--- MITgcm_contrib/articles/ceaice/ceaice_adjoint.tex 2008/03/25 22:04:31 1.3 +++ MITgcm_contrib/articles/ceaice/ceaice_adjoint.tex 2008/07/28 12:34:27 1.7 @@ -1,59 +1,55 @@ -\section{Adjoint sensiivities of the MITsim} +\section{Adjoint sensitivities of the MITsim} \label{sec:adjoint} \subsection{The adjoint of MITsim} - -The ability to generate tangent linear and adjoint components -of a coupled ocean sea-ice system was one of the main drivers -behind the MITsim development. -For the ocean the adjoint capability has proven to be an -invaluable tool for sensitivity analysis as well as state estimation, -as evidenced by various adjoint-based studies -(for a recent summary, see \cite{heim:08}). - -The adjoint model operator (ADM) is the transpose of the tangent linear -model operator (TLM) -of the full (in general nonlinear) forward model, i.e. the MITsim. -It enables very efficient computation of 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. Performing parameter perturbations -to assess model sensitivities quickly becomes prohibitive at these scales. -Alternatively, 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 efficient adjoint model is available. - -Following closely the development and maintenance of the -TLM and ADM components of the MITgcm we have relied heavily on the -autmomatic differentiation (AD) tool -"Transformation of Algorithms in Fortran" (TAF) -developed by Fastopt \citep{gier-kami:98}. -to derive TLM and ADM code of the MITsim -(for details see \cite{maro-etal:99}, \cite{heim-etal:05}). -Briefly, the nonlinear parent model is fed to the AD tool which produces -derivative code for the specified control space and objective function. -Apart from its evident success, advantages of this approach have been -pointed out, e.g. by \cite{gier-kami:98}. - -Many issues underlying the efficient exact adjoint sea-ice code generation -are similar to those arising for the ocean model's adjoint. -Linearizing the model around the exact nonlinear model trajectory, -as we do, is a crucial aspect in the presence of different -regimes (e.g. effect of the seaice growth term at or away from the -freezing point of the ocean surface). -Adjusting the (parent) model code to support the AD tool in -providing exact and efficient adjoint code is the main initial work. -This may be substantial for legacy code, but fairly straightforward -when coding with "AD application in mind". -Once in place, an adjoint model of a new model configuration -may be derived in about 10 minutes. +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!] @@ -69,288 +65,460 @@ \subsection{An example: sensitivities of sea-ice export through -the Lancaster and Jones Sound} - -We demonstrate the power of the adjoint method -in the context of investigating sea-ice export sensitivities through -Lancaster and Jones 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 [ARE WE RESOLVING THIS?], -see e.g. \cite{mell:02, mich-etal:06,muen-etal:06}. - -The model domain is a coarsened version of the Arctic face of the -high-resolution cubed-sphere configuration of the ECCO2 project -\citep[see][]{menemenlis05}. It covers the entire Arctic, -extends into the North Pacific such as to cover the entire -ice-covered regions, and comprises parts of the North Atlantic -down to XXN to enable analysis of remote influences of the -North Atlantic current to sea-ice variability and export. -The horizontal resolution varies between XX and YY km -with 50 unevenly spaced vertical levels. -The adjoint models run efficiently on 80 processors -(benchmarks have been performed both on an SGI Altix as well as an -IBM SP5 at NASA/ARC). - -Following a 3-year spinup, the model has been integrated for four -years and five months between January 1989 and May 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}. Derivation of air-sea fluxes in the presence of -sea-ice is handled by the ice model as described in \refsec{model}. -The objective function chosen is -sea-ice export through -Lancaster Sound at XX$^{\circ}$W -averaged over an 8-month period between October 1992 and May 1993. - -The adjoint model computes sensitivities -to sea-ice 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 behaviour. -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 - -\subsection{Sensitivities to the sea-ice state} +the Lancaster Sound} -\paragraph{Sensitivities to the sea-ice thickness} - -The most readily interpretable ice-export sensitivity is that -to ice thickness, $\partial J / \partial heff$. -Fig. XXX depcits transient $\partial J / \partial heff$ 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. -The dominant features are in accordance with expectations: - -(*) -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 \\ +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 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 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 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, % -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} +\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; ML: what did you actually compute? I did not scale +anything, yet. Please insert what is actually done.} 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?} +\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. \ml{[Here we can say that the export through + Lancaster Sound is highly uncertain, making is a perfect candidate + for sensitivity, bla bla?]} + +The adjoint model is the transpose of the tangent linear (or Jacobian) model +operator. It runs backwards in time, from September 1993 to +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 +areas, the sea-ice adjoint converts surface ocean sensitivities to +atmospheric sensitivities. -T.B.D. +DISCUSS FORWARD STATE, INCLUDING SOME NUMBERS ON SEA-ICE EXPORT -\subsection{Sensitivities to the atmospheric state} +\subsubsection{Adjoint sensitivities} -\begin{itemize} -% -\item -plot of ATEMP for 12, 24, 36, 48 months +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 beginning of October 1992, +that is 12 months before 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 + 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*} + +The sensitivity patterns for effective ice thickness 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. +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. +\ml{PH: The first explanation (East) I buy, the second (West) I + don't.} \ml{[ML: unfortunately, I don't have anything better to + offer, do you? Keep in mind that these sensitivites are very small + and only show up, because of the colorscale. In Fig6, they are + hardly visible.]} + +The temporal evolution of several ice export sensitivities (eqn. XX, +\ml{[which equation do you mean?]}) along a zonal axis through +Lancaster Sound, Barrow Strait, and Melville Sound (115\degW\ to +80\degW, averaged across the passages) are depicted as Hovmueller +diagrams in \reffig{lancaster}. These are, from top to bottom, the +sensitivities 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. % -\item -plot of HEFF for 12, 24, 36, 48 months +\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*} % -\end{itemize} - +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 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 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 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. + +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, 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?]} -\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}} -} +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.} % -\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} +the sensitivities 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 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. +To me it seems that the "rain" effect doesn't really play a role +because the neg. sensitivities are too late in the fall, +probably mostly falling as snow.} \ml{[ML: correct, I looked at +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 +85\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 + PRECIP it would be great to have two perturbation experiments, one + where ADJprecip is posivite and one where ADJprecip is negative]} + + +%(*) +%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. %%% Local Variables: