/[MITgcm]/MITgcm_contrib/articles/ceaice/ceaice_adjoint.tex
ViewVC logotype

Diff of /MITgcm_contrib/articles/ceaice/ceaice_adjoint.tex

Parent Directory Parent Directory | Revision Log Revision Log | View Revision Graph Revision Graph | View Patch Patch

revision 1.1 by dimitri, Tue Feb 26 19:27:26 2008 UTC revision 1.5 by mlosch, Fri Jul 25 15:01:19 2008 UTC
# Line 1  Line 1 
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 outflow paths of sea-ice flowing through the Canadian Arctic
75    Archipelago (CAA).  Export sensitivities reflect dominant pathways
76    through the CAA as resolved by the model.  Sensitivity maps can shed a
77    very detailed light on various quantities affecting the sea-ice export
78    (and thus the underlying pathways).  Note that while the dominant
79    circulation through Lancaster Sound is toward the East, there is a
80    small Westward flow to the North, hugging the coast of Devon Island
81    \citep{mell:02, mich-etal:06,muen-etal:06}, which is not resolved in
82    our simulation.
83    
84    The model domain is a coarsened version of the Arctic face of the
85  high-resolution cubed-sphere configuration of the ECCO2 project  high-resolution cubed-sphere configuration of the ECCO2 project
86  \citep[see][]{menemenlis05}. It covers the entire Arctic,  \citep{menemenlis05} as described in \refsec{forward}.  The horizontal
87  extends into the North Pacific such as to cover the entire  resolution is half of that in \refsec{forward} while the vertical grid
88  ice-covered regions, and comprises parts of the North Atlantic  is the same. \ml{[Is this important? Do we need to be more specific?:
89  down to XXN to enable analysis of remote influences of the    ]} The adjoint models run efficiently on 80 processors (as validated
90  North Atlantic current to sea-ice variability and export.  by benchmarks on both an SGI Altix and an IBM SP5 at NASA/ARC).
91  The horizontal resolution varies between XX and YY km  
92  with 50 unevenly spaced vertical levels.  Following a 3-year spinup, the model is integrated for four
93  The adjoint models run efficiently on 80 processors  years and five months between January 1989 and September 1993.
94  (benchmarks have been performed both on an SGI Altix as well as an  \ml{[Patrick: to what extent is this different from section 3?]}
95  IBM SP5 at NASA/ARC).  It is forced using realistic 6-hourly NCEP/NCAR atmospheric state variables.
96    %Over the open ocean these are
97  Following a 1-year spinup, the model has been integrated for four  %converted into air-sea fluxes via the bulk formulae of
98  years between 1992 and 1995. It is forced using realistic 6-hourly  %\citet{large04}.  The air-sea fluxes in the presence of
99  NCEP/NCAR atmospheric state variables. Over the open ocean these are  %sea-ice are handled by the ice model as described in \refsec{model}.
100  converted into air-sea fluxes via the bulk formulae of  The objective function $J$ is chosen as the ``solid'' fresh water
101  \citet{large04}.  Derivation of air-sea fluxes in the presence of  export, that is the export of ice and snow converted to units of fresh
102  sea-ice is handled by the ice model as described in \refsec{model}.  water $(\rho_{i} h_{i}c + \rho_{s} h_{s}c)\,u$, through Lancaster
103  The objective function chosen is sea-ice export through Fram Strait  Sound at approximately 82\degW\ (cross-section G in
104  computed for December 1995.  The adjoint model computes sensitivities  \reffig{arctic_topog}) averaged over a 12-month period between October
105  to sea-ice export back in time from 1995 to 1992 along this  1992 and September 1993.
106  trajectory.  In principle all adjoint model variable (i.e., Lagrange  
107  multipliers) of the coupled ocean/sea-ice model are available to  The forward trajectory of the model integration resembles broadly that
108  analyze the transient sensitivity behaviour of the ocean and sea-ice  of the model in \refsec{forward}. Many details are different, owning
109  state.  Over the open ocean, the adjoint of the bulk formula scheme  to different resolution and integration period; for example, the solid
110  computes sensitivities to the time-varying atmospheric state.  Over  fresh water transport through Lancaster Sound is
111  ice-covered parts, the sea-ice adjoint converts surface ocean  $116\pm101\text{\,km$^{3}$\,y$^{-1}$}$ for a free slip simulation with
112  sensitivities to atmospheric sensitivities.  the C-LSOR solver, but only $39\pm64\text{\,km$^{3}$\,y$^{-1}$}$ for a
113    no slip simulation.
114  \reffig{4yradjheff}(a--d) depict sensitivities of sea-ice export  
115  through Fram Strait in December 1995 to changes in sea-ice thickness  The adjoint model computes sensitivities of this export back in time
116  12, 24, 36, 48 months back in time. Corresponding sensitivities to  from 1993 to 1989 along this trajectory.  In principle all adjoint
117  ocean surface temperature are depicted in  model variable (i.e., Lagrange multipliers) of the coupled
118  \reffig{4yradjthetalev1}(a--d).  The main characteristics is  ocean/sea-ice model as well as the surface atmospheric state are
119  consistency with expected advection of sea-ice over the relevant time  available to analyze the transient sensitivity behavior.  Over the
120  scales considered.  The general positive pattern means that an  open ocean, the adjoint of the bulk formula scheme computes
121  increase in sea-ice thickness at location $(x,y)$ and time $t$ will  sensitivities to the time-varying atmospheric state.  Over ice-covered
122  increase sea-ice export through Fram Strait at time $T_e$.  Largest  parts, the sea-ice adjoint converts surface ocean sensitivities to
123  distances from Fram Strait indicate fastest sea-ice advection over the  atmospheric sensitivities.
124  time span considered.  The ice thickness sensitivities are in close  
125  correspondence to ocean surface sentivitites, but of opposite sign.  DISCUSS FORWARD STATE, INCLUDING SOME NUMBERS ON SEA-ICE EXPORT
126  An increase in temperature will incur ice melting, decrease in ice  
127  thickness, and therefore decrease in sea-ice export at time $T_e$.  \subsubsection{Adjoint sensitivities}
128    
129  The picture is fundamentally different and much more complex  The most readily interpretable ice-export sensitivity is that to
130  for sensitivities to ocean temperatures away from the surface.  effective ice thickness, $\partial{J} / \partial{(hc)}$.
131  \reffig{4yradjthetalev10??}(a--d) depicts ice export sensitivities to  \reffig{adjheff} shows transient $\partial{J} / \partial{(hc)}$ using
132  temperatures at roughly 400 m depth.  free-slip (left column) and no-slip (right column) boundary
133  Primary features are the effect of the heat transport of the North  conditions. Sensitivity snapshots are depicted for 12 months prior to
134  Atlantic current which feeds into the West Spitsbergen current,  September 1993 (at the beginning of the averaging period for the objective
135  the circulation around Svalbard, and ...  function $J$, top) and at the beginning of the integration in January
136    1989 (bottom).
137  \begin{figure}[t!]  \begin{figure*}[t]
138  \centerline{    \includegraphics*[width=\textwidth]{\fpath/adjheff}
139  \subfigure[{\footnotesize -12 months}]    \caption{Sensitivity $\partial{J}/\partial{(hc)}$ in
140  {\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim072_cmax2.0E+02.eps}}      m$^2$\,s$^{-1}$/m for two different times (rows) and two different
141  %\includegraphics*[width=.3\textwidth]{H_c.bin_res_100_lev1.pdf}      boundary conditions for sea ice drift. The color scale is chosen
142  %      to illustrate the patterns of the sensitivities; the maximum and
143  \subfigure[{\footnotesize -24 months}]      minimum values are given above the figures.
144  {\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim145_cmax2.0E+02.eps}}      \label{fig:adjheff}}
145  }  \end{figure*}
146    
147  \centerline{  At the beginning of October 1992, the positive sensitivities in
148  \subfigure[{\footnotesize  the Lancaster Sound mean that an increase of ice volume increase the
149  -36 months}]  solid fresh water export. The negative sensivities to the East and to the
150  {\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim218_cmax2.0E+02.eps}}  West can be explained by indirect effects: less ice to the East means
151  %  less resistance to eastward drift and thus more export; similarly, less ice to
152  \subfigure[{\footnotesize  the West means that more ice can be moved eastwards from the Barrow Strait
153  -48 months}]  into the Lancaster Sound leading to more ice export. The sensitivities
154  {\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim292_cmax2.0E+02.eps}}  are similar for both no slip and free slip solutions with a slightly larger
155  }  area covered by non-zero sensitivities in the free slip solution. At
156  \caption{Sensitivity of sea-ice export through Fram Strait in December 2005 to  the beginning of the integration (the end of the backward adjoint
157  sea-ice thickness at various prior times.  integration) the free and no slip solutions are very different. The
158  \label{fig:4yradjheff}}  sensitivities of the free slip solution extend through the enitre
159  \end{figure}  Canadian Archipelago and into the Arctic while in the no slip solution
160    they still are confined to the Lancaster Sound and the Barrow
161    Strait. This implies that in the free slip solution ice can drift more
162  \begin{figure}[t!]  easily through the narrow straits of the Canadian Archipelago, so that
163  \centerline{  a positive ice volume anomaly anywhere in the Canadian Archipelago is
164  \subfigure[{\footnotesize -12 months}]  moved through the Lancaster Sound within 4 years thus increasing the
165  {\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJtheta_arc_lev1_tim072_cmax5.0E+01.eps}}  ice export.
166  %\includegraphics*[width=.3\textwidth]{H_c.bin_res_100_lev1.pdf}  
167  %  The temporal evolution of several sensitivities along the zonal axis
168  \subfigure[{\footnotesize -24 months}]  Lancaster Sound-Barrow Strait-Melville Sound are shown in
169  {\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJtheta_arc_lev1_tim145_cmax5.0E+01.eps}}  \reffig{lancaster}.
170  }  \begin{figure*}
171      \includegraphics*[height=.8\textheight]{\fpath/lancaster_adj}
172  \centerline{    \caption{Hovermoeller diagrams of sensitivities (derivatives) of the
173  \subfigure[{\footnotesize      ``solid'' fresh water (i.e., ice and snow) export $J$ through Lancaster sound
174  -36 months}]      (\reffig{arctic_topog}, cross-section G) with respect to effective
175  {\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJtheta_arc_lev1_tim218_cmax5.0E+01.eps}}      ice thickness ($hc$), ocean surface temperature (SST) and
176  %      precipitation ($p$) for two runs with free slip and no slip boundary
177  \subfigure[{\footnotesize      conditions for the sea ice drift. Also shown it the normalized ice
178  -48 months}]      strengh $P/P^*=(hc)\,\exp[-C\,(1-c)]$ (bottom panel); each plot is
179  {\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJtheta_arc_lev1_tim292_cmax5.0E+01.eps}}      overlaid with the contours 1 and 3 of the normalized ice strength
180  }      for orientation.
181  \caption{Same as \reffig{4yradjheff} but for sea surface temperature      \label{fig:lancaster}}
182  \label{fig:4yradjthetalev1}}  \end{figure*}
183  \end{figure}  \reffig{lancaster} shows the sensitivities of ``solid'' fresh water
184    export, that is ice and snow, through Lancaster sound (cross-section G
185    in \reffig{arctic_topog}) with respect to effective ice thickness
186    ($hc$), ocean surface temperature (SST) and precipitation ($p$) for
187    two runs with free slip and no slip boundary conditions for the sea
188    ice drift. The Hovmoeller diagrams of sensitivities (derivatives) with
189    respect to effective ice thickness (top) and ocean surface temperature
190    (second from top) are coherent: more ice in the Lancaster Sound leads
191    to more export and one way to get more ice is by colder surface
192    temperatures (less melting from below). In the free slip case the
193    sensitivities can propagate westwards (backwards in time) when the ice
194    strength is low in late summer. In the no slip case the (normalized)
195    ice strength does not fall below 1 during the winters of 1991 to 1993
196    (mainly because the ice concentrations remain nearly 100\%, not
197    shown), so that ice is blocked and cannot drift eastwards (forward in
198    time) in the Melville Sound-Barrow Strait-Lancaster Sound channel.
199    Consequently the sensitivies do not propagate westwards (backwards in
200    time) and the export through Lancaster Sound is only affected by
201    local ice formation and melting.
202    
203    The sensitivities to precipitation are negative (more precipitation
204    leads to less export) before January and mostly positive after
205    January. Further they are mostly positive for normalized ice strengths
206    over 3. Assuming that most precipation is snow in this area---in the
207    current implementation the model differentiates between snow and rain
208    depending on the thermodynamic growth rate; when it is cold enough for
209    ice to grow, all precipitation is assumed to be snow---the
210    sensitivities can be interpreted in terms of the model physics.  Short
211    wave radiation cannot penetrate a snow cover and has a higer albedo
212    than ice (0.85 for dry snow and 0.75 for dry ice in our case); thus it
213    protects the ice against melting in spring (after January).  On the
214    other hand, snow reduces the effective conductivity and thus the heat
215    flux through the ice. This insulating effect slows down the cooling of
216    the surface water underneath the ice and limits the ice growth from
217    below, so that less snow in the ice-growing season leads to more new
218    ice and thus more ice export.
219    
220    %Und jetzt weiss ich nicht mehr weiter, aber nun kann folgendes passiert sein:
221    %1. snow insulates against melting from above during spring: more precip (snow) -> more export
222    %2. less snow during fall -> more ice -> more export
223    %3. precip is both snow and rain, depending on the sign of "FICE" (thermodynamic growth rate), with probably different implications
224    
225    
226    \subsubsection{Forward sensitivities}
227    
228    \ml{[Here we need for integrations to show that the adjoint
229      sensitivites are not just academic. I suggest to perturb HEFF
230      and THETA initial conditions, and PRECIP somewhere in the Melville
231      Sound and then produce plots similar to reffig{lancaster}. For
232      PRECIP it would be great to have two perturbation experiments, one
233      where ADJprecip is posivite and one where ADJprecip is negative]}
234    %The dominant features are\ml{ in accordance with expectations/as expected}:
235    
236    %(*)
237    %Dominant pattern (for the free-slip run) is that of positive sensitivities, i.e.
238    %a unit increase in sea-ice thickness in most places upstream
239    %of Lancaster Sound will increase sea-ice export through Lancaster Sound.
240    %The dominant pathway follows (backward in time) through Barrow Strait
241    %into Viscount Melville Sound, and from there trough M'Clure Strait
242    %into the Arctic Ocean (the "Northwest Passage").
243    %Secondary paths are Northward from
244    %Viscount Melville Sound through Byam Martin Channel into
245    %Prince Gustav Adolf Sea and through Penny Strait into MacLean Strait.
246    
247    %(*)
248    %As expected, at any given time the
249    %region of influence is larger for the free-slip than no-slip simulation.
250    %For the no-slip run, the region of influence is confined, after four years,
251    %to just West of Barrow Strait (North of Prince of Wales Island),
252    %and to the South of Penny Strait.
253    %In contrast, sensitivities of the free-slip run extend
254    %all the way to the Arctic interior both to the West
255    %(M'Clure St.) and to the North (Ballantyne St., Prince Gustav Adolf Sea,
256    %Massey Sound).
257    
258    %(*)
259    %sensitivities seem to spread out in "pulses" (seasonal cycle)
260    %[PLOT A TIME SERIES OF ADJheff in Barrow Strait)
261    
262    %(*)
263    %The sensitivity in Baffin Bay are more complex.
264    %The pattern evolves along the Western boundary, connecting
265    %the Lancaster Sound Polynya, the Coburg Island Polynya, and the
266    %North Water Polynya, and reaches into Nares Strait and the Kennedy Channel.
267    %The sign of sensitivities has an oscillatory character
268    %[AT FREQUENCY OF SEASONAL CYCLE?].
269    %First, we need to establish whether forward perturbation runs
270    %corroborate the oscillatory behaviour.
271    %Then, several possible explanations:
272    %(i) connection established through Nares Strait throughflow
273    %which extends into Western boundary current in Northern Baffin Bay.
274    %(ii) sea-ice concentration there is seasonal, i.e. partly
275    %ice-free during the year. Seasonal cycle in sensitivity likely
276    %connected to ice-free vs. ice-covered parts of the year.
277    %Negative sensitivities can potentially be attributed
278    %to blocking of Lancaster Sound ice export by Western boundary ice
279    %in Baffin Bay.
280    %(iii) Alternatively to (ii), flow reversal in Lancaster Sound is a possibility
281    %(in reality there's a Northern counter current hugging the coast of
282    %Devon Island which we probably don't resolve).
283    
284    %Remote control of Kennedy Channel on Lancaster Sound ice export
285    %seems a nice test for appropriateness of free-slip vs. no-slip BCs.
286    
287    %\paragraph{Sensitivities to the sea-ice area}
288    
289    %Fig. XXX depcits transient sea-ice export sensitivities
290    %to changes in sea-ice concentration
291    % $\partial J / \partial area$ using free-slip
292    %(left column) and no-slip (right column) boundary conditions.
293    %Sensitivity snapshots are depicted for (from top to bottom)
294    %12, 24, 36, and 48 months prior to May 2003.
295    %Contrary to the steady patterns seen for thickness sensitivities,
296    %the ice-concentration sensitivities exhibit a strong seasonal cycle
297    %in large parts of the domain (but synchronized on large scale).
298    %The following discussion is w.r.t. free-slip run.
299    
300    %(*)
301    %Months, during which sensitivities are negative:
302    %\\
303    %0 to 5   Db=N/A, Dr=5 (May-Jan) \\
304    %10 to 17 Db=7, Dr=5 (Jul-Jan) \\
305    %22 to 29 Db=7, Dr=5 (Jul-Jan) \\
306    %34 to 41 Db=7, Dr=5 (Jul-Jan) \\
307    %46 to 49 D=N/A \\
308    %%
309    %These negative sensitivities seem to be connected to months
310    %during which main parts of the CAA are essentially entirely ice-covered.
311    %This means that increase in ice concentration during this period
312    %will likely reduce ice export due to blocking
313    %[NEED TO EXPLAIN WHY THIS IS NOT THE CASE FOR dJ/dHEFF].
314    %Only during periods where substantial parts of the CAA are
315    %ice free (i.e. sea-ice concentration is less than one in larger parts of
316    %the CAA) will an increase in ice-concentration increase ice export.
317    
318    %(*)
319    %Sensitivities peak about 2-3 months before sign reversal, i.e.
320    %max. negative sensitivities are expected end of July
321    %[DOUBLE CHECK THIS].
322    
323    %(*)
324    %Peaks/bursts of sensitivities for months
325    %14-17, 19-21, 27-29, 30-33, 38-40, 42-45
326    
327    %(*)
328    %Spatial "anti-correlation" (in sign) between main sensitivity branch
329    %(essentially Northwest Passage and immediate connecting channels),
330    %and remote places.
331    %For example: month 20, 28, 31.5, 40, 43.
332    %The timings of max. sensitivity extent are similar between
333    %free-slip and no-slip run; and patterns are similar within CAA,
334    %but differ in the Arctic Ocean interior.
335    
336    %(*)
337    %Interesting (but real?) patterns in Arctic Ocean interior.
338    
339    %\paragraph{Sensitivities to the sea-ice velocity}
340    
341    %(*)
342    %Patterns of ADJuice at almost any point in time are rather complicated
343    %(in particular with respect to spatial structure of signs).
344    %Might warrant perturbation tests.
345    %Patterns of ADJvice, on the other hand, are more spatially coherent,
346    %but still hard to interpret (or even counter-intuitive
347    %in many places).
348    
349    %(*)
350    %"Growth in extent of sensitivities" goes in clear pulses:
351    %almost no change between months: 0-5, 10-20, 24-32, 36-44
352    %These essentially correspond to months of
353    
354    
355    %\subsection{Sensitivities to the oceanic state}
356    
357    %\paragraph{Sensitivities to theta}
358    
359    %\textit{Sensitivities at the surface (z = 5 m)}
360    
361    %(*)
362    %mabye redo with caxmax=0.02 or even 0.05
363    
364    %(*)
365    %Core of negative sensitivities spreading through the CAA as
366    %one might expect [TEST]:
367    %Increase in SST will decrease ice thickness and therefore ice export.
368    
369    %(*)
370    %What's maybe unexpected is patterns of positive sensitivities
371    %at the fringes of the "core", e.g. in the Southern channels
372    %(Bellot St., Peel Sound, M'Clintock Channel), and to the North
373    %(initially MacLean St., Prince Gustav Adolf Sea, Hazen St.,
374    %then shifting Northward into the Arctic interior).
375    
376    %(*)
377    %Marked sensitivity from the Arctic interior roughly along 60$^{\circ}$W
378    %propagating into Lincoln Sea, then
379    %entering Nares Strait and Smith Sound, periodically
380    %warming or cooling[???] the Lancaster Sound exit.
381    
382    %\textit{Sensitivities at depth (z = 200 m)}
383    
384    %(*)
385    %Negative sensitivities almost everywhere, as might be expected.
386    
387    %(*)
388    %Sensitivity patterns between free-slip and no-slip BCs
389    %are quite similar, except in Lincoln Sea (North of Nares St),
390    %where the sign is reversed (but pattern remains similar).
391    
392    %\paragraph{Sensitivities to salt}
393    
394    %T.B.D.
395    
396    %\paragraph{Sensitivities to velocity}
397    
398    %T.B.D.
399    
400    %\subsection{Sensitivities to the atmospheric state}
401    
402    %\begin{itemize}
403    %%
404    %\item
405    %plot of ATEMP for 12, 24, 36, 48 months
406    %%
407    %\item
408    %plot of HEFF for 12, 24, 36, 48 months
409    %%
410    %\end{itemize}
411    
412    
413    
414    %\reffig{4yradjheff}(a--d) depict sensitivities of sea-ice export
415    %through Fram Strait in December 1995 to changes in sea-ice thickness
416    %12, 24, 36, 48 months back in time. Corresponding sensitivities to
417    %ocean surface temperature are depicted in
418    %\reffig{4yradjthetalev1}(a--d).  The main characteristics is
419    %consistency with expected advection of sea-ice over the relevant time
420    %scales considered.  The general positive pattern means that an
421    %increase in sea-ice thickness at location $(x,y)$ and time $t$ will
422    %increase sea-ice export through Fram Strait at time $T_e$.  Largest
423    %distances from Fram Strait indicate fastest sea-ice advection over the
424    %time span considered.  The ice thickness sensitivities are in close
425    %correspondence to ocean surface sentivitites, but of opposite sign.
426    %An increase in temperature will incur ice melting, decrease in ice
427    %thickness, and therefore decrease in sea-ice export at time $T_e$.
428    
429    %The picture is fundamentally different and much more complex
430    %for sensitivities to ocean temperatures away from the surface.
431    %\reffig{4yradjthetalev10??}(a--d) depicts ice export sensitivities to
432    %temperatures at roughly 400 m depth.
433    %Primary features are the effect of the heat transport of the North
434    %Atlantic current which feeds into the West Spitsbergen current,
435    %the circulation around Svalbard, and ...
436    
437    
438    %%\begin{figure}[t!]
439    %%\centerline{
440    %%\subfigure[{\footnotesize -12 months}]
441    %%{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim072_cmax2.0E+02.eps}}
442    %%\includegraphics*[width=.3\textwidth]{H_c.bin_res_100_lev1.pdf}
443    %%
444    %%\subfigure[{\footnotesize -24 months}]
445    %%{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim145_cmax2.0E+02.eps}}
446    %%}
447    %%
448    %%\caption{Sensitivity of sea-ice export through Fram Strait in December 2005 to
449    %%sea-ice thickness at various prior times.
450    %%\label{fig:4yradjheff}}
451    %%\end{figure}
452    
453    
454    %\ml{[based on the movie series
455    %  zzz\_run\_export\_canarch\_freeslip\_4yr\_1989\_ADJ*:]} The ice
456    %export through the Canadian Archipelag is highly sensitive to the
457    %previous state of the ocean-ice system in the Archipelago and the
458    %Western Arctic. According to the \ml{(adjoint)} senstivities of the
459    %eastward ice transport through Lancaster Sound (\reffig{arctic_topog},
460    %cross-section G) with respect to ice volume (effective thickness), ocean
461    %surface temperature, and vertical diffusivity near the surface
462    %(\reffig{fouryearadj}) after 4 years of integration the following
463    %mechanisms can be identified: near the ``observation'' (cross-section
464    %G), smaller vertical diffusivities lead to lower surface temperatures
465    %and hence to more ice that is available for export. Further away from
466    %cross-section G, the sensitivity to vertical diffusivity has the
467    %opposite sign, but temperature and ice volume sensitivities have the
468    %same sign as close to the observation.
469    
470    
471    %%% Local Variables:
472    %%% mode: latex
473    %%% TeX-master: "ceaice"
474    %%% End:
Legend:
Removed from v.1.1  
changed lines
  Added in v.1.5
  ViewVC Help
Powered by ViewVC 1.1.22