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\section{Adjoint sensiivities of the MITsim} |
\section{Adjoint sensitivities of the MITsim} |
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\label{sec:adjoint} |
\label{sec:adjoint} |
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\subsection{The adjoint of MITsim} |
\subsection{The adjoint of MITsim} |
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The ability to generate tangent linear and adjoint model components |
The adjoint model of the MITgcm has become an invaluable |
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of the MITsim has been a main design task. |
tool for sensitivity analysis as well as state estimation \citep[for a |
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For the ocean the adjoint capability has proven to be an |
recent summary, see][]{heim:08}. The code has been developed and |
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invaluable tool for sensitivity analysis as well as state estimation. |
tailored to be readily used with automatic differentiation tools for |
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In short, the adjoint enables very efficient computation of the gradient |
adjoint code generation. This route was also taken in developing and |
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of scalar-valued model diagnostics (called cost function or objective function) |
adapting the sea-ice compontent MITsim, so that tangent linear and |
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with respect to many model "variables". |
adjoint components can be obtained and kept up to date without |
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These variables can be two- or three-dimensional fields of initial |
excessive effort. |
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conditions, model parameters such as mixing coefficients, or |
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time-varying surface or lateral (open) boundary conditions. |
The adjoint model operator (ADM) is the transpose of the tangent |
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When combined, these variables span a potentially high-dimensional |
linear model operator (TLM) of the full (in general nonlinear) forward |
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(e.g. O(10$^8$)) so-called control space. Performing parameter perturbations |
model, in this case the MITsim. This operator computes the gradients |
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to assess model sensitivities quickly becomes prohibitive at these scales. |
of scalar-valued model diagnostics (so-called cost function or |
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Alternatively, (time-varying) sensitivities of the objective function |
objective function) with respect to many model inputs (so-called |
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to any element of the control space can be computed very efficiently in |
independent or control variables). These inputs can be two- or |
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one single adjoint |
three-dimensional fields of initial conditions of the ocean or sea-ice |
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model integration, provided an efficient adjoint model is available. |
state, model parameters such as mixing coefficients, or time-varying |
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surface or lateral (open) boundary conditions. When combined, these |
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[REFERENCES] |
variables span a potentially high-dimensional (e.g. O(10$^8$)) |
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so-called control space. At this problem dimension, perturbing |
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individual parameters to assess model sensitivities quickly becomes |
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The adjoint operator (ADM) is the transpose of the tangent linear operator (TLM) |
prohibitive. By contrast, transient sensitivities of the objective |
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of the full (in general nonlinear) forward model, i.e. the MITsim. |
function to any element of the control and model state space can be |
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The TLM maps perturbations of elements of the control space |
computed very efficiently in one single adjoint model integration, |
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(e.g. initial ice thickness distribution) |
provided an adjoint model is available. |
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via the model Jacobian |
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to a perturbation in the objective function |
In anology to the TLM and ADM components of the MITgcm we rely on the |
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(e.g. sea-ice export at the end of the integration interval). |
autmomatic differentiation (AD) tool ``Transformation of Algorithms in |
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\textit{Tangent} linearity ensures that the derivatives are evaluated |
Fortran'' (TAF) developed by Fastopt \citep{gier-kami:98} to generate |
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with respect to the underlying model trajectory at each point in time. |
TLM and ADM code of the MITsim \citep[for details see][]{maro-etal:99, |
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This is crucial for nonlinear trajectories and the presence of different |
heim-etal:05}. In short, the AD tool uses the nonlinear parent |
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regimes (e.g. effect of the seaice growth term at or away from the |
model code to generate derivative code for the specified control space |
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freezing point of the ocean surface). |
and objective function. Advantages of this approach have been pointed |
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Ensuring tangent linearity can be easily achieved by integrating |
out, for example by \cite{gier-kami:98}. |
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the full model in sync with the TLM to provide the underlying model state. |
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Ensuring \textit{tangent} adjoints is equally crucial, but much more |
Many issues of generating efficient exact adjoint sea-ice code are |
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difficult to achieve because of the reverse nature of the integration: |
similar to those for the ocean model's adjoint. Linearizing the model |
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the adjoint accumulates sensitivities backward in time, |
around the exact nonlinear model trajectory is a crucial aspect in the |
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starting from a unit perturbation of the objective function. |
presence of different regimes (e.g., is the thermodynamic growth term |
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The adjoint model requires the model state in reverse order. |
for sea-ice evaluated near or far away from the freezing point of the |
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This presents one of the major complications in deriving an |
ocean surface?). Adapting the (parent) model code to support the AD |
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exact, i.e. \textit{tangent} adjoint model. |
tool in providing exact and efficient adjoint code represents the main |
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work load initially. For legacy code, this task may become |
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Following closely the development and maintenance of TLM and ADM |
substantial, but it is fairly straightforward when writing new code |
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components of the MITgcm we have relied heavily on the |
with an AD tool in mind. Once this initial task is completed, |
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autmomatic differentiation (AD) tool |
generating the adjoint code of a new model configuration takes about |
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"Transformation of Algorithms in Fortran" (TAF) |
10 minutes. |
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developed by Fastopt (Giering and Kaminski, 1998) |
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to derive TLM and ADM code of the MITsim. |
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Briefly, the nonlinear parent model is fed to the AD tool which produces |
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derivative code for the specified control space and objective function. |
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Following this approach has (apart from its evident success) |
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several advantages: |
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(1) the adjoint model is the exact adjoint operator of the parent model, |
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(2) the adjoint model can be kept up to date with respect to ongoing |
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development of the parent model, and adjustments to the parent model |
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to extend the automatically generated adjoint are incremental changes |
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only, rather than extensive re-developments, |
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(3) the parallel structure of the parent model is preserved |
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by the adjoint model, ensuring efficient use in high performance |
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computing environments. |
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Some initial code adjustments are required to support dependency analysis |
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of the flow reversal and certain language limitations which may lead |
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to irreducible flow graphs (e.g. GOTO statements). |
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The problem of providing the required model state in reverse order |
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at the time of evaluating nonlinear or conditional |
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derivatives is solved via balancing |
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storing vs. recomputation of the model state in a multi-level |
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checkpointing loop. |
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Again, an initial code adjustment is required to support TAFs |
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checkpointing capability. |
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The code adjustments are sufficiently simple so as not to cause |
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major limitations to the full nonlinear parent model. |
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Once in place, an adjoint model of a new model configuration |
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may be derived in about 10 minutes. |
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[HIGHLIGHT COUPLED NATURE OF THE ADJOINT!] |
[HIGHLIGHT COUPLED NATURE OF THE ADJOINT!] |
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* approximate adjoints |
* approximate adjoints |
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\subsection{An example: sensitivities of sea-ice export through Fram Strait} |
\subsection{An example: sensitivities of sea-ice export through |
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the Lancaster Sound} |
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We demonstrate the power of the adjoint method in the context of |
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investigating sea-ice export sensitivities through Lancaster Sound. |
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The rationale for doing so is to complement the analysis of sea-ice |
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dynamics in the presence of narrow straits. Lancaster Sound is one of |
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the main paths of sea-ice flowing through the Canadian Arctic |
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Archipelago (CAA). Export sensitivities reflect dominant pathways |
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through the CAA as resolved by the model. Sensitivity maps can shed a |
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very detailed light on various quantities affecting the sea-ice export |
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(and thus the underlying pathways). Note that while the dominant |
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circulation through Lancaster Sound is toward the East, there is a |
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small Westward flow to the North, hugging the coast of Devon Island |
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\citep{mell:02, mich-etal:06,muen-etal:06}, which is not resolved in |
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our simulation. |
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The model domain is the same as the one described in \refsec{forward}, |
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but with halved horizontal resolution. |
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The adjoint models run efficiently on 80 processors (as validated |
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by benchmarks on both an SGI Altix and an IBM SP5 at NASA/ARC). |
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Following a 4-year spinup (1985 to 1988), the model is integrated for four |
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years and nine months between January 1989 and September 1993. |
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It is forced using realistic 6-hourly NCEP/NCAR atmospheric state variables. |
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%Over the open ocean these are |
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%converted into air-sea fluxes via the bulk formulae of |
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%\citet{large04}. The air-sea fluxes in the presence of |
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%sea-ice are handled by the ice model as described in \refsec{model}. |
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The objective function $J$ is chosen as the ``solid'' fresh water |
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export, that is the export of ice and snow converted to units of fresh |
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water, |
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% |
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\begin{equation} |
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J \, = \, (\rho_{i} h_{i}c + \rho_{s} h_{s}c)\,u |
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\end{equation} |
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% |
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through Lancaster Sound at approximately 82\degW\ (cross-section G in |
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\reffig{arctic_topog}) averaged \ml{PH: Maybe integrated quantity is |
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more physical} over the final 12-month of the integration between October |
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1992 and September 1993. |
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The forward trajectory of the model integration resembles broadly that |
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of the model in \refsec{forward}. Many details are different, owning |
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to different resolution and integration period; for example, the solid |
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fresh water transport through Lancaster Sound is |
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% |
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\ml{PH: Martin, where did you get these numbers from?} |
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% |
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$116\pm101\text{\,km$^{3}$\,y$^{-1}$}$ for a free slip simulation with |
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the C-LSOR solver, but only $39\pm64\text{\,km$^{3}$\,y$^{-1}$}$ for a |
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no slip simulation. |
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The adjoint model is the transpose of the tangent linear (or Jacobian) model |
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operator. It runs backward in time, from September 1993 to |
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January 1989. Along its integration it accumulates the Lagrange multipliers |
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of the model subject to the objective function (solid freshwater export), |
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which can be interpreted as sensitivities of the objective function |
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to each control variable and each element of the intermediate |
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coupled model state variables. |
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Thus, all sensitivity elements of the coupled |
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ocean/sea-ice model state as well as the surface atmospheric state are |
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available for analysis of the transient sensitivity behavior. Over the |
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open ocean, the adjoint of the bulk formula scheme computes |
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sensitivities to the time-varying atmospheric state. Over ice-covered |
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parts, the sea-ice adjoint converts surface ocean sensitivities to |
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atmospheric sensitivities. |
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DISCUSS FORWARD STATE, INCLUDING SOME NUMBERS ON SEA-ICE EXPORT |
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\subsubsection{Adjoint sensitivities} |
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The most readily interpretable ice-export sensitivity is that to |
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effective ice thickness, $\partial{J} / \partial{(hc)}$. |
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\reffig{adjheff} shows transient $\partial{J} / \partial{(hc)}$ using |
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free-slip (left column) and no-slip (right column) boundary |
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conditions. Sensitivity snapshots are depicted for beginning of October 2002, |
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i.e. 12 months back in time from September 1993 |
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(the beginning of the averaging period for the objective |
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function $J$, top), |
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and for Jannuary 1989, the beginning of the forward integration (bottom). |
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\begin{figure*}[t] |
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\includegraphics*[width=\textwidth]{\fpath/adjheff} |
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\caption{Sensitivity $\partial{J}/\partial{(hc)}$ in |
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m$^2$\,s$^{-1}$/m for two different times (rows) and two different |
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boundary conditions for sea ice drift. The color scale is chosen |
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to illustrate the patterns of the sensitivities; the maximum and |
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minimum values are given above the figures. |
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\label{fig:adjheff}} |
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\end{figure*} |
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As expected, the sensitivity patterns are predominantly positive, |
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an increase in ice volume in most places ``upstream'' of |
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Lancaster sound increases the solid fresh water export at the exit section. |
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Also obvious is the transient nature of the sensitivity patterns |
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(top panels vs. bottom panels), |
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i.e. as time moves backward, an increasing area upstream of Lancaster Sound |
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contributes to the export sensitivity. |
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The dominant pathway (free slip case) follows (backward in time) |
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through Barrow Strait |
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into Viscount Melville Sound, and from there trough M'Clure Strait |
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into the Arctic Ocean (the ``Northwest Passage''). |
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Secondary paths are Northward from |
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Viscount Melville Sound through Byam Martin Channel into |
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Prince Gustav Adolf Sea and through Penny Strait into MacLean Strait. |
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The difference between the free slip and no slip solution is evident: |
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by the end of the adjoint integration, in January 1989 |
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the free-slip sensitivities (bottom left) extend through most of the CAA |
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and all the way into the Arctic interior, both to the West (M'Clure St.) |
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and to the North |
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(Ballantyne St., Prince Gustav Adolf Sea, Massey Sound), |
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whereas the no slip sensitivities (bottom right) are overall weaker |
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and remain mostly confined to Lancaster Sound and just West of Barrow Strait. |
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In the free slip solution ice can drift more |
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easily through narrow straits, and |
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a positive ice volume anomaly further upstream in the CAA may increase |
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ice export through the Lancaster Sound within a 4 year period. |
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One peculiar feature in the October 1992 sensitivity maps (top panels) |
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are the negative sensivities to the East and to the West. |
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These can be explained by indirect effects: less ice to the East means |
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less resistance to eastward drift and thus more export; similarly, less ice to |
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the West means that more ice can be moved eastwards from the Barrow Strait |
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into the Lancaster Sound leading to more ice export. |
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\ml{PH: The first explanation (East) I buy, the second (West) I don't}. |
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The temporal evolution of several ice export sensitivities |
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(eqn. XX) along a zonal axis through |
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Lancaster Sound, Barrow Strait,and Melville Sound |
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(115\degW\ to 80\degW\ ), |
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are depicted as Hovmueller diagrams in \reffig{lancaster}. |
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From top to bottom, sensitivities are with respect to effective |
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ice thickness ($hc$), |
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ocean surface temperature ($SST$) and precipitation ($p$) for free slip |
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(left column) and no slip (right column) ice drift boundary conditions. |
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% |
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\begin{figure*} |
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\includegraphics*[height=.8\textheight]{\fpath/lancaster_adj} |
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\caption{Hovermoeller diagrams of sensitivities (derivatives) of the |
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``solid'' fresh water (i.e., ice and snow) export $J$ through Lancaster sound |
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(\reffig{arctic_topog}, cross-section G) with respect to effective |
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ice thickness ($hc$), ocean surface temperature (SST) and |
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precipitation ($p$) for two runs with free slip and no slip boundary |
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conditions for the sea ice drift. Also shown it the normalized ice |
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strengh $P/P^*=(hc)\,\exp[-C\,(1-c)]$ (bottom panel); each plot is |
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overlaid with the contours 1 and 3 of the normalized ice strength |
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for orientation. |
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\label{fig:lancaster}} |
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\end{figure*} |
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% |
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The Hovmoeller diagrams of ice thickness (top row) and sea surface temperature |
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(second row) sensitivities are coherent: |
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more ice in the Lancaster Sound leads |
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to more export, and one way to get more ice is by colder surface |
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temperatures (less melting from below). In the free slip case the |
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sensitivities spread out in "pulses" following a seasonal cycle: |
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can propagate westwards (backwards in time) when the ice |
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strength is low in late summer, early autumn. |
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In contrast, during winter, the sensitivities show little to now |
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westward propagation. |
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In the no slip case the (normalized) |
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ice strength does not fall below 1 during the winters of 1991 to 1993 |
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(mainly because the ice concentrations remain nearly 100\%, not |
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shown). Ice is therefore blocked and cannot drift eastwards |
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(forward in time) through the |
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Melville Sound, Barrow Strait, Lancaster Sound channel system. |
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Consequently, the sensitivities do not propagate westwards (backwards in |
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time) and the export through Lancaster Sound is only affected by |
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local ice formation and melting for the entire integration period. |
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The sensitivities to precipitation exhibit an oscillatory behaviour: |
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they are negative (more precipitation leads to less export) |
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before January (more precisely, late fall) and mostly positive after January |
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(more precisely, January through July). |
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Times of positive sensitivities coincide with times of |
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normalized ice strengths exceeding values of 3. |
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% |
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\ml{PH: Problem is, that's not true for the first two years (backward), |
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East of 95\degW\ , i.e. in Lancaster Sound. |
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For example, at 90\degW\ the sensitivities are negative throughout 1992, |
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and no clear correlation to ice strength is apparent there.}. |
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% |
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Assuming that most precipation is snow in this area |
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% |
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\footnote{ |
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In the |
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current implementation the model differentiates between snow and rain |
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depending on the thermodynamic growth rate; when it is cold enough for |
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ice to grow, all precipitation is assumed to be snow.} |
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% |
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the sensitivities can be interpreted in terms of the model physics. Short |
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wave radiation cannot penetrate the snow cover and has a higer albedo |
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than ice (0.85 for dry snow and 0.75 for dry ice in our case); thus it |
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protects the ice against melting in spring (after January). |
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\ml{PH: what about the direct effect of accumulation of precip. as snow |
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which directly increases the volume.}. |
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On the other hand, snow reduces the effective conductivity and thus the heat |
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flux through the ice. This insulating effect slows down the cooling of |
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the surface water underneath the ice and limits the ice growth from |
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below, so that less snow in the ice-growing season leads to more new |
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ice and thus more ice export. |
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\ml{PH: Should probably discuss the effect of snow vs. rain. |
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To me it seems that the "rain" effect doesn't really play |
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because the neg. sensitivities are too late in the fall, |
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probably mostly falling as snow.}. |
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%Und jetzt weiss ich nicht mehr weiter, aber nun kann folgendes passiert sein: |
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%1. snow insulates against melting from above during spring: more precip (snow) -> more export |
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%2. less snow during fall -> more ice -> more export |
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%3. precip is both snow and rain, depending on the sign of "FICE" (thermodynamic growth rate), with probably different implications |
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\subsubsection{Forward sensitivities} |
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\ml{[Here we need for integrations to show that the adjoint |
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sensitivites are not just academic. I suggest to perturb HEFF |
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and THETA initial conditions, and PRECIP somewhere in the Melville |
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Sound and then produce plots similar to reffig{lancaster}. For |
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PRECIP it would be great to have two perturbation experiments, one |
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where ADJprecip is posivite and one where ADJprecip is negative]} |
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%(*) |
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%The sensitivity in Baffin Bay are more complex. |
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%The pattern evolves along the Western boundary, connecting |
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%the Lancaster Sound Polynya, the Coburg Island Polynya, and the |
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%North Water Polynya, and reaches into Nares Strait and the Kennedy Channel. |
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%The sign of sensitivities has an oscillatory character |
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%[AT FREQUENCY OF SEASONAL CYCLE?]. |
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%First, we need to establish whether forward perturbation runs |
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%corroborate the oscillatory behaviour. |
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%Then, several possible explanations: |
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%(i) connection established through Nares Strait throughflow |
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%which extends into Western boundary current in Northern Baffin Bay. |
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%(ii) sea-ice concentration there is seasonal, i.e. partly |
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%ice-free during the year. Seasonal cycle in sensitivity likely |
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%connected to ice-free vs. ice-covered parts of the year. |
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%Negative sensitivities can potentially be attributed |
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%to blocking of Lancaster Sound ice export by Western boundary ice |
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%in Baffin Bay. |
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%(iii) Alternatively to (ii), flow reversal in Lancaster Sound is a possibility |
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%(in reality there's a Northern counter current hugging the coast of |
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%Devon Island which we probably don't resolve). |
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|
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%Remote control of Kennedy Channel on Lancaster Sound ice export |
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%seems a nice test for appropriateness of free-slip vs. no-slip BCs. |
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|
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%\paragraph{Sensitivities to the sea-ice area} |
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|
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%Fig. XXX depcits transient sea-ice export sensitivities |
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%to changes in sea-ice concentration |
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% $\partial J / \partial area$ using free-slip |
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%(left column) and no-slip (right column) boundary conditions. |
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%Sensitivity snapshots are depicted for (from top to bottom) |
| 323 |
|
%12, 24, 36, and 48 months prior to May 2003. |
| 324 |
|
%Contrary to the steady patterns seen for thickness sensitivities, |
| 325 |
|
%the ice-concentration sensitivities exhibit a strong seasonal cycle |
| 326 |
|
%in large parts of the domain (but synchronized on large scale). |
| 327 |
|
%The following discussion is w.r.t. free-slip run. |
| 328 |
|
|
| 329 |
|
%(*) |
| 330 |
|
%Months, during which sensitivities are negative: |
| 331 |
|
%\\ |
| 332 |
|
%0 to 5 Db=N/A, Dr=5 (May-Jan) \\ |
| 333 |
|
%10 to 17 Db=7, Dr=5 (Jul-Jan) \\ |
| 334 |
|
%22 to 29 Db=7, Dr=5 (Jul-Jan) \\ |
| 335 |
|
%34 to 41 Db=7, Dr=5 (Jul-Jan) \\ |
| 336 |
|
%46 to 49 D=N/A \\ |
| 337 |
|
%% |
| 338 |
|
%These negative sensitivities seem to be connected to months |
| 339 |
|
%during which main parts of the CAA are essentially entirely ice-covered. |
| 340 |
|
%This means that increase in ice concentration during this period |
| 341 |
|
%will likely reduce ice export due to blocking |
| 342 |
|
%[NEED TO EXPLAIN WHY THIS IS NOT THE CASE FOR dJ/dHEFF]. |
| 343 |
|
%Only during periods where substantial parts of the CAA are |
| 344 |
|
%ice free (i.e. sea-ice concentration is less than one in larger parts of |
| 345 |
|
%the CAA) will an increase in ice-concentration increase ice export. |
| 346 |
|
|
| 347 |
|
%(*) |
| 348 |
|
%Sensitivities peak about 2-3 months before sign reversal, i.e. |
| 349 |
|
%max. negative sensitivities are expected end of July |
| 350 |
|
%[DOUBLE CHECK THIS]. |
| 351 |
|
|
| 352 |
|
%(*) |
| 353 |
|
%Peaks/bursts of sensitivities for months |
| 354 |
|
%14-17, 19-21, 27-29, 30-33, 38-40, 42-45 |
| 355 |
|
|
| 356 |
|
%(*) |
| 357 |
|
%Spatial "anti-correlation" (in sign) between main sensitivity branch |
| 358 |
|
%(essentially Northwest Passage and immediate connecting channels), |
| 359 |
|
%and remote places. |
| 360 |
|
%For example: month 20, 28, 31.5, 40, 43. |
| 361 |
|
%The timings of max. sensitivity extent are similar between |
| 362 |
|
%free-slip and no-slip run; and patterns are similar within CAA, |
| 363 |
|
%but differ in the Arctic Ocean interior. |
| 364 |
|
|
| 365 |
|
%(*) |
| 366 |
|
%Interesting (but real?) patterns in Arctic Ocean interior. |
| 367 |
|
|
| 368 |
|
%\paragraph{Sensitivities to the sea-ice velocity} |
| 369 |
|
|
| 370 |
|
%(*) |
| 371 |
|
%Patterns of ADJuice at almost any point in time are rather complicated |
| 372 |
|
%(in particular with respect to spatial structure of signs). |
| 373 |
|
%Might warrant perturbation tests. |
| 374 |
|
%Patterns of ADJvice, on the other hand, are more spatially coherent, |
| 375 |
|
%but still hard to interpret (or even counter-intuitive |
| 376 |
|
%in many places). |
| 377 |
|
|
| 378 |
|
%(*) |
| 379 |
|
%"Growth in extent of sensitivities" goes in clear pulses: |
| 380 |
|
%almost no change between months: 0-5, 10-20, 24-32, 36-44 |
| 381 |
|
%These essentially correspond to months of |
| 382 |
|
|
| 383 |
|
|
| 384 |
|
%\subsection{Sensitivities to the oceanic state} |
| 385 |
|
|
| 386 |
|
%\paragraph{Sensitivities to theta} |
| 387 |
|
|
| 388 |
|
%\textit{Sensitivities at the surface (z = 5 m)} |
| 389 |
|
|
| 390 |
|
%(*) |
| 391 |
|
%mabye redo with caxmax=0.02 or even 0.05 |
| 392 |
|
|
| 393 |
|
%(*) |
| 394 |
|
%Core of negative sensitivities spreading through the CAA as |
| 395 |
|
%one might expect [TEST]: |
| 396 |
|
%Increase in SST will decrease ice thickness and therefore ice export. |
| 397 |
|
|
| 398 |
|
%(*) |
| 399 |
|
%What's maybe unexpected is patterns of positive sensitivities |
| 400 |
|
%at the fringes of the "core", e.g. in the Southern channels |
| 401 |
|
%(Bellot St., Peel Sound, M'Clintock Channel), and to the North |
| 402 |
|
%(initially MacLean St., Prince Gustav Adolf Sea, Hazen St., |
| 403 |
|
%then shifting Northward into the Arctic interior). |
| 404 |
|
|
| 405 |
|
%(*) |
| 406 |
|
%Marked sensitivity from the Arctic interior roughly along 60$^{\circ}$W |
| 407 |
|
%propagating into Lincoln Sea, then |
| 408 |
|
%entering Nares Strait and Smith Sound, periodically |
| 409 |
|
%warming or cooling[???] the Lancaster Sound exit. |
| 410 |
|
|
| 411 |
|
%\textit{Sensitivities at depth (z = 200 m)} |
| 412 |
|
|
| 413 |
|
%(*) |
| 414 |
|
%Negative sensitivities almost everywhere, as might be expected. |
| 415 |
|
|
| 416 |
|
%(*) |
| 417 |
|
%Sensitivity patterns between free-slip and no-slip BCs |
| 418 |
|
%are quite similar, except in Lincoln Sea (North of Nares St), |
| 419 |
|
%where the sign is reversed (but pattern remains similar). |
| 420 |
|
|
| 421 |
|
%\paragraph{Sensitivities to salt} |
| 422 |
|
|
| 423 |
|
%T.B.D. |
| 424 |
|
|
| 425 |
|
%\paragraph{Sensitivities to velocity} |
| 426 |
|
|
| 427 |
|
%T.B.D. |
| 428 |
|
|
| 429 |
|
%\subsection{Sensitivities to the atmospheric state} |
| 430 |
|
|
| 431 |
|
%\begin{itemize} |
| 432 |
|
%% |
| 433 |
|
%\item |
| 434 |
|
%plot of ATEMP for 12, 24, 36, 48 months |
| 435 |
|
%% |
| 436 |
|
%\item |
| 437 |
|
%plot of HEFF for 12, 24, 36, 48 months |
| 438 |
|
%% |
| 439 |
|
%\end{itemize} |
| 440 |
|
|
| 441 |
|
|
| 442 |
|
|
| 443 |
|
%\reffig{4yradjheff}(a--d) depict sensitivities of sea-ice export |
| 444 |
|
%through Fram Strait in December 1995 to changes in sea-ice thickness |
| 445 |
|
%12, 24, 36, 48 months back in time. Corresponding sensitivities to |
| 446 |
|
%ocean surface temperature are depicted in |
| 447 |
|
%\reffig{4yradjthetalev1}(a--d). The main characteristics is |
| 448 |
|
%consistency with expected advection of sea-ice over the relevant time |
| 449 |
|
%scales considered. The general positive pattern means that an |
| 450 |
|
%increase in sea-ice thickness at location $(x,y)$ and time $t$ will |
| 451 |
|
%increase sea-ice export through Fram Strait at time $T_e$. Largest |
| 452 |
|
%distances from Fram Strait indicate fastest sea-ice advection over the |
| 453 |
|
%time span considered. The ice thickness sensitivities are in close |
| 454 |
|
%correspondence to ocean surface sentivitites, but of opposite sign. |
| 455 |
|
%An increase in temperature will incur ice melting, decrease in ice |
| 456 |
|
%thickness, and therefore decrease in sea-ice export at time $T_e$. |
| 457 |
|
|
| 458 |
|
%The picture is fundamentally different and much more complex |
| 459 |
|
%for sensitivities to ocean temperatures away from the surface. |
| 460 |
|
%\reffig{4yradjthetalev10??}(a--d) depicts ice export sensitivities to |
| 461 |
|
%temperatures at roughly 400 m depth. |
| 462 |
|
%Primary features are the effect of the heat transport of the North |
| 463 |
|
%Atlantic current which feeds into the West Spitsbergen current, |
| 464 |
|
%the circulation around Svalbard, and ... |
| 465 |
|
|
| 466 |
|
|
| 467 |
|
%%\begin{figure}[t!] |
| 468 |
|
%%\centerline{ |
| 469 |
|
%%\subfigure[{\footnotesize -12 months}] |
| 470 |
|
%%{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim072_cmax2.0E+02.eps}} |
| 471 |
|
%%\includegraphics*[width=.3\textwidth]{H_c.bin_res_100_lev1.pdf} |
| 472 |
|
%% |
| 473 |
|
%%\subfigure[{\footnotesize -24 months}] |
| 474 |
|
%%{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim145_cmax2.0E+02.eps}} |
| 475 |
|
%%} |
| 476 |
|
%% |
| 477 |
|
%%\caption{Sensitivity of sea-ice export through Fram Strait in December 2005 to |
| 478 |
|
%%sea-ice thickness at various prior times. |
| 479 |
|
%%\label{fig:4yradjheff}} |
| 480 |
|
%%\end{figure} |
| 481 |
|
|
| 482 |
|
|
| 483 |
|
%\ml{[based on the movie series |
| 484 |
|
% zzz\_run\_export\_canarch\_freeslip\_4yr\_1989\_ADJ*:]} The ice |
| 485 |
|
%export through the Canadian Archipelag is highly sensitive to the |
| 486 |
|
%previous state of the ocean-ice system in the Archipelago and the |
| 487 |
|
%Western Arctic. According to the \ml{(adjoint)} senstivities of the |
| 488 |
|
%eastward ice transport through Lancaster Sound (\reffig{arctic_topog}, |
| 489 |
|
%cross-section G) with respect to ice volume (effective thickness), ocean |
| 490 |
|
%surface temperature, and vertical diffusivity near the surface |
| 491 |
|
%(\reffig{fouryearadj}) after 4 years of integration the following |
| 492 |
|
%mechanisms can be identified: near the ``observation'' (cross-section |
| 493 |
|
%G), smaller vertical diffusivities lead to lower surface temperatures |
| 494 |
|
%and hence to more ice that is available for export. Further away from |
| 495 |
|
%cross-section G, the sensitivity to vertical diffusivity has the |
| 496 |
|
%opposite sign, but temperature and ice volume sensitivities have the |
| 497 |
|
%same sign as close to the observation. |
| 498 |
|
|
|
We demonstrate the power of the adjoint method |
|
|
in the context of investigating sea-ice export sensitivities through Fram Strait |
|
|
(for details of this study see Heimbach et al., 2007). |
|
|
%\citep[for details of this study see][]{heimbach07}. %Heimbach et al., 2007). |
|
|
The domain chosen 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 1-year spinup, the model has been integrated for four |
|
|
years between 1992 and 1995. 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 Fram Strait |
|
|
computed for December 1995. The adjoint model computes sensitivities |
|
|
to sea-ice export back in time from 1995 to 1992 along this |
|
|
trajectory. In principle all adjoint model variable (i.e., Lagrange |
|
|
multipliers) of the coupled ocean/sea-ice model are available to |
|
|
analyze the transient sensitivity behaviour of the ocean and sea-ice |
|
|
state. 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. |
|
|
|
|
|
\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}} |
|
|
} |
|
|
|
|
|
\centerline{ |
|
|
\subfigure[{\footnotesize |
|
|
-36 months}] |
|
|
{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim218_cmax2.0E+02.eps}} |
|
|
% |
|
|
\subfigure[{\footnotesize |
|
|
-48 months}] |
|
|
{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim292_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} |
|
|
|
|
|
|
|
|
\begin{figure}[t!] |
|
|
\centerline{ |
|
|
\subfigure[{\footnotesize -12 months}] |
|
|
{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJtheta_arc_lev1_tim072_cmax5.0E+01.eps}} |
|
|
%\includegraphics*[width=.3\textwidth]{H_c.bin_res_100_lev1} |
|
|
% |
|
|
\subfigure[{\footnotesize -24 months}] |
|
|
{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJtheta_arc_lev1_tim145_cmax5.0E+01.eps}} |
|
|
} |
|
|
|
|
|
\centerline{ |
|
|
\subfigure[{\footnotesize |
|
|
-36 months}] |
|
|
{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJtheta_arc_lev1_tim218_cmax5.0E+01.eps}} |
|
|
% |
|
|
\subfigure[{\footnotesize |
|
|
-48 months}] |
|
|
{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJtheta_arc_lev1_tim292_cmax5.0E+01.eps}} |
|
|
} |
|
|
\caption{Same as \reffig{4yradjheff} but for sea surface temperature |
|
|
\label{fig:4yradjthetalev1}} |
|
|
\end{figure} |
|
| 499 |
|
|
| 500 |
%%% Local Variables: |
%%% Local Variables: |
| 501 |
%%% mode: latex |
%%% mode: latex |
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