articles/ceaice/ceaice_adjoint.tex

\section{Adjoint sensitivities of the MITsim}
\label{sec:adjoint}

\subsection{The adjoint of MITsim}

The adjoint model of the MITgcm has become an invaluable
tool for sensitivity analysis as well as state estimation \citep[for a
recent summary, see][]{heim:08}. The code has been developed and
tailored to be readily used with automatic differentiation tools for
adjoint code generation. This route was also taken in developing and
adapting the sea-ice compontent MITsim, so that tangent linear and
adjoint components can be obtained and kept up to date without
excessive effort.

The adjoint model operator (ADM) is the transpose of the tangent
linear model operator (TLM) of the full (in general nonlinear) forward
model, in this case the MITsim. This operator computes the gradients
of scalar-valued model diagnostics (so-called cost function or
objective function) with respect to many model inputs (so-called
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$^8$))
so-called 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.

In anology to the TLM and ADM components of the 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}.  In short, the AD tool uses the nonlinear parent
model code to generate derivative code for the specified control space
and objective function. Advantages of this approach have been pointed
out, for example by \cite{gier-kami:98}.

Many issues of generating efficient exact adjoint sea-ice code are
similar to those for the ocean model's adjoint.  Linearizing the model
around the exact nonlinear model trajectory is a crucial aspect in the
presence of different regimes (e.g., is the thermodynamic growth term
for sea-ice evaluated near or far away from the freezing point of the
ocean surface?). Adapting the (parent) model code to support the AD
tool in providing exact and efficient adjoint code represents the main
work load initially. For legacy code, this task may become
substantial, but it is fairly straightforward when writing new code
with an AD tool in mind. Once this initial task is completed,
generating the adjoint code of a new model configuration takes about
10 minutes.

[HIGHLIGHT COUPLED NATURE OF THE ADJOINT!]

\subsection{Special considerations}

* growth term(?)

* small active denominators

* dynamic solver (implicit function theorem)

* approximate adjoints


\subsection{An example: 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.  Lancaster Sound is one of
the main outflow 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
(and thus the underlying pathways).  Note that 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.

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
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?]}
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 $(\rho_{i} h_{i}c + \rho_{s} h_{s}c)\,u$, through Lancaster
Sound at approximately 82\degW\ (cross-section G in
\reffig{arctic_topog}) averaged over a 12-month period between October
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
$116\pm101\text{\,km$^{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
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
atmospheric sensitivities.

DISCUSS FORWARD STATE, INCLUDING SOME NUMBERS ON SEA-ICE EXPORT

\subsubsection{Adjoint sensitivities}

The most readily interpretable ice-export sensitivity is that to
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).
\begin{figure*}[t]
  \includegraphics*[width=\textwidth]{\fpath/adjheff}
  \caption{Sensitivity $\partial{J}/\partial{(hc)}$ in
    m$^2$\,s$^{-1}$/m for two different times (rows) and two different
    boundary conditions for sea ice drift. The color scale is chosen
    to illustrate the patterns of the sensitivities; the maximum and
    minimum values are given above the figures.
    \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
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}.
\begin{figure*}
  \includegraphics*[height=.8\textheight]{\fpath/lancaster_adj}
  \caption{Hovermoeller diagrams of 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
    conditions for the sea ice drift. Also shown it the normalized ice
    strengh $P/P^*=(hc)\,\exp[-C\,(1-c)]$ (bottom panel); each plot is
    overlaid with the contours 1 and 3 of the normalized ice strength
    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
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)
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
time) and the export through Lancaster Sound is only affected by
local ice formation and melting.

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
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
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.

%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
%2. less snow during fall -> more ice -> more export
%3. precip is both snow and rain, depending on the sign of "FICE" (thermodynamic growth rate), with probably different implications


\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
  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.
%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 (effective 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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