--- MITgcm_contrib/articles/ceaice/ceaice_adjoint.tex	2008/07/25 15:01:19	1.5
+++ MITgcm_contrib/articles/ceaice/ceaice_adjoint.tex	2008/07/28 04:34:37	1.6
@@ -71,7 +71,7 @@
 investigating sea-ice export sensitivities through Lancaster Sound.
 The rationale for doing so is to complement the analysis of sea-ice
 dynamics in the presence of narrow straits.  Lancaster Sound is one of
-the main outflow paths of sea-ice flowing through the Canadian Arctic
+the main paths of sea-ice flowing through the Canadian Arctic
 Archipelago (CAA).  Export sensitivities reflect dominant pathways
 through the CAA as resolved by the model.  Sensitivity maps can shed a
 very detailed light on various quantities affecting the sea-ice export
@@ -81,17 +81,12 @@
 \citep{mell:02, mich-etal:06,muen-etal:06}, which is not resolved in
 our simulation.
 
-The model domain is a coarsened version of the Arctic face of the
-high-resolution cubed-sphere configuration of the ECCO2 project
-\citep{menemenlis05} as described in \refsec{forward}.  The horizontal
-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
+The model domain is the same as the one described in \refsec{forward},
+but with halved horizontal resolution.
+The adjoint models run efficiently on 80 processors (as validated
 by benchmarks on both an SGI Altix and an IBM SP5 at NASA/ARC).
-
-Following a 3-year spinup, the model is integrated for four
-years and five months between January 1989 and September 1993.
-\ml{[Patrick: to what extent is this different from section 3?]}
+Following a 4-year spinup (1985 to 1988), the model is integrated for four
+years and nine months between January 1989 and September 1993.
 It is forced using realistic 6-hourly NCEP/NCAR atmospheric state variables. 
 %Over the open ocean these are
 %converted into air-sea fluxes via the bulk formulae of
@@ -99,24 +94,38 @@
 %sea-ice are handled by the ice model as described in \refsec{model}.
 The objective function $J$ is chosen as the ``solid'' fresh water
 export, that is the export of ice and snow converted to units of fresh
-water $(\rho_{i} h_{i}c + \rho_{s} h_{s}c)\,u$, through Lancaster
-Sound at approximately 82\degW\ (cross-section G in
-\reffig{arctic_topog}) averaged over a 12-month period between October
+water,
+%
+\begin{equation}
+J \, = \, (\rho_{i} h_{i}c + \rho_{s} h_{s}c)\,u
+\end{equation} 
+%
+through Lancaster Sound at approximately 82\degW\ (cross-section G in
+\reffig{arctic_topog}) averaged \ml{PH: Maybe integrated quantity is
+more physical} over the final 12-month of the integration between October
 1992 and September 1993.
 
 The forward trajectory of the model integration resembles broadly that
 of the model in \refsec{forward}. Many details are different, owning
 to different resolution and integration period; for example, the solid
 fresh water transport through Lancaster Sound is
+%
+\ml{PH: Martin, where did you get these numbers from?}
+%
 $116\pm101\text{\,km$^{3}$\,y$^{-1}$}$ for a free slip simulation with
 the C-LSOR solver, but only $39\pm64\text{\,km$^{3}$\,y$^{-1}$}$ for a
 no slip simulation.
 
-The adjoint model computes sensitivities of this export back in time
-from 1993 to 1989 along this trajectory.  In principle all adjoint
-model variable (i.e., Lagrange multipliers) of the coupled
-ocean/sea-ice model as well as the surface atmospheric state are
-available to analyze the transient sensitivity behavior.  Over the
+The adjoint model is the transpose of the tangent linear (or Jacobian) model
+operator. It runs backward in time, from September 1993 to
+January 1989. Along its integration it accumulates the Lagrange multipliers
+of the model subject to the objective function (solid freshwater export),
+which can be interpreted as sensitivities of the objective function
+to each control variable and each element of the intermediate 
+coupled model state variables.
+Thus, all sensitivity elements of the coupled
+ocean/sea-ice model state as well as the surface atmospheric state are
+available for analysis of the transient sensitivity behavior.  Over the
 open ocean, the adjoint of the bulk formula scheme computes
 sensitivities to the time-varying atmospheric state.  Over ice-covered
 parts, the sea-ice adjoint converts surface ocean sensitivities to
@@ -130,10 +139,11 @@
 effective ice thickness, $\partial{J} / \partial{(hc)}$.
 \reffig{adjheff} shows transient $\partial{J} / \partial{(hc)}$ using
 free-slip (left column) and no-slip (right column) boundary
-conditions. Sensitivity snapshots are depicted for 12 months prior to
-September 1993 (at the beginning of the averaging period for the objective
-function $J$, top) and at the beginning of the integration in January
-1989 (bottom).
+conditions. Sensitivity snapshots are depicted for beginning of October 2002,
+i.e. 12 months back in time from September 1993 
+(the beginning of the averaging period for the objective
+function $J$, top),
+and for Jannuary 1989, the beginning of the forward integration (bottom).
 \begin{figure*}[t]
   \includegraphics*[width=\textwidth]{\fpath/adjheff}
   \caption{Sensitivity $\partial{J}/\partial{(hc)}$ in
@@ -144,29 +154,52 @@
     \label{fig:adjheff}}
 \end{figure*}
 
-At the beginning of October 1992, the positive sensitivities in
-the Lancaster Sound mean that an increase of ice volume increase the
-solid fresh water export. The negative sensivities to the East and to the
-West can be explained by indirect effects: less ice to the East means
+As expected, the sensitivity patterns are predominantly positive,
+an increase in ice volume in most places ``upstream'' of
+Lancaster sound increases the solid fresh water export at the exit section.
+Also obvious is the transient nature of the sensitivity patterns
+(top panels vs. bottom panels),
+i.e. as time moves backward, an increasing area upstream of Lancaster Sound 
+contributes to the export sensitivity.
+The dominant pathway (free slip case) 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.
+
+The difference between the free slip and no slip solution is evident:
+by the end of the adjoint integration, in January 1989
+the free-slip sensitivities (bottom left) extend through most of the CAA
+and all the way into the Arctic interior, both to the West (M'Clure St.) 
+and to the North 
+(Ballantyne St., Prince Gustav Adolf Sea, Massey Sound), 
+whereas the no slip sensitivities (bottom right) are overall weaker 
+and remain mostly confined to Lancaster Sound and just West of Barrow Strait.
+In the free slip solution ice can drift more
+easily through narrow straits, and
+a positive ice volume anomaly further upstream in the CAA may increase
+ice export through the Lancaster Sound within a 4 year period.
+
+One peculiar feature in the October 1992 sensitivity maps (top panels)
+are the negative sensivities to the East and to the West.
+These can be explained by indirect effects: less ice to the East means
 less resistance to eastward drift and thus more export; similarly, less ice to
 the West means that more ice can be moved eastwards from the Barrow Strait
-into the Lancaster Sound leading to more ice export. The sensitivities
-are similar for both no slip and free slip solutions with a slightly larger
-area covered by non-zero sensitivities in the free slip solution. At
-the beginning of the integration (the end of the backward adjoint
-integration) the free and no slip solutions are very different. The
-sensitivities of the free slip solution extend through the enitre
-Canadian Archipelago and into the Arctic while in the no slip solution
-they still are confined to the Lancaster Sound and the Barrow
-Strait. This implies that in the free slip solution ice can drift more
-easily through the narrow straits of the Canadian Archipelago, so that
-a positive ice volume anomaly anywhere in the Canadian Archipelago is
-moved through the Lancaster Sound within 4 years thus increasing the
-ice export.
-
-The temporal evolution of several sensitivities along the zonal axis
-Lancaster Sound-Barrow Strait-Melville Sound are shown in
-\reffig{lancaster}.
+into the Lancaster Sound leading to more ice export. 
+\ml{PH: The first explanation (East) I buy, the second (West) I don't}.
+
+The temporal evolution of several ice export sensitivities 
+(eqn. XX) along a zonal axis through
+Lancaster Sound, Barrow Strait,and  Melville Sound 
+(115\degW\ to 80\degW\ ), 
+are depicted as Hovmueller diagrams in \reffig{lancaster}.
+From top to bottom, sensitivities are with respect to effective 
+ice thickness ($hc$), 
+ocean surface temperature ($SST$) and precipitation ($p$) for free slip 
+(left column) and no slip (right column) ice drift boundary conditions.
+%
 \begin{figure*}
   \includegraphics*[height=.8\textheight]{\fpath/lancaster_adj}
   \caption{Hovermoeller diagrams of sensitivities (derivatives) of the
@@ -180,42 +213,64 @@
     for orientation.
     \label{fig:lancaster}}
 \end{figure*}
-\reffig{lancaster} shows the sensitivities of ``solid'' fresh water
-export, that is ice and snow, through Lancaster sound (cross-section G
-in \reffig{arctic_topog}) with respect to effective ice thickness
-($hc$), ocean surface temperature (SST) and precipitation ($p$) for
-two runs with free slip and no slip boundary conditions for the sea
-ice drift. The Hovmoeller diagrams of sensitivities (derivatives) with
-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
+%
+
+The Hovmoeller diagrams of ice thickness (top row) and sea surface temperature
+(second row) sensitivities are coherent: 
+more ice in the Lancaster Sound leads
+to more export, and one way to get more ice is by colder surface
 temperatures (less melting from below). In the free slip case the
-sensitivities can propagate westwards (backwards in time) when the ice
-strength is low in late summer. In the no slip case the (normalized)
+sensitivities spread out in "pulses" following a seasonal cycle:
+can propagate westwards (backwards in time) when the ice
+strength is low in late summer, early autumn. 
+In contrast, during winter, the sensitivities show little to now
+westward propagation.
+In the no slip case the (normalized)
 ice strength does not fall below 1 during the winters of 1991 to 1993
 (mainly because the ice concentrations remain nearly 100\%, not
-shown), so that ice is blocked and cannot drift eastwards (forward in
-time) in the Melville Sound-Barrow Strait-Lancaster Sound channel.
-Consequently the sensitivies do not propagate westwards (backwards in
+shown). Ice is therefore blocked and cannot drift eastwards 
+(forward in time) through the 
+Melville Sound, Barrow Strait, Lancaster Sound channel system.
+Consequently, the sensitivities do not propagate westwards (backwards in
 time) and the export through Lancaster Sound is only affected by
-local ice formation and melting.
+local ice formation and melting for the entire integration period.
 
-The sensitivities to precipitation are negative (more precipitation
-leads to less export) before January and mostly positive after
-January. Further they are mostly positive for normalized ice strengths
-over 3. Assuming that most precipation is snow in this area---in the
+The sensitivities to precipitation exhibit an oscillatory behaviour:
+they are negative (more precipitation leads to less export) 
+before January (more precisely, late fall) and mostly positive after January
+(more precisely, January through July). 
+Times of positive sensitivities coincide with times of
+normalized ice strengths exceeding values of 3.
+%
+\ml{PH: Problem is, that's not true for the first two years (backward),
+East of 95\degW\ , i.e. in Lancaster Sound.
+For example, at 90\degW\ the sensitivities are negative throughout 1992,
+and no clear correlation to ice strength is apparent there.}.
+%
+Assuming that most precipation is snow in this area
+%
+\footnote{
+In the
 current implementation the model differentiates between snow and rain
 depending on the thermodynamic growth rate; when it is cold enough for
-ice to grow, all precipitation is assumed to be snow---the
-sensitivities can be interpreted in terms of the model physics.  Short
-wave radiation cannot penetrate a snow cover and has a higer albedo
+ice to grow, all precipitation is assumed to be snow.}
+%
+the sensitivities can be interpreted in terms of the model physics.  Short
+wave radiation cannot penetrate the snow cover and has a higer albedo
 than ice (0.85 for dry snow and 0.75 for dry ice in our case); thus it
-protects the ice against melting in spring (after January).  On the
-other hand, snow reduces the effective conductivity and thus the heat
+protects the ice against melting in spring (after January).  
+\ml{PH: what about the direct effect of accumulation of precip. as snow
+which directly increases the volume.}.
+
+On the other hand, snow reduces the effective conductivity and thus the heat
 flux through the ice. This insulating effect slows down the cooling of
 the surface water underneath the ice and limits the ice growth from
 below, so that less snow in the ice-growing season leads to more new
 ice and thus more ice export.
+\ml{PH: Should probably discuss the effect of snow vs. rain.
+To me it seems that the "rain" effect doesn't really play
+because the neg. sensitivities are too late in the fall,
+probably mostly falling as snow.}.
 
 %Und jetzt weiss ich nicht mehr weiter, aber nun kann folgendes passiert sein:
 %1. snow insulates against melting from above during spring: more precip (snow) -> more export
@@ -231,33 +286,7 @@
   Sound and then produce plots similar to reffig{lancaster}. For
   PRECIP it would be great to have two perturbation experiments, one
   where ADJprecip is posivite and one where ADJprecip is negative]}
-%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)
+  
 
 %(*)
 %The sensitivity in Baffin Bay are more complex.
