ceaice_split_version/ceaice_part2/ceaice_adjoint.tex

\section{MITgcm/sim adjoint code generation}
\label{sec:adjoint}

\begin{figure}
\newcommand{\textinfigure}[1]{{\normalsize\textbf{\textsf{#1}}}} 
\newcommand{\mathinfigure}[1]{\normalsize\ensuremath{{#1}}}
\psfrag{delS}{\mathinfigure{\delta S}}
\psfrag{delT}{\mathinfigure{\delta \Theta}}
\psfrag{delc}{\mathinfigure{\delta c}}
\psfrag{delh}{\mathinfigure{\delta h}}
\psfrag{delAT}{\mathinfigure{\delta T_a}}
\psfrag{delP}{\mathinfigure{\delta p}}
\psfrag{delJ}{\mathinfigure{\delta J}}
%
\psfrag{addS}{\mathinfigure{\delta^{\ast} S}}
\psfrag{addT}{\mathinfigure{\delta^{\ast} \Theta}}
\psfrag{addc}{\mathinfigure{\delta^{\ast} c}}
\psfrag{addh}{\mathinfigure{\delta^{\ast} h}}
\psfrag{addAT}{\mathinfigure{\delta^{\ast} T_a}}
\psfrag{addP}{\mathinfigure{\delta^{\ast} p}}
\psfrag{addJ}{\mathinfigure{\delta^{\ast} J}}
  \centerline{
  \includegraphics*[width=.5\textwidth]{\fpath/coupling_schematic}
  }
  \caption{
  A schematatic serve to distinguish between the effect of 
  perturbing \textit{individual} variables 
  (e.g. ocean temperature $\delta \Theta$)
  on the cost function, $\delta J$ (left), and how a (unit) change
  in cost function sensitivity $\delta^{\ast} J$ is affected by \textit{all}
  sensitivities.
  For a cost function of the coupled problem
  $J \, = \, J( \, \mathrm{ice[atm,oce]} \, )$,
  the sensitivities spread through the coupled adjoint.
    \label{fig:couplingschematic}}
\end{figure}


There is now a growing body of literature on adjoint applications
in oceanography and adjoint code generation via AD.
We therefore limit the description of the method to a brief summary.
The adjoint model operator (ADM) is the transpose of the 
Jacobian or tangent
linear model operator (TLM) of the full (in general nonlinear) forward
model, in this case the MITgcm/sim. 
The TLM computes directional derivatives for a given perturbation direction.
In contrast, for scalar-valued model diagnostics (cost function or
objective function), the ADM computes the the full gradient
of the cost function with respect to all model inputs
(independent or control variables).  These inputs can be two- or
three-dimensional fields of initial conditions of the ocean or sea-ice
state, model parameters such as mixing coefficients, or time-varying
surface or lateral (open) boundary conditions.  When combined, these
variables span a potentially high-dimensional (e.g.  O(10$^8$))
control space. At this problem dimension, perturbing
individual parameters to assess model sensitivities quickly becomes
prohibitive. By contrast, transient sensitivities of the objective
function to any element of the control and model state space can be
computed very efficiently in one single adjoint model integration,
provided an adjoint model is available.

The burden of developing ``by hand''
an adjoint model in general matches that of
the forward model development. The substantial extra investment
often prevents serious attempts of making available adjoint 
components of sophisticated models.
The alternative route of rigorous application of AD has proven
very successful in the context of MITgcm ocean modeling applications.
The model has been tailored to be readily used with AD 
tools for adjoint code generation.
The adjoint model of the MITgcm has become an invaluable
tool for sensitivity analysis as well as state estimation \citep[for a
recent overview and summary, see][]{heim:08}. 
AD also enables the largest possible variety of configurations 
and studies to be conducted with adjoint methods.
\cite{gier-kami:98} discuss in detail the advantages of the AD approach.

The AD route was also taken in developing and adapting the sea-ice 
component, so that tangent linear and adjoint components can be obtained 
and kept up to date without excessive effort.
As for the TLM and ADM components of MITgcm we rely on the
autmomatic differentiation (AD) tool ``Transformation of Algorithms in
Fortran'' (TAF) developed by Fastopt \citep{gier-kami:98} to generate
TLM and ADM code of the MITsim \citep[for details see][]{maro-etal:99,heim-etal:05} (for the ocean component we are now also able to generate
  efficient derivative code using the new open-source tool OpenAD
  \citep{utke-etal:08}).
Appendix \ref{app:adissues} provides details of
adjoint code generation of MITgcm/sim.

\begin{figure*}[t]
  \centering
  \includegraphics*[width=0.9\textwidth]{\fpath/map_part2}
%  \includegraphics*[width=0.9\textwidth]{\fpath/map_sverdrup_basin_melling_2002}
  \caption{Map of the Canadian Arctic Archipelago with model
    coastlines and grid (filled grey boxes are land). The black
    contours are the true coastlines as taken from the GSHHS data base
    \citep{wessel96}. 
    \label{fig:sverdrupbasin}}
\end{figure*}

To conclude, we emphasize the coupled nature of the MITgcm/sim adjoint.
Fig. \ref{fig:couplingschematic} illustrates how sensitivities of the 
objective function (sea-ice export)
that depends solely on the sea-ice state nevetheless
propagates both into the time-varying ocean state as well 
as atmospheric boundary conditions.

\begin{figure*}[t]
  \includegraphics*[width=\textwidth]{\fpath/adj_canarch_freeslip_ADJheff}
  \caption{Sensitivity $\partial{J}/\partial{(hc)}$ in
    m$^2$\,s$^{-1}$/m for four different different times using 
    \textbf{free-slip}
    lateral boundary conditions for sea ice drift. The color scale is chosen
    to illustrate the patterns of the sensitivities.
    \ml{[What's wrong with the figure titles: 01-Oct and 02-Oct???]}
    \label{fig:adjhefffreeslip}}
\end{figure*}

\begin{figure*}[t]
  \includegraphics*[width=\textwidth]{\fpath/adj_canarch_noslip_ADJheff}
  \caption{Same as Fig. \ref{fig:adjhefffreeslip}, but for \textbf{no-slip}
  lateral boundary conditions for sea ice drift.
    \label{fig:adjheffnoslip}}
\end{figure*}

\section{A case study: Sensitivities of sea-ice export through
the Lancaster Sound}

We demonstrate the power of the adjoint method in the context of
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 presented in Part 1
(see also \cite{losc-dani:09}).  Lancaster Sound is one of
the main paths of sea ice through the Canadian Arctic
Archipelago (CAA).  Fig. \ref{fig:sverdrupbasin} %taken from \cite{mell:02} 
reflects the intricate local geopgraphy of
straits, sounds, and islands.
Export sensitivities reflect dominant pathways
through the CAA as resolved by the model.  Sensitivity maps can \ml{provide a very detailed view of}
%shed a very detailed light on
various quantities affecting the sea-ice export
(and thus the underlying \ml{propagation} pathways).  
A caveat of the present study is the limited resolution which
is not adequate to realistically simulate the CAA.
For example, while the dominant
circulation through Lancaster Sound is toward the East, there is a
small Westward flow to the North, hugging the coast of Devon Island
\citep{mell:02, mich-etal:06,muen-etal:06}, which is not resolved in
our simulation.
Nevertheless, the focus here is on elucidating model sensitivities in a 
general way. For any given simulation, whether deemed
``realistic'' or not, the adjoint provides exact model sensitivities,
and supports understanding whether hypothesized processes are actually
borne out by the model dynamics.

\subsection{The model configuration}

The model domain is \ml{similar to} the one described in Part 1,
i.e., \ml{it is carved out} from the Arcitc face of the global eddy-permitting
cubed-sphere \ml{simulation} \citep{menemenlis05},
but with \ml{half the} horizontal resolution \ml{of 36~km grid cell width}.
%, now amounting to roughly 36 km..
The adjoint models run efficiently on 80 processors (as validated
by benchmarks on both an SGI Altix and an IBM SP5 at NASA/ARC and NCAR/CSL).
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
%\citet{large04}.  The air-sea fluxes in the presence of
%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,
%
\begin{equation}
\label{eq:costls}
J \, = \int_\mathrm{one year} \, (\rho_{i} h_{i}c + \rho_{s} h_{s}c)\,u \,dt
\end{equation} 
%
through Lancaster Sound at approximately 82\degW\ (cross-section G in
\reffig{arctic_topog}) integrated over the final
12-month \ml{period} of the integration between October 1992 and September 1993.
\ml{$c$ is the fractional ice cover, $u$ is the along channel ice drift velocity, and $h_X$ and $\rho_X$ are the ice ($X=i$) and snow ($X=s$) thickness, respectively.}

The forward trajectory of the model integration resembles broadly that
of the model in Part~1. Many details are different, owning
to different resolution and integration period.
%
%\ml{PH: Martin, please confirm/double-check following sentence:}
%
For example, the differences in solid
fresh water transport through Lancaster Sound are smaller
between no-slip and
free-slip lateral boundary conditions at higher resolution
($91\pm85\text{\,km$^{3}$\,y$^{-1}$}$ and 
$77\pm110\text{\,km$^{3}$\,y$^{-1}$}$ for free-slip and no-slip, respectively,
and for a C-grid LSR solver) than at lower resolution
($116\pm101\text{\,km$^{3}$\,y$^{-1}$}$ and 
$39\pm64\text{\,km$^{3}$\,y$^{-1}$}$ for free-slip and no-slip, respectively).
\ml{The large range of these estimates alone emphasizes the need to
  better understand the model sensitivities to lateral boundary
  conditions an different configuration details. We aim to explore
  these sensitivities across the entire model state space in a
  comprehensive manner by means of the adjoint model.}
%The large discrepancy between all these numbers underlines the need to
%better understand the model sensitivities across the entire model state space
%resulting from different lateral boundary conditions and different
%configurations, and which we aim to explore in a more
%comprehensive manner through the adjoint.

The adjoint model is the transpose of the tangent linear model
operator. It runs backwards in time, from September 1993 to
January 1989. During its integration \ml{the Lagrange multipliers
of the model subject to the objective function \refeq{costls} (solid
freshwater export) are accumulated. These Langrangian multipliers}
are the sensitivities (derivatives) of the objective function with respect
%ML 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
areas, the sea-ice adjoint converts surface ocean sensitivities to
atmospheric sensitivities. \ml{[ML: maybe we need to stress this
  analogy more, ...]}

\subsection{Adjoint sensitivities}

\begin{figure*}
  \centerline{
  \includegraphics*[height=.9\textheight]{\fpath/lancaster_adj}
  }
  \caption{Hovmoeller diagrams along the axis Viscount Melville
    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
    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. Each plot is overlaid
    with the contours 1 and 3 of the normalized ice strengh
    $P/P^*=(hc)\,\exp[-C\,(1-c)]$ for orientation.
    \label{fig:lancasteradj}}
\end{figure*}
%
\begin{figure*}
  \centerline{
  \includegraphics*[height=.9\textheight]{\fpath/lancaster_fwd_1}
  }
  \caption{Hovmoeller diagrams along the axis Viscount Melville
    Sound/Barrow Strait/Lancaster Sound of ice thickness
    ($hc$), 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:lancasterfwd1}}
\end{figure*}
%
\begin{figure*}
  \centerline{
  \includegraphics*[height=.9\textheight]{\fpath/lancaster_fwd_2}
  }
  \caption{Same as Fig. \ref{fig:lancasterfwd1}, but for SST, SSS,
  and precipitation.
    \label{fig:lancasterfwd2}}
\end{figure*}
%

The most readily interpretable ice-export sensitivity is that to
ice thickness, $\partial{J} / \partial{(hc)}$.
Maps of transient sensitivities 
$\partial{J} / \partial{(hc)}$ are depicted using
free-slip (\reffig{adjhefffreeslip}) and no-slip (\reffig{adjheffnoslip}) boundary conditions.
Each Figure depicts four sensitivity snapshots from 1~October 1992
(i.e. \ml{at the beginning of the 12-month averaging period for the
  objective function $J$}), 
%and 12 months prior to the end of the integration, September 1993),
going back in time to 1~October 1989
(beginning of model integration is 1~January 1989).

The sensitivity patterns for ice thickness are predominantly positive.
An increase in ice volume in most places \ml{west of the}
%``upstream'' of
Lancaster Sound increases the solid fresh water export at the exit section.
The transient nature of the sensitivity patterns is 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 branch of the ``Northwest Passage'' first 
discovered by Robert McClure during his 1850 to 1854 expedition\ml{;
  McClure lost his research vessel in the Viscount Melville Sound}). 
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. 

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 (\reffig{adjheffnoslip}) 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 (\reffig{adjhefffreeslip})
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
are the negative sensivities to the East and, albeit much weaker,
to the West of the Lancaster Sound.
The former can be explained by indirect effects: less ice \ml{eastward
  of the Lancaster Sound} means
less resistance to eastward drift and thus more export.
A similar mechanism might account for the latter,
albeit more speculative: 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. \ml{[This
  paragraph is very weak, need to think of something else, longer
  fetch maybe?]}

\begin{figure}
  %\centerline{
  \subfigure %[$hc$]
  {\includegraphics*[width=.5\textwidth]{\fpath/lanc_pert_heff}}

  \subfigure %[SST]
  {\includegraphics*[width=.5\textwidth]{\fpath/lanc_pert_theta}}

  \subfigure %[$p$]
  {\includegraphics*[width=.5\textwidth]{\fpath/lanc_pert_precip}}
  %}
  \caption{~
    \label{fig:lancpert}}
\end{figure}

The temporal evolution of several ice export sensitivities 
along a zonal axis through
Lancaster Sound, Barrow Strait, and Melville Sound (115\degW\ to
80\degW, averaged across the passages) are depicted as Hovmoeller
diagrams in \reffig{lancasteradj}. 
\ml{Serving as an example for}
%In order to represent sensitivities to elements of the state of
each component of the coupled ocean/sea-ice/atmosphere control space, we
depict, from top to bottom, the
sensitivities to ice thickness ($hc$), ocean
surface temperature ($SST$) and precipitation ($p$) for free slip
(left column) and no slip (right column) ice drift boundary
conditions.

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 \ml{form} more ice is by colder surface
temperatures (less melting from below). In the free slip case the
sensitivities spread out in ``pulses'' following a seasonal cycle:
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 no
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 near 100\%, not
shown). Ice is therefore blocked and cannot drift eastwards 
(forward in time) through the Viscount
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 for the entire integration period.

\ml{[Somewhere here we should refer to \reffig{lancasterfwd1},
  \reffig{lancasterfwd2}, and say that it's much more straitforward
  to see ice motion in the adjoint hovmoeller, another advantage of
  the adjoint; or cut the figures]}

The sensitivities to precipitation \ml{are more complex.}
%exhibit a more complex behaviour.
\ml{To first order, they have an oscillatory pattern
with negative sensitivity (more precipitation leads to less export) 
between roughly August and December and mostly positive sensitivity
from January through July.}
%A fairly accurate description would note an oscillatory behaviour:
%they are negative (more precipitation leads to less export) 
%before January (more precisely, between roughly August and December) 
%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{This pattern is broken only immediatly preceding the evaluation
  period of the ice export cost function: In contrast to previous
  years, the sensitivity is negative between January and August~1992
  and east of 95\degW.}
\ml{PH: Could it be that this portion goes past Lancaster Sound,
and is connected with the strong blocking downstream of LS?
If so, the negative sensitivity would make sense:
the blocking, initiated through ice emanating Nares Strait
is re-inforced by strong ice export through LS
Some evidence for this in Fig. 1, upper left panel??? 
Are the Figs consistent???} %
\ml{ML: plausible, but in order to see that we have to regenerate
fig5 and extend it a little further east, shouldn't be hard}.

\begin{table*}
\caption{Blabla... All perturbations were applied on a patch around
101.24$^{\circ}$W, 75.76$^{\circ}$N. Reference value for export is
$J_0$ = 69.6 km$^3$.
}
\label{tab:pertexp}
\centering
\begin{tabular}{cc@{\hspace{3ex}}c@{\hspace{3ex}}cr@{\hspace{3ex}}r}
\hline
variable & time & $\Delta t$ & $\epsilon$ & 
$\delta J$(adj.) [km$^3$/yr] & $\delta J$(f.d.) [km$^3$/yr] \\
\hline \hline
$hc$ & 1-Jan-1989 & init. & 0.5 m & 0.98 & 1.1 \\
SST & 1-Jan-1989 & init. & 0.5$^{\circ}$C & -0.125 & -0.108 \\
$p$ & 1-Apr-1991 & 10 days & 1.6$\cdot10^{-7}$ m/s & 0.185 & 0.191 \\
$p$ & 1-Apr-1991 & 10 days & -1.6$\cdot10^{-7}$ m/s & -0.185 & -0.071 \\
$p$ & 1-Nov-1991 & 10 days & 1.6$\cdot10^{-7}$ m/s & -0.435 & -1.016 \\
$p$ & 1-Nov-1991 & 10 days & -1.6$\cdot10^{-7}$ m/s & 0.435 & 0.259 \\
\hline
\end{tabular}
\end{table*}

\ml{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 atmospheric conditions (i.e., the atmospheric forcing
  fields) most of the year in the Lancaster 
  Sound region mean that almost all precipitation is treated as snow
  except for a short period in July and August; even then air
  temperatures are only slightly above freezing. Given that most
  precipitation is snow}
% 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 \ml{to precipitation} can be interpreted in terms of the model physics.
The accumulation of snow directly increases the exported volume.
Further, short wave radiation cannot penetrate the snow cover and has
a higher 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 \ml{also} 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{This interpretation is further supported by
  \reffig{lancasterperthov} in \refsec{forwardpert}.} %
\ml{[Repetition:]} %
We note that the effect of snow vs.\ rain is not relevant in explaining
positive vs.\ negative sensitivity patterns\ml{, because the atmospheric
conditions during phases of negative sensitivities to precipitation
turn all precipitation into snow}. 
% Negative sensitivities occur too late in the fall,
% as evidenced by both NCEP/NCAR and CORE air temperatures.
% They are hardly above freezing even in Jul/Aug, and otherwise
% consistently below freezing, implying snowfall during most of the year.

The negative sensitivities to precipitation between 95\degW\ and
85\degW\ in spring 1992 may also be explained \ml{by the presence of
  snow}: in an area of thick snow (almost 50\,cm), ice cannot melt and
tends to block the channel so that ice coming from the West cannot
pass thus leading to less ice export in the next season.
%
%\ml{PH: Why is this true for 1992 but not 1991?}
\ml{This explanation does not hold for 1991 because by then the high
  sensitivites have propagated westward out of the area of thick 
  snow and ice around 90\degW.}

\ml{[This paragraph is very incoherent, we need to get the story
  strait. I think that's now possible with \reffig{lancasterperthov},
  however, we might want to reorganize the adjoint and forward
  senstitivy analysis?]}

\subsection{Forward perturbation experiments}
\label{sec:forwardpert}

\ml{Applying an adjoint model}
%Using an adjoint model obtained via automatic differentiation
%and applied 
under potentially highly nonlinear conditions \ml{stipulates} the question 
to what extent the adjoint sensitivities are ``reliable".
Adjoint sensitivities that are physically interpretable provide
%Obtaining adjoint fields that are physically interpretable provides
some support, but \ml{a more} quantitative \ml{test} is required to
\ml{gain confidence in} the calculations.
%credence to the calculations.
\ml{[Do we need to set this appart from gradient checks?]}
Such a verification can be achieved by comparing the adjoint-derived gradient
with the one obtained from finite-difference perturbation experiments.
More specifically, for a control variable of interest $\mathbf{u}$
we can readily calculate an expected change $\delta J$ in the objective function
from an applied perturbation $\mathbf{\delta u}$ over the domain $A$ via
%
\begin{equation}
\delta J \, = \, \int_A \frac{\partial J}{\partial \mathbf{u}} \,
\mathbf{\delta u} \, dA
\label{eqn:adjpert}
\end{equation}
%
Alternatively we can infer the magnitude of the cost perturbation
without use of the adjoint, but instead by applying the same
perturbation $\epsilon = | \mathbf{\delta u} |$ to the control space over 
the same domain $A$ and run the
forward model. We obtain the perturbed cost by calculating
%
\begin{equation}
\delta J \, = \, 
J(\mathbf{u}+\mathbf{\delta u}) - J(\mathbf{u})
\label{eqn:fdpert}
\end{equation}

The degree to which eqn. (\ref{eqn:adjpert}) and (\ref{eqn:fdpert}) agree
depends both on the magnitude of the perturbation 
$\epsilon = | \mathbf{\delta u} |$
and on the integration period (note that forward and adjoint models are
evaluated over the same period).
For nonlinear models they are expected to diverge both with 
perturbation magnitude as well as with integration time.
Bearing this in mind, we perform several such experiments
for various control variables, summarized in Table \ref{tab:pertexp}.

Comparison between f.d. and adjoint-derived ice-export perturbations
show remarkable agreement for both initial value perturbations
(ice thickness, sea surface temperature).
Deviations between perturbed cost function values remain below roughly 50 \%.
Fig. \ref{fig:lancpert} depicts the temporal evolution of
perturbed minus un-perturbed ice export \ml{through Lancaster Sound} for initial ice thickness 
(top panel) and SST (middle panel) perturbation.
In both cases, \ml{differences are confined} to the melting season during which
the ice \ml{unlocks which} %gets ``unstuck'' and
can lead to significant export.
As ``predicted'' by the adjoint, the two curves are of opposite sign,
and scales differ by almost an order of magnitude.
%
\ml{PH: Tja, was soll man da noch sagen...}
%

A challenging test \ml{[why is this challenging?]} is ascertaining the sign changes through time 
(and magnitude) of the transient precipitation sensitivities.
To investigate this, we have performed two perturbation experiments:
one, in which we perturb precipitation over a 10-day period
between April 1st and 10th, 1991 (coincident with a period of
positive adjoint sensitivities),
and one in which we apply the same perturbation over the 10-day period
November 1st to 10th, 1991 (coincident with a period of
negative adjoint sensitivities).
The perturbation magnitude chosen is $\epsilon = 1.6 \times 10^{-7}$ m/s
as a measure of spatial mean standard deviation of precipitation
variability. %The results are as follows:
First, the perturbation experiments confirm the sign change
when perturbing in different seasons.
Second, we observe good quantitative agreement for the Apr. 1991 case, 
and a 50 \% deviation for the Nov. 1991 case.
%
While the latter discrepancy seems discouraging,
we recall that the perturbation experiments are performed 
over a multi-year period, and under likely nonlinear model behaviour.
To support this view, we \ml{repeated} the perturbation experiments by
applying the same the same perturbation \ml{with} opposite sign,
$\epsilon = -1.6 \times 10^{-7}$ m/s.
At this point both perturbation periods lead to about
50 \% discrepancies between finite-difference and adjoint-derived
ice export differences.
%
\ml{Note that the export changes are different in amplitude for
  positive and negative perturbations pointing to the strong nonlinearity of
  the problem.}

In this light, and given that these experiments constitute very
severe tests \ml{[why ``severe''?]} on the adjoint, the results can be regarded as useful in 
obtaining qualitative, and within certain limits quantitative
information of comprehensive model sensitivities
that cannot realistically be computed otherwise.


\begin{figure*}
  \centerline{
  \includegraphics*[width=.9\textwidth]{\fpath/lancaster_pert_hov}
  }
  \caption{Same as Fig. \ref{fig:lancasterfwd1}, but for the anomalies
    of ice thickness $(hc)$, snow thickness $(h_\mathrm{snow}c)$ and sea-surface
    temperature (SST) after a perturbation of precipitation of
    $1.6\cdot10^{-1}\text{\,m\,s$^{-1}$}$ on 1-Nov-1991 (left) and
    1-Apr-1991 (right). Each plot is overlaid
    with the contours 1 and 3 of the normalized ice strengh
    $P/P^*=(hc)\,\exp[-C\,(1-c)]$ for orientation.
    \label{fig:lancasterperthov}}
\end{figure*}
%
Revisiting the interpretation of the sensitivities to precipation
$\partial{J}/\partial{p}$, \reffig{lancasterperthov} shows how the
small localized perturbations of precipitation are propagated in ice,
snow in SST. This analysis nicely supports the interpretion of the
adjoint sensitivities to precipitation as follows. More precipation
leads to more snow on the ice. The same perturbation in different
seasons has an opposite effect on the (solid) fresh water export
through the Lancaster Sound: More snow in November (on thin ice)
insulates the ice and reduces ice formation in the following winter
(growing season), but more snow in April (on thick ice) insulates the
ice against melting and leads to more ice in the following growing
season. The feedback with the SST appears to be negligible because
there is no connection of anomalies between seasons.

%(*)
%The sensitivity in Baffin Bay are more complex.
%The pattern evolves along the Western boundary, connecting
%the Lancaster Sound Polynya, the Coburg Island Polynya, and the
%North Water Polynya, and reaches into Nares Strait and the Kennedy Channel.
%The sign of sensitivities has an oscillatory character
%[AT FREQUENCY OF SEASONAL CYCLE?].
%First, we need to establish whether forward perturbation runs
%corroborate the oscillatory behaviour.
%Then, several possible explanations:
%(i) connection established through Nares Strait throughflow
%which extends into Western boundary current in Northern Baffin Bay.
%(ii) sea-ice concentration there is seasonal, i.e. partly
%ice-free during the year. Seasonal cycle in sensitivity likely
%connected to ice-free vs. ice-covered parts of the year.
%Negative sensitivities can potentially be attributed
%to blocking of Lancaster Sound ice export by Western boundary ice
%in Baffin Bay.
%(iii) Alternatively to (ii), flow reversal in Lancaster Sound is a possibility
%(in reality there's a Northern counter current hugging the coast of
%Devon Island which we probably don't resolve).

%Remote control of Kennedy Channel on Lancaster Sound ice export
%seems a nice test for appropriateness of free-slip vs. no-slip BCs.

%\paragraph{Sensitivities to the sea-ice area}

%Fig. XXX depcits transient sea-ice export sensitivities
%to changes in sea-ice concentration
% $\partial J / \partial area$ using free-slip
%(left column) and no-slip (right column) boundary conditions.
%Sensitivity snapshots are depicted for (from top to bottom) 
%12, 24, 36, and 48 months prior to May 2003.
%Contrary to the steady patterns seen for thickness sensitivities,
%the ice-concentration sensitivities exhibit a strong seasonal cycle
%in large parts of the domain (but synchronized on large scale).
%The following discussion is w.r.t. free-slip run.

%(*)
%Months, during which sensitivities are negative:
%\\
%0 to 5   Db=N/A, Dr=5 (May-Jan) \\
%10 to 17 Db=7, Dr=5 (Jul-Jan) \\
%22 to 29 Db=7, Dr=5 (Jul-Jan) \\
%34 to 41 Db=7, Dr=5 (Jul-Jan) \\
%46 to 49 D=N/A \\
%%
%These negative sensitivities seem to be connected to months
%during which main parts of the CAA are essentially entirely ice-covered.
%This means that increase in ice concentration during this period
%will likely reduce ice export due to blocking
%[NEED TO EXPLAIN WHY THIS IS NOT THE CASE FOR dJ/dHEFF].
%Only during periods where substantial parts of the CAA are
%ice free (i.e. sea-ice concentration is less than one in larger parts of
%the CAA) will an increase in ice-concentration increase ice export.

%(*)
%Sensitivities peak about 2-3 months before sign reversal, i.e.
%max. negative sensitivities are expected end of July
%[DOUBLE CHECK THIS].

%(*) 
%Peaks/bursts of sensitivities for months
%14-17, 19-21, 27-29, 30-33, 38-40, 42-45

%(*) 
%Spatial ``anti-correlation'' (in sign) between main sensitivity branch
%(essentially Northwest Passage and immediate connecting channels),
%and remote places.
%For example: month 20, 28, 31.5, 40, 43.
%The timings of max. sensitivity extent are similar between
%free-slip and no-slip run; and patterns are similar within CAA,
%but differ in the Arctic Ocean interior.

%(*) 
%Interesting (but real?) patterns in Arctic Ocean interior.

%\paragraph{Sensitivities to the sea-ice velocity}

%(*)
%Patterns of ADJuice at almost any point in time are rather complicated
%(in particular with respect to spatial structure of signs).
%Might warrant perturbation tests.
%Patterns of ADJvice, on the other hand, are more spatially coherent,
%but still hard to interpret (or even counter-intuitive
%in many places).

%(*)
%``Growth in extent of sensitivities'' goes in clear pulses:
%almost no change between months: 0-5, 10-20, 24-32, 36-44
%These essentially correspond to months of


%\subsection{Sensitivities to the oceanic state}

%\paragraph{Sensitivities to theta}

%\textit{Sensitivities at the surface (z = 5 m)}

%(*)
%mabye redo with caxmax=0.02 or even 0.05

%(*)
%Core of negative sensitivities spreading through the CAA as 
%one might expect [TEST]:
%Increase in SST will decrease ice thickness and therefore ice export.

%(*)
%What's maybe unexpected is patterns of positive sensitivities
%at the fringes of the ``core'', e.g. in the Southern channels
%(Bellot St., Peel Sound, M'Clintock Channel), and to the North
%(initially MacLean St., Prince Gustav Adolf Sea, Hazen St.,
%then shifting Northward into the Arctic interior).

%(*)
%Marked sensitivity from the Arctic interior roughly along 60$^{\circ}$W
%propagating into Lincoln Sea, then
%entering Nares Strait and Smith Sound, periodically
%warming or cooling[???] the Lancaster Sound exit.

%\textit{Sensitivities at depth (z = 200 m)}

%(*)
%Negative sensitivities almost everywhere, as might be expected.

%(*)
%Sensitivity patterns between free-slip and no-slip BCs
%are quite similar, except in Lincoln Sea (North of Nares St),
%where the sign is reversed (but pattern remains similar).

%\paragraph{Sensitivities to salt}

%T.B.D.

%\paragraph{Sensitivities to velocity}

%T.B.D.

%\subsection{Sensitivities to the atmospheric state}

%\begin{itemize}
%%
%\item
%plot of ATEMP for 12, 24, 36, 48 months
%%
%\item
%plot of HEFF for 12, 24, 36, 48 months
%%
%\end{itemize}


%\reffig{4yradjheff}(a--d) depict sensitivities of sea-ice export
%through Fram Strait in December 1995 to changes in sea-ice thickness
%12, 24, 36, 48 months back in time. Corresponding sensitivities to
%ocean surface temperature are depicted in
%\reffig{4yradjthetalev1}(a--d).  The main characteristics is
%consistency with expected advection of sea-ice over the relevant time
%scales considered.  The general positive pattern means that an
%increase in sea-ice thickness at location $(x,y)$ and time $t$ will
%increase sea-ice export through Fram Strait at time $T_e$.  Largest
%distances from Fram Strait indicate fastest sea-ice advection over the
%time span considered.  The ice thickness sensitivities are in close
%correspondence to ocean surface sentivitites, but of opposite sign.
%An increase in temperature will incur ice melting, decrease in ice
%thickness, and therefore decrease in sea-ice export at time $T_e$.

%The picture is fundamentally different and much more complex
%for sensitivities to ocean temperatures away from the surface.
%\reffig{4yradjthetalev10??}(a--d) depicts ice export sensitivities to
%temperatures at roughly 400 m depth.
%Primary features are the effect of the heat transport of the North
%Atlantic current which feeds into the West Spitsbergen current,
%the circulation around Svalbard, and ...


%%\begin{figure}[t!]
%%\centerline{
%%\subfigure[{\footnotesize -12 months}]
%%{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim072_cmax2.0E+02.eps}}
%%\includegraphics*[width=.3\textwidth]{H_c.bin_res_100_lev1.pdf}
%%
%%\subfigure[{\footnotesize -24 months}]
%%{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim145_cmax2.0E+02.eps}}
%%}
%%
%%\caption{Sensitivity of sea-ice export through Fram Strait in December 2005 to
%%sea-ice thickness at various prior times.
%%\label{fig:4yradjheff}}
%%\end{figure}


%\ml{[based on the movie series
%  zzz\_run\_export\_canarch\_freeslip\_4yr\_1989\_ADJ*:]} The ice
%export through the Canadian Archipelag is highly sensitive to the
%previous state of the ocean-ice system in the Archipelago and the
%Western Arctic. According to the \ml{(adjoint)} senstivities of the
%eastward ice transport through Lancaster Sound (\reffig{arctic_topog},
%cross-section G) with respect to ice volume (thickness), ocean
%surface temperature, and vertical diffusivity near the surface
%(\reffig{fouryearadj}) after 4 years of integration the following
%mechanisms can be identified: near the ``observation'' (cross-section
%G), smaller vertical diffusivities lead to lower surface temperatures
%and hence to more ice that is available for export. Further away from
%cross-section G, the sensitivity to vertical diffusivity has the
%opposite sign, but temperature and ice volume sensitivities have the
%same sign as close to the observation.


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