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-revision 1.5 by mlosch,
Fri Jul 25 15:01:19 2008 UTC
+revision 1.8 by mlosch,
Tue Jul 29 08:34:22 2008 UTC
 Line 71 
 We demonstrate the power of the adjoint
  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
 Line 81 
 small Westward flow to the North, huggin
  \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
+ The model domain is the same as the one described in \refsec{forward},
- high-resolution cubed-sphere configuration of the ECCO2 project
+ but with halved horizontal resolution.
- \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
  by benchmarks on both an SGI Altix and an IBM SP5 at NASA/ARC).
+ Following a 4-year spinup (1985 to 1988), the model is integrated for four
- 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?]}
  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
-Line 99 
 It is forced using realistic 6-hourly NC
+Line 94 
 It is forced using realistic 6-hourly NC
  %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
+ water,
- Sound at approximately 82\degW\ (cross-section G in
+ %
- \reffig{arctic_topog}) averaged over a 12-month period between October
+ \begin{equation}
-and September 1993.
+ 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; ML: what did you actually compute? I did not scale
+ anything, yet. Please insert what is actually done.} over the final
+-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?}
+ \ml{[ML: I computed hu = -sum((SIheff+SIhsnow)*SIuice*area)/sum(area) at
+ $i=100,j=116:122$, and then took mean(hu) and std(hu). What are your numbers?]}
+ %
  $116\pm101\text{\,km$^{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.
+ no slip simulation. \ml{[Here we can say that the export through
+   Lancaster Sound is highly uncertain, making is a perfect candidate
- The adjoint model computes sensitivities of this export back in time
+   for sensitivity, bla bla?]}
- from 1993 to 1989 along this trajectory.  In principle all adjoint
- model variable (i.e., Lagrange multipliers) of the coupled
+ The adjoint model is the transpose of the tangent linear (or Jacobian) model
- ocean/sea-ice model as well as the surface atmospheric state are
+ operator. It runs backwards in time, from September 1993 to
- available to analyze the transient sensitivity behavior.  Over the
+ January 1989. During 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
+ areas, the sea-ice adjoint converts surface ocean sensitivities to
  atmospheric sensitivities.
  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
  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
+ conditions. Sensitivity snapshots are depicted for beginning of October 1992,
- September 1993 (at the beginning of the averaging period for the objective
+ that is 12 months before September 1993
- function $J$, top) and at the beginning of the integration in January
+ (the beginning of the averaging period for the objective
-(bottom).
+ 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
-Line 144 
 function $J$, top) and at the beginning
+Line 159 
 function $J$, top) and at the beginning
      \label{fig:adjheff}}
  \end{figure*}
- At the beginning of October 1992, the positive sensitivities in
+ The sensitivity patterns for effective ice thickness are predominantly positive.
- the Lancaster Sound mean that an increase of ice volume increase the
+ An increase in ice volume in most places ``upstream'' of
- 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.
- West can be explained by indirect effects: less ice to the East means
+ The transient nature of the sensitivity patterns
+ (top panels vs. bottom panels) is also obvious:
+ the area upstream of the Lancaster Sound that
+ contributes to the export sensitivity is larger in the earlier snapshot.
+ In the free slip case, the sensivity follows (backwards in time) the dominant pathway
+ through the Barrow Strait
+ into the Viscount Melville Sound, and from there trough the M'Clure Strait
+ into the Arctic Ocean (the ``Northwest Passage''). \ml{[Is that really
+   the Northwest Passage? I thought it would turn south in Barrow
+   Strait, but I am easily convinced because it makes a nicer story.]}
+ Secondary paths are northward from the
+ Viscount Melville Sound through the Byam Martin Channel into
+ the Prince Gustav Adolf Sea and through the Penny Strait into the
+ MacLean Strait. \ml{[Patrick, all these names, if mentioned in the
+   text need to be included somewhere in a figure (i.e. fig1). Can you
+   either do this in fig1 (based on martins\_figs.m) or send me a map
+   where these names are visible so I can do this unambiguously. I
+   don't know where  Byam
+   Martin Channel, Prince Gustav Adolf Sea, Penny Strait, MacLean
+   Strait, Ballantyne St., Massey Sound are.]}
+ There are large differences between the free slip and no slip
+ solution.  By the end of the adjoint integration in January 1989, the
+ no slip sensitivities (bottom right) are generally weaker than the
+ free slip sensitivities and hardly reach beyond the western end of the
+ Barrow Strait. In contrast, the free-slip sensitivities (bottom left)
+ extend through most of the CAA and into the Arctic interior, both to
+ the West (M'Clure St.)  and to the North (Ballantyne St., Prince
+ Gustav Adolf Sea, Massey Sound), because in this case the ice can
+ drift more easily through narrow straits, so that a positive ice
+ volume anomaly anywhere upstream in the CAA increases ice export
+ through the Lancaster Sound within the simulated 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 of the
+ Lancaster Sound.
+ 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
+ into the Lancaster Sound leading to more ice export.
- 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
- area covered by non-zero sensitivities in the free slip solution. At
+   don't.} \ml{[ML: unfortunately, I don't have anything better to
- the beginning of the integration (the end of the backward adjoint
+   offer, do you? Keep in mind that these sensitivites are very small
- integration) the free and no slip solutions are very different. The
+   and only show up, because of the colorscale. In Fig6, they are
- sensitivities of the free slip solution extend through the enitre
+   hardly visible.]}
- Canadian Archipelago and into the Arctic while in the no slip solution
- they still are confined to the Lancaster Sound and the Barrow
+ The temporal evolution of several ice export sensitivities (eqn. XX,
- Strait. This implies that in the free slip solution ice can drift more
+ \ml{[which equation do you mean?]}) along a zonal axis through
- easily through the narrow straits of the Canadian Archipelago, so that
+ Lancaster Sound, Barrow Strait, and Melville Sound (115\degW\ to
- a positive ice volume anomaly anywhere in the Canadian Archipelago is
+\degW, averaged across the passages) are depicted as Hovmueller
- moved through the Lancaster Sound within 4 years thus increasing the
+ diagrams in \reffig{lancasteradj}. These are, from top to bottom, the
- ice export.
+ sensitivities with respect to effective ice thickness ($hc$), ocean
+ surface temperature ($SST$) and precipitation ($p$) for free slip
- The temporal evolution of several sensitivities along the zonal axis
+ (left column) and no slip (right column) ice drift boundary
- Lancaster Sound-Barrow Strait-Melville Sound are shown in
+ conditions.
- \reffig{lancaster}.
+ %
  \begin{figure*}
    \includegraphics*[height=.8\textheight]{\fpath/lancaster_adj}
-   \caption{Hovermoeller diagrams of sensitivities (derivatives) of the
+   \caption{Hovermoeller diagrams along the axis Viscount Melville
-     ``solid'' fresh water (i.e., ice and snow) export $J$ through Lancaster sound
+     Sound/Barrow Strait/Lancaster Sound. The diagrams show the
+     sensitivities (derivatives) of the ``solid'' fresh water (i.e.,
+     ice and snow) export $J$ through Lancaster sound
      (\reffig{arctic_topog}, cross-section G) with respect to effective
      ice thickness ($hc$), ocean surface temperature (SST) and
-     precipitation ($p$) for two runs with free slip and no slip boundary
+     precipitation ($p$) for two runs with free slip and no slip
-     conditions for the sea ice drift. Also shown it the normalized ice
+     boundary conditions for the sea ice drift. Each plot is overlaid
-     strengh $P/P^*=(hc)\,\exp[-C\,(1-c)]$ (bottom panel); each plot is
+     with the contours 1 and 3 of the normalized ice strengh
-     overlaid with the contours 1 and 3 of the normalized ice strength
+     $P/P^*=(hc)\,\exp[-C\,(1-c)]$ for orientation.
-     for orientation.
+     \label{fig:lancasteradj}}
-     \label{fig:lancaster}}
+ \end{figure*}
+ %
+ \begin{figure*}
+   \includegraphics*[height=.8\textheight]{\fpath/lancaster_fwd}
+   \caption{Hovermoeller diagrams along the axis Viscount Melville
+     Sound/Barrow Strait/Lancaster Sound of effective ice thickness
+     ($hc$), effective snow thickness ($h_{s}c$) and normalized ice
+     strengh $P/P^*=(hc)\,\exp[-C\,(1-c)]$ for two runs with free slip
+     and no slip boundary conditions for the sea ice drift. Each plot
+     is overlaid with the contours 1 and 3 of the normalized ice
+     strength for orientation.
+     \label{fig:lancasterfwd}}
  \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
+ The Hovmoeller diagrams of ice thickness (top row) and sea surface temperature
- ($hc$), ocean surface temperature (SST) and precipitation ($p$) for
+ (second row) sensitivities are coherent:
- two runs with free slip and no slip boundary conditions for the sea
+ more ice in the Lancaster Sound leads
- 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
  temperatures (less melting from below). In the free slip case the
- sensitivities can propagate westwards (backwards in time) when the ice
+ sensitivities spread out in "pulses" following a seasonal cycle:
- strength is low in late summer. In the no slip case the (normalized)
+ ice can propagate eastwards (forward in time and thus sensitivites can
+ propagate westwards (backwards in time) when the ice strength is low
+ in late summer to early autumn.
+ In contrast, during winter, the sensitivities show little to now
+ westward propagation, as the ice is frozen solid and does not move.
+ 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
+ (mainly because the ice concentrations remain near 100\%, not
- shown), so that ice is blocked and cannot drift eastwards (forward in
+ shown). Ice is therefore blocked and cannot drift eastwards
- time) in the Melville Sound-Barrow Strait-Lancaster Sound channel.
+ (forward in time) through the Viscount
- Consequently the sensitivies do not propagate westwards (backwards in
+ 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
+ The sensitivities to precipitation exhibit an oscillatory behaviour:
- leads to less export) before January and mostly positive after
+ they are negative (more precipitation leads to less export)
- January. Further they are mostly positive for normalized ice strengths
+ before January (more precisely, late fall) and mostly positive after January
- over 3. Assuming that most precipation is snow in this area---in the
+ (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, that is, in the Lancaster Sound.
+ For example, at 90\degW\ the sensitivities are negative throughout 1992,
+ and no clear correlation to ice strength is apparent there.}
+ except between 95\degW\ and 85\degW, which is an area of
+ increased snow cover in spring. \ml{[ML: and no, I cannot explain
+   that. Can you?]}
+ %
+ 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
+ ice to grow, all precipitation is assumed to be snow.}
- sensitivities can be interpreted in terms of the model physics.  Short
+ %
- wave radiation cannot penetrate a snow cover and has a higer albedo
+ the sensitivities can be interpreted in terms of the model physics.
- 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.
- protects the ice against melting in spring (after January).  On the
+ Further, short wave radiation cannot penetrate the snow cover and has
- 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
+ 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
  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.
- %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
- %1. snow insulates against melting from above during spring: more precip (snow) -> more export
+ because the neg. sensitivities are too late in the fall,
- %2. less snow during fall -> more ice -> more export
+ probably mostly falling as snow.} \ml{[ML: correct, I looked at
- %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
+ Jul/Aug, but otherwise below freezing, that why I can assume that most
+ precip is snow. ]} \ml{[this is not very good but do you have anything
+ better?:]}
+ The negative sensitivities to precipitation between 95\degW\ and
+\degW\ in spring 1992 may be explained by a similar mechanism: in an
+ area of thick snow (almost 50\,cm), ice cannot melt and tends to block
+ the channel so that ice coming in from the West cannot pass thus
+ leading to less ice export in the next season.
  \subsubsection{Forward sensitivities}
  \ml{[Here we need for integrations to show that the adjoint
    sensitivites are not just academic. I suggest to perturb HEFF
    and THETA initial conditions, and PRECIP somewhere in the Melville
-   Sound and then produce plots similar to reffig{lancaster}. For
+   Sound and then produce plots similar to reffig{lancasteradj}. 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.

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