \gmcomment{ % add a subsection here |
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328 | |
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329 | %------------------------------------------------------------------------------------------------------------- |
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330 | % Time Domain |
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331 | % ------------------------------------------------------------------------------------------------------------- |
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332 | \subsection{Time domain} |
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333 | \label{subsec:STP_time} |
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334 | %--------------------------------------------namrun------------------------------------------- |
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335 | |
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336 | \nlst{namdom} |
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337 | %-------------------------------------------------------------------------------------------------------------- |
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338 | |
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339 | Options are defined through the \ngn{namdom} namelist variables. |
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340 | \colorbox{yellow}{add here a few word on nit000 and nitend} |
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341 | |
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342 | \colorbox{yellow}{Write documentation on the calendar and the key variable adatrj} |
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343 | |
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344 | add a description of daymod, and the model calandar (leap-year and co) |
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345 | |
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346 | } %% end add |
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347 | |
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348 | |
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349 | |
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350 | %% |
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351 | \gmcomment{ % add implicit in vvl case and Crant-Nicholson scheme |
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352 | |
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353 | Implicit time stepping in case of variable volume thickness. |
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354 | |
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355 | Tracer case (NB for momentum in vector invariant form take care!) |
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356 | |
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357 | \begin{flalign*} |
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358 | &\frac{\lt( e_{3t}\,T \rt)_k^{t+1}-\lt( e_{3t}\,T \rt)_k^{t-1}}{2\rdt} |
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359 | \equiv \text{RHS}+ \delta_k \lt[ {\frac{A_w^{vt} }{e_{3w}^{t+1} }\delta_{k + 1/2} \lt[ {T^{t+1}} \rt]} |
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360 | \rt] \\ |
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361 | &\lt( e_{3t}\,T \rt)_k^{t+1}-\lt( e_{3t}\,T \rt)_k^{t-1} |
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362 | \equiv {2\rdt} \ \text{RHS}+ {2\rdt} \ \delta_k \lt[ {\frac{A_w^{vt} }{e_{3w}^{t+1} }\delta_{k + 1/2} \lt[ {T^{t+1}} \rt]} |
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363 | \rt] \\ |
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364 | &\lt( e_{3t}\,T \rt)_k^{t+1}-\lt( e_{3t}\,T \rt)_k^{t-1} |
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365 | \equiv 2\rdt \ \text{RHS} |
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366 | + 2\rdt \ \lt\{ \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k + 1/2} [ T_{k +1}^{t+1} - T_k ^{t+1} ] |
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367 | - \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k - 1/2} [ T_k ^{t+1} - T_{k -1}^{t+1} ] \rt\} \\ |
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368 | &\\ |
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369 | &\lt( e_{3t}\,T \rt)_k^{t+1} |
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370 | - {2\rdt} \ \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k + 1/2} T_{k +1}^{t+1} |
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371 | + {2\rdt} \ \lt\{ \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k + 1/2} |
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372 | + \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k - 1/2} \rt\} T_{k }^{t+1} |
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373 | - {2\rdt} \ \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k - 1/2} T_{k -1}^{t+1} \\ |
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374 | &\equiv \lt( e_{3t}\,T \rt)_k^{t-1} + {2\rdt} \ \text{RHS} \\ |
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375 | % |
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376 | \end{flalign*} |
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377 | \begin{flalign*} |
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378 | \allowdisplaybreaks |
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379 | \intertext{ Tracer case } |
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380 | % |
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381 | & \qquad \qquad \quad - {2\rdt} \ \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k + 1/2} |
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382 | \qquad \qquad \qquad \qquad T_{k +1}^{t+1} \\ |
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383 | &+ {2\rdt} \ \biggl\{ (e_{3t})_{k }^{t+1} \bigg. + \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k + 1/2} |
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384 | + \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k - 1/2} \bigg. \biggr\} \ \ \ T_{k }^{t+1} &&\\ |
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385 | & \qquad \qquad \qquad \qquad \qquad \quad \ \ - {2\rdt} \ \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k - 1/2} \quad \ \ T_{k -1}^{t+1} |
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386 | \ \equiv \ \lt( e_{3t}\,T \rt)_k^{t-1} + {2\rdt} \ \text{RHS} \\ |
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387 | % |
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388 | \end{flalign*} |
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389 | \begin{flalign*} |
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390 | \allowdisplaybreaks |
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391 | \intertext{ Tracer content case } |
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392 | % |
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393 | & - {2\rdt} \ & \frac{(A_w^{vt})_{k + 1/2}} {(e_{3w})_{k + 1/2}^{t+1}\;(e_{3t})_{k +1}^{t+1}} && \ \lt( e_{3t}\,T \rt)_{k +1}^{t+1} &\\ |
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394 | & + {2\rdt} \ \lt[ 1 \rt.+ & \frac{(A_w^{vt})_{k + 1/2}} {(e_{3w})_{k + 1/2}^{t+1}\;(e_{3t})_k^{t+1}} |
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395 | + & \frac{(A_w^{vt})_{k - 1/2}} {(e_{3w})_{k - 1/2}^{t+1}\;(e_{3t})_k^{t+1}} \lt. \rt] & \lt( e_{3t}\,T \rt)_{k }^{t+1} &\\ |
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396 | & - {2\rdt} \ & \frac{(A_w^{vt})_{k - 1/2}} {(e_{3w})_{k - 1/2}^{t+1}\;(e_{3t})_{k -1}^{t+1}} &\ \lt( e_{3t}\,T \rt)_{k -1}^{t+1} |
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397 | \equiv \lt( e_{3t}\,T \rt)_k^{t-1} + {2\rdt} \ \text{RHS} & |
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398 | \end{flalign*} |
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399 | |
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400 | %% |
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401 | } |
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402 | |
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403 | \biblio |
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404 | |
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405 | \pindex |
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406 | |
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407 | \end{document} |
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