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} |
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9 | \label{chap:TD} |
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10 | \chaptertoc |
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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 and 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:MB}, 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:TD_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:TD} |
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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 stepping depending on the physics with which it is associated. |
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43 | |
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44 | The choice of the time stepping 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:TD}) prior to time-stepping the equation. |
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55 | The time stepping itself is performed once at each time step where implicit vertical diffusion is computed, \ie\ in the \mdl{trazdf} and \mdl{dynzdf} modules. |
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56 | |
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57 | % ------------------------------------------------------------------------------------------------------------- |
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58 | % Non-Diffusive Part---Leapfrog Scheme |
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59 | % ------------------------------------------------------------------------------------------------------------- |
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60 | \section{Non-diffusive part --- Leapfrog scheme} |
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61 | \label{sec:TD_leap_frog} |
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62 | |
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63 | The time stepping used for processes other than diffusion is the well-known leapfrog scheme |
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64 | \citep{mesinger.arakawa_bk76}. |
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65 | This scheme is widely used for advection processes in low-viscosity fluids. |
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66 | It is a time centred scheme, \ie\ the RHS in \autoref{eq:TD} is evaluated at time step $t$, the now time step. |
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67 | It may be used for momentum and tracer advection, pressure gradient, and Coriolis terms, |
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68 | but not for diffusion terms. |
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69 | It is an efficient method that achieves second-order accuracy with |
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70 | just one right hand side evaluation per time step. |
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71 | Moreover, it does not artificially damp linear oscillatory motion nor does it produce instability by |
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72 | amplifying the oscillations. |
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73 | These advantages are somewhat diminished by the large phase-speed error of the leapfrog scheme, |
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74 | and the unsuitability of leapfrog differencing for the representation of diffusion and Rayleigh damping processes. |
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75 | However, the scheme allows the coexistence of a numerical and a physical mode due to |
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76 | its leading third order dispersive error. |
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77 | In other words a divergence of odd and even time steps may occur. |
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78 | To prevent it, the leapfrog scheme is often used in association with a Robert-Asselin time filter |
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79 | (hereafter the LF-RA scheme). |
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80 | This filter, first designed by \citet{robert_JMSJ66} and more comprehensively studied by \citet{asselin_MWR72}, |
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81 | is a kind of laplacian diffusion in time that mixes odd and even time steps: |
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82 | \begin{equation} |
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83 | \label{eq:TD_asselin} |
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84 | x_F^t = x^t + \gamma \, \lt[ x_F^{t - \rdt} - 2 x^t + x^{t + \rdt} \rt] |
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85 | \end{equation} |
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86 | where the subscript $F$ denotes filtered values and $\gamma$ is the Asselin coefficient. |
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87 | $\gamma$ is initialized as \np{rn\_atfp} (namelist parameter). |
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88 | Its default value is \np{rn\_atfp}\forcode{ = 10.e-3} (see \autoref{sec:TD_mLF}), |
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89 | causing only a weak dissipation of high frequency motions (\citep{farge-coulombier_phd87}). |
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90 | The addition of a time filter degrades the accuracy of the calculation from second to first order. |
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91 | However, the second order truncation error is proportional to $\gamma$, which is small compared to 1. |
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92 | Therefore, the LF-RA is a quasi second order accurate scheme. |
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93 | The LF-RA scheme is preferred to other time differencing schemes such as predictor corrector or trapezoidal schemes, |
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94 | because the user has an explicit and simple control of the magnitude of the time diffusion of the scheme. |
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95 | When used with the 2nd order space centred discretisation of the advection terms in |
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96 | the momentum and tracer equations, LF-RA avoids implicit numerical diffusion: |
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97 | diffusion is set explicitly by the user through the Robert-Asselin |
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98 | filter parameter and the viscosity and diffusion coefficients. |
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99 | |
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100 | % ------------------------------------------------------------------------------------------------------------- |
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101 | % Diffusive Part---Forward or Backward Scheme |
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102 | % ------------------------------------------------------------------------------------------------------------- |
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103 | \section{Diffusive part --- Forward or backward scheme} |
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104 | \label{sec:TD_forward_imp} |
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105 | |
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106 | The leapfrog differencing scheme is unsuitable for the representation of diffusion and damping processes. |
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107 | For a tendency $D_x$, representing a diffusion term or a restoring term to a tracer climatology |
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108 | (when present, see \autoref{sec:TRA_dmp}), a forward time differencing scheme is used : |
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109 | \[ |
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110 | %\label{eq:TD_euler} |
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111 | x^{t + \rdt} = x^{t - \rdt} + 2 \, \rdt \ D_x^{t - \rdt} |
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112 | \] |
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113 | |
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114 | This is diffusive in time and conditionally stable. |
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115 | The conditions for stability of second and fourth order horizontal diffusion schemes are \citep{griffies_bk04}: |
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116 | \begin{equation} |
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117 | \label{eq:TD_euler_stability} |
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118 | A^h < |
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119 | \begin{cases} |
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120 | \frac{e^2}{ 8 \, \rdt} & \text{laplacian diffusion} \\ |
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121 | \frac{e^4}{64 \, \rdt} & \text{bilaplacian diffusion} |
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122 | \end{cases} |
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123 | \end{equation} |
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124 | where $e$ is the smallest grid size in the two horizontal directions and $A^h$ is the mixing coefficient. |
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125 | The linear constraint \autoref{eq:TD_euler_stability} is a necessary condition, but not sufficient. |
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126 | If it is not satisfied, even mildly, then the model soon becomes wildly unstable. |
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127 | The instability can be removed by either reducing the length of the time steps or reducing the mixing coefficient. |
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128 | |
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129 | For the vertical diffusion terms, a forward time differencing scheme can be used, |
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130 | but usually the numerical stability condition imposes a strong constraint on the time step. To overcome the stability constraint, a |
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131 | backward (or implicit) time differencing scheme is used. This scheme is unconditionally stable but diffusive and can be written as follows: |
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132 | \begin{equation} |
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133 | \label{eq:TD_imp} |
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134 | x^{t + \rdt} = x^{t - \rdt} + 2 \, \rdt \ \text{RHS}_x^{t + \rdt} |
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135 | \end{equation} |
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136 | |
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137 | %%gm |
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138 | %%gm UPDATE the next paragraphs with time varying thickness ... |
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139 | %%gm |
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140 | |
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141 | This scheme is rather time consuming since it requires a matrix inversion. For example, the finite difference approximation of the temperature equation is: |
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142 | \[ |
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143 | % \label{eq:TD_imp_zdf} |
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144 | \frac{T(k)^{t + 1} - T(k)^{t - 1}}{2 \; \rdt} |
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145 | \equiv |
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146 | \text{RHS} + \frac{1}{e_{3t}} \delta_k \lt[ \frac{A_w^{vT}}{e_{3w} } \delta_{k + 1/2} \lt[ T^{t + 1} \rt] \rt] |
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147 | \] |
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148 | where RHS is the right hand side of the equation except for the vertical diffusion term. |
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149 | We rewrite \autoref{eq:TD_imp} as: |
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150 | \begin{equation} |
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151 | \label{eq:TD_imp_mat} |
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152 | -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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153 | \end{equation} |
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154 | where |
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155 | \begin{align*} |
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156 | c(k) &= A_w^{vT} (k) \, / \, e_{3w} (k) \\ |
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157 | d(k) &= e_{3t} (k) \, / \, (2 \rdt) + c_k + c_{k + 1} \\ |
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158 | b(k) &= e_{3t} (k) \; \lt( T^{t - 1}(k) \, / \, (2 \rdt) + \text{RHS} \rt) |
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159 | \end{align*} |
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160 | |
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161 | \autoref{eq:TD_imp_mat} is a linear system of equations with an associated matrix which is tridiagonal. |
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162 | Moreover, |
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163 | $c(k)$ and $d(k)$ are positive and the diagonal term is greater than the sum of the two extra-diagonal terms, |
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164 | therefore a special adaptation of the Gauss elimination procedure is used to find the solution |
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165 | (see for example \citet{richtmyer.morton_bk67}). |
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166 | |
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167 | % ------------------------------------------------------------------------------------------------------------- |
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168 | % Surface Pressure gradient |
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169 | % ------------------------------------------------------------------------------------------------------------- |
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170 | \section{Surface pressure gradient} |
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171 | \label{sec:TD_spg_ts} |
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172 | |
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173 | The leapfrog environment supports a centred in time computation of the surface pressure, \ie\ evaluated |
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174 | at \textit{now} time step. This refers to as the explicit free surface case in the code (\np{ln\_dynspg\_exp}\forcode{=.true.}). |
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175 | This choice however imposes a strong constraint on the time step which should be small enough to resolve the propagation |
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176 | of external gravity waves. As a matter of fact, one rather use in a realistic setup, a split-explicit free surface |
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177 | (\np{ln\_dynspg\_ts}\forcode{=.true.}) in which barotropic and baroclinic dynamical equations are solved separately with ad-hoc |
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178 | time steps. The use of the time-splitting (in combination with non-linear free surface) imposes some constraints on the design of |
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179 | the overall flowchart, in particular to ensure exact tracer conservation (see \autoref{fig:TD_TimeStep_flowchart}). |
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180 | |
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181 | Compared to the former use of the filtered free surface in \NEMO\ v3.6 (\citet{roullet.madec_JGR00}), the use of a split-explicit free surface is advantageous |
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182 | on massively parallel computers. Indeed, no global computations are anymore required by the elliptic solver which saves a substantial amount of communication |
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183 | time. Fast barotropic motions (such as tides) are also simulated with a better accuracy. |
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184 | |
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185 | %\gmcomment{ |
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186 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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187 | \begin{figure}[!t] |
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188 | \centering |
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189 | \includegraphics[width=0.66\textwidth]{Fig_TimeStepping_flowchart_v4} |
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190 | \caption[Leapfrog time stepping sequence with split-explicit free surface]{ |
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191 | Sketch of the leapfrog time stepping sequence in \NEMO\ with split-explicit free surface. |
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192 | The latter combined with non-linear free surface requires the dynamical tendency being |
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193 | updated prior tracers tendency to ensure conservation. |
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194 | Note the use of time integrated fluxes issued from the barotropic loop in |
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195 | subsequent calculations of tracer advection and in the continuity equation. |
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196 | Details about the time-splitting scheme can be found in \autoref{subsec:DYN_spg_ts}.} |
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197 | \label{fig:TD_TimeStep_flowchart} |
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198 | \end{figure} |
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199 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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200 | %} |
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201 | |
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202 | % ------------------------------------------------------------------------------------------------------------- |
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203 | % The Modified Leapfrog -- Asselin Filter scheme |
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204 | % ------------------------------------------------------------------------------------------------------------- |
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205 | \section{Modified Leapfrog -- Asselin filter scheme} |
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206 | \label{sec:TD_mLF} |
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207 | |
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208 | Significant changes have been introduced by \cite{leclair.madec_OM09} in the LF-RA scheme in order to |
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209 | ensure tracer conservation and to allow the use of a much smaller value of the Asselin filter parameter. |
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210 | The modifications affect both the forcing and filtering treatments in the LF-RA scheme. |
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211 | |
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212 | In a classical LF-RA environment, the forcing term is centred in time, |
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213 | \ie\ it is time-stepped over a $2 \rdt$ period: |
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214 | $x^t = x^t + 2 \rdt Q^t$ where $Q$ is the forcing applied to $x$, |
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215 | and the time filter is given by \autoref{eq:TD_asselin} so that $Q$ is redistributed over several time step. |
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216 | In the modified LF-RA environment, these two formulations have been replaced by: |
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217 | \begin{gather} |
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218 | \label{eq:TD_forcing} |
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219 | x^{t + \rdt} = x^{t - \rdt} + \rdt \lt( Q^{t - \rdt / 2} + Q^{t + \rdt / 2} \rt) \\ |
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220 | \label{eq:TD_RA} |
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221 | x_F^t = x^t + \gamma \, \lt( x_F^{t - \rdt} - 2 x^t + x^{t + \rdt} \rt) |
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222 | - \gamma \, \rdt \, \lt( Q^{t + \rdt / 2} - Q^{t - \rdt / 2} \rt) |
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223 | \end{gather} |
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224 | The change in the forcing formulation given by \autoref{eq:TD_forcing} (see \autoref{fig:TD_MLF_forcing}) |
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225 | has a significant effect: |
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226 | the forcing term no longer excites the divergence of odd and even time steps \citep{leclair.madec_OM09}. |
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227 | % forcing seen by the model.... |
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228 | This property improves the LF-RA scheme in two aspects. |
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229 | First, the LF-RA can now ensure the local and global conservation of tracers. |
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230 | Indeed, time filtering is no longer required on the forcing part. |
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231 | The influence of the Asselin filter on the forcing is explicitly removed by adding a new term in the filter |
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232 | (last term in \autoref{eq:TD_RA} compared to \autoref{eq:TD_asselin}). |
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233 | Since the filtering of the forcing was the source of non-conservation in the classical LF-RA scheme, |
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234 | the modified formulation becomes conservative \citep{leclair.madec_OM09}. |
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235 | Second, the LF-RA becomes a truly quasi -second order scheme. |
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236 | Indeed, \autoref{eq:TD_forcing} used in combination with a careful treatment of static instability |
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237 | (\autoref{subsec:ZDF_evd}) and of the TKE physics (\autoref{subsec:ZDF_tke_ene}) |
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238 | (the two other main sources of time step divergence), |
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239 | allows a reduction by two orders of magnitude of the Asselin filter parameter. |
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240 | |
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241 | Note that the forcing is now provided at the middle of a time step: |
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242 | $Q^{t + \rdt / 2}$ is the forcing applied over the $[t,t + \rdt]$ time interval. |
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243 | This and the change in the time filter, \autoref{eq:TD_RA}, |
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244 | allows for an exact evaluation of the contribution due to the forcing term between any two time steps, |
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245 | even if separated by only $\rdt$ since the time filter is no longer applied to the forcing term. |
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246 | |
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247 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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248 | \begin{figure}[!t] |
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249 | \centering |
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250 | \includegraphics[width=0.66\textwidth]{Fig_MLF_forcing} |
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251 | \caption[Forcing integration methods for modified leapfrog (top and bottom)]{ |
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252 | Illustration of forcing integration methods. |
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253 | (top) ''Traditional'' formulation: |
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254 | the forcing is defined at the same time as the variable to which it is applied |
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255 | (integer value of the time step index) and it is applied over a $2 \rdt$ period. |
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256 | (bottom) modified formulation: |
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257 | the forcing is defined in the middle of the time |
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258 | (integer and a half value of the time step index) and |
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259 | the mean of two successive forcing values ($n - 1 / 2$, $n + 1 / 2$) is applied over |
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260 | a $2 \rdt$ period.} |
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261 | \label{fig:TD_MLF_forcing} |
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262 | \end{figure} |
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263 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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264 | |
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265 | % ------------------------------------------------------------------------------------------------------------- |
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266 | % Start/Restart strategy |
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267 | % ------------------------------------------------------------------------------------------------------------- |
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268 | \section{Start/Restart strategy} |
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269 | \label{sec:TD_rst} |
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270 | |
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271 | %--------------------------------------------namrun------------------------------------------- |
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272 | \begin{listing} |
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273 | \nlst{namrun} |
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274 | \caption{\forcode{&namrun}} |
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275 | \label{lst:namrun} |
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276 | \end{listing} |
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277 | %-------------------------------------------------------------------------------------------------------------- |
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278 | |
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279 | The first time step of this three level scheme when starting from initial conditions is a forward step |
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280 | (Euler time integration): |
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281 | \[ |
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282 | % \label{eq:TD_DOM_euler} |
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283 | x^1 = x^0 + \rdt \ \text{RHS}^0 |
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284 | \] |
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285 | This is done simply by keeping the leapfrog environment (\ie\ the \autoref{eq:TD} three level time stepping) but |
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286 | setting all $x^0$ (\textit{before}) and $x^1$ (\textit{now}) fields equal at the first time step and |
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287 | using half the value of a leapfrog time step ($2 \rdt$). |
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288 | |
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289 | It is also possible to restart from a previous computation, by using a restart file. |
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290 | The restart strategy is designed to ensure perfect restartability of the code: |
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291 | the user should obtain the same results to machine precision either by |
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292 | running the model for $2N$ time steps in one go, |
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293 | or by performing two consecutive experiments of $N$ steps with a restart. |
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294 | This requires saving two time levels and many auxiliary data in the restart files in machine precision. |
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295 | |
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296 | Note that the time step $\rdt$, is also saved in the restart file. |
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297 | When restarting, if the time step has been changed, or one of the prognostic variables at \textit{before} time step |
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298 | is missing, an Euler time stepping scheme is imposed. A forward initial step can still be enforced by the user by setting |
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299 | the namelist variable \np{nn\_euler}\forcode{=0}. Other options to control the time integration of the model |
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300 | are defined through the \nam{run} namelist variables. |
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301 | %%% |
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302 | \gmcomment{ |
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303 | add here how to force the restart to contain only one time step for operational purposes |
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304 | |
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305 | add also the idea of writing several restart for seasonal forecast : how is it done ? |
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306 | |
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307 | verify that all namelist pararmeters are truly described |
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308 | |
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309 | a word on the check of restart ..... |
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310 | } |
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311 | %%% |
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312 | |
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313 | \gmcomment{ % add a subsection here |
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314 | |
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315 | %------------------------------------------------------------------------------------------------------------- |
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316 | % Time Domain |
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317 | % ------------------------------------------------------------------------------------------------------------- |
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318 | \subsection{Time domain} |
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319 | \label{subsec:TD_time} |
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320 | %--------------------------------------------namrun------------------------------------------- |
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321 | |
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322 | %-------------------------------------------------------------------------------------------------------------- |
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323 | |
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324 | Options are defined through the \nam{dom} namelist variables. |
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325 | \colorbox{yellow}{add here a few word on nit000 and nitend} |
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326 | |
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327 | \colorbox{yellow}{Write documentation on the calendar and the key variable adatrj} |
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328 | |
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329 | add a description of daymod, and the model calandar (leap-year and co) |
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330 | |
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331 | } %% end add |
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332 | |
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333 | |
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334 | |
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335 | %% |
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336 | \gmcomment{ % add implicit in vvl case and Crant-Nicholson scheme |
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337 | |
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338 | Implicit time stepping in case of variable volume thickness. |
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339 | |
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340 | Tracer case (NB for momentum in vector invariant form take care!) |
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341 | |
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342 | \begin{flalign*} |
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343 | &\frac{\lt( e_{3t}\,T \rt)_k^{t+1}-\lt( e_{3t}\,T \rt)_k^{t-1}}{2\rdt} |
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344 | \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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345 | \rt] \\ |
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346 | &\lt( e_{3t}\,T \rt)_k^{t+1}-\lt( e_{3t}\,T \rt)_k^{t-1} |
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347 | \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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348 | \rt] \\ |
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349 | &\lt( e_{3t}\,T \rt)_k^{t+1}-\lt( e_{3t}\,T \rt)_k^{t-1} |
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350 | \equiv 2\rdt \ \text{RHS} |
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351 | + 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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352 | - \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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353 | &\\ |
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354 | &\lt( e_{3t}\,T \rt)_k^{t+1} |
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355 | - {2\rdt} \ \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k + 1/2} T_{k +1}^{t+1} |
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356 | + {2\rdt} \ \lt\{ \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k + 1/2} |
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357 | + \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k - 1/2} \rt\} T_{k }^{t+1} |
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358 | - {2\rdt} \ \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k - 1/2} T_{k -1}^{t+1} \\ |
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359 | &\equiv \lt( e_{3t}\,T \rt)_k^{t-1} + {2\rdt} \ \text{RHS} \\ |
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360 | % |
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361 | \end{flalign*} |
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362 | \begin{flalign*} |
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363 | \allowdisplaybreaks |
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364 | \intertext{ Tracer case } |
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365 | % |
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366 | & \qquad \qquad \quad - {2\rdt} \ \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k + 1/2} |
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367 | \qquad \qquad \qquad \qquad T_{k +1}^{t+1} \\ |
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368 | &+ {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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369 | + \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k - 1/2} \bigg. \biggr\} \ \ \ T_{k }^{t+1} &&\\ |
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370 | & \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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371 | \ \equiv \ \lt( e_{3t}\,T \rt)_k^{t-1} + {2\rdt} \ \text{RHS} \\ |
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372 | % |
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373 | \end{flalign*} |
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374 | \begin{flalign*} |
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375 | \allowdisplaybreaks |
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376 | \intertext{ Tracer content case } |
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377 | % |
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378 | & - {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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379 | & + {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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380 | + & \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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381 | & - {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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382 | \equiv \lt( e_{3t}\,T \rt)_k^{t-1} + {2\rdt} \ \text{RHS} & |
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383 | \end{flalign*} |
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384 | |
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385 | %% |
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386 | } |
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387 | |
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388 | \biblio |
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389 | |
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390 | \pindex |
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391 | |
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392 | \end{document} |
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