[10414] | 1 | \documentclass[../main/NEMO_manual]{subfiles} |
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| 2 | |
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[6997] | 3 | \begin{document} |
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[2298] | 4 | % ================================================================ |
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| 5 | % Chapter Assimilation increments (ASM) |
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| 6 | % ================================================================ |
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[9393] | 7 | \chapter{Apply Assimilation Increments (ASM)} |
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[9407] | 8 | \label{chap:ASM} |
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[2298] | 9 | |
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[2474] | 10 | Authors: D. Lea, M. Martin, K. Mogensen, A. Weaver, ... % do we keep |
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[2298] | 11 | |
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| 12 | \minitoc |
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| 13 | |
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| 14 | \newpage |
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| 15 | |
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[10354] | 16 | The ASM code adds the functionality to apply increments to the model variables: temperature, salinity, |
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| 17 | sea surface height, velocity and sea ice concentration. |
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| 18 | These are read into the model from a NetCDF file which may be produced by separate data assimilation code. |
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| 19 | The code can also output model background fields which are used as an input to data assimilation code. |
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| 20 | This is all controlled by the namelist \textit{\ngn{nam\_asminc} }. |
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| 21 | There is a brief description of all the namelist options provided. |
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| 22 | To build the ASM code \key{asminc} must be set. |
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[2298] | 23 | |
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| 24 | %=============================================================== |
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| 25 | |
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[2349] | 26 | \section{Direct initialization} |
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[9407] | 27 | \label{sec:ASM_DI} |
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[2298] | 28 | |
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[10354] | 29 | Direct initialization (DI) refers to the instantaneous correction of the model background state using |
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| 30 | the analysis increment. |
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[9393] | 31 | DI is used when \np{ln\_asmdin} is set to true. |
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[2298] | 32 | |
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[9393] | 33 | \section{Incremental analysis updates} |
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[9407] | 34 | \label{sec:ASM_IAU} |
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[2298] | 35 | |
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| 36 | Rather than updating the model state directly with the analysis increment, |
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[10354] | 37 | it may be preferable to introduce the increment gradually into the ocean model in order to |
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| 38 | minimize spurious adjustment processes. |
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| 39 | This technique is referred to as Incremental Analysis Updates (IAU) \citep{Bloom_al_MWR96}. |
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[2298] | 40 | IAU is a common technique used with 3D assimilation methods such as 3D-Var or OI. |
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[9393] | 41 | IAU is used when \np{ln\_asmiau} is set to true. |
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[2298] | 42 | |
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[10354] | 43 | With IAU, the model state trajectory ${\bf x}$ in the assimilation window ($t_{0} \leq t_{i} \leq t_{N}$) |
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| 44 | is corrected by adding the analysis increments for temperature, salinity, horizontal velocity and SSH as |
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| 45 | additional tendency terms to the prognostic equations: |
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[10414] | 46 | \begin{align*} |
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| 47 | % \label{eq:wa_traj_iau} |
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| 48 | {\bf x}^{a}(t_{i}) = M(t_{i}, t_{0})[{\bf x}^{b}(t_{0})] \; + \; F_{i} \delta \tilde{\bf x}^{a} |
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| 49 | \end{align*} |
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[10354] | 50 | where $F_{i}$ is a weighting function for applying the increments $\delta\tilde{\bf x}^{a}$ defined such that |
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| 51 | $\sum_{i=1}^{N} F_{i}=1$. |
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| 52 | ${\bf x}^b$ denotes the model initial state and ${\bf x}^a$ is the model state after the increments are applied. |
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[2298] | 53 | To control the adjustment time of the model to the increment, |
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[10354] | 54 | the increment can be applied over an arbitrary sub-window, $t_{m} \leq t_{i} \leq t_{n}$, |
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| 55 | of the main assimilation window, where $t_{0} \leq t_{m} \leq t_{i}$ and $t_{i} \leq t_{n} \leq t_{N}$. |
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[2298] | 56 | Typically the increments are spread evenly over the full window. |
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| 57 | In addition, two different weighting functions have been implemented. |
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| 58 | The first function employs constant weights, |
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[10414] | 59 | \begin{align} |
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| 60 | \label{eq:F1_i} |
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| 61 | F^{(1)}_{i} |
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| 62 | =\left\{ |
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| 63 | \begin{array}{ll} |
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| 64 | 0 & {\rm if} \; \; \; t_{i} < t_{m} \\ |
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| 65 | 1/M & {\rm if} \; \; \; t_{m} < t_{i} \leq t_{n} \\ |
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| 66 | 0 & {\rm if} \; \; \; t_{i} > t_{n} |
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| 67 | \end{array} |
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| 68 | \right. |
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[10406] | 69 | \end{align} |
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[2298] | 70 | where $M = m-n$. |
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[10354] | 71 | The second function employs peaked hat-like weights in order to give maximum weight in the centre of the sub-window, |
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| 72 | with the weighting reduced linearly to a small value at the window end-points: |
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[10414] | 73 | \begin{align} |
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| 74 | \label{eq:F2_i} |
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| 75 | F^{(2)}_{i} |
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| 76 | =\left\{ |
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| 77 | \begin{array}{ll} |
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| 78 | 0 & {\rm if} \; \; \; t_{i} < t_{m} \\ |
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| 79 | \alpha \, i & {\rm if} \; \; \; t_{m} \leq t_{i} \leq t_{M/2} \\ |
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| 80 | \alpha \, (M - i +1) & {\rm if} \; \; \; t_{M/2} < t_{i} \leq t_{n} \\ |
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| 81 | 0 & {\rm if} \; \; \; t_{i} > t_{n} |
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| 82 | \end{array} |
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| 83 | \right. |
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[10406] | 84 | \end{align} |
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[2298] | 85 | where $\alpha^{-1} = \sum_{i=1}^{M/2} 2i$ and $M$ is assumed to be even. |
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[10354] | 86 | The weights described by \autoref{eq:F2_i} provide a smoother transition of the analysis trajectory from |
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| 87 | one assimilation cycle to the next than that described by \autoref{eq:F1_i}. |
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[2298] | 88 | |
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| 89 | %========================================================================== |
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[3294] | 90 | % Divergence damping description %%% |
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| 91 | \section{Divergence damping initialisation} |
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[9414] | 92 | \label{sec:ASM_div_dmp} |
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[2298] | 93 | |
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[10354] | 94 | The velocity increments may be initialized by the iterative application of a divergence damping operator. |
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| 95 | In iteration step $n$ new estimates of velocity increments $u^{n}_I$ and $v^{n}_I$ are updated by: |
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[10414] | 96 | \begin{equation} |
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| 97 | \label{eq:asm_dmp} |
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| 98 | \left\{ |
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| 99 | \begin{aligned} |
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| 100 | u^{n}_I = u^{n-1}_I + \frac{1}{e_{1u} } \delta_{i+1/2} \left( {A_D |
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| 101 | \;\chi^{n-1}_I } \right) \\ \\ |
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| 102 | v^{n}_I = v^{n-1}_I + \frac{1}{e_{2v} } \delta_{j+1/2} \left( {A_D |
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| 103 | \;\chi^{n-1}_I } \right) \\ |
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| 104 | \end{aligned} |
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| 105 | \right., |
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[3294] | 106 | \end{equation} |
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| 107 | where |
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[10414] | 108 | \[ |
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| 109 | % \label{eq:asm_div} |
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| 110 | \chi^{n-1}_I = \frac{1}{e_{1t}\,e_{2t}\,e_{3t} } |
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| 111 | \left( {\delta_i \left[ {e_{2u}\,e_{3u}\,u^{n-1}_I} \right] |
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| 112 | +\delta_j \left[ {e_{1v}\,e_{3v}\,v^{n-1}_I} \right]} \right). |
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| 113 | \] |
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[10354] | 114 | By the application of \autoref{eq:asm_dmp} and \autoref{eq:asm_dmp} the divergence is filtered in each iteration, |
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| 115 | and the vorticity is left unchanged. |
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| 116 | In the presence of coastal boundaries with zero velocity increments perpendicular to the coast |
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| 117 | the divergence is strongly damped. |
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| 118 | This type of the initialisation reduces the vertical velocity magnitude and |
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| 119 | alleviates the problem of the excessive unphysical vertical mixing in the first steps of the model integration |
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| 120 | \citep{Talagrand_JAS72, Dobricic_al_OS07}. |
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| 121 | Diffusion coefficients are defined as $A_D = \alpha e_{1t} e_{2t}$, where $\alpha = 0.2$. |
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| 122 | The divergence damping is activated by assigning to \np{nn\_divdmp} in the \textit{nam\_asminc} namelist |
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| 123 | a value greater than zero. |
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| 124 | By choosing this value to be of the order of 100 the increments in |
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| 125 | the vertical velocity will be significantly reduced. |
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[3294] | 126 | |
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| 127 | |
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| 128 | %========================================================================== |
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| 129 | |
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[2298] | 130 | \section{Implementation details} |
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[9407] | 131 | \label{sec:ASM_details} |
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[2298] | 132 | |
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[10354] | 133 | Here we show an example \ngn{namasm} namelist and the header of an example assimilation increments file on |
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| 134 | the ORCA2 grid. |
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[2298] | 135 | |
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| 136 | %------------------------------------------namasm----------------------------------------------------- |
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[10146] | 137 | % |
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| 138 | \nlst{nam_asminc} |
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[2298] | 139 | %------------------------------------------------------------------------------------------------------------- |
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| 140 | |
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[2474] | 141 | The header of an assimilation increments file produced using the NetCDF tool |
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| 142 | \mbox{\textit{ncdump~-h}} is shown below |
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[2298] | 143 | |
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[9388] | 144 | \begin{clines} |
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[2298] | 145 | netcdf assim_background_increments { |
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| 146 | dimensions: |
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| 147 | x = 182 ; |
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| 148 | y = 149 ; |
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| 149 | z = 31 ; |
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| 150 | t = UNLIMITED ; // (1 currently) |
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| 151 | variables: |
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| 152 | float nav_lon(y, x) ; |
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| 153 | float nav_lat(y, x) ; |
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| 154 | float nav_lev(z) ; |
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| 155 | double time_counter(t) ; |
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| 156 | double time ; |
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| 157 | double z_inc_dateb ; |
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| 158 | double z_inc_datef ; |
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| 159 | double bckint(t, z, y, x) ; |
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| 160 | double bckins(t, z, y, x) ; |
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| 161 | double bckinu(t, z, y, x) ; |
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| 162 | double bckinv(t, z, y, x) ; |
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| 163 | double bckineta(t, y, x) ; |
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| 164 | |
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| 165 | // global attributes: |
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| 166 | :DOMAIN_number_total = 1 ; |
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| 167 | :DOMAIN_number = 0 ; |
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| 168 | :DOMAIN_dimensions_ids = 1, 2 ; |
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| 169 | :DOMAIN_size_global = 182, 149 ; |
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| 170 | :DOMAIN_size_local = 182, 149 ; |
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| 171 | :DOMAIN_position_first = 1, 1 ; |
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| 172 | :DOMAIN_position_last = 182, 149 ; |
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| 173 | :DOMAIN_halo_size_start = 0, 0 ; |
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| 174 | :DOMAIN_halo_size_end = 0, 0 ; |
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| 175 | :DOMAIN_type = "BOX" ; |
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| 176 | } |
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[9388] | 177 | \end{clines} |
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[9376] | 178 | |
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[10414] | 179 | \biblio |
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| 180 | |
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[10442] | 181 | \pindex |
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| 182 | |
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[6997] | 183 | \end{document} |
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