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1\documentclass[../main/NEMO_manual]{subfiles}
2
3\begin{document}
4
5\chapter{Apply Assimilation Increments (ASM)}
6\label{chap:ASM}
7
8%    {\em 4.0} & {\em D. J. Lea} & {\em \NEMO\ 4.0 updates}  \\
9%    {\em 3.4} & {\em D. J. Lea, M. Martin, K. Mogensen, A. Weaver} & {\em Initial version}  \\
10
11\thispagestyle{plain}
12
13\chaptertoc
14
15\paragraph{Changes record} ~\\
16
17{\footnotesize
18  \begin{tabularx}{\textwidth}{l||X|X}
19    Release & Author(s) & Modifications \\
20    \hline
21    {\em   4.0} & {\em ...} & {\em ...} \\
22    {\em   3.6} & {\em ...} & {\em ...} \\
23    {\em   3.4} & {\em ...} & {\em ...} \\
24    {\em <=3.4} & {\em ...} & {\em ...}
25  \end{tabularx}
26}
27
28\clearpage
29
30The ASM code adds the functionality to apply increments to the model variables: temperature, salinity,
31sea surface height, velocity and sea ice concentration.
32These are read into the model from a NetCDF file which may be produced by separate data assimilation code.
33The code can also output model background fields which are used as an input to data assimilation code.
34This is all controlled by the namelist \nam{_asminc}{\_asminc}.
35There is a brief description of all the namelist options provided.
36To build the ASM code \key{asminc} must be set.
37
38%% =================================================================================================
39\section{Direct initialization}
40\label{sec:ASM_DI}
41
42Direct initialization (DI) refers to the instantaneous correction of the model background state using
43the analysis increment.
44DI is used when \np{ln_asmdin}{ln\_asmdin} is set to true.
45
46%% =================================================================================================
47\section{Incremental analysis updates}
48\label{sec:ASM_IAU}
49
50Rather than updating the model state directly with the analysis increment,
51it may be preferable to introduce the increment gradually into the ocean model in order to
52minimize spurious adjustment processes.
53This technique is referred to as Incremental Analysis Updates (IAU) \citep{bloom.takacs.ea_MWR96}.
54IAU is a common technique used with 3D assimilation methods such as 3D-Var or OI.
55IAU is used when \np{ln_asmiau}{ln\_asmiau} is set to true.
56
57With IAU, the model state trajectory ${\mathbf x}$ in the assimilation window ($t_{0} \leq t_{i} \leq t_{N}$)
58is corrected by adding the analysis increments for temperature, salinity, horizontal velocity and SSH as
59additional tendency terms to the prognostic equations:
60\begin{align*}
61  % \label{eq:ASM_wa_traj_iau}
62  {\mathbf x}^{a}(t_{i}) = M(t_{i}, t_{0})[{\mathbf x}^{b}(t_{0})] \; + \; F_{i} \delta \tilde{\mathbf x}^{a}
63\end{align*}
64where $F_{i}$ is a weighting function for applying the increments $\delta\tilde{\mathbf x}^{a}$ defined such that
65$\sum_{i=1}^{N} F_{i}=1$.
66${\mathbf x}^b$ denotes the model initial state and ${\mathbf x}^a$ is the model state after the increments are applied.
67To control the adjustment time of the model to the increment,
68the increment can be applied over an arbitrary sub-window, $t_{m} \leq t_{i} \leq t_{n}$,
69of the main assimilation window, where $t_{0} \leq t_{m} \leq t_{i}$ and $t_{i} \leq t_{n} \leq t_{N}$.
70Typically the increments are spread evenly over the full window.
71In addition, two different weighting functions have been implemented.
72The first function (namelist option \np{niaufn}{niaufn}=0) employs constant weights,
73\begin{align}
74  \label{eq:ASM_F1_i}
75  F^{(1)}_{i}
76  =\left\{
77  \begin{array}{ll}
78    0     &    {\mathrm if} \; \; \; t_{i} < t_{m}                \\
79    1/M &    {\mathrm if} \; \; \; t_{m} < t_{i} \leq t_{n} \\
80    0     &    {\mathrm if} \; \; \; t_{i} > t_{n}
81  \end{array}
82            \right.
83\end{align}
84where $M = m-n$.
85The second function (namelist option \np{niaufn}{niaufn}=1) employs peaked hat-like weights in order to give maximum weight in the centre of the sub-window,
86with the weighting reduced linearly to a small value at the window end-points:
87\begin{align}
88  \label{eq:ASM_F2_i}
89  F^{(2)}_{i}
90  =\left\{
91  \begin{array}{ll}
92    0                           &    {\mathrm if} \; \; \; t_{i}       <     t_{m}                        \\
93    \alpha \, i               &    {\mathrm if} \; \; \; t_{m}    \leq t_{i}    \leq   t_{M/2}   \\
94    \alpha \, (M - i +1) &    {\mathrm if} \; \; \; t_{M/2}  <    t_{i}    \leq   t_{n}       \\
95    0                            &   {\mathrm if} \; \; \; t_{i}        >    t_{n}
96  \end{array}
97                                   \right.
98\end{align}
99where $\alpha^{-1} = \sum_{i=1}^{M/2} 2i$ and $M$ is assumed to be even.
100The weights described by \autoref{eq:ASM_F2_i} provide a smoother transition of the analysis trajectory from
101one assimilation cycle to the next than that described by \autoref{eq:ASM_F1_i}.
102
103%% =================================================================================================
104\section{Divergence damping initialisation}
105\label{sec:ASM_div_dmp}
106
107It is quite challenging for data assimilation systems to provide non-divergent velocity increments.
108Applying 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}.
109
110In iteration step $n$ (starting at $n=1$) new estimates of velocity increments $u^{n}_I$ and $v^{n}_I$ are updated by:
111
112\begin{equation}
113  \label{eq:ASM_dmp}
114  \left\{
115    \begin{aligned}
116      u^{n}_I = u^{n-1}_I + \frac{1}{e_{1u} } \delta_{i+1/2} \left( {A_D
117          \;\chi^{n-1}_I } \right) \\ \\
118      v^{n}_I = v^{n-1}_I + \frac{1}{e_{2v} } \delta_{j+1/2} \left( {A_D
119          \;\chi^{n-1}_I } \right) \\
120    \end{aligned}
121  \right.,
122\end{equation}
123
124where the divergence is defined as
125
126\[
127  % \label{eq:ASM_div}
128  \chi^{n-1}_I = \frac{1}{e_{1t}\,e_{2t}\,e_{3t} }
129  \left( {\delta_i \left[ {e_{2u}\,e_{3u}\,u^{n-1}_I} \right]
130      +\delta_j \left[ {e_{1v}\,e_{3v}\,v^{n-1}_I} \right]} \right).
131\]
132
133By the application of \autoref{eq:ASM_dmp} the divergence is filtered in each iteration,
134and the vorticity is left unchanged.
135In the presence of coastal boundaries with zero velocity increments perpendicular to the coast
136the divergence is strongly damped.
137This type of the initialisation reduces the vertical velocity magnitude and
138alleviates the problem of the excessive unphysical vertical mixing in the first steps of the model integration
139\citep{talagrand_JAS72, dobricic.pinardi.ea_OS07}.
140Diffusion coefficients are defined as $A_D = \alpha e_{1t} e_{2t}$, where $\alpha = 0.2$.
141The divergence damping is activated by assigning to \np{nn_divdmp}{nn\_divdmp} in the \nam{_asminc}{\_asminc} namelist
142a value greater than zero.
143This 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.
144
145%% =================================================================================================
146\section{Implementation details}
147\label{sec:ASM_details}
148
149Here we show an example \nam{_asminc}{\_asminc} namelist and the header of an example assimilation increments file on
150the ORCA2 grid.
151
152\begin{listing}
153  \nlst{nam_asminc}
154  \caption{\forcode{&nam_asminc}}
155  \label{lst:nam_asminc}
156\end{listing}
157
158The header of an assimilation increments file produced using the NetCDF tool
159\mbox{\textit{ncdump~-h}} is shown below
160
161\begin{clines}
162netcdf assim_background_increments {
163dimensions:
164        x = 182 ;
165        y = 149 ;
166        z = 31 ;
167        t = UNLIMITED ; // (1 currently)
168variables:
169        float nav_lon(y, x) ;
170        float nav_lat(y, x) ;
171        float nav_lev(z) ;
172        double time_counter(t) ;
173        double time ;
174        double z_inc_dateb ;
175        double z_inc_datef ;
176        double bckint(t, z, y, x) ;
177        double bckins(t, z, y, x) ;
178        double bckinu(t, z, y, x) ;
179        double bckinv(t, z, y, x) ;
180        double bckineta(t, y, x) ;
181
182// global attributes:
183                :DOMAIN_number_total = 1 ;
184                :DOMAIN_number = 0 ;
185                :DOMAIN_dimensions_ids = 1, 2 ;
186                :DOMAIN_size_global = 182, 149 ;
187                :DOMAIN_size_local = 182, 149 ;
188                :DOMAIN_position_first = 1, 1 ;
189                :DOMAIN_position_last = 182, 149 ;
190                :DOMAIN_halo_size_start = 0, 0 ;
191                :DOMAIN_halo_size_end = 0, 0 ;
192                :DOMAIN_type = "BOX" ;
193}
194\end{clines}
195
196\onlyinsubfile{\input{../../global/epilogue}}
197
198\end{document}
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