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revision 1.5 by mlosch, Fri Jul 25 15:01:19 2008 UTC revision 1.7 by mlosch, Mon Jul 28 12:34:27 2008 UTC
# Line 71  We demonstrate the power of the adjoint Line 71  We demonstrate the power of the adjoint
71  investigating sea-ice export sensitivities through Lancaster Sound.  investigating sea-ice export sensitivities through Lancaster Sound.
72  The rationale for doing so is to complement the analysis of sea-ice  The rationale for doing so is to complement the analysis of sea-ice
73  dynamics in the presence of narrow straits.  Lancaster Sound is one of  dynamics in the presence of narrow straits.  Lancaster Sound is one of
74  the main outflow paths of sea-ice flowing through the Canadian Arctic  the main paths of sea-ice flowing through the Canadian Arctic
75  Archipelago (CAA).  Export sensitivities reflect dominant pathways  Archipelago (CAA).  Export sensitivities reflect dominant pathways
76  through the CAA as resolved by the model.  Sensitivity maps can shed a  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  very detailed light on various quantities affecting the sea-ice export
# Line 81  small Westward flow to the North, huggin Line 81  small Westward flow to the North, huggin
81  \citep{mell:02, mich-etal:06,muen-etal:06}, which is not resolved in  \citep{mell:02, mich-etal:06,muen-etal:06}, which is not resolved in
82  our simulation.  our simulation.
83    
84  The model domain is a coarsened version of the Arctic face of the  The model domain is the same as the one described in \refsec{forward},
85  high-resolution cubed-sphere configuration of the ECCO2 project  but with halved horizontal resolution.
86  \citep{menemenlis05} as described in \refsec{forward}.  The horizontal  The adjoint models run efficiently on 80 processors (as validated
 resolution is half of that in \refsec{forward} while the vertical grid  
 is the same. \ml{[Is this important? Do we need to be more specific?:  
   ]} The adjoint models run efficiently on 80 processors (as validated  
87  by benchmarks on both an SGI Altix and an IBM SP5 at NASA/ARC).  by benchmarks on both an SGI Altix and an IBM SP5 at NASA/ARC).
88    Following a 4-year spinup (1985 to 1988), the model is integrated for four
89  Following a 3-year spinup, the model is integrated for four  years and nine months between January 1989 and September 1993.
 years and five months between January 1989 and September 1993.  
 \ml{[Patrick: to what extent is this different from section 3?]}  
90  It is forced using realistic 6-hourly NCEP/NCAR atmospheric state variables.  It is forced using realistic 6-hourly NCEP/NCAR atmospheric state variables.
91  %Over the open ocean these are  %Over the open ocean these are
92  %converted into air-sea fluxes via the bulk formulae of  %converted into air-sea fluxes via the bulk formulae of
# Line 99  It is forced using realistic 6-hourly NC Line 94  It is forced using realistic 6-hourly NC
94  %sea-ice are handled by the ice model as described in \refsec{model}.  %sea-ice are handled by the ice model as described in \refsec{model}.
95  The objective function $J$ is chosen as the ``solid'' fresh water  The objective function $J$ is chosen as the ``solid'' fresh water
96  export, that is the export of ice and snow converted to units of fresh  export, that is the export of ice and snow converted to units of fresh
97  water $(\rho_{i} h_{i}c + \rho_{s} h_{s}c)\,u$, through Lancaster  water,
98  Sound at approximately 82\degW\ (cross-section G in  %
99  \reffig{arctic_topog}) averaged over a 12-month period between October  \begin{equation}
100  1992 and September 1993.  J \, = \, (\rho_{i} h_{i}c + \rho_{s} h_{s}c)\,u
101    \end{equation}
102    %
103    through Lancaster Sound at approximately 82\degW\ (cross-section G in
104    \reffig{arctic_topog}) averaged \ml{PH: Maybe integrated quantity is
105    more physical; ML: what did you actually compute? I did not scale
106    anything, yet. Please insert what is actually done.} over the final
107    12-month of the integration between October 1992 and September 1993.
108    
109  The forward trajectory of the model integration resembles broadly that  The forward trajectory of the model integration resembles broadly that
110  of the model in \refsec{forward}. Many details are different, owning  of the model in \refsec{forward}. Many details are different, owning
111  to different resolution and integration period; for example, the solid  to different resolution and integration period; for example, the solid
112  fresh water transport through Lancaster Sound is  fresh water transport through Lancaster Sound is
113    %
114    \ml{PH: Martin, where did you get these numbers from?}
115    \ml{[ML: I computed hu = -sum((SIheff+SIhsnow)*SIuice*area)/sum(area) at
116    $i=100,j=116:122$, and then took mean(hu) and std(hu). What are your numbers?]}
117    %
118  $116\pm101\text{\,km$^{3}$\,y$^{-1}$}$ for a free slip simulation with  $116\pm101\text{\,km$^{3}$\,y$^{-1}$}$ for a free slip simulation with
119  the C-LSOR solver, but only $39\pm64\text{\,km$^{3}$\,y$^{-1}$}$ for a  the C-LSOR solver, but only $39\pm64\text{\,km$^{3}$\,y$^{-1}$}$ for a
120  no slip simulation.  no slip simulation. \ml{[Here we can say that the export through
121      Lancaster Sound is highly uncertain, making is a perfect candidate
122  The adjoint model computes sensitivities of this export back in time    for sensitivity, bla bla?]}
123  from 1993 to 1989 along this trajectory.  In principle all adjoint  
124  model variable (i.e., Lagrange multipliers) of the coupled  The adjoint model is the transpose of the tangent linear (or Jacobian) model
125  ocean/sea-ice model as well as the surface atmospheric state are  operator. It runs backwards in time, from September 1993 to
126  available to analyze the transient sensitivity behavior.  Over the  January 1989. During its integration it accumulates the Lagrange multipliers
127    of the model subject to the objective function (solid freshwater export),
128    which can be interpreted as sensitivities of the objective function
129    to each control variable and each element of the intermediate
130    coupled model state variables.
131    Thus, all sensitivity elements of the coupled
132    ocean/sea-ice model state as well as the surface atmospheric state are
133    available for analysis of the transient sensitivity behavior.  Over the
134  open ocean, the adjoint of the bulk formula scheme computes  open ocean, the adjoint of the bulk formula scheme computes
135  sensitivities to the time-varying atmospheric state.  Over ice-covered  sensitivities to the time-varying atmospheric state.  Over ice-covered
136  parts, the sea-ice adjoint converts surface ocean sensitivities to  areas, the sea-ice adjoint converts surface ocean sensitivities to
137  atmospheric sensitivities.  atmospheric sensitivities.
138    
139  DISCUSS FORWARD STATE, INCLUDING SOME NUMBERS ON SEA-ICE EXPORT  DISCUSS FORWARD STATE, INCLUDING SOME NUMBERS ON SEA-ICE EXPORT
# Line 130  The most readily interpretable ice-expor Line 144  The most readily interpretable ice-expor
144  effective ice thickness, $\partial{J} / \partial{(hc)}$.  effective ice thickness, $\partial{J} / \partial{(hc)}$.
145  \reffig{adjheff} shows transient $\partial{J} / \partial{(hc)}$ using  \reffig{adjheff} shows transient $\partial{J} / \partial{(hc)}$ using
146  free-slip (left column) and no-slip (right column) boundary  free-slip (left column) and no-slip (right column) boundary
147  conditions. Sensitivity snapshots are depicted for 12 months prior to  conditions. Sensitivity snapshots are depicted for beginning of October 1992,
148  September 1993 (at the beginning of the averaging period for the objective  that is 12 months before September 1993
149  function $J$, top) and at the beginning of the integration in January  (the beginning of the averaging period for the objective
150  1989 (bottom).  function $J$, top),
151    and for Jannuary 1989, the beginning of the forward integration (bottom).
152  \begin{figure*}[t]  \begin{figure*}[t]
153    \includegraphics*[width=\textwidth]{\fpath/adjheff}    \includegraphics*[width=\textwidth]{\fpath/adjheff}
154    \caption{Sensitivity $\partial{J}/\partial{(hc)}$ in    \caption{Sensitivity $\partial{J}/\partial{(hc)}$ in
# Line 144  function $J$, top) and at the beginning Line 159  function $J$, top) and at the beginning
159      \label{fig:adjheff}}      \label{fig:adjheff}}
160  \end{figure*}  \end{figure*}
161    
162  At the beginning of October 1992, the positive sensitivities in  The sensitivity patterns for effective ice thickness are predominantly positive.
163  the Lancaster Sound mean that an increase of ice volume increase the  An increase in ice volume in most places ``upstream'' of
164  solid fresh water export. The negative sensivities to the East and to the  Lancaster sound increases the solid fresh water export at the exit section.
165  West can be explained by indirect effects: less ice to the East means  The transient nature of the sensitivity patterns
166    (top panels vs. bottom panels) is also obvious:
167    the area upstream of the Lancaster Sound that
168    contributes to the export sensitivity is larger in the earlier snapshot.
169    In the free slip case, the sensivity follows (backwards in time) the dominant pathway
170    through the Barrow Strait
171    into the Viscount Melville Sound, and from there trough the M'Clure Strait
172    into the Arctic Ocean (the ``Northwest Passage''). \ml{[Is that really
173      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  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  the West means that more ice can be moved eastwards from the Barrow Strait
204  into the Lancaster Sound leading to more ice export. The sensitivities  into the Lancaster Sound leading to more ice export.
205  are similar for both no slip and free slip solutions with a slightly larger  \ml{PH: The first explanation (East) I buy, the second (West) I
206  area covered by non-zero sensitivities in the free slip solution. At    don't.} \ml{[ML: unfortunately, I don't have anything better to
207  the beginning of the integration (the end of the backward adjoint    offer, do you? Keep in mind that these sensitivites are very small
208  integration) the free and no slip solutions are very different. The    and only show up, because of the colorscale. In Fig6, they are
209  sensitivities of the free slip solution extend through the enitre    hardly visible.]}
210  Canadian Archipelago and into the Arctic while in the no slip solution  
211  they still are confined to the Lancaster Sound and the Barrow  The temporal evolution of several ice export sensitivities (eqn. XX,
212  Strait. This implies that in the free slip solution ice can drift more  \ml{[which equation do you mean?]}) along a zonal axis through
213  easily through the narrow straits of the Canadian Archipelago, so that  Lancaster Sound, Barrow Strait, and Melville Sound (115\degW\ to
214  a positive ice volume anomaly anywhere in the Canadian Archipelago is  80\degW, averaged across the passages) are depicted as Hovmueller
215  moved through the Lancaster Sound within 4 years thus increasing the  diagrams in \reffig{lancaster}. These are, from top to bottom, the
216  ice export.  sensitivities with respect to effective ice thickness ($hc$), ocean
217    surface temperature ($SST$) and precipitation ($p$) for free slip
218  The temporal evolution of several sensitivities along the zonal axis  (left column) and no slip (right column) ice drift boundary
219  Lancaster Sound-Barrow Strait-Melville Sound are shown in  conditions.
220  \reffig{lancaster}.  %
221  \begin{figure*}  \begin{figure*}
222    \includegraphics*[height=.8\textheight]{\fpath/lancaster_adj}    \includegraphics*[height=.8\textheight]{\fpath/lancaster_adj}
223    \caption{Hovermoeller diagrams of sensitivities (derivatives) of the    \caption{Hovermoeller diagrams of sensitivities (derivatives) of the
# Line 180  Lancaster Sound-Barrow Strait-Melville S Line 231  Lancaster Sound-Barrow Strait-Melville S
231      for orientation.      for orientation.
232      \label{fig:lancaster}}      \label{fig:lancaster}}
233  \end{figure*}  \end{figure*}
234  \reffig{lancaster} shows the sensitivities of ``solid'' fresh water  %
235  export, that is ice and snow, through Lancaster sound (cross-section G  
236  in \reffig{arctic_topog}) with respect to effective ice thickness  The Hovmoeller diagrams of ice thickness (top row) and sea surface temperature
237  ($hc$), ocean surface temperature (SST) and precipitation ($p$) for  (second row) sensitivities are coherent:
238  two runs with free slip and no slip boundary conditions for the sea  more ice in the Lancaster Sound leads
239  ice drift. The Hovmoeller diagrams of sensitivities (derivatives) with  to more export, and one way to get more ice is by colder surface
 respect to effective ice thickness (top) and ocean surface temperature  
 (second from top) are coherent: more ice in the Lancaster Sound leads  
 to more export and one way to get more ice is by colder surface  
240  temperatures (less melting from below). In the free slip case the  temperatures (less melting from below). In the free slip case the
241  sensitivities can propagate westwards (backwards in time) when the ice  sensitivities spread out in "pulses" following a seasonal cycle:
242  strength is low in late summer. In the no slip case the (normalized)  ice can propagate eastwards (forward in time and thus sensitivites can
243    propagate westwards (backwards in time) when the ice strength is low
244    in late summer to early autumn.  
245    In contrast, during winter, the sensitivities show little to now
246    westward propagation, as the ice is frozen solid and does not move.
247    In the no slip case the (normalized)
248  ice strength does not fall below 1 during the winters of 1991 to 1993  ice strength does not fall below 1 during the winters of 1991 to 1993
249  (mainly because the ice concentrations remain nearly 100\%, not  (mainly because the ice concentrations remain near 100\%, not
250  shown), so that ice is blocked and cannot drift eastwards (forward in  shown). Ice is therefore blocked and cannot drift eastwards
251  time) in the Melville Sound-Barrow Strait-Lancaster Sound channel.  (forward in time) through the Viscount
252  Consequently the sensitivies do not propagate westwards (backwards in  Melville Sound, Barrow Strait, Lancaster Sound channel system.
253    Consequently, the sensitivities do not propagate westwards (backwards in
254  time) and the export through Lancaster Sound is only affected by  time) and the export through Lancaster Sound is only affected by
255  local ice formation and melting.  local ice formation and melting for the entire integration period.
256    
257  The sensitivities to precipitation are negative (more precipitation  The sensitivities to precipitation exhibit an oscillatory behaviour:
258  leads to less export) before January and mostly positive after  they are negative (more precipitation leads to less export)
259  January. Further they are mostly positive for normalized ice strengths  before January (more precisely, late fall) and mostly positive after January
260  over 3. Assuming that most precipation is snow in this area---in the  (more precisely, January through July).
261    Times of positive sensitivities coincide with times of
262    normalized ice strengths exceeding values of 3
263    %
264    \ml{PH: Problem is, that's not true for the first two years (backward),
265    east of 95\degW, that is, in the Lancaster Sound.
266    For example, at 90\degW\ the sensitivities are negative throughout 1992,
267    and no clear correlation to ice strength is apparent there.}
268    except between 95\degW\ and 85\degW, which is an area of
269    increased snow cover in spring. \ml{[ML: and no, I cannot explain
270      that. Can you?]}
271    
272    %
273    Assuming that most precipation is snow in this area\footnote{
274    In the
275  current implementation the model differentiates between snow and rain  current implementation the model differentiates between snow and rain
276  depending on the thermodynamic growth rate; when it is cold enough for  depending on the thermodynamic growth rate; when it is cold enough for
277  ice to grow, all precipitation is assumed to be snow---the  ice to grow, all precipitation is assumed to be snow.}
278  sensitivities can be interpreted in terms of the model physics.  Short  %
279  wave radiation cannot penetrate a snow cover and has a higer albedo  the sensitivities can be interpreted in terms of the model physics.
280  than ice (0.85 for dry snow and 0.75 for dry ice in our case); thus it  The accumulation of snow directly increases the exported volume.
281  protects the ice against melting in spring (after January).  On the  Further, short wave radiation cannot penetrate the snow cover and has
282  other hand, snow reduces the effective conductivity and thus the heat  a higer albedo than ice (0.85 for dry snow and 0.75 for dry ice in our
283    case); thus it protects the ice against melting in spring (after
284    January).
285    
286    On the other hand, snow reduces the effective conductivity and thus the heat
287  flux through the ice. This insulating effect slows down the cooling of  flux through the ice. This insulating effect slows down the cooling of
288  the surface water underneath the ice and limits the ice growth from  the surface water underneath the ice and limits the ice growth from
289  below, so that less snow in the ice-growing season leads to more new  below, so that less snow in the ice-growing season leads to more new
290  ice and thus more ice export.  ice and thus more ice export.
291    \ml{PH: Should probably discuss the effect of snow vs. rain.
292  %Und jetzt weiss ich nicht mehr weiter, aber nun kann folgendes passiert sein:  To me it seems that the "rain" effect doesn't really play a role
293  %1. snow insulates against melting from above during spring: more precip (snow) -> more export  because the neg. sensitivities are too late in the fall,
294  %2. less snow during fall -> more ice -> more export  probably mostly falling as snow.} \ml{[ML: correct, I looked at
295  %3. precip is both snow and rain, depending on the sign of "FICE" (thermodynamic growth rate), with probably different implications  NCEP/CORE air temperatures, and they are hardly above freezing in
296    Jul/Aug, but otherwise below freezing, that why I can assume that most
297    precip is snow. ]} \ml{[this is not very good but do you have anything
298    better?:]}
299    The negative sensitivities to precipitation between 95\degW\ and
300    85\degW\ in spring 1992 may be explained by a similar mechanism: in an
301    area of thick snow (almost 50\,cm), ice cannot melt and tends to block
302    the channel so that ice coming in from the West cannot pass thus
303    leading to less ice export in the next season.
304    
305  \subsubsection{Forward sensitivities}  \subsubsection{Forward sensitivities}
306    
# Line 231  ice and thus more ice export. Line 310  ice and thus more ice export.
310    Sound and then produce plots similar to reffig{lancaster}. For    Sound and then produce plots similar to reffig{lancaster}. For
311    PRECIP it would be great to have two perturbation experiments, one    PRECIP it would be great to have two perturbation experiments, one
312    where ADJprecip is posivite and one where ADJprecip is negative]}    where ADJprecip is posivite and one where ADJprecip is negative]}
313  %The dominant features are\ml{ in accordance with expectations/as expected}:    
   
 %(*)  
 %Dominant pattern (for the free-slip run) is that of positive sensitivities, i.e.  
 %a unit increase in sea-ice thickness in most places upstream  
 %of Lancaster Sound will increase sea-ice export through Lancaster Sound.  
 %The dominant pathway follows (backward in time) through Barrow Strait  
 %into Viscount Melville Sound, and from there trough M'Clure Strait  
 %into the Arctic Ocean (the "Northwest Passage").  
 %Secondary paths are Northward from  
 %Viscount Melville Sound through Byam Martin Channel into  
 %Prince Gustav Adolf Sea and through Penny Strait into MacLean Strait.  
   
 %(*)  
 %As expected, at any given time the  
 %region of influence is larger for the free-slip than no-slip simulation.  
 %For the no-slip run, the region of influence is confined, after four years,  
 %to just West of Barrow Strait (North of Prince of Wales Island),  
 %and to the South of Penny Strait.  
 %In contrast, sensitivities of the free-slip run extend  
 %all the way to the Arctic interior both to the West  
 %(M'Clure St.) and to the North (Ballantyne St., Prince Gustav Adolf Sea,  
 %Massey Sound).  
   
 %(*)  
 %sensitivities seem to spread out in "pulses" (seasonal cycle)  
 %[PLOT A TIME SERIES OF ADJheff in Barrow Strait)  
314    
315  %(*)  %(*)
316  %The sensitivity in Baffin Bay are more complex.  %The sensitivity in Baffin Bay are more complex.
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