1 | \documentclass[NEMO_book]{subfiles} |
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2 | \begin{document} |
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3 | |
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4 | % ================================================================ |
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5 | % Chapter 2 ——— Time Domain (step.F90) |
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6 | % ================================================================ |
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7 | \chapter{Time Domain (STP) } |
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8 | \label{STP} |
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9 | \minitoc |
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10 | |
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11 | % Missing things: |
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12 | % - daymod: definition of the time domain (nit000, nitend andd the calendar) |
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13 | |
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14 | |
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15 | \gmcomment{STEVEN :maybe a picture of the directory structure in the introduction |
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16 | which could be referred to here, would help ==> to be added} |
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17 | %%%% |
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18 | |
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19 | |
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20 | \newpage |
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21 | $\ $\newline % force a new ligne |
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22 | |
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23 | |
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24 | Having defined the continuous equations in Chap.~\ref{PE}, we need now to choose |
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25 | a time discretization, a key feature of an ocean model as it exerts a strong influence |
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26 | on the structure of the computer code ($i.e.$ on its flowchart). |
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27 | In the present chapter, we provide a general description of the \NEMO |
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28 | time stepping strategy and the consequences for the order in which the equations are |
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29 | solved. |
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30 | |
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31 | $\ $\newline % force a new ligne |
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32 | % ================================================================ |
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33 | % Time Discretisation |
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34 | % ================================================================ |
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35 | \section{Time stepping environment} |
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36 | \label{STP_environment} |
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37 | |
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38 | The time stepping used in \NEMO is a three level scheme that can be |
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39 | represented as follows: |
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40 | \begin{equation} \label{Eq_STP} |
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41 | x^{t+\rdt} = x^{t-\rdt} + 2 \, \rdt \ \text{RHS}_x^{t-\rdt,\,t,\,t+\rdt} |
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42 | \end{equation} |
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43 | where $x$ stands for $u$, $v$, $T$ or $S$; RHS is the Right-Hand-Side of the |
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44 | corresponding time evolution equation; $\rdt$ is the time step; and the |
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45 | superscripts indicate the time at which a quantity is evaluated. Each term of the |
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46 | RHS is evaluated at a specific time step depending on the physics with which |
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47 | it is associated. |
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48 | |
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49 | The choice of the time step used for this evaluation is discussed below as |
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50 | well as the implications for starting or restarting a model |
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51 | simulation. Note that the time stepping calculation is generally performed in a single |
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52 | operation. With such a complex and nonlinear system of equations it would be |
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53 | dangerous to let a prognostic variable evolve in time for each term separately. |
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54 | |
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55 | The three level scheme requires three arrays for each prognostic variable. |
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56 | For each variable $x$ there is $x_b$ (before), $x_n$ (now) and $x_a$. The third array, |
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57 | although referred to as $x_a$ (after) in the code, is usually not the variable at |
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58 | the after time step; but rather it is used to store the time derivative (RHS in |
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59 | \eqref{Eq_STP}) prior to time-stepping the equation. Generally, the time |
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60 | stepping is performed once at each time step in the \mdl{tranxt} and \mdl{dynnxt} |
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61 | modules, except when using implicit vertical diffusion or calculating sea surface height |
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62 | in which case time-splitting options are used. |
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63 | |
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64 | % ------------------------------------------------------------------------------------------------------------- |
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65 | % Non-Diffusive Part---Leapfrog Scheme |
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66 | % ------------------------------------------------------------------------------------------------------------- |
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67 | \section{Non-Diffusive Part --- Leapfrog Scheme} |
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68 | \label{STP_leap_frog} |
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69 | |
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70 | The time stepping used for processes other than diffusion is the well-known leapfrog |
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71 | scheme \citep{Mesinger_Arakawa_Bk76}. This scheme is widely used for advection |
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72 | processes in low-viscosity fluids. It is a time centred scheme, $i.e.$ |
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73 | the RHS in \eqref{Eq_STP} is evaluated at time step $t$, the now time step. |
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74 | It may be used for momentum and tracer advection, |
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75 | pressure gradient, and Coriolis terms, but not for diffusion terms. |
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76 | It is an efficient method that achieves |
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77 | second-order accuracy with just one right hand side evaluation per time step. |
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78 | Moreover, it does not artificially damp linear oscillatory motion nor does it produce |
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79 | instability by amplifying the oscillations. These advantages are somewhat diminished |
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80 | by the large phase-speed error of the leapfrog scheme, and the unsuitability |
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81 | of leapfrog differencing for the representation of diffusion and Rayleigh |
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82 | damping processes. However, the scheme allows the coexistence of a numerical |
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83 | and a physical mode due to its leading third order dispersive error. In other words a |
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84 | divergence of odd and even time steps may occur. To prevent it, the leapfrog scheme |
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85 | is often used in association with a Robert-Asselin time filter (hereafter the LF-RA scheme). |
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86 | This filter, first designed by \citet{Robert_JMSJ66} and more comprehensively studied |
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87 | by \citet{Asselin_MWR72}, is a kind of laplacian diffusion in time that mixes odd and |
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88 | even time steps: |
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89 | \begin{equation} \label{Eq_STP_asselin} |
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90 | x_F^t = x^t + \gamma \, \left[ x_F^{t-\rdt} - 2 x^t + x^{t+\rdt} \right] |
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91 | \end{equation} |
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92 | where the subscript $F$ denotes filtered values and $\gamma$ is the Asselin |
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93 | coefficient. $\gamma$ is initialized as \np{rn\_atfp} (namelist parameter). |
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94 | Its default value is \np{rn\_atfp}=$10^{-3}$ (see \S~\ref{STP_mLF}), |
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95 | causing only a weak dissipation of high frequency motions (\citep{Farge1987}). |
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96 | The addition of a time filter degrades the accuracy of the |
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97 | calculation from second to first order. However, the second order truncation |
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98 | error is proportional to $\gamma$, which is small compared to 1. Therefore, |
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99 | the LF-RA is a quasi second order accurate scheme. The LF-RA scheme |
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100 | is preferred to other time differencing schemes such as |
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101 | predictor corrector or trapezoidal schemes, because the user has an explicit |
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102 | and simple control of the magnitude of the time diffusion of the scheme. |
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103 | When used with the 2nd order space centred discretisation of the |
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104 | advection terms in the momentum and tracer equations, LF-RA avoids implicit |
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105 | numerical diffusion: diffusion is set explicitly by the user through the Robert-Asselin |
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106 | filter parameter and the viscosity and diffusion coefficients. |
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107 | |
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108 | % ------------------------------------------------------------------------------------------------------------- |
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109 | % Diffusive Part---Forward or Backward Scheme |
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110 | % ------------------------------------------------------------------------------------------------------------- |
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111 | \section{Diffusive Part --- Forward or Backward Scheme} |
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112 | \label{STP_forward_imp} |
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113 | |
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114 | The leapfrog differencing scheme is unsuitable for the representation of |
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115 | diffusion and damping processes. For a tendancy $D_x$, representing a |
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116 | diffusion term or a restoring term to a tracer climatology |
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117 | (when present, see \S~\ref{TRA_dmp}), a forward time differencing scheme |
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118 | is used : |
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119 | \begin{equation} \label{Eq_STP_euler} |
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120 | x^{t+\rdt} = x^{t-\rdt} + 2 \, \rdt \ {D_x}^{t-\rdt} |
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121 | \end{equation} |
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122 | |
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123 | This is diffusive in time and conditionally stable. The |
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124 | conditions for stability of second and fourth order horizontal diffusion schemes are \citep{Griffies_Bk04}: |
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125 | \begin{equation} \label{Eq_STP_euler_stability} |
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126 | A^h < \left\{ |
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127 | \begin{aligned} |
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128 | &\frac{e^2}{ 8 \; \rdt } &&\quad \text{laplacian diffusion} \\ |
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129 | &\frac{e^4}{64 \; \rdt } &&\quad \text{bilaplacian diffusion} |
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130 | \end{aligned} |
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131 | \right. |
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132 | \end{equation} |
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133 | where $e$ is the smallest grid size in the two horizontal directions and $A^h$ is |
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134 | the mixing coefficient. The linear constraint \eqref{Eq_STP_euler_stability} |
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135 | is a necessary condition, but not sufficient. If it is not satisfied, even mildly, |
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136 | then the model soon becomes wildly unstable. The instability can be removed |
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137 | by either reducing the length of the time steps or reducing the mixing coefficient. |
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138 | |
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139 | For the vertical diffusion terms, a forward time differencing scheme can be |
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140 | used, but usually the numerical stability condition imposes a strong |
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141 | constraint on the time step. Two solutions are available in \NEMO to overcome |
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142 | the stability constraint: $(a)$ a forward time differencing scheme using a |
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143 | time splitting technique (\np{ln\_zdfexp} = true) or $(b)$ a backward (or implicit) |
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144 | time differencing scheme (\np{ln\_zdfexp} = false). In $(a)$, the master |
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145 | time step $\Delta $t is cut into $N$ fractional time steps so that the |
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146 | stability criterion is reduced by a factor of $N$. The computation is performed as |
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147 | follows: |
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148 | \begin{equation} \label{Eq_STP_ts} |
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149 | \begin{split} |
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150 | & x_\ast ^{t-\rdt} = x^{t-\rdt} \\ |
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151 | & x_\ast ^{t-\rdt+L\frac{2\rdt}{N}}=x_\ast ^{t-\rdt+\left( {L-1} |
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152 | \right)\frac{2\rdt}{N}}+\frac{2\rdt}{N}\;\text{DF}^{t-\rdt+\left( {L-1} \right)\frac{2\rdt}{N}} |
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153 | \quad \text{for $L=1$ to $N$} \\ |
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154 | & x^{t+\rdt} = x_\ast^{t+\rdt} |
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155 | \end{split} |
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156 | \end{equation} |
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157 | with DF a vertical diffusion term. The number of fractional time steps, $N$, is given |
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158 | by setting \np{nn\_zdfexp}, (namelist parameter). The scheme $(b)$ is unconditionally |
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159 | stable but diffusive. It can be written as follows: |
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160 | \begin{equation} \label{Eq_STP_imp} |
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161 | x^{t+\rdt} = x^{t-\rdt} + 2 \, \rdt \ \text{RHS}_x^{t+\rdt} |
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162 | \end{equation} |
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163 | |
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164 | %%gm |
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165 | %%gm UPDATE the next paragraphs with time varying thickness ... |
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166 | %%gm |
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167 | |
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168 | This scheme is rather time consuming since it requires a matrix inversion, |
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169 | but it becomes attractive since a value of 3 or more is needed for N in |
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170 | the forward time differencing scheme. For example, the finite difference |
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171 | approximation of the temperature equation is: |
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172 | \begin{equation} \label{Eq_STP_imp_zdf} |
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173 | \frac{T(k)^{t+1}-T(k)^{t-1}}{2\;\rdt}\equiv \text{RHS}+\frac{1}{e_{3t} }\delta |
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174 | _k \left[ {\frac{A_w^{vT} }{e_{3w} }\delta _{k+1/2} \left[ {T^{t+1}} \right]} |
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175 | \right] |
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176 | \end{equation} |
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177 | where RHS is the right hand side of the equation except for the vertical diffusion term. |
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178 | We rewrite \eqref{Eq_STP_imp} as: |
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179 | \begin{equation} \label{Eq_STP_imp_mat} |
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180 | -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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181 | \end{equation} |
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182 | where |
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183 | \begin{align*} |
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184 | c(k) &= A_w^{vT} (k) \, / \, e_{3w} (k) \\ |
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185 | d(k) &= e_{3t} (k) \, / \, (2\rdt) + c_k + c_{k+1} \\ |
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186 | b(k) &= e_{3t} (k) \; \left( T^{t-1}(k) \, / \, (2\rdt) + \text{RHS} \right) |
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187 | \end{align*} |
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188 | |
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189 | \eqref{Eq_STP_imp_mat} is a linear system of equations with an associated |
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190 | matrix which is tridiagonal. Moreover, $c(k)$ and $d(k)$ are positive and the diagonal |
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191 | term is greater than the sum of the two extra-diagonal terms, therefore a special |
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192 | adaptation of the Gauss elimination procedure is used to find the solution |
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193 | (see for example \citet{Richtmyer1967}). |
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194 | |
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195 | |
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196 | |
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197 | % ------------------------------------------------------------------------------------------------------------- |
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198 | % Surface Pressure gradient |
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199 | % ------------------------------------------------------------------------------------------------------------- |
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200 | \section{Surface Pressure Gradient} |
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201 | \label{STP_spg_ts} |
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202 | |
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203 | ===>>>> TO BE written.... :-) |
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204 | |
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205 | %\gmcomment{ |
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206 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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207 | \begin{figure}[!t] \begin{center} |
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208 | \includegraphics[width=0.7\textwidth]{Fig_TimeStepping_flowchart} |
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209 | \caption{ \protect\label{Fig_TimeStep_flowchart} |
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210 | Sketch of the leapfrog time stepping sequence in \NEMO from \citet{Leclair_Madec_OM09}. |
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211 | The use of a semi-implicit computation of the hydrostatic pressure gradient requires |
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212 | the tracer equation to be stepped forward prior to the momentum equation. |
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213 | The need for knowledge of the vertical scale factor (here denoted as $h$) |
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214 | requires the sea surface height and the continuity equation to be stepped forward |
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215 | prior to the computation of the tracer equation. |
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216 | Note that the method for the evaluation of the surface pressure gradient $\nabla p_s$ is not presented here |
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217 | (see \S~\ref{DYN_spg}). } |
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218 | \end{center} \end{figure} |
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219 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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220 | %} |
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221 | |
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222 | % ------------------------------------------------------------------------------------------------------------- |
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223 | % The Modified Leapfrog -- Asselin Filter scheme |
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224 | % ------------------------------------------------------------------------------------------------------------- |
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225 | \section{The Modified Leapfrog -- Asselin Filter scheme} |
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226 | \label{STP_mLF} |
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227 | |
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228 | Significant changes have been introduced by \cite{Leclair_Madec_OM09} in the |
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229 | LF-RA scheme in order to ensure tracer conservation and to allow the use of |
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230 | a much smaller value of the Asselin filter parameter. The modifications affect |
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231 | both the forcing and filtering treatments in the LF-RA scheme. |
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232 | |
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233 | In a classical LF-RA environment, the forcing term is centred in time, $i.e.$ |
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234 | it is time-stepped over a $2\rdt$ period: $x^t = x^t + 2\rdt Q^t $ where $Q$ |
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235 | is the forcing applied to $x$, and the time filter is given by \eqref{Eq_STP_asselin} |
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236 | so that $Q$ is redistributed over several time step. |
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237 | In the modified LF-RA environment, these two formulations have been replaced by: |
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238 | \begin{align} |
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239 | x^{t+\rdt} &= x^{t-\rdt} + \rdt \left( Q^{t-\rdt/2} + Q^{t+\rdt/2} \right) \label{Eq_STP_forcing} \\ |
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240 | % |
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241 | x_F^t &= x^t + \gamma \, \left[ x_F^{t-\rdt} - 2 x^t + x^{t+\rdt} \right] |
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242 | - \gamma\,\rdt \, \left[ Q^{t+\rdt/2} - Q^{t-\rdt/2} \right] \label{Eq_STP_RA} |
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243 | \end{align} |
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244 | The change in the forcing formulation given by \eqref{Eq_STP_forcing} |
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245 | (see Fig.\ref{Fig_MLF_forcing}) has a significant effect: the forcing term no |
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246 | longer excites the divergence of odd and even time steps \citep{Leclair_Madec_OM09}. |
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247 | % forcing seen by the model.... |
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248 | This property improves the LF-RA scheme in two respects. |
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249 | First, the LF-RA can now ensure the local and global conservation of tracers. |
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250 | Indeed, time filtering is no longer required on the forcing part. The influence of |
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251 | the Asselin filter on the forcing is be removed by adding a new term in the filter |
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252 | (last term in \eqref{Eq_STP_RA} compared to \eqref{Eq_STP_asselin}). Since |
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253 | the filtering of the forcing was the source of non-conservation in the classical |
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254 | LF-RA scheme, the modified formulation becomes conservative \citep{Leclair_Madec_OM09}. |
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255 | Second, the LF-RA becomes a truly quasi-second order scheme. Indeed, |
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256 | \eqref{Eq_STP_forcing} used in combination with a careful treatment of static |
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257 | instability (\S\ref{ZDF_evd}) and of the TKE physics (\S\ref{ZDF_tke_ene}), |
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258 | the two other main sources of time step divergence, allows a reduction by |
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259 | two orders of magnitude of the Asselin filter parameter. |
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260 | |
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261 | Note that the forcing is now provided at the middle of a time step: $Q^{t+\rdt/2}$ |
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262 | is the forcing applied over the $[t,t+\rdt]$ time interval. This and the change |
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263 | in the time filter, \eqref{Eq_STP_RA}, allows an exact evaluation of the |
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264 | contribution due to the forcing term between any two time steps, |
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265 | even if separated by only $\rdt$ since the time filter is no longer applied to the |
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266 | forcing term. |
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267 | |
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268 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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269 | \begin{figure}[!t] \begin{center} |
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270 | \includegraphics[width=0.90\textwidth]{Fig_MLF_forcing} |
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271 | \caption{ \protect\label{Fig_MLF_forcing} |
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272 | Illustration of forcing integration methods. |
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273 | (top) ''Traditional'' formulation : the forcing is defined at the same time as the variable |
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274 | to which it is applied (integer value of the time step index) and it is applied over a $2\rdt$ period. |
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275 | (bottom) modified formulation : the forcing is defined in the middle of the time (integer and a half |
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276 | value of the time step index) and the mean of two successive forcing values ($n-1/2$, $n+1/2$). |
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277 | is applied over a $2\rdt$ period.} |
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278 | \end{center} \end{figure} |
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279 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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280 | |
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281 | % ------------------------------------------------------------------------------------------------------------- |
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282 | % Start/Restart strategy |
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283 | % ------------------------------------------------------------------------------------------------------------- |
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284 | \section{Start/Restart strategy} |
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285 | \label{STP_rst} |
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286 | |
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287 | %--------------------------------------------namrun------------------------------------------- |
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288 | \fortranfile{namelists/namrun} |
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289 | %-------------------------------------------------------------------------------------------------------------- |
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290 | |
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291 | The first time step of this three level scheme when starting from initial conditions |
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292 | is a forward step (Euler time integration): |
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293 | \begin{equation} \label{Eq_DOM_euler} |
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294 | x^1 = x^0 + \rdt \ \text{RHS}^0 |
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295 | \end{equation} |
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296 | This is done simply by keeping the leapfrog environment ($i.e.$ the \eqref{Eq_STP} |
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297 | three level time stepping) but setting all $x^0$ (\textit{before}) and $x^{1}$ (\textit{now}) fields |
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298 | equal at the first time step and using half the value of $\rdt$. |
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299 | |
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300 | It is also possible to restart from a previous computation, by using a |
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301 | restart file. The restart strategy is designed to ensure perfect |
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302 | restartability of the code: the user should obtain the same results to |
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303 | machine precision either by running the model for $2N$ time steps in one go, |
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304 | or by performing two consecutive experiments of $N$ steps with a restart. |
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305 | This requires saving two time levels and many auxiliary data in the restart |
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306 | files in machine precision. |
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307 | |
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308 | Note that when a semi-implicit scheme is used to evaluate the hydrostatic pressure |
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309 | gradient (see \S\ref{DYN_hpg_imp}), an extra three-dimensional field has to be |
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310 | added to the restart file to ensure an exact restartability. This is done optionally |
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311 | via the \np{nn\_dynhpg\_rst} namelist parameter, so that the size of the |
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312 | restart file can be reduced when restartability is not a key issue (operational |
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313 | oceanography or in ensemble simulations for seasonal forecasting). |
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314 | |
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315 | Note the size of the time step used, $\rdt$, is also saved in the restart file. |
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316 | When restarting, if the the time step has been changed, a restart using an Euler time |
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317 | stepping scheme is imposed. |
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318 | Options are defined through the \ngn{namrun} namelist variables. |
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319 | %%% |
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320 | \gmcomment{ |
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321 | add here how to force the restart to contain only one time step for operational purposes |
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322 | |
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323 | add also the idea of writing several restart for seasonal forecast : how is it done ? |
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324 | |
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325 | verify that all namelist pararmeters are truly described |
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326 | |
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327 | a word on the check of restart ..... |
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328 | } |
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329 | %%% |
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330 | |
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331 | \gmcomment{ % add a subsection here |
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332 | |
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333 | %------------------------------------------------------------------------------------------------------------- |
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334 | % Time Domain |
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335 | % ------------------------------------------------------------------------------------------------------------- |
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336 | \subsection{Time domain} |
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337 | \label{STP_time} |
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338 | %--------------------------------------------namrun------------------------------------------- |
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339 | \fortranfile{namelists/namdom} |
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340 | %-------------------------------------------------------------------------------------------------------------- |
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341 | |
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342 | Options are defined through the \ngn{namdom} namelist variables. |
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343 | \colorbox{yellow}{add here a few word on nit000 and nitend} |
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344 | |
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345 | \colorbox{yellow}{Write documentation on the calendar and the key variable adatrj} |
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346 | |
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347 | add a description of daymod, and the model calandar (leap-year and co) |
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348 | |
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349 | } %% end add |
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350 | |
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351 | |
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352 | |
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353 | %% |
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354 | \gmcomment{ % add implicit in vvl case and Crant-Nicholson scheme |
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355 | |
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356 | Implicit time stepping in case of variable volume thickness. |
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357 | |
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358 | Tracer case (NB for momentum in vector invariant form take care!) |
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359 | |
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360 | \begin{flalign*} |
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361 | &\frac{\left( e_{3t}\,T \right)_k^{t+1}-\left( e_{3t}\,T \right)_k^{t-1}}{2\rdt} |
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362 | \equiv \text{RHS}+ \delta _k \left[ {\frac{A_w^{vt} }{e_{3w}^{t+1} }\delta _{k+1/2} \left[ {T^{t+1}} \right]} |
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363 | \right] \\ |
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364 | &\left( e_{3t}\,T \right)_k^{t+1}-\left( e_{3t}\,T \right)_k^{t-1} |
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365 | \equiv {2\rdt} \ \text{RHS}+ {2\rdt} \ \delta _k \left[ {\frac{A_w^{vt} }{e_{3w}^{t+1} }\delta _{k+1/2} \left[ {T^{t+1}} \right]} |
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366 | \right] \\ |
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367 | &\left( e_{3t}\,T \right)_k^{t+1}-\left( e_{3t}\,T \right)_k^{t-1} |
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368 | \equiv 2\rdt \ \text{RHS} |
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369 | + 2\rdt \ \left\{ \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k+1/2} [ T_{k+1}^{t+1} - T_k ^{t+1} ] |
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370 | - \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k-1/2} [ T_k ^{t+1} - T_{k-1}^{t+1} ] \right\} \\ |
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371 | &\\ |
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372 | &\left( e_{3t}\,T \right)_k^{t+1} |
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373 | - {2\rdt} \ \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k+1/2} T_{k+1}^{t+1} |
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374 | + {2\rdt} \ \left\{ \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k+1/2} |
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375 | + \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k-1/2} \right\} T_{k }^{t+1} |
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376 | - {2\rdt} \ \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k-1/2} T_{k-1}^{t+1} \\ |
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377 | &\equiv \left( e_{3t}\,T \right)_k^{t-1} + {2\rdt} \ \text{RHS} \\ |
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378 | % |
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379 | \end{flalign*} |
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380 | |
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381 | \begin{flalign*} |
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382 | \allowdisplaybreaks |
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383 | \intertext{ Tracer case } |
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384 | % |
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385 | & \qquad \qquad \quad - {2\rdt} \ \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k+1/2} |
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386 | \qquad \qquad \qquad \qquad T_{k+1}^{t+1} \\ |
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387 | &+ {2\rdt} \ \biggl\{ (e_{3t})_{k }^{t+1} \bigg. + \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k+1/2} |
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388 | + \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k-1/2} \bigg. \biggr\} \ \ \ T_{k }^{t+1} &&\\ |
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389 | & \qquad \qquad \qquad \qquad \qquad \quad \ \ - {2\rdt} \ \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k-1/2} \quad \ \ T_{k-1}^{t+1} |
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390 | \ \equiv \ \left( e_{3t}\,T \right)_k^{t-1} + {2\rdt} \ \text{RHS} \\ |
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391 | % |
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392 | \end{flalign*} |
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393 | \begin{flalign*} |
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394 | \allowdisplaybreaks |
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395 | \intertext{ Tracer content case } |
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396 | % |
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397 | & - {2\rdt} \ & \frac{(A_w^{vt})_{k+1/2}} {(e_{3w})_{k+1/2}^{t+1}\;(e_{3t})_{k+1}^{t+1}} && \ \left( e_{3t}\,T \right)_{k+1}^{t+1} &\\ |
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398 | & + {2\rdt} \ \left[ 1 \right.+ & \frac{(A_w^{vt})_{k+1/2}} {(e_{3w})_{k+1/2}^{t+1}\;(e_{3t})_k^{t+1}} |
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399 | + & \frac{(A_w^{vt})_{k -1/2}} {(e_{3w})_{k -1/2}^{t+1}\;(e_{3t})_k^{t+1}} \left. \right] & \left( e_{3t}\,T \right)_{k }^{t+1} &\\ |
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400 | & - {2\rdt} \ & \frac{(A_w^{vt})_{k -1/2}} {(e_{3w})_{k -1/2}^{t+1}\;(e_{3t})_{k-1}^{t+1}} &\ \left( e_{3t}\,T \right)_{k-1}^{t+1} |
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401 | \equiv \left( e_{3t}\,T \right)_k^{t-1} + {2\rdt} \ \text{RHS} & |
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402 | \end{flalign*} |
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403 | |
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404 | %% |
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405 | } |
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406 | %% |
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407 | \end{document} |
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