[10414] | 1 | \documentclass[../main/NEMO_manual]{subfiles} |
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| 2 | |
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[6997] | 3 | \begin{document} |
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[2282] | 4 | |
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| 5 | % ================================================================ |
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[10414] | 6 | % Chapter 2 Time Domain (step.F90) |
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[2282] | 7 | % ================================================================ |
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| 8 | \chapter{Time Domain (STP) } |
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[9407] | 9 | \label{chap:STP} |
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[10414] | 10 | |
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[2282] | 11 | \minitoc |
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| 12 | |
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| 13 | % Missing things: |
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| 14 | % - daymod: definition of the time domain (nit000, nitend andd the calendar) |
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| 15 | |
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[10354] | 16 | \gmcomment{STEVEN :maybe a picture of the directory structure in the introduction which could be referred to here, |
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| 17 | would help ==> to be added} |
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[2282] | 18 | %%%% |
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| 19 | |
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| 20 | \newpage |
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| 21 | |
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[10354] | 22 | Having defined the continuous equations in \autoref{chap:PE}, we need now to choose a time discretization, |
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| 23 | a key feature of an ocean model as it exerts a strong influence on the structure of the computer code |
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| 24 | ($i.e.$ on its flowchart). |
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| 25 | In the present chapter, we provide a general description of the \NEMO time stepping strategy and |
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| 26 | the consequences for the order in which the equations are solved. |
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[2282] | 27 | |
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| 28 | % ================================================================ |
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| 29 | % Time Discretisation |
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| 30 | % ================================================================ |
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| 31 | \section{Time stepping environment} |
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[9407] | 32 | \label{sec:STP_environment} |
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[2282] | 33 | |
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[10354] | 34 | The time stepping used in \NEMO is a three level scheme that can be represented as follows: |
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[10414] | 35 | \begin{equation} |
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| 36 | \label{eq:STP} |
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| 37 | x^{t+\rdt} = x^{t-\rdt} + 2 \, \rdt \ \text{RHS}_x^{t-\rdt,\,t,\,t+\rdt} |
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[2282] | 38 | \end{equation} |
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[10354] | 39 | where $x$ stands for $u$, $v$, $T$ or $S$; |
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| 40 | RHS is the Right-Hand-Side of the corresponding time evolution equation; |
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| 41 | $\rdt$ is the time step; |
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| 42 | and the superscripts indicate the time at which a quantity is evaluated. |
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| 43 | 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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[2282] | 44 | |
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[10354] | 45 | The choice of the time step used for this evaluation is discussed below as well as |
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| 46 | the implications for starting or restarting a model simulation. |
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| 47 | Note that the time stepping calculation is generally performed in a single operation. |
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| 48 | With such a complex and nonlinear system of equations it would be dangerous to let a prognostic variable evolve in |
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| 49 | time for each term separately. |
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[2282] | 50 | |
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[10354] | 51 | The three level scheme requires three arrays for each prognostic variable. |
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| 52 | For each variable $x$ there is $x_b$ (before), $x_n$ (now) and $x_a$. |
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| 53 | The third array, although referred to as $x_a$ (after) in the code, |
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| 54 | is usually not the variable at the after time step; |
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| 55 | 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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| 56 | Generally, the time stepping is performed once at each time step in the \mdl{tranxt} and \mdl{dynnxt} modules, |
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| 57 | except when using implicit vertical diffusion or calculating sea surface height in which |
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| 58 | case time-splitting options are used. |
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[2282] | 59 | |
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| 60 | % ------------------------------------------------------------------------------------------------------------- |
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| 61 | % Non-Diffusive Part---Leapfrog Scheme |
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| 62 | % ------------------------------------------------------------------------------------------------------------- |
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[9393] | 63 | \section{Non-diffusive part --- Leapfrog scheme} |
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[9407] | 64 | \label{sec:STP_leap_frog} |
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[2282] | 65 | |
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[10354] | 66 | The time stepping used for processes other than diffusion is the well-known leapfrog scheme |
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| 67 | \citep{Mesinger_Arakawa_Bk76}. |
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| 68 | This scheme is widely used for advection processes in low-viscosity fluids. |
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| 69 | It is a time centred scheme, $i.e.$ the RHS in \autoref{eq:STP} is evaluated at time step $t$, the now time step. |
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| 70 | It may be used for momentum and tracer advection, pressure gradient, and Coriolis terms, |
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| 71 | but not for diffusion terms. |
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| 72 | It is an efficient method that achieves second-order accuracy with |
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| 73 | just one right hand side evaluation per time step. |
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| 74 | Moreover, it does not artificially damp linear oscillatory motion nor does it produce instability by |
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| 75 | amplifying the oscillations. |
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| 76 | These advantages are somewhat diminished by the large phase-speed error of the leapfrog scheme, |
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| 77 | and the unsuitability of leapfrog differencing for the representation of diffusion and Rayleigh damping processes. |
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| 78 | However, the scheme allows the coexistence of a numerical and a physical mode due to |
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| 79 | its leading third order dispersive error. |
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| 80 | In other words a divergence of odd and even time steps may occur. |
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| 81 | To prevent it, the leapfrog scheme is often used in association with a Robert-Asselin time filter |
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| 82 | (hereafter the LF-RA scheme). |
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| 83 | This filter, first designed by \citet{Robert_JMSJ66} and more comprehensively studied by \citet{Asselin_MWR72}, |
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| 84 | is a kind of laplacian diffusion in time that mixes odd and even time steps: |
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[10414] | 85 | \begin{equation} |
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| 86 | \label{eq:STP_asselin} |
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| 87 | x_F^t = x^t + \gamma \, \left[ x_F^{t-\rdt} - 2 x^t + x^{t+\rdt} \right] |
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[2282] | 88 | \end{equation} |
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[10354] | 89 | where the subscript $F$ denotes filtered values and $\gamma$ is the Asselin coefficient. |
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| 90 | $\gamma$ is initialized as \np{rn\_atfp} (namelist parameter). |
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| 91 | Its default value is \np{rn\_atfp}\forcode{ = 10.e-3} (see \autoref{sec:STP_mLF}), |
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| 92 | causing only a weak dissipation of high frequency motions (\citep{Farge1987}). |
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| 93 | The addition of a time filter degrades the accuracy of the calculation from second to first order. |
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| 94 | However, the second order truncation error is proportional to $\gamma$, which is small compared to 1. |
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| 95 | Therefore, the LF-RA is a quasi second order accurate scheme. |
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| 96 | The LF-RA scheme is preferred to other time differencing schemes such as predictor corrector or trapezoidal schemes, |
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| 97 | because the user has an explicit and simple control of the magnitude of the time diffusion of the scheme. |
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| 98 | When used with the 2nd order space centred discretisation of the advection terms in |
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| 99 | the momentum and tracer equations, LF-RA avoids implicit numerical diffusion: |
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| 100 | diffusion is set explicitly by the user through the Robert-Asselin |
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[2282] | 101 | filter parameter and the viscosity and diffusion coefficients. |
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| 102 | |
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| 103 | % ------------------------------------------------------------------------------------------------------------- |
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| 104 | % Diffusive Part---Forward or Backward Scheme |
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| 105 | % ------------------------------------------------------------------------------------------------------------- |
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[9393] | 106 | \section{Diffusive part --- Forward or backward scheme} |
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[9407] | 107 | \label{sec:STP_forward_imp} |
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[2282] | 108 | |
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[10354] | 109 | The leapfrog differencing scheme is unsuitable for the representation of diffusion and damping processes. |
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| 110 | For a tendancy $D_x$, representing a diffusion term or a restoring term to a tracer climatology |
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| 111 | (when present, see \autoref{sec:TRA_dmp}), a forward time differencing scheme is used : |
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[10414] | 112 | \[ |
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| 113 | % \label{eq:STP_euler} |
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[2282] | 114 | x^{t+\rdt} = x^{t-\rdt} + 2 \, \rdt \ {D_x}^{t-\rdt} |
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[10414] | 115 | \] |
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[2282] | 116 | |
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[10354] | 117 | This is diffusive in time and conditionally stable. |
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| 118 | The conditions for stability of second and fourth order horizontal diffusion schemes are \citep{Griffies_Bk04}: |
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[10414] | 119 | \begin{equation} |
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| 120 | \label{eq:STP_euler_stability} |
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| 121 | A^h < \left\{ |
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| 122 | \begin{aligned} |
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| 123 | &\frac{e^2}{ 8 \; \rdt } &&\quad \text{laplacian diffusion} \\ |
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| 124 | &\frac{e^4}{64 \; \rdt } &&\quad \text{bilaplacian diffusion} |
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| 125 | \end{aligned} |
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| 126 | \right. |
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[2282] | 127 | \end{equation} |
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[10354] | 128 | where $e$ is the smallest grid size in the two horizontal directions and $A^h$ is the mixing coefficient. |
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| 129 | The linear constraint \autoref{eq:STP_euler_stability} is a necessary condition, but not sufficient. |
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| 130 | If it is not satisfied, even mildly, then the model soon becomes wildly unstable. |
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| 131 | The instability can be removed by either reducing the length of the time steps or reducing the mixing coefficient. |
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[2282] | 132 | |
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[10354] | 133 | For the vertical diffusion terms, a forward time differencing scheme can be used, |
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| 134 | but usually the numerical stability condition imposes a strong constraint on the time step. |
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| 135 | Two solutions are available in \NEMO to overcome the stability constraint: |
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| 136 | $(a)$ a forward time differencing scheme using a time splitting technique (\np{ln\_zdfexp}\forcode{ = .true.}) or |
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| 137 | $(b)$ a backward (or implicit) time differencing scheme (\np{ln\_zdfexp}\forcode{ = .false.}). |
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| 138 | In $(a)$, the master time step $\Delta $t is cut into $N$ fractional time steps so that |
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| 139 | the stability criterion is reduced by a factor of $N$. |
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| 140 | The computation is performed as follows: |
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[10414] | 141 | \[ |
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| 142 | % \label{eq:STP_ts} |
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| 143 | \begin{split} |
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| 144 | & x_\ast ^{t-\rdt} = x^{t-\rdt} \\ |
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| 145 | & x_\ast ^{t-\rdt+L\frac{2\rdt}{N}}=x_\ast ^{t-\rdt+\left( {L-1} |
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| 146 | \right)\frac{2\rdt}{N}}+\frac{2\rdt}{N}\;\text{DF}^{t-\rdt+\left( {L-1} \right)\frac{2\rdt}{N}} |
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| 147 | \quad \text{for $L=1$ to $N$} \\ |
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| 148 | & x^{t+\rdt} = x_\ast^{t+\rdt} |
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| 149 | \end{split} |
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| 150 | \] |
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[10354] | 151 | with DF a vertical diffusion term. |
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| 152 | The number of fractional time steps, $N$, is given by setting \np{nn\_zdfexp}, (namelist parameter). |
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| 153 | The scheme $(b)$ is unconditionally stable but diffusive. It can be written as follows: |
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[10414] | 154 | \begin{equation} |
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| 155 | \label{eq:STP_imp} |
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| 156 | x^{t+\rdt} = x^{t-\rdt} + 2 \, \rdt \ \text{RHS}_x^{t+\rdt} |
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[2282] | 157 | \end{equation} |
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| 158 | |
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[6140] | 159 | %%gm |
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| 160 | %%gm UPDATE the next paragraphs with time varying thickness ... |
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| 161 | %%gm |
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| 162 | |
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[10354] | 163 | This scheme is rather time consuming since it requires a matrix inversion, |
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| 164 | 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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| 165 | For example, the finite difference approximation of the temperature equation is: |
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[10414] | 166 | \[ |
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| 167 | % \label{eq:STP_imp_zdf} |
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| 168 | \frac{T(k)^{t+1}-T(k)^{t-1}}{2\;\rdt}\equiv \text{RHS}+\frac{1}{e_{3t} }\delta |
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| 169 | _k \left[ {\frac{A_w^{vT} }{e_{3w} }\delta_{k+1/2} \left[ {T^{t+1}} \right]} |
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| 170 | \right] |
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| 171 | \] |
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[10354] | 172 | where RHS is the right hand side of the equation except for the vertical diffusion term. |
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[9407] | 173 | We rewrite \autoref{eq:STP_imp} as: |
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[10414] | 174 | \begin{equation} |
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| 175 | \label{eq:STP_imp_mat} |
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| 176 | -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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[2282] | 177 | \end{equation} |
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| 178 | where |
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[10414] | 179 | \begin{align*} |
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| 180 | c(k) &= A_w^{vT} (k) \, / \, e_{3w} (k) \\ |
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| 181 | d(k) &= e_{3t} (k) \, / \, (2\rdt) + c_k + c_{k+1} \\ |
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| 182 | b(k) &= e_{3t} (k) \; \left( T^{t-1}(k) \, / \, (2\rdt) + \text{RHS} \right) |
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[2282] | 183 | \end{align*} |
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| 184 | |
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[10354] | 185 | \autoref{eq:STP_imp_mat} is a linear system of equations with an associated matrix which is tridiagonal. |
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| 186 | Moreover, |
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| 187 | $c(k)$ and $d(k)$ are positive and the diagonal term is greater than the sum of the two extra-diagonal terms, |
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| 188 | therefore a special adaptation of the Gauss elimination procedure is used to find the solution |
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[2282] | 189 | (see for example \citet{Richtmyer1967}). |
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| 190 | |
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| 191 | |
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| 192 | |
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| 193 | % ------------------------------------------------------------------------------------------------------------- |
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[6140] | 194 | % Surface Pressure gradient |
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[2282] | 195 | % ------------------------------------------------------------------------------------------------------------- |
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[9393] | 196 | \section{Surface pressure gradient} |
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[9407] | 197 | \label{sec:STP_spg_ts} |
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[2282] | 198 | |
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[6140] | 199 | ===>>>> TO BE written.... :-) |
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| 200 | |
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[2282] | 201 | %\gmcomment{ |
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| 202 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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[10354] | 203 | \begin{figure}[!t] |
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| 204 | \begin{center} |
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| 205 | \includegraphics[width=0.7\textwidth]{Fig_TimeStepping_flowchart} |
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[10414] | 206 | \caption{ |
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| 207 | \protect\label{fig:TimeStep_flowchart} |
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[10354] | 208 | Sketch of the leapfrog time stepping sequence in \NEMO from \citet{Leclair_Madec_OM09}. |
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| 209 | The use of a semi-implicit computation of the hydrostatic pressure gradient requires the tracer equation to |
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| 210 | be stepped forward prior to the momentum equation. |
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| 211 | The need for knowledge of the vertical scale factor (here denoted as $h$) requires the sea surface height and |
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| 212 | the continuity equation to be stepped forward prior to the computation of the tracer equation. |
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| 213 | Note that the method for the evaluation of the surface pressure gradient $\nabla p_s$ is not presented here |
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| 214 | (see \autoref{sec:DYN_spg}). |
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| 215 | } |
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[10414] | 216 | \end{center} |
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| 217 | \end{figure} |
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[2282] | 218 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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| 219 | %} |
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| 220 | |
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| 221 | % ------------------------------------------------------------------------------------------------------------- |
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| 222 | % The Modified Leapfrog -- Asselin Filter scheme |
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| 223 | % ------------------------------------------------------------------------------------------------------------- |
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[9393] | 224 | \section{Modified Leapfrog -- Asselin filter scheme} |
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[9407] | 225 | \label{sec:STP_mLF} |
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[2282] | 226 | |
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[10354] | 227 | Significant changes have been introduced by \cite{Leclair_Madec_OM09} in the LF-RA scheme in order to ensure tracer conservation and to allow the use of a much smaller value of the Asselin filter parameter. |
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| 228 | The modifications affect both the forcing and filtering treatments in the LF-RA scheme. |
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[2282] | 229 | |
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[10354] | 230 | In a classical LF-RA environment, the forcing term is centred in time, |
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| 231 | $i.e.$ it is time-stepped over a $2\rdt$ period: |
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| 232 | $x^t = x^t + 2\rdt Q^t $ where $Q$ is the forcing applied to $x$, |
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| 233 | and the time filter is given by \autoref{eq:STP_asselin} so that $Q$ is redistributed over several time step. |
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[2282] | 234 | In the modified LF-RA environment, these two formulations have been replaced by: |
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[10414] | 235 | \begin{align} |
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| 236 | 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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| 237 | % |
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| 238 | x_F^t &= x^t + \gamma \, \left[ x_F^{t-\rdt} - 2 x^t + x^{t+\rdt} \right] |
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[9407] | 239 | - \gamma\,\rdt \, \left[ Q^{t+\rdt/2} - Q^{t-\rdt/2} \right] \label{eq:STP_RA} |
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[2282] | 240 | \end{align} |
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[10354] | 241 | The change in the forcing formulation given by \autoref{eq:STP_forcing} (see \autoref{fig:MLF_forcing}) |
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| 242 | has a significant effect: |
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| 243 | the forcing term no longer excites the divergence of odd and even time steps \citep{Leclair_Madec_OM09}. |
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[2282] | 244 | % forcing seen by the model.... |
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[10354] | 245 | This property improves the LF-RA scheme in two respects. |
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[2282] | 246 | First, the LF-RA can now ensure the local and global conservation of tracers. |
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[10354] | 247 | Indeed, time filtering is no longer required on the forcing part. |
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| 248 | The influence of the Asselin filter on the forcing is be removed by adding a new term in the filter |
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| 249 | (last term in \autoref{eq:STP_RA} compared to \autoref{eq:STP_asselin}). |
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| 250 | Since the filtering of the forcing was the source of non-conservation in the classical LF-RA scheme, |
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| 251 | the modified formulation becomes conservative \citep{Leclair_Madec_OM09}. |
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| 252 | Second, the LF-RA becomes a truly quasi-second order scheme. |
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| 253 | Indeed, \autoref{eq:STP_forcing} used in combination with a careful treatment of static instability |
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| 254 | (\autoref{subsec:ZDF_evd}) and of the TKE physics (\autoref{subsec:ZDF_tke_ene}), |
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| 255 | the two other main sources of time step divergence, |
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| 256 | allows a reduction by two orders of magnitude of the Asselin filter parameter. |
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[2282] | 257 | |
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[10354] | 258 | Note that the forcing is now provided at the middle of a time step: |
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| 259 | $Q^{t+\rdt/2}$ is the forcing applied over the $[t,t+\rdt]$ time interval. |
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| 260 | This and the change in the time filter, \autoref{eq:STP_RA}, |
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| 261 | allows an exact evaluation of the contribution due to the forcing term between any two time steps, |
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| 262 | even if separated by only $\rdt$ since the time filter is no longer applied to the forcing term. |
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[2282] | 263 | |
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| 264 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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[10354] | 265 | \begin{figure}[!t] |
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| 266 | \begin{center} |
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| 267 | \includegraphics[width=0.90\textwidth]{Fig_MLF_forcing} |
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[10414] | 268 | \caption{ |
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| 269 | \protect\label{fig:MLF_forcing} |
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[10354] | 270 | Illustration of forcing integration methods. |
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| 271 | (top) ''Traditional'' formulation: |
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| 272 | the forcing is defined at the same time as the variable to which it is applied |
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| 273 | (integer value of the time step index) and it is applied over a $2\rdt$ period. |
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| 274 | (bottom) modified formulation: |
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| 275 | 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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| 276 | the mean of two successive forcing values ($n-1/2$, $n+1/2$) is applied over a $2\rdt$ period. |
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| 277 | } |
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| 278 | \end{center} |
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| 279 | \end{figure} |
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[2282] | 280 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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| 281 | |
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| 282 | % ------------------------------------------------------------------------------------------------------------- |
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| 283 | % Start/Restart strategy |
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| 284 | % ------------------------------------------------------------------------------------------------------------- |
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| 285 | \section{Start/Restart strategy} |
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[9407] | 286 | \label{sec:STP_rst} |
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[9376] | 287 | |
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[2282] | 288 | %--------------------------------------------namrun------------------------------------------- |
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[10146] | 289 | \nlst{namrun} |
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[2282] | 290 | %-------------------------------------------------------------------------------------------------------------- |
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| 291 | |
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[10354] | 292 | The first time step of this three level scheme when starting from initial conditions is a forward step |
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| 293 | (Euler time integration): |
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[10414] | 294 | \[ |
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| 295 | % \label{eq:DOM_euler} |
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| 296 | x^1 = x^0 + \rdt \ \text{RHS}^0 |
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| 297 | \] |
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[10354] | 298 | This is done simply by keeping the leapfrog environment ($i.e.$ the \autoref{eq:STP} three level time stepping) but |
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| 299 | setting all $x^0$ (\textit{before}) and $x^{1}$ (\textit{now}) fields equal at the first time step and |
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| 300 | using half the value of $\rdt$. |
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[2282] | 301 | |
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[10354] | 302 | It is also possible to restart from a previous computation, by using a restart file. |
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| 303 | The restart strategy is designed to ensure perfect restartability of the code: |
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| 304 | the user should obtain the same results to machine precision either by |
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| 305 | running the model for $2N$ time steps in one go, |
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| 306 | or by performing two consecutive experiments of $N$ steps with a restart. |
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| 307 | This requires saving two time levels and many auxiliary data in the restart files in machine precision. |
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[2282] | 308 | |
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[10354] | 309 | Note that when a semi-implicit scheme is used to evaluate the hydrostatic pressure gradient |
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| 310 | (see \autoref{subsec:DYN_hpg_imp}), an extra three-dimensional field has to |
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| 311 | be added to the restart file to ensure an exact restartability. |
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[10414] | 312 | This is done optionally via the \np{nn\_dynhpg\_rst} namelist parameter, |
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[10354] | 313 | so that the size of the restart file can be reduced when restartability is not a key issue |
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| 314 | (operational oceanography or in ensemble simulations for seasonal forecasting). |
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[2282] | 315 | |
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[10354] | 316 | Note the size of the time step used, $\rdt$, is also saved in the restart file. |
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| 317 | When restarting, if the the time step has been changed, a restart using an Euler time stepping scheme is imposed. |
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[10414] | 318 | Options are defined through the \ngn{namrun} namelist variables. |
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[2282] | 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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[9407] | 337 | \label{subsec:STP_time} |
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[2282] | 338 | %--------------------------------------------namrun------------------------------------------- |
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[10146] | 339 | |
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| 340 | \nlst{namdom} |
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[2282] | 341 | %-------------------------------------------------------------------------------------------------------------- |
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| 342 | |
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[4147] | 343 | Options are defined through the \ngn{namdom} namelist variables. |
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[2282] | 344 | \colorbox{yellow}{add here a few word on nit000 and nitend} |
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| 345 | |
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| 346 | \colorbox{yellow}{Write documentation on the calendar and the key variable adatrj} |
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| 347 | |
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| 348 | add a description of daymod, and the model calandar (leap-year and co) |
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| 349 | |
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| 350 | } %% end add |
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| 351 | |
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| 352 | |
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| 353 | |
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| 354 | %% |
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| 355 | \gmcomment{ % add implicit in vvl case and Crant-Nicholson scheme |
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| 356 | |
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| 357 | Implicit time stepping in case of variable volume thickness. |
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| 358 | |
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| 359 | Tracer case (NB for momentum in vector invariant form take care!) |
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| 360 | |
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| 361 | \begin{flalign*} |
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[10414] | 362 | &\frac{\left( e_{3t}\,T \right)_k^{t+1}-\left( e_{3t}\,T \right)_k^{t-1}}{2\rdt} |
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| 363 | \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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| 364 | \right] \\ |
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| 365 | &\left( e_{3t}\,T \right)_k^{t+1}-\left( e_{3t}\,T \right)_k^{t-1} |
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| 366 | \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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| 367 | \right] \\ |
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| 368 | &\left( e_{3t}\,T \right)_k^{t+1}-\left( e_{3t}\,T \right)_k^{t-1} |
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| 369 | \equiv 2\rdt \ \text{RHS} |
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| 370 | + 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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| 371 | - \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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| 372 | &\\ |
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| 373 | &\left( e_{3t}\,T \right)_k^{t+1} |
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| 374 | - {2\rdt} \ \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k+1/2} T_{k+1}^{t+1} |
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| 375 | + {2\rdt} \ \left\{ \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k+1/2} |
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| 376 | + \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k-1/2} \right\} T_{k }^{t+1} |
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| 377 | - {2\rdt} \ \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k-1/2} T_{k-1}^{t+1} \\ |
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| 378 | &\equiv \left( e_{3t}\,T \right)_k^{t-1} + {2\rdt} \ \text{RHS} \\ |
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| 379 | % |
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[2282] | 380 | \end{flalign*} |
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| 381 | |
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| 382 | \begin{flalign*} |
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[10414] | 383 | \allowdisplaybreaks |
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| 384 | \intertext{ Tracer case } |
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| 385 | % |
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| 386 | & \qquad \qquad \quad - {2\rdt} \ \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k+1/2} |
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| 387 | \qquad \qquad \qquad \qquad T_{k+1}^{t+1} \\ |
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| 388 | &+ {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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| 389 | + \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k-1/2} \bigg. \biggr\} \ \ \ T_{k }^{t+1} &&\\ |
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| 390 | & \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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| 391 | \ \equiv \ \left( e_{3t}\,T \right)_k^{t-1} + {2\rdt} \ \text{RHS} \\ |
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| 392 | % |
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[2282] | 393 | \end{flalign*} |
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| 394 | \begin{flalign*} |
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[10414] | 395 | \allowdisplaybreaks |
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| 396 | \intertext{ Tracer content case } |
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| 397 | % |
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| 398 | & - {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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| 399 | & + {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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| 400 | + & \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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| 401 | & - {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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| 402 | \equiv \left( e_{3t}\,T \right)_k^{t-1} + {2\rdt} \ \text{RHS} & |
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[2282] | 403 | \end{flalign*} |
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| 404 | |
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| 405 | %% |
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| 406 | } |
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| 407 | %% |
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[10414] | 408 | \biblio |
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| 409 | |
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[6997] | 410 | \end{document} |
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