[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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[11435] | 10 | \chaptertoc |
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[2298] | 11 | |
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[11316] | 12 | \vfill |
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| 13 | \begin{figure}[b] |
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| 14 | \subsubsection*{Changes record} |
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| 15 | \begin{tabular}{l||l|m{0.65\linewidth}} |
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| 16 | Release & Author & Modifications \\ |
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[11435] | 17 | {\em 4.0} & {\em D. J. Lea} & {\em \NEMO\ 4.0 updates} \\ |
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[11316] | 18 | {\em 3.4} & {\em D. J. Lea, M. Martin, K. Mogensen, A. Weaver} & {\em Initial version} \\ |
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| 19 | \end{tabular} |
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| 20 | \end{figure} |
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| 21 | |
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[2298] | 22 | \newpage |
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| 23 | |
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[10354] | 24 | The ASM code adds the functionality to apply increments to the model variables: temperature, salinity, |
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[11316] | 25 | sea surface height, velocity and sea ice concentration. |
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[10354] | 26 | These are read into the model from a NetCDF file which may be produced by separate data assimilation code. |
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| 27 | The code can also output model background fields which are used as an input to data assimilation code. |
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[11435] | 28 | This is all controlled by the namelist \nam{\_asminc}. |
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[10354] | 29 | There is a brief description of all the namelist options provided. |
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| 30 | To build the ASM code \key{asminc} must be set. |
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[2298] | 31 | |
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| 32 | %=============================================================== |
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| 33 | |
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[2349] | 34 | \section{Direct initialization} |
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[9407] | 35 | \label{sec:ASM_DI} |
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[2298] | 36 | |
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[10354] | 37 | Direct initialization (DI) refers to the instantaneous correction of the model background state using |
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| 38 | the analysis increment. |
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[9393] | 39 | DI is used when \np{ln\_asmdin} is set to true. |
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[2298] | 40 | |
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[9393] | 41 | \section{Incremental analysis updates} |
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[9407] | 42 | \label{sec:ASM_IAU} |
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[2298] | 43 | |
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| 44 | Rather than updating the model state directly with the analysis increment, |
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[10354] | 45 | it may be preferable to introduce the increment gradually into the ocean model in order to |
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| 46 | minimize spurious adjustment processes. |
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[11123] | 47 | This technique is referred to as Incremental Analysis Updates (IAU) \citep{bloom.takacs.ea_MWR96}. |
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[2298] | 48 | IAU is a common technique used with 3D assimilation methods such as 3D-Var or OI. |
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[9393] | 49 | IAU is used when \np{ln\_asmiau} is set to true. |
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[2298] | 50 | |
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[11151] | 51 | With IAU, the model state trajectory ${\mathbf x}$ in the assimilation window ($t_{0} \leq t_{i} \leq t_{N}$) |
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[10354] | 52 | is corrected by adding the analysis increments for temperature, salinity, horizontal velocity and SSH as |
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| 53 | additional tendency terms to the prognostic equations: |
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[10414] | 54 | \begin{align*} |
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[11543] | 55 | % \label{eq:ASM_wa_traj_iau} |
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[11151] | 56 | {\mathbf x}^{a}(t_{i}) = M(t_{i}, t_{0})[{\mathbf x}^{b}(t_{0})] \; + \; F_{i} \delta \tilde{\mathbf x}^{a} |
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[10414] | 57 | \end{align*} |
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[11151] | 58 | where $F_{i}$ is a weighting function for applying the increments $\delta\tilde{\mathbf x}^{a}$ defined such that |
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[10354] | 59 | $\sum_{i=1}^{N} F_{i}=1$. |
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[11151] | 60 | ${\mathbf x}^b$ denotes the model initial state and ${\mathbf x}^a$ is the model state after the increments are applied. |
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[2298] | 61 | To control the adjustment time of the model to the increment, |
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[10354] | 62 | the increment can be applied over an arbitrary sub-window, $t_{m} \leq t_{i} \leq t_{n}$, |
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| 63 | 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] | 64 | Typically the increments are spread evenly over the full window. |
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| 65 | In addition, two different weighting functions have been implemented. |
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[11537] | 66 | The first function (namelist option \np{niaufn}=0) employs constant weights, |
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[10414] | 67 | \begin{align} |
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[11543] | 68 | \label{eq:ASM_F1_i} |
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[10414] | 69 | F^{(1)}_{i} |
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| 70 | =\left\{ |
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| 71 | \begin{array}{ll} |
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[11151] | 72 | 0 & {\mathrm if} \; \; \; t_{i} < t_{m} \\ |
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| 73 | 1/M & {\mathrm if} \; \; \; t_{m} < t_{i} \leq t_{n} \\ |
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| 74 | 0 & {\mathrm if} \; \; \; t_{i} > t_{n} |
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[10414] | 75 | \end{array} |
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[11316] | 76 | \right. |
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[10406] | 77 | \end{align} |
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[2298] | 78 | where $M = m-n$. |
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[11537] | 79 | The second function (namelist option \np{niaufn}=1) employs peaked hat-like weights in order to give maximum weight in the centre of the sub-window, |
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[10354] | 80 | with the weighting reduced linearly to a small value at the window end-points: |
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[10414] | 81 | \begin{align} |
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[11543] | 82 | \label{eq:ASM_F2_i} |
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[10414] | 83 | F^{(2)}_{i} |
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| 84 | =\left\{ |
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| 85 | \begin{array}{ll} |
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[11151] | 86 | 0 & {\mathrm if} \; \; \; t_{i} < t_{m} \\ |
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| 87 | \alpha \, i & {\mathrm if} \; \; \; t_{m} \leq t_{i} \leq t_{M/2} \\ |
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| 88 | \alpha \, (M - i +1) & {\mathrm if} \; \; \; t_{M/2} < t_{i} \leq t_{n} \\ |
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| 89 | 0 & {\mathrm if} \; \; \; t_{i} > t_{n} |
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[10414] | 90 | \end{array} |
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| 91 | \right. |
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[10406] | 92 | \end{align} |
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[11316] | 93 | where $\alpha^{-1} = \sum_{i=1}^{M/2} 2i$ and $M$ is assumed to be even. |
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[11543] | 94 | The weights described by \autoref{eq:ASM_F2_i} provide a smoother transition of the analysis trajectory from |
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| 95 | one assimilation cycle to the next than that described by \autoref{eq:ASM_F1_i}. |
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[2298] | 96 | |
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| 97 | %========================================================================== |
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[3294] | 98 | % Divergence damping description %%% |
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| 99 | \section{Divergence damping initialisation} |
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[9414] | 100 | \label{sec:ASM_div_dmp} |
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[2298] | 101 | |
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[11316] | 102 | It is quite challenging for data assimilation systems to provide non-divergent velocity increments. |
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[11435] | 103 | Applying divergent velocity increments will likely cause spurious vertical velocities in the model. This section describes a method to take velocity increments provided to \NEMO\ ($u^0_I$ and $v^0_I$) and adjust them by the iterative application of a divergence damping operator. The method is also described in \citet{dobricic.pinardi.ea_OS07}. |
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[11316] | 104 | |
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| 105 | In iteration step $n$ (starting at $n=1$) new estimates of velocity increments $u^{n}_I$ and $v^{n}_I$ are updated by: |
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| 106 | |
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[10414] | 107 | \begin{equation} |
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[11543] | 108 | \label{eq:ASM_dmp} |
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[10414] | 109 | \left\{ |
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| 110 | \begin{aligned} |
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| 111 | u^{n}_I = u^{n-1}_I + \frac{1}{e_{1u} } \delta_{i+1/2} \left( {A_D |
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| 112 | \;\chi^{n-1}_I } \right) \\ \\ |
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| 113 | v^{n}_I = v^{n-1}_I + \frac{1}{e_{2v} } \delta_{j+1/2} \left( {A_D |
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| 114 | \;\chi^{n-1}_I } \right) \\ |
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| 115 | \end{aligned} |
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| 116 | \right., |
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[3294] | 117 | \end{equation} |
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[11316] | 118 | |
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| 119 | where the divergence is defined as |
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| 120 | |
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[10414] | 121 | \[ |
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[11543] | 122 | % \label{eq:ASM_div} |
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[10414] | 123 | \chi^{n-1}_I = \frac{1}{e_{1t}\,e_{2t}\,e_{3t} } |
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| 124 | \left( {\delta_i \left[ {e_{2u}\,e_{3u}\,u^{n-1}_I} \right] |
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| 125 | +\delta_j \left[ {e_{1v}\,e_{3v}\,v^{n-1}_I} \right]} \right). |
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| 126 | \] |
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[11316] | 127 | |
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[11543] | 128 | By the application of \autoref{eq:ASM_dmp} the divergence is filtered in each iteration, |
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[10354] | 129 | and the vorticity is left unchanged. |
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| 130 | In the presence of coastal boundaries with zero velocity increments perpendicular to the coast |
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| 131 | the divergence is strongly damped. |
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| 132 | This type of the initialisation reduces the vertical velocity magnitude and |
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| 133 | alleviates the problem of the excessive unphysical vertical mixing in the first steps of the model integration |
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[11123] | 134 | \citep{talagrand_JAS72, dobricic.pinardi.ea_OS07}. |
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[10354] | 135 | Diffusion coefficients are defined as $A_D = \alpha e_{1t} e_{2t}$, where $\alpha = 0.2$. |
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[11435] | 136 | The divergence damping is activated by assigning to \np{nn\_divdmp} in the \nam{\_asminc} namelist |
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[10354] | 137 | a value greater than zero. |
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[11316] | 138 | This specifies the number of iterations of the divergence damping. Setting a value of the order of 100 will result in a significant reduction in the vertical velocity induced by the increments. |
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[3294] | 139 | |
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| 140 | |
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| 141 | %========================================================================== |
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| 142 | |
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[2298] | 143 | \section{Implementation details} |
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[9407] | 144 | \label{sec:ASM_details} |
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[2298] | 145 | |
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[11435] | 146 | Here we show an example \nam{\_asminc} namelist and the header of an example assimilation increments file on |
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[10354] | 147 | the ORCA2 grid. |
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[2298] | 148 | |
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[11316] | 149 | %------------------------------------------nam_asminc----------------------------------------------------- |
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[10146] | 150 | % |
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[11558] | 151 | \begin{listing} |
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| 152 | \nlst{nam_asminc} |
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[11567] | 153 | \caption{\forcode{&nam_asminc}} |
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[11558] | 154 | \label{lst:nam_asminc} |
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| 155 | \end{listing} |
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[2298] | 156 | %------------------------------------------------------------------------------------------------------------- |
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| 157 | |
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[2474] | 158 | The header of an assimilation increments file produced using the NetCDF tool |
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| 159 | \mbox{\textit{ncdump~-h}} is shown below |
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[2298] | 160 | |
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[9388] | 161 | \begin{clines} |
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[2298] | 162 | netcdf assim_background_increments { |
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| 163 | dimensions: |
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| 164 | x = 182 ; |
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| 165 | y = 149 ; |
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| 166 | z = 31 ; |
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| 167 | t = UNLIMITED ; // (1 currently) |
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| 168 | variables: |
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| 169 | float nav_lon(y, x) ; |
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| 170 | float nav_lat(y, x) ; |
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| 171 | float nav_lev(z) ; |
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| 172 | double time_counter(t) ; |
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| 173 | double time ; |
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| 174 | double z_inc_dateb ; |
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| 175 | double z_inc_datef ; |
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| 176 | double bckint(t, z, y, x) ; |
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| 177 | double bckins(t, z, y, x) ; |
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| 178 | double bckinu(t, z, y, x) ; |
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| 179 | double bckinv(t, z, y, x) ; |
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| 180 | double bckineta(t, y, x) ; |
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| 181 | |
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| 182 | // global attributes: |
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| 183 | :DOMAIN_number_total = 1 ; |
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| 184 | :DOMAIN_number = 0 ; |
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| 185 | :DOMAIN_dimensions_ids = 1, 2 ; |
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| 186 | :DOMAIN_size_global = 182, 149 ; |
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| 187 | :DOMAIN_size_local = 182, 149 ; |
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| 188 | :DOMAIN_position_first = 1, 1 ; |
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| 189 | :DOMAIN_position_last = 182, 149 ; |
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| 190 | :DOMAIN_halo_size_start = 0, 0 ; |
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| 191 | :DOMAIN_halo_size_end = 0, 0 ; |
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| 192 | :DOMAIN_type = "BOX" ; |
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| 193 | } |
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[9388] | 194 | \end{clines} |
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[9376] | 195 | |
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[10414] | 196 | \biblio |
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| 197 | |
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[10442] | 198 | \pindex |
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| 199 | |
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[6997] | 200 | \end{document} |
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