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revision 1.5 by mlosch, Fri Jul 25 15:01:19 2008 UTC revision 1.6 by heimbach, Mon Jul 28 04:34:37 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    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} over the final 12-month of the integration between October
106  1992 and September 1993.  1992 and September 1993.
107    
108  The forward trajectory of the model integration resembles broadly that  The forward trajectory of the model integration resembles broadly that
109  of the model in \refsec{forward}. Many details are different, owning  of the model in \refsec{forward}. Many details are different, owning
110  to different resolution and integration period; for example, the solid  to different resolution and integration period; for example, the solid
111  fresh water transport through Lancaster Sound is  fresh water transport through Lancaster Sound is
112    %
113    \ml{PH: Martin, where did you get these numbers from?}
114    %
115  $116\pm101\text{\,km$^{3}$\,y$^{-1}$}$ for a free slip simulation with  $116\pm101\text{\,km$^{3}$\,y$^{-1}$}$ for a free slip simulation with
116  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
117  no slip simulation.  no slip simulation.
118    
119  The adjoint model computes sensitivities of this export back in time  The adjoint model is the transpose of the tangent linear (or Jacobian) model
120  from 1993 to 1989 along this trajectory.  In principle all adjoint  operator. It runs backward in time, from September 1993 to
121  model variable (i.e., Lagrange multipliers) of the coupled  January 1989. Along its integration it accumulates the Lagrange multipliers
122  ocean/sea-ice model as well as the surface atmospheric state are  of the model subject to the objective function (solid freshwater export),
123  available to analyze the transient sensitivity behavior.  Over the  which can be interpreted as sensitivities of the objective function
124    to each control variable and each element of the intermediate
125    coupled model state variables.
126    Thus, all sensitivity elements of the coupled
127    ocean/sea-ice model state as well as the surface atmospheric state are
128    available for analysis of the transient sensitivity behavior.  Over the
129  open ocean, the adjoint of the bulk formula scheme computes  open ocean, the adjoint of the bulk formula scheme computes
130  sensitivities to the time-varying atmospheric state.  Over ice-covered  sensitivities to the time-varying atmospheric state.  Over ice-covered
131  parts, the sea-ice adjoint converts surface ocean sensitivities to  parts, the sea-ice adjoint converts surface ocean sensitivities to
# Line 130  The most readily interpretable ice-expor Line 139  The most readily interpretable ice-expor
139  effective ice thickness, $\partial{J} / \partial{(hc)}$.  effective ice thickness, $\partial{J} / \partial{(hc)}$.
140  \reffig{adjheff} shows transient $\partial{J} / \partial{(hc)}$ using  \reffig{adjheff} shows transient $\partial{J} / \partial{(hc)}$ using
141  free-slip (left column) and no-slip (right column) boundary  free-slip (left column) and no-slip (right column) boundary
142  conditions. Sensitivity snapshots are depicted for 12 months prior to  conditions. Sensitivity snapshots are depicted for beginning of October 2002,
143  September 1993 (at the beginning of the averaging period for the objective  i.e. 12 months back in time from September 1993
144  function $J$, top) and at the beginning of the integration in January  (the beginning of the averaging period for the objective
145  1989 (bottom).  function $J$, top),
146    and for Jannuary 1989, the beginning of the forward integration (bottom).
147  \begin{figure*}[t]  \begin{figure*}[t]
148    \includegraphics*[width=\textwidth]{\fpath/adjheff}    \includegraphics*[width=\textwidth]{\fpath/adjheff}
149    \caption{Sensitivity $\partial{J}/\partial{(hc)}$ in    \caption{Sensitivity $\partial{J}/\partial{(hc)}$ in
# Line 144  function $J$, top) and at the beginning Line 154  function $J$, top) and at the beginning
154      \label{fig:adjheff}}      \label{fig:adjheff}}
155  \end{figure*}  \end{figure*}
156    
157  At the beginning of October 1992, the positive sensitivities in  As expected, the sensitivity patterns are predominantly positive,
158  the Lancaster Sound mean that an increase of ice volume increase the  an increase in ice volume in most places ``upstream'' of
159  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.
160  West can be explained by indirect effects: less ice to the East means  Also obvious is the transient nature of the sensitivity patterns
161    (top panels vs. bottom panels),
162    i.e. as time moves backward, an increasing area upstream of Lancaster Sound
163    contributes to the export sensitivity.
164    The dominant pathway (free slip case) follows (backward in time)
165    through Barrow Strait
166    into Viscount Melville Sound, and from there trough M'Clure Strait
167    into the Arctic Ocean (the ``Northwest Passage'').
168    Secondary paths are Northward from
169    Viscount Melville Sound through Byam Martin Channel into
170    Prince Gustav Adolf Sea and through Penny Strait into MacLean Strait.
171    
172    The difference between the free slip and no slip solution is evident:
173    by the end of the adjoint integration, in January 1989
174    the free-slip sensitivities (bottom left) extend through most of the CAA
175    and all the way into the Arctic interior, both to the West (M'Clure St.)
176    and to the North
177    (Ballantyne St., Prince Gustav Adolf Sea, Massey Sound),
178    whereas the no slip sensitivities (bottom right) are overall weaker
179    and remain mostly confined to Lancaster Sound and just West of Barrow Strait.
180    In the free slip solution ice can drift more
181    easily through narrow straits, and
182    a positive ice volume anomaly further upstream in the CAA may increase
183    ice export through the Lancaster Sound within a 4 year period.
184    
185    One peculiar feature in the October 1992 sensitivity maps (top panels)
186    are the negative sensivities to the East and to the West.
187    These can be explained by indirect effects: less ice to the East means
188  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
189  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
190  into the Lancaster Sound leading to more ice export. The sensitivities  into the Lancaster Sound leading to more ice export.
191  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 don't}.
192  area covered by non-zero sensitivities in the free slip solution. At  
193  the beginning of the integration (the end of the backward adjoint  The temporal evolution of several ice export sensitivities
194  integration) the free and no slip solutions are very different. The  (eqn. XX) along a zonal axis through
195  sensitivities of the free slip solution extend through the enitre  Lancaster Sound, Barrow Strait,and  Melville Sound
196  Canadian Archipelago and into the Arctic while in the no slip solution  (115\degW\ to 80\degW\ ),
197  they still are confined to the Lancaster Sound and the Barrow  are depicted as Hovmueller diagrams in \reffig{lancaster}.
198  Strait. This implies that in the free slip solution ice can drift more  From top to bottom, sensitivities are with respect to effective
199  easily through the narrow straits of the Canadian Archipelago, so that  ice thickness ($hc$),
200  a positive ice volume anomaly anywhere in the Canadian Archipelago is  ocean surface temperature ($SST$) and precipitation ($p$) for free slip
201  moved through the Lancaster Sound within 4 years thus increasing the  (left column) and no slip (right column) ice drift boundary conditions.
202  ice export.  %
   
 The temporal evolution of several sensitivities along the zonal axis  
 Lancaster Sound-Barrow Strait-Melville Sound are shown in  
 \reffig{lancaster}.  
203  \begin{figure*}  \begin{figure*}
204    \includegraphics*[height=.8\textheight]{\fpath/lancaster_adj}    \includegraphics*[height=.8\textheight]{\fpath/lancaster_adj}
205    \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 213  Lancaster Sound-Barrow Strait-Melville S
213      for orientation.      for orientation.
214      \label{fig:lancaster}}      \label{fig:lancaster}}
215  \end{figure*}  \end{figure*}
216  \reffig{lancaster} shows the sensitivities of ``solid'' fresh water  %
217  export, that is ice and snow, through Lancaster sound (cross-section G  
218  in \reffig{arctic_topog}) with respect to effective ice thickness  The Hovmoeller diagrams of ice thickness (top row) and sea surface temperature
219  ($hc$), ocean surface temperature (SST) and precipitation ($p$) for  (second row) sensitivities are coherent:
220  two runs with free slip and no slip boundary conditions for the sea  more ice in the Lancaster Sound leads
221  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  
222  temperatures (less melting from below). In the free slip case the  temperatures (less melting from below). In the free slip case the
223  sensitivities can propagate westwards (backwards in time) when the ice  sensitivities spread out in "pulses" following a seasonal cycle:
224  strength is low in late summer. In the no slip case the (normalized)  can propagate westwards (backwards in time) when the ice
225    strength is low in late summer, early autumn.
226    In contrast, during winter, the sensitivities show little to now
227    westward propagation.
228    In the no slip case the (normalized)
229  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
230  (mainly because the ice concentrations remain nearly 100\%, not  (mainly because the ice concentrations remain nearly 100\%, not
231  shown), so that ice is blocked and cannot drift eastwards (forward in  shown). Ice is therefore blocked and cannot drift eastwards
232  time) in the Melville Sound-Barrow Strait-Lancaster Sound channel.  (forward in time) through the
233  Consequently the sensitivies do not propagate westwards (backwards in  Melville Sound, Barrow Strait, Lancaster Sound channel system.
234    Consequently, the sensitivities do not propagate westwards (backwards in
235  time) and the export through Lancaster Sound is only affected by  time) and the export through Lancaster Sound is only affected by
236  local ice formation and melting.  local ice formation and melting for the entire integration period.
237    
238  The sensitivities to precipitation are negative (more precipitation  The sensitivities to precipitation exhibit an oscillatory behaviour:
239  leads to less export) before January and mostly positive after  they are negative (more precipitation leads to less export)
240  January. Further they are mostly positive for normalized ice strengths  before January (more precisely, late fall) and mostly positive after January
241  over 3. Assuming that most precipation is snow in this area---in the  (more precisely, January through July).
242    Times of positive sensitivities coincide with times of
243    normalized ice strengths exceeding values of 3.
244    %
245    \ml{PH: Problem is, that's not true for the first two years (backward),
246    East of 95\degW\ , i.e. in Lancaster Sound.
247    For example, at 90\degW\ the sensitivities are negative throughout 1992,
248    and no clear correlation to ice strength is apparent there.}.
249    %
250    Assuming that most precipation is snow in this area
251    %
252    \footnote{
253    In the
254  current implementation the model differentiates between snow and rain  current implementation the model differentiates between snow and rain
255  depending on the thermodynamic growth rate; when it is cold enough for  depending on the thermodynamic growth rate; when it is cold enough for
256  ice to grow, all precipitation is assumed to be snow---the  ice to grow, all precipitation is assumed to be snow.}
257  sensitivities can be interpreted in terms of the model physics.  Short  %
258  wave radiation cannot penetrate a snow cover and has a higer albedo  the sensitivities can be interpreted in terms of the model physics.  Short
259    wave radiation cannot penetrate the snow cover and has a higer albedo
260  than ice (0.85 for dry snow and 0.75 for dry ice in our case); thus it  than ice (0.85 for dry snow and 0.75 for dry ice in our case); thus it
261  protects the ice against melting in spring (after January).  On the  protects the ice against melting in spring (after January).  
262  other hand, snow reduces the effective conductivity and thus the heat  \ml{PH: what about the direct effect of accumulation of precip. as snow
263    which directly increases the volume.}.
264    
265    On the other hand, snow reduces the effective conductivity and thus the heat
266  flux through the ice. This insulating effect slows down the cooling of  flux through the ice. This insulating effect slows down the cooling of
267  the surface water underneath the ice and limits the ice growth from  the surface water underneath the ice and limits the ice growth from
268  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
269  ice and thus more ice export.  ice and thus more ice export.
270    \ml{PH: Should probably discuss the effect of snow vs. rain.
271    To me it seems that the "rain" effect doesn't really play
272    because the neg. sensitivities are too late in the fall,
273    probably mostly falling as snow.}.
274    
275  %Und jetzt weiss ich nicht mehr weiter, aber nun kann folgendes passiert sein:  %Und jetzt weiss ich nicht mehr weiter, aber nun kann folgendes passiert sein:
276  %1. snow insulates against melting from above during spring: more precip (snow) -> more export  %1. snow insulates against melting from above during spring: more precip (snow) -> more export
# Line 231  ice and thus more ice export. Line 286  ice and thus more ice export.
286    Sound and then produce plots similar to reffig{lancaster}. For    Sound and then produce plots similar to reffig{lancaster}. For
287    PRECIP it would be great to have two perturbation experiments, one    PRECIP it would be great to have two perturbation experiments, one
288    where ADJprecip is posivite and one where ADJprecip is negative]}    where ADJprecip is posivite and one where ADJprecip is negative]}
289  %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)  
290    
291  %(*)  %(*)
292  %The sensitivity in Baffin Bay are more complex.  %The sensitivity in Baffin Bay are more complex.
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