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1  \section{Adjoint sensiivities of the MITsim}  \section{Adjoint sensitivities of the MITsim}
2  \label{sec:adjoint}  \label{sec:adjoint}
3    
4  \subsection{The adjoint of MITsim}  \subsection{The adjoint of MITsim}
5    
6  The ability to generate tangent linear and adjoint model components  The adjoint model of the MITgcm has become an invaluable
7  of the MITsim has been a main design task.  tool for sensitivity analysis as well as state estimation \citep[for a
8  For the ocean the adjoint capability has proven to be an  recent summary, see][]{heim:08}. The code has been developed and
9  invaluable tool for sensitivity analysis as well as state estimation.  tailored to be readily used with automatic differentiation tools for
10  In short, the adjoint enables very efficient computation of the gradient  adjoint code generation. This route was also taken in developing and
11  of scalar-valued model diagnostics (called cost function or objective function)  adapting the sea-ice compontent MITsim, so that tangent linear and
12  with respect to many model "variables".  adjoint components can be obtained and kept up to date without
13  These variables can be two- or three-dimensional fields of initial  excessive effort.
14  conditions, model parameters such as mixing coefficients, or  
15  time-varying surface or lateral (open) boundary conditions.  The adjoint model operator (ADM) is the transpose of the tangent
16  When combined, these variables span a potentially high-dimensional  linear model operator (TLM) of the full (in general nonlinear) forward
17  (e.g. O(10$^8$)) so-called control space. Performing parameter perturbations  model, in this case the MITsim. This operator computes the gradients
18  to assess model sensitivities quickly becomes prohibitive at these scales.  of scalar-valued model diagnostics (so-called cost function or
19  Alternatively, (time-varying) sensitivities of the objective function  objective function) with respect to many model inputs (so-called
20  to any element of the  control space can be computed very efficiently in  independent or control variables).  These inputs can be two- or
21  one single adjoint  three-dimensional fields of initial conditions of the ocean or sea-ice
22  model integration, provided an efficient adjoint model is available.  state, model parameters such as mixing coefficients, or time-varying
23    surface or lateral (open) boundary conditions.  When combined, these
24  [REFERENCES]  variables span a potentially high-dimensional (e.g.  O(10$^8$))
25    so-called control space. At this problem dimension, perturbing
26    individual parameters to assess model sensitivities quickly becomes
27  The adjoint operator (ADM) is the transpose of the tangent linear operator (TLM)  prohibitive. By contrast, transient sensitivities of the objective
28  of the full (in general nonlinear) forward model, i.e. the MITsim.  function to any element of the control and model state space can be
29  The TLM maps perturbations of elements of the control space  computed very efficiently in one single adjoint model integration,
30  (e.g. initial ice thickness distribution)  provided an adjoint model is available.
31  via the model Jacobian  
32  to a perturbation in the objective function  In anology to the TLM and ADM components of the MITgcm we rely on the
33  (e.g. sea-ice export at the end of the integration interval).  autmomatic differentiation (AD) tool ``Transformation of Algorithms in
34  \textit{Tangent} linearity ensures that the derivatives are evaluated  Fortran'' (TAF) developed by Fastopt \citep{gier-kami:98} to generate
35  with respect to the underlying model trajectory at each point in time.  TLM and ADM code of the MITsim \citep[for details see][]{maro-etal:99,
36  This is crucial for nonlinear trajectories and the presence of different    heim-etal:05}.  In short, the AD tool uses the nonlinear parent
37  regimes (e.g. effect of the seaice growth term at or away from the  model code to generate derivative code for the specified control space
38  freezing point of the ocean surface).  and objective function. Advantages of this approach have been pointed
39  Ensuring tangent linearity can be easily achieved by integrating  out, for example by \cite{gier-kami:98}.
40  the full model in sync with the TLM to provide the underlying model state.  
41  Ensuring \textit{tangent} adjoints is equally crucial, but much more  Many issues of generating efficient exact adjoint sea-ice code are
42  difficult to achieve because of the reverse nature of the integration:  similar to those for the ocean model's adjoint.  Linearizing the model
43  the adjoint accumulates sensitivities backward in time,  around the exact nonlinear model trajectory is a crucial aspect in the
44  starting from a unit perturbation of the objective function.  presence of different regimes (e.g., is the thermodynamic growth term
45  The adjoint model requires the model state in reverse order.  for sea-ice evaluated near or far away from the freezing point of the
46  This presents one of the major complications in deriving an  ocean surface?). Adapting the (parent) model code to support the AD
47  exact, i.e. \textit{tangent} adjoint model.  tool in providing exact and efficient adjoint code represents the main
48    work load initially. For legacy code, this task may become
49  Following closely the development and maintenance of TLM and ADM  substantial, but it is fairly straightforward when writing new code
50  components of the MITgcm we have relied heavily on the  with an AD tool in mind. Once this initial task is completed,
51  autmomatic differentiation (AD) tool  generating the adjoint code of a new model configuration takes about
52  "Transformation of Algorithms in Fortran" (TAF)  10 minutes.
 developed by Fastopt (Giering and Kaminski, 1998)  
 to derive TLM and ADM code of the MITsim.  
 Briefly, the nonlinear parent model is fed to the AD tool which produces  
 derivative code for the specified control space and objective function.  
 Following this approach has (apart from its evident success)  
 several advantages:  
 (1) the adjoint model is the exact adjoint operator of the parent model,  
 (2) the adjoint model can be kept up to date with respect to ongoing  
 development of the parent model, and adjustments to the parent model  
 to extend the automatically generated adjoint are incremental changes  
 only, rather than extensive re-developments,  
 (3) the parallel structure of the parent model is preserved  
 by the adjoint model, ensuring efficient use in high performance  
 computing environments.  
   
 Some initial code adjustments are required to support dependency analysis  
 of the flow reversal and certain language limitations which may lead  
 to irreducible flow graphs (e.g. GOTO statements).  
 The problem of providing the required model state in reverse order  
 at the time of evaluating nonlinear or conditional  
 derivatives is solved via balancing  
 storing vs. recomputation of the model state in a multi-level  
 checkpointing loop.  
 Again, an initial code adjustment is required to support TAFs  
 checkpointing capability.  
 The code adjustments are sufficiently simple so as not to cause  
 major limitations to the full nonlinear parent model.  
 Once in place, an adjoint model of a new model configuration  
 may be derived in about 10 minutes.  
53    
54  [HIGHLIGHT COUPLED NATURE OF THE ADJOINT!]  [HIGHLIGHT COUPLED NATURE OF THE ADJOINT!]
55    
# Line 93  may be derived in about 10 minutes. Line 64  may be derived in about 10 minutes.
64  * approximate adjoints  * approximate adjoints
65    
66    
67  \subsection{An example: sensitivities of sea-ice export through Fram Strait}  \subsection{An example: sensitivities of sea-ice export through
68    the Lancaster Sound}
69    
70  We demonstrate the power of the adjoint method  We demonstrate the power of the adjoint method in the context of
71  in the context of investigating sea-ice export sensitivities through Fram Strait  investigating sea-ice export sensitivities through Lancaster Sound.
72  (for details of this study see Heimbach et al., 2007).  The rationale for doing so is to complement the analysis of sea-ice
73  %\citep[for details of this study see][]{heimbach07}. %Heimbach et al., 2007).  dynamics in the presence of narrow straits.  Lancaster Sound is one of
74  The domain chosen is a coarsened version of the Arctic face of the  the main paths of sea-ice flowing through the Canadian Arctic
75  high-resolution cubed-sphere configuration of the ECCO2 project  Archipelago (CAA).  Export sensitivities reflect dominant pathways
76  \citep[see][]{menemenlis05}. It covers the entire Arctic,  through the CAA as resolved by the model.  Sensitivity maps can shed a
77  extends into the North Pacific such as to cover the entire  very detailed light on various quantities affecting the sea-ice export
78  ice-covered regions, and comprises parts of the North Atlantic  (and thus the underlying pathways).  Note that while the dominant
79  down to XXN to enable analysis of remote influences of the  circulation through Lancaster Sound is toward the East, there is a
80  North Atlantic current to sea-ice variability and export.  small Westward flow to the North, hugging the coast of Devon Island
81  The horizontal resolution varies between XX and YY km  \citep{mell:02, mich-etal:06,muen-etal:06}, which is not resolved in
82  with 50 unevenly spaced vertical levels.  our simulation.
83  The adjoint models run efficiently on 80 processors  
84  (benchmarks have been performed both on an SGI Altix as well as an  The model domain is the same as the one described in \refsec{forward},
85  IBM SP5 at NASA/ARC).  but with halved horizontal resolution.
86    The adjoint models run efficiently on 80 processors (as validated
87  Following a 1-year spinup, the model has been integrated for four  by benchmarks on both an SGI Altix and an IBM SP5 at NASA/ARC).
88  years between 1992 and 1995. It is forced using realistic 6-hourly  Following a 4-year spinup (1985 to 1988), the model is integrated for four
89  NCEP/NCAR atmospheric state variables. Over the open ocean these are  years and nine months between January 1989 and September 1993.
90  converted into air-sea fluxes via the bulk formulae of  It is forced using realistic 6-hourly NCEP/NCAR atmospheric state variables.
91  \citet{large04}.  Derivation of air-sea fluxes in the presence of  %Over the open ocean these are
92  sea-ice is handled by the ice model as described in \refsec{model}.  %converted into air-sea fluxes via the bulk formulae of
93  The objective function chosen is sea-ice export through Fram Strait  %\citet{large04}.  The air-sea fluxes in the presence of
94  computed for December 1995.  The adjoint model computes sensitivities  %sea-ice are handled by the ice model as described in \refsec{model}.
95  to sea-ice export back in time from 1995 to 1992 along this  The objective function $J$ is chosen as the ``solid'' fresh water
96  trajectory.  In principle all adjoint model variable (i.e., Lagrange  export, that is the export of ice and snow converted to units of fresh
97  multipliers) of the coupled ocean/sea-ice model are available to  water,
98  analyze the transient sensitivity behaviour of the ocean and sea-ice  %
99  state.  Over the open ocean, the adjoint of the bulk formula scheme  \begin{equation}
100  computes sensitivities to the time-varying atmospheric state.  Over  J \, = \, (\rho_{i} h_{i}c + \rho_{s} h_{s}c)\,u
101  ice-covered parts, the sea-ice adjoint converts surface ocean  \end{equation}
102  sensitivities to atmospheric sensitivities.  %
103    through Lancaster Sound at approximately 82\degW\ (cross-section G in
104  \reffig{4yradjheff}(a--d) depict sensitivities of sea-ice export  \reffig{arctic_topog}) averaged \ml{PH: Maybe integrated quantity is
105  through Fram Strait in December 1995 to changes in sea-ice thickness  more physical; ML: what did you actually compute? I did not scale
106  12, 24, 36, 48 months back in time. Corresponding sensitivities to  anything, yet. Please insert what is actually done.} over the final
107  ocean surface temperature are depicted in  12-month of the integration between October 1992 and September 1993.
108  \reffig{4yradjthetalev1}(a--d).  The main characteristics is  
109  consistency with expected advection of sea-ice over the relevant time  The forward trajectory of the model integration resembles broadly that
110  scales considered.  The general positive pattern means that an  of the model in \refsec{forward}. Many details are different, owning
111  increase in sea-ice thickness at location $(x,y)$ and time $t$ will  to different resolution and integration period; for example, the solid
112  increase sea-ice export through Fram Strait at time $T_e$.  Largest  fresh water transport through Lancaster Sound is
113  distances from Fram Strait indicate fastest sea-ice advection over the  %
114  time span considered.  The ice thickness sensitivities are in close  \ml{PH: Martin, where did you get these numbers from?}
115  correspondence to ocean surface sentivitites, but of opposite sign.  \ml{[ML: I computed hu = -sum((SIheff+SIhsnow)*SIuice*area)/sum(area) at
116  An increase in temperature will incur ice melting, decrease in ice  $i=100,j=116:122$, and then took mean(hu) and std(hu). What are your numbers?]}
117  thickness, and therefore decrease in sea-ice export at time $T_e$.  %
118    $116\pm101\text{\,km$^{3}$\,y$^{-1}$}$ for a free slip simulation with
119  The picture is fundamentally different and much more complex  the C-LSOR solver, but only $39\pm64\text{\,km$^{3}$\,y$^{-1}$}$ for a
120  for sensitivities to ocean temperatures away from the surface.  no slip simulation. \ml{[Here we can say that the export through
121  \reffig{4yradjthetalev10??}(a--d) depicts ice export sensitivities to    Lancaster Sound is highly uncertain, making is a perfect candidate
122  temperatures at roughly 400 m depth.    for sensitivity, bla bla?]}
123  Primary features are the effect of the heat transport of the North  
124  Atlantic current which feeds into the West Spitsbergen current,  The adjoint model is the transpose of the tangent linear (or Jacobian) model
125  the circulation around Svalbard, and ...  operator. It runs backwards in time, from September 1993 to
126    January 1989. During its integration it accumulates the Lagrange multipliers
127  \begin{figure}[t!]  of the model subject to the objective function (solid freshwater export),
128  \centerline{  which can be interpreted as sensitivities of the objective function
129  \subfigure[{\footnotesize -12 months}]  to each control variable and each element of the intermediate
130  {\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim072_cmax2.0E+02.eps}}  coupled model state variables.
131  %\includegraphics*[width=.3\textwidth]{H_c.bin_res_100_lev1.pdf}  Thus, all sensitivity elements of the coupled
132  %  ocean/sea-ice model state as well as the surface atmospheric state are
133  \subfigure[{\footnotesize -24 months}]  available for analysis of the transient sensitivity behavior.  Over the
134  {\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim145_cmax2.0E+02.eps}}  open ocean, the adjoint of the bulk formula scheme computes
135  }  sensitivities to the time-varying atmospheric state.  Over ice-covered
136    areas, the sea-ice adjoint converts surface ocean sensitivities to
137  \centerline{  atmospheric sensitivities.
138  \subfigure[{\footnotesize  
139  -36 months}]  DISCUSS FORWARD STATE, INCLUDING SOME NUMBERS ON SEA-ICE EXPORT
140  {\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim218_cmax2.0E+02.eps}}  
141  %  \subsubsection{Adjoint sensitivities}
142  \subfigure[{\footnotesize  
143  -48 months}]  The most readily interpretable ice-export sensitivity is that to
144  {\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim292_cmax2.0E+02.eps}}  effective ice thickness, $\partial{J} / \partial{(hc)}$.
145  }  \reffig{adjheff} shows transient $\partial{J} / \partial{(hc)}$ using
146  \caption{Sensitivity of sea-ice export through Fram Strait in December 2005 to  free-slip (left column) and no-slip (right column) boundary
147  sea-ice thickness at various prior times.  conditions. Sensitivity snapshots are depicted for beginning of October 1992,
148  \label{fig:4yradjheff}}  that is 12 months before September 1993
149  \end{figure}  (the beginning of the averaging period for the objective
150    function $J$, top),
151    and for Jannuary 1989, the beginning of the forward integration (bottom).
152  \begin{figure}[t!]  \begin{figure*}[t]
153  \centerline{    \includegraphics*[width=\textwidth]{\fpath/adjheff}
154  \subfigure[{\footnotesize -12 months}]    \caption{Sensitivity $\partial{J}/\partial{(hc)}$ in
155  {\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJtheta_arc_lev1_tim072_cmax5.0E+01.eps}}      m$^2$\,s$^{-1}$/m for two different times (rows) and two different
156  %\includegraphics*[width=.3\textwidth]{H_c.bin_res_100_lev1.pdf}      boundary conditions for sea ice drift. The color scale is chosen
157  %      to illustrate the patterns of the sensitivities; the maximum and
158  \subfigure[{\footnotesize -24 months}]      minimum values are given above the figures.
159  {\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJtheta_arc_lev1_tim145_cmax5.0E+01.eps}}      \label{fig:adjheff}}
160  }  \end{figure*}
161    
162  \centerline{  The sensitivity patterns for effective ice thickness are predominantly positive.
163  \subfigure[{\footnotesize  An increase in ice volume in most places ``upstream'' of
164  -36 months}]  Lancaster sound increases the solid fresh water export at the exit section.
165  {\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJtheta_arc_lev1_tim218_cmax5.0E+01.eps}}  The transient nature of the sensitivity patterns
166  %  (top panels vs. bottom panels) is also obvious:
167  \subfigure[{\footnotesize  the area upstream of the Lancaster Sound that
168  -48 months}]  contributes to the export sensitivity is larger in the earlier snapshot.
169  {\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJtheta_arc_lev1_tim292_cmax5.0E+01.eps}}  In the free slip case, the sensivity follows (backwards in time) the dominant pathway
170  }  through the Barrow Strait
171  \caption{Same as \reffig{4yradjheff} but for sea surface temperature  into the Viscount Melville Sound, and from there trough the M'Clure Strait
172  \label{fig:4yradjthetalev1}}  into the Arctic Ocean (the ``Northwest Passage''). \ml{[Is that really
173  \end{figure}    the Northwest Passage? I thought it would turn south in Barrow
174      Strait, but I am easily convinced because it makes a nicer story.]}
175    Secondary paths are northward from the
176    Viscount Melville Sound through the Byam Martin Channel into
177    the Prince Gustav Adolf Sea and through the Penny Strait into the
178    MacLean Strait. \ml{[Patrick, all these names, if mentioned in the
179      text need to be included somewhere in a figure (i.e. fig1). Can you
180      either do this in fig1 (based on martins\_figs.m) or send me a map
181      where these names are visible so I can do this unambiguously. I
182      don't know where  Byam
183      Martin Channel, Prince Gustav Adolf Sea, Penny Strait, MacLean
184      Strait, Ballantyne St., Massey Sound are.]}
185    
186    There are large differences between the free slip and no slip
187    solution.  By the end of the adjoint integration in January 1989, the
188    no slip sensitivities (bottom right) are generally weaker than the
189    free slip sensitivities and hardly reach beyond the western end of the
190    Barrow Strait. In contrast, the free-slip sensitivities (bottom left)
191    extend through most of the CAA and into the Arctic interior, both to
192    the West (M'Clure St.)  and to the North (Ballantyne St., Prince
193    Gustav Adolf Sea, Massey Sound), because in this case the ice can
194    drift more easily through narrow straits, so that a positive ice
195    volume anomaly anywhere upstream in the CAA increases ice export
196    through the Lancaster Sound within the simulated 4 year period.
197    
198    One peculiar feature in the October 1992 sensitivity maps (top panels)
199    are the negative sensivities to the East and to the West of the
200    Lancaster Sound.
201    These can be explained by indirect effects: less ice to the East means
202    less resistance to eastward drift and thus more export; similarly, less ice to
203    the West means that more ice can be moved eastwards from the Barrow Strait
204    into the Lancaster Sound leading to more ice export.
205    \ml{PH: The first explanation (East) I buy, the second (West) I
206      don't.} \ml{[ML: unfortunately, I don't have anything better to
207      offer, do you? Keep in mind that these sensitivites are very small
208      and only show up, because of the colorscale. In Fig6, they are
209      hardly visible.]}
210    
211    The temporal evolution of several ice export sensitivities (eqn. XX,
212    \ml{[which equation do you mean?]}) along a zonal axis through
213    Lancaster Sound, Barrow Strait, and Melville Sound (115\degW\ to
214    80\degW, averaged across the passages) are depicted as Hovmueller
215    diagrams in \reffig{lancasteradj}. These are, from top to bottom, the
216    sensitivities with respect to effective ice thickness ($hc$), ocean
217    surface temperature ($SST$) and precipitation ($p$) for free slip
218    (left column) and no slip (right column) ice drift boundary
219    conditions.
220    %
221    \begin{figure*}
222      \includegraphics*[height=.8\textheight]{\fpath/lancaster_adj}
223      \caption{Hovermoeller diagrams along the axis Viscount Melville
224        Sound/Barrow Strait/Lancaster Sound. The diagrams show the
225        sensitivities (derivatives) of the ``solid'' fresh water (i.e.,
226        ice and snow) export $J$ through Lancaster sound
227        (\reffig{arctic_topog}, cross-section G) with respect to effective
228        ice thickness ($hc$), ocean surface temperature (SST) and
229        precipitation ($p$) for two runs with free slip and no slip
230        boundary conditions for the sea ice drift. Each plot is overlaid
231        with the contours 1 and 3 of the normalized ice strengh
232        $P/P^*=(hc)\,\exp[-C\,(1-c)]$ for orientation.
233        \label{fig:lancasteradj}}
234    \end{figure*}
235    %
236    \begin{figure*}
237      \includegraphics*[height=.8\textheight]{\fpath/lancaster_fwd}
238      \caption{Hovermoeller diagrams along the axis Viscount Melville
239        Sound/Barrow Strait/Lancaster Sound of effective ice thickness
240        ($hc$), effective snow thickness ($h_{s}c$) and normalized ice
241        strengh $P/P^*=(hc)\,\exp[-C\,(1-c)]$ for two runs with free slip
242        and no slip boundary conditions for the sea ice drift. Each plot
243        is overlaid with the contours 1 and 3 of the normalized ice
244        strength for orientation.
245        \label{fig:lancasterfwd}}
246    \end{figure*}
247    %
248    
249    The Hovmoeller diagrams of ice thickness (top row) and sea surface temperature
250    (second row) sensitivities are coherent:
251    more ice in the Lancaster Sound leads
252    to more export, and one way to get more ice is by colder surface
253    temperatures (less melting from below). In the free slip case the
254    sensitivities spread out in "pulses" following a seasonal cycle:
255    ice can propagate eastwards (forward in time and thus sensitivites can
256    propagate westwards (backwards in time) when the ice strength is low
257    in late summer to early autumn.  
258    In contrast, during winter, the sensitivities show little to now
259    westward propagation, as the ice is frozen solid and does not move.
260    In the no slip case the (normalized)
261    ice strength does not fall below 1 during the winters of 1991 to 1993
262    (mainly because the ice concentrations remain near 100\%, not
263    shown). Ice is therefore blocked and cannot drift eastwards
264    (forward in time) through the Viscount
265    Melville Sound, Barrow Strait, Lancaster Sound channel system.
266    Consequently, the sensitivities do not propagate westwards (backwards in
267    time) and the export through Lancaster Sound is only affected by
268    local ice formation and melting for the entire integration period.
269    
270    The sensitivities to precipitation exhibit an oscillatory behaviour:
271    they are negative (more precipitation leads to less export)
272    before January (more precisely, late fall) and mostly positive after January
273    (more precisely, January through July).
274    Times of positive sensitivities coincide with times of
275    normalized ice strengths exceeding values of 3
276    %
277    \ml{PH: Problem is, that's not true for the first two years (backward),
278    east of 95\degW, that is, in the Lancaster Sound.
279    For example, at 90\degW\ the sensitivities are negative throughout 1992,
280    and no clear correlation to ice strength is apparent there.}
281    except between 95\degW\ and 85\degW, which is an area of
282    increased snow cover in spring. \ml{[ML: and no, I cannot explain
283      that. Can you?]}
284    
285    %
286    Assuming that most precipation is snow in this area\footnote{
287    In the
288    current implementation the model differentiates between snow and rain
289    depending on the thermodynamic growth rate; when it is cold enough for
290    ice to grow, all precipitation is assumed to be snow.}
291    %
292    the sensitivities can be interpreted in terms of the model physics.
293    The accumulation of snow directly increases the exported volume.
294    Further, short wave radiation cannot penetrate the snow cover and has
295    a higer albedo than ice (0.85 for dry snow and 0.75 for dry ice in our
296    case); thus it protects the ice against melting in spring (after
297    January).
298    
299    On the other hand, snow reduces the effective conductivity and thus the heat
300    flux through the ice. This insulating effect slows down the cooling of
301    the surface water underneath the ice and limits the ice growth from
302    below, so that less snow in the ice-growing season leads to more new
303    ice and thus more ice export.
304    \ml{PH: Should probably discuss the effect of snow vs. rain.
305    To me it seems that the "rain" effect doesn't really play a role
306    because the neg. sensitivities are too late in the fall,
307    probably mostly falling as snow.} \ml{[ML: correct, I looked at
308    NCEP/CORE air temperatures, and they are hardly above freezing in
309    Jul/Aug, but otherwise below freezing, that why I can assume that most
310    precip is snow. ]} \ml{[this is not very good but do you have anything
311    better?:]}
312    The negative sensitivities to precipitation between 95\degW\ and
313    85\degW\ in spring 1992 may be explained by a similar mechanism: in an
314    area of thick snow (almost 50\,cm), ice cannot melt and tends to block
315    the channel so that ice coming in from the West cannot pass thus
316    leading to less ice export in the next season.
317    
318    \subsubsection{Forward sensitivities}
319    
320    \ml{[Here we need for integrations to show that the adjoint
321      sensitivites are not just academic. I suggest to perturb HEFF
322      and THETA initial conditions, and PRECIP somewhere in the Melville
323      Sound and then produce plots similar to reffig{lancasteradj}. For
324      PRECIP it would be great to have two perturbation experiments, one
325      where ADJprecip is posivite and one where ADJprecip is negative]}
326      
327    
328    %(*)
329    %The sensitivity in Baffin Bay are more complex.
330    %The pattern evolves along the Western boundary, connecting
331    %the Lancaster Sound Polynya, the Coburg Island Polynya, and the
332    %North Water Polynya, and reaches into Nares Strait and the Kennedy Channel.
333    %The sign of sensitivities has an oscillatory character
334    %[AT FREQUENCY OF SEASONAL CYCLE?].
335    %First, we need to establish whether forward perturbation runs
336    %corroborate the oscillatory behaviour.
337    %Then, several possible explanations:
338    %(i) connection established through Nares Strait throughflow
339    %which extends into Western boundary current in Northern Baffin Bay.
340    %(ii) sea-ice concentration there is seasonal, i.e. partly
341    %ice-free during the year. Seasonal cycle in sensitivity likely
342    %connected to ice-free vs. ice-covered parts of the year.
343    %Negative sensitivities can potentially be attributed
344    %to blocking of Lancaster Sound ice export by Western boundary ice
345    %in Baffin Bay.
346    %(iii) Alternatively to (ii), flow reversal in Lancaster Sound is a possibility
347    %(in reality there's a Northern counter current hugging the coast of
348    %Devon Island which we probably don't resolve).
349    
350    %Remote control of Kennedy Channel on Lancaster Sound ice export
351    %seems a nice test for appropriateness of free-slip vs. no-slip BCs.
352    
353    %\paragraph{Sensitivities to the sea-ice area}
354    
355    %Fig. XXX depcits transient sea-ice export sensitivities
356    %to changes in sea-ice concentration
357    % $\partial J / \partial area$ using free-slip
358    %(left column) and no-slip (right column) boundary conditions.
359    %Sensitivity snapshots are depicted for (from top to bottom)
360    %12, 24, 36, and 48 months prior to May 2003.
361    %Contrary to the steady patterns seen for thickness sensitivities,
362    %the ice-concentration sensitivities exhibit a strong seasonal cycle
363    %in large parts of the domain (but synchronized on large scale).
364    %The following discussion is w.r.t. free-slip run.
365    
366    %(*)
367    %Months, during which sensitivities are negative:
368    %\\
369    %0 to 5   Db=N/A, Dr=5 (May-Jan) \\
370    %10 to 17 Db=7, Dr=5 (Jul-Jan) \\
371    %22 to 29 Db=7, Dr=5 (Jul-Jan) \\
372    %34 to 41 Db=7, Dr=5 (Jul-Jan) \\
373    %46 to 49 D=N/A \\
374    %%
375    %These negative sensitivities seem to be connected to months
376    %during which main parts of the CAA are essentially entirely ice-covered.
377    %This means that increase in ice concentration during this period
378    %will likely reduce ice export due to blocking
379    %[NEED TO EXPLAIN WHY THIS IS NOT THE CASE FOR dJ/dHEFF].
380    %Only during periods where substantial parts of the CAA are
381    %ice free (i.e. sea-ice concentration is less than one in larger parts of
382    %the CAA) will an increase in ice-concentration increase ice export.
383    
384    %(*)
385    %Sensitivities peak about 2-3 months before sign reversal, i.e.
386    %max. negative sensitivities are expected end of July
387    %[DOUBLE CHECK THIS].
388    
389    %(*)
390    %Peaks/bursts of sensitivities for months
391    %14-17, 19-21, 27-29, 30-33, 38-40, 42-45
392    
393    %(*)
394    %Spatial "anti-correlation" (in sign) between main sensitivity branch
395    %(essentially Northwest Passage and immediate connecting channels),
396    %and remote places.
397    %For example: month 20, 28, 31.5, 40, 43.
398    %The timings of max. sensitivity extent are similar between
399    %free-slip and no-slip run; and patterns are similar within CAA,
400    %but differ in the Arctic Ocean interior.
401    
402    %(*)
403    %Interesting (but real?) patterns in Arctic Ocean interior.
404    
405    %\paragraph{Sensitivities to the sea-ice velocity}
406    
407    %(*)
408    %Patterns of ADJuice at almost any point in time are rather complicated
409    %(in particular with respect to spatial structure of signs).
410    %Might warrant perturbation tests.
411    %Patterns of ADJvice, on the other hand, are more spatially coherent,
412    %but still hard to interpret (or even counter-intuitive
413    %in many places).
414    
415    %(*)
416    %"Growth in extent of sensitivities" goes in clear pulses:
417    %almost no change between months: 0-5, 10-20, 24-32, 36-44
418    %These essentially correspond to months of
419    
420    
421    %\subsection{Sensitivities to the oceanic state}
422    
423    %\paragraph{Sensitivities to theta}
424    
425    %\textit{Sensitivities at the surface (z = 5 m)}
426    
427    %(*)
428    %mabye redo with caxmax=0.02 or even 0.05
429    
430    %(*)
431    %Core of negative sensitivities spreading through the CAA as
432    %one might expect [TEST]:
433    %Increase in SST will decrease ice thickness and therefore ice export.
434    
435    %(*)
436    %What's maybe unexpected is patterns of positive sensitivities
437    %at the fringes of the "core", e.g. in the Southern channels
438    %(Bellot St., Peel Sound, M'Clintock Channel), and to the North
439    %(initially MacLean St., Prince Gustav Adolf Sea, Hazen St.,
440    %then shifting Northward into the Arctic interior).
441    
442    %(*)
443    %Marked sensitivity from the Arctic interior roughly along 60$^{\circ}$W
444    %propagating into Lincoln Sea, then
445    %entering Nares Strait and Smith Sound, periodically
446    %warming or cooling[???] the Lancaster Sound exit.
447    
448    %\textit{Sensitivities at depth (z = 200 m)}
449    
450    %(*)
451    %Negative sensitivities almost everywhere, as might be expected.
452    
453    %(*)
454    %Sensitivity patterns between free-slip and no-slip BCs
455    %are quite similar, except in Lincoln Sea (North of Nares St),
456    %where the sign is reversed (but pattern remains similar).
457    
458    %\paragraph{Sensitivities to salt}
459    
460    %T.B.D.
461    
462    %\paragraph{Sensitivities to velocity}
463    
464    %T.B.D.
465    
466    %\subsection{Sensitivities to the atmospheric state}
467    
468    %\begin{itemize}
469    %%
470    %\item
471    %plot of ATEMP for 12, 24, 36, 48 months
472    %%
473    %\item
474    %plot of HEFF for 12, 24, 36, 48 months
475    %%
476    %\end{itemize}
477    
478    
479    
480    %\reffig{4yradjheff}(a--d) depict sensitivities of sea-ice export
481    %through Fram Strait in December 1995 to changes in sea-ice thickness
482    %12, 24, 36, 48 months back in time. Corresponding sensitivities to
483    %ocean surface temperature are depicted in
484    %\reffig{4yradjthetalev1}(a--d).  The main characteristics is
485    %consistency with expected advection of sea-ice over the relevant time
486    %scales considered.  The general positive pattern means that an
487    %increase in sea-ice thickness at location $(x,y)$ and time $t$ will
488    %increase sea-ice export through Fram Strait at time $T_e$.  Largest
489    %distances from Fram Strait indicate fastest sea-ice advection over the
490    %time span considered.  The ice thickness sensitivities are in close
491    %correspondence to ocean surface sentivitites, but of opposite sign.
492    %An increase in temperature will incur ice melting, decrease in ice
493    %thickness, and therefore decrease in sea-ice export at time $T_e$.
494    
495    %The picture is fundamentally different and much more complex
496    %for sensitivities to ocean temperatures away from the surface.
497    %\reffig{4yradjthetalev10??}(a--d) depicts ice export sensitivities to
498    %temperatures at roughly 400 m depth.
499    %Primary features are the effect of the heat transport of the North
500    %Atlantic current which feeds into the West Spitsbergen current,
501    %the circulation around Svalbard, and ...
502    
503    
504    %%\begin{figure}[t!]
505    %%\centerline{
506    %%\subfigure[{\footnotesize -12 months}]
507    %%{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim072_cmax2.0E+02.eps}}
508    %%\includegraphics*[width=.3\textwidth]{H_c.bin_res_100_lev1.pdf}
509    %%
510    %%\subfigure[{\footnotesize -24 months}]
511    %%{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim145_cmax2.0E+02.eps}}
512    %%}
513    %%
514    %%\caption{Sensitivity of sea-ice export through Fram Strait in December 2005 to
515    %%sea-ice thickness at various prior times.
516    %%\label{fig:4yradjheff}}
517    %%\end{figure}
518    
519    
520    %\ml{[based on the movie series
521    %  zzz\_run\_export\_canarch\_freeslip\_4yr\_1989\_ADJ*:]} The ice
522    %export through the Canadian Archipelag is highly sensitive to the
523    %previous state of the ocean-ice system in the Archipelago and the
524    %Western Arctic. According to the \ml{(adjoint)} senstivities of the
525    %eastward ice transport through Lancaster Sound (\reffig{arctic_topog},
526    %cross-section G) with respect to ice volume (effective thickness), ocean
527    %surface temperature, and vertical diffusivity near the surface
528    %(\reffig{fouryearadj}) after 4 years of integration the following
529    %mechanisms can be identified: near the ``observation'' (cross-section
530    %G), smaller vertical diffusivities lead to lower surface temperatures
531    %and hence to more ice that is available for export. Further away from
532    %cross-section G, the sensitivity to vertical diffusivity has the
533    %opposite sign, but temperature and ice volume sensitivities have the
534    %same sign as close to the observation.
535    
536    
537    %%% Local Variables:
538    %%% mode: latex
539    %%% TeX-master: "ceaice"
540    %%% End:
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