--- MITgcm_contrib/articles/ceaice/ceaice.tex 2007/11/07 14:38:57 1.2 +++ MITgcm_contrib/articles/ceaice/ceaice.tex 2008/03/25 22:04:31 1.20 @@ -1,6 +1,8 @@ +% $Header: /home/ubuntu/mnt/e9_copy/MITgcm_contrib/articles/ceaice/ceaice.tex,v 1.20 2008/03/25 22:04:31 heimbach Exp $ +% $Name: $ \documentclass[12pt]{article} -\usepackage{epsfig} -\usepackage{graphics} + +\usepackage[]{graphicx} \usepackage{subfigure} \usepackage[round,comma]{natbib} @@ -10,6 +12,8 @@ \newcommand\bmmax{10} \newcommand\hmmax{10} \usepackage{bm} +\usepackage{url} + % math abbreviations \newcommand{\vek}[1]{\ensuremath{\mathbf{#1}}} \newcommand{\mat}[1]{\ensuremath{\mathbf{#1}}} @@ -35,7 +39,10 @@ \newlength{\mediumfigwidth}\setlength{\mediumfigwidth}{39pc} %\newlength{\widefigwidth}\setlength{\widefigwidth}{39pc} \newlength{\widefigwidth}\setlength{\widefigwidth}{\textwidth} -\newcommand{\fpath}{.} +\newcommand{\fpath}{figs} + +% commenting scheme +\newcommand{\ml}[1]{\textsf{\slshape #1}} \title{A Dynamic-Thermodynamic Sea ice Model for Ocean Climate Estimation on an Arakawa C-Grid} @@ -46,629 +53,30 @@ \maketitle -\begin{abstract} - Some blabla -\end{abstract} - -\section{Introduction} -\label{sec:intro} - -more blabla - -\section{Model} -\label{sec:model} - -Traditionally, probably for historical reasons and the ease of -treating the Coriolis term, most standard sea-ice models are -discretized on Arakawa-B-grids \citep[e.g.,][]{hibler79, harder99, - kreyscher00, zhang98, hunke97}. From the perspective of coupling a -sea ice-model to a C-grid ocean model, the exchange of fluxes of heat -and fresh-water pose no difficulty for a B-grid sea-ice model -\citep[e.g.,][]{timmermann02a}. However, surface stress is defined at -velocities points and thus needs to be interpolated between a B-grid -sea-ice model and a C-grid ocean model. While the smoothing implicitly -associated with this interpolation may mask grid scale noise, it may -in two-way coupling lead to a computational mode as will be shown. By -choosing a C-grid for the sea-ice model, we circumvene this difficulty -altogether and render the stress coupling as consistent as the -buoyancy coupling. - -A further advantage of the C-grid formulation is apparent in narrow -straits. In the limit of only one grid cell between coasts there is no -flux allowed for a B-grid (with no-slip lateral boundary counditions), -whereas the C-grid formulation allows a flux of sea-ice through this -passage for all types of lateral boundary conditions. We (will) -demonstrate this effect in the Candian archipelago. - -\subsection{Dynamics} -\label{sec:dynamics} - -The momentum equations of the sea-ice model are standard with -\begin{equation} - \label{eq:momseaice} - m \frac{D\vek{u}}{Dt} = -mf\vek{k}\times\vek{u} + \vtau_{air} + - \vtau_{ocean} - m \nabla{\phi(0)} + \vek{F}, -\end{equation} -where $\vek{u} = u\vek{i}+v\vek{j}$ is the ice velocity vectory, $m$ -the ice mass per unit area, $f$ the Coriolis parameter, $g$ is the -gravity accelation, $\nabla\phi$ is the gradient (tilt) of the sea -surface height potential beneath the ice. $\phi$ is the sum of -atmpheric pressure $p_{a}$ and loading due to ice and snow -$(m_{i}+m_{s})g$. $\vtau_{air}$ and $\vtau_{ocean}$ are the wind and -ice-ocean stresses, respectively. $\vek{F}$ is the interaction force -and $\vek{i}$, $\vek{j}$, and $\vek{k}$ are the unit vectors in the -$x$, $y$, and $z$ directions. Advection of sea-ice momentum is -neglected. The wind and ice-ocean stress terms are given by -\begin{align*} - \vtau_{air} =& \rho_{air} |\vek{U}_{air}|R_{air}(\vek{U}_{air}) \\ - \vtau_{ocean} =& \rho_{ocean} |\vek{U}_{ocean}-\vek{u}| - R_{ocean}(\vek{U}_{ocean}-\vek{u}), \\ -\end{align*} -where $\vek{U}_{air/ocean}$ are the surface winds of the atmosphere -and surface currents of the ocean, respectively. $C_{air/ocean}$ are -air and ocean drag coefficients, $\rho_{air/ocean}$ reference -densities, and $R_{air/ocean}$ rotation matrices that act on the -wind/current vectors. $\vek{F} = \nabla\cdot\sigma$ is the divergence -of the interal stress tensor $\sigma_{ij}$. - -For an isotropic system this stress tensor can be related to the ice -strain rate and strength by a nonlinear viscous-plastic (VP) -constitutive law \citep{hibler79, zhang98}: -\begin{equation} - \label{eq:vpequation} - \sigma_{ij}=2\eta(\dot{\epsilon}_{ij},P)\dot{\epsilon}_{ij} - + \left[\zeta(\dot{\epsilon}_{ij},P) - - \eta(\dot{\epsilon}_{ij},P)\right]\dot{\epsilon}_{kk}\delta_{ij} - - \frac{P}{2}\delta_{ij}. -\end{equation} -The ice strain rate is given by -\begin{equation*} - \dot{\epsilon}_{ij} = \frac{1}{2}\left( - \frac{\partial{u_{i}}}{\partial{x_{j}}} + - \frac{\partial{u_{j}}}{\partial{x_{i}}}\right). -\end{equation*} -The pressure $P$, a measure of ice strength, depends on both thickness -$h$ and compactness (concentration) $c$: \[P = -P^{*}c\,h\,e^{[C^{*}\cdot(1-c)]},\] with the constants $P^{*}$ and -$C^{*}$. The nonlinear bulk and shear viscosities $\eta$ and $\zeta$ -are functions of ice strain rate invariants and ice strength such that -the principal components of the stress lie on an elliptical yield -curve with the ratio of major to minor axis $e$ equal to $2$; they are -given by: -\begin{align*} - \zeta =& \frac{P}{2\Delta} \\ - \eta =& \frac{P}{2\Delta{e}^2} \\ - \intertext{with the abbreviation} - \Delta = & \left[ - \left(\dot{\epsilon}_{11}^2+\dot{\epsilon}_{22}^2\right) - (1+e^{-2}) + 4e^{-2}\dot{\epsilon}_{12}^2 + - 2\dot{\epsilon}_{11}\dot{\epsilon}_{22} (1-e^{-2}) - \right]^{-\frac{1}{2}} -\end{align*} -In the current implementation, the VP-model is integrated with the -semi-implicit line successive over relaxation (LSOR)-solver of -\citet{zhang98}, which allows for long time steps that, in our case, -is limited by the explicit treatment of the Coriolis term. The -explicit treatment of the Coriolis term does not represent a severe -limitation because it restricts the time step to approximately the -same length as in the ocean model where the Coriolis term is also -treated explicitly. - -\citet{hunke97}'s introduced an elastic contribution to the strain -rate elatic-viscous-plastic in order to regularize -Eq.\refeq{vpequation} in such a way that the resulting -elatic-viscous-plastic (EVP) and VP models are identical at steady -state, -\begin{equation} - \label{eq:evpequation} - \frac{1}{E}\frac{\partial\sigma_{ij}}{\partial{t}} + - \frac{1}{2\eta}\sigma_{ij} - + \frac{\eta - \zeta}{4\zeta\eta}\sigma_{kk}\delta_{ij} - + \frac{P}{4\zeta}\delta_{ij} - = \dot{\epsilon}_{ij}. -\end{equation} -%In the EVP model, equations for the components of the stress tensor -%$\sigma_{ij}$ are solved explicitly. Both model formulations will be -%used and compared the present sea-ice model study. -The EVP-model uses an explicit time stepping scheme with a short -timestep. According to the recommendation of \citet{hunke97}, the -EVP-model is stepped forward in time 120 times within the physical -ocean model time step (although this parameter is under debate), to -allow for elastic waves to disappear. Because the scheme does not -require a matrix inversion it is fast in spite of the small timestep -\citep{hunke97}. For completeness, we repeat the equations for the -components of the stress tensor $\sigma_{1} = -\sigma_{11}+\sigma_{22}$, $\sigma_{2}= \sigma_{11}-\sigma_{22}$, and -$\sigma_{12}$. Introducing the divergence $D_D = -\dot{\epsilon}_{11}+\dot{\epsilon}_{22}$, and the horizontal tension -and shearing strain rates, $D_T = -\dot{\epsilon}_{11}-\dot{\epsilon}_{22}$ and $D_S = -2\dot{\epsilon}_{12}$, respectively and using the above abbreviations, -the equations can be written as: -\begin{align} - \label{eq:evpstresstensor1} - \frac{\partial\sigma_{1}}{\partial{t}} + \frac{\sigma_{1}}{2T} + - \frac{P}{2T} &= \frac{P}{2T\Delta} D_D \\ - \label{eq:evpstresstensor2} - \frac{\partial\sigma_{2}}{\partial{t}} + \frac{\sigma_{2} e^{2}}{2T} - &= \frac{P}{2T\Delta} D_T \\ - \label{eq:evpstresstensor12} - \frac{\partial\sigma_{12}}{\partial{t}} + \frac{\sigma_{12} e^{2}}{2T} - &= \frac{P}{4T\Delta} D_S -\end{align} -Here, the elastic parameter $E$ is redefined in terms of a damping timescale -$T$ for elastic waves \[E=\frac{\zeta}{T}.\] -$T=E_{0}\Delta{t}$ with the tunable parameter $E_0<1$ and -the external (long) timestep $\Delta{t}$. \citet{hunke97} recommend -$E_{0} = \frac{1}{3}$. - -For details of the spatial discretization, the reader is referred to -\citet{zhang98, zhang03}. Our discretization differs only (but -importantly) in the underlying grid, namely the Arakawa C-grid, but is -otherwise straightforward. The EVP model in particular is discretized -naturally on the C-grid with $\sigma_{1}$ and $\sigma_{2}$ on the -center points and $\sigma_{12}$ on the corner (or vorticity) points of -the grid. With this choice all derivatives are discretized as central -differences and averaging is only involved in computing $\Delta$ and -$P$ at vorticity points. - -For a general curvilinear grid, one needs in principle to take metric -terms into account that arise in the transformation a curvilinear grid -on the sphere. However, for now we can neglect these metric terms -because they are very small on the cubed sphere grids used in this -paper; in particular, only near the edges of the cubed sphere grid, we -expect them to be non-zero, but these edges are at approximately -35\degS\ or 35\degN\ which are never covered by sea-ice in our -simulations. Everywhere else the coordinate system is locally nearly -cartesian. However, for last-glacial-maximum or snowball-earth-like -simulations the question of metric terms needs to be reconsidered. -Either, one includes these terms as in \citet{zhang03}, or one finds a -vector-invariant formulation fo the sea-ice internal stress term that -does not require any metric terms, as it is done in the ocean dynamics -of the MITgcm \citep{adcroft04:_cubed_sphere}. - -Moving sea ice exerts a stress on the ocean which is the opposite of -the stress $\vtau_{ocean}$ in Eq.\refeq{momseaice}. This stess is -applied directly to the surface layer of the ocean model. An -alternative ocean stress formulation is given by \citet{hibler87}. -Rather than applying $\vtau_{ocean}$ directly, the stress is derived -from integrating over the ice thickness to the bottom of the oceanic -surface layer. In the resulting equation for the \emph{combined} -ocean-ice momentum, the interfacial stress cancels and the total -stress appears as the sum of windstress and divergence of internal ice -stresses: $\delta(z) (\vtau_{air} + \vek{F})/\rho_0$, \citep[see also -Eq.\,2 of][]{hibler87}. The disadvantage of this formulation is that -now the velocity in the surface layer of the ocean that is used to -advect tracers, is really an average over the ocean surface -velocity and the ice velocity leading to an inconsistency as the ice -temperature and salinity are different from the oceanic variables. - -Sea ice distributions are characterized by sharp gradients and edges. -For this reason, we employ a positive 3rd-order advection scheme -\citep{hundsdorfer94} for the thermodynamic variables discussed in the -next section. - -\subparagraph{boundary conditions: no-slip, free-slip, half-slip} - -\begin{itemize} -\item transition from existing B-Grid to C-Grid -\item boundary conditions: no-slip, free-slip, half-slip -\item fancy (multi dimensional) advection schemes -\item VP vs.\ EVP \citep{hunke97} -\item ocean stress formulation \citep{hibler87} -\end{itemize} - -\subsection{Thermodynamics} -\label{sec:thermodynamics} - -In the original formulation the sea ice model \citep{menemenlis05} -uses simple thermodynamics following the appendix of -\citet{semtner76}. This formulation does not allow storage of heat -(heat capacity of ice is zero, and this type of model is often refered -to as a ``zero-layer'' model). Upward heat flux is parameterized -assuming a linear temperature profile and together with a constant ice -conductivity. It is expressed as $(K/h)(T_{w}-T_{0})$, where $K$ is -the ice conductivity, $h$ the ice thickness, and $T_{w}-T_{0}$ the -difference between water and ice surface temperatures. The surface -heat budget is computed in a similar way to that of -\citet{parkinson79} and \citet{manabe79}. - -There is considerable doubt about the reliability of such a simple -thermodynamic model---\citet{semtner84} found significant errors in -phase (one month lead) and amplitude ($\approx$50\%\,overestimate) in -such models---, so that today many sea ice models employ more complex -thermodynamics. A popular thermodynamics model of \citet{winton00} is -based on the 3-layer model of \citet{semtner76} and treats brine -content by means of enthalphy conservation. This model requires in -addition to ice-thickness and compactness (fractional area) additional -state variables to be advected by ice velocities, namely enthalphy of -the two ice layers and the thickness of the overlying snow layer. - -\section{Funnel Experiments} -\label{sec:funnel} - -\begin{itemize} -\item B-grid LSR no-slip -\item C-grid LSR no-slip -\item C-grid LSR slip -\item C-grid EVP no-slip -\item C-grid EVP slip -\end{itemize} - -\subsection{B-grid vs.\ C-grid} -Comparison between: -\begin{itemize} -\item B-grid, lsr, no-slip -\item C-grid, lsr, no-slip -\item C-grid, evp, no-slip -\end{itemize} -all without ice-ocean stress, because ice-ocean stress does not work -for B-grid. - -\subsection{C-grid} -\begin{itemize} -\item no-slip vs. free-slip for both lsr and evp; - "diagnostics" to look at and use for comparison - \begin{itemize} - \item ice transport through Fram Strait/Denmark Strait/Davis - Strait/Bering strait (these are general) - \item ice transport through narrow passages, e.g.\ Nares-Strait - \end{itemize} -\item compare different advection schemes (if lsr turns out to be more - effective, then with lsr otherwise I prefer evp), eg. - \begin{itemize} - \item default 2nd-order with diff1=0.002 - \item 1st-order upwind with diff1=0. - \item DST3FL (SEAICEadvScheme=33 with diff1=0., should work, works for me) - \item 2nd-order wit flux limiter (SEAICEadvScheme=77 with diff1=0.) - \end{itemize} - That should be enough. Here, total ice mass and location of ice edge - is interesting. However, this comparison can be done in an idealized - domain, may not require full Arctic Domain? -\item -Do a little study on the parameters of LSR and EVP -\begin{enumerate} -\item convergence of LSR, how many iterations do you need to get a - true elliptic yield curve -\item vary deltaTevp and the relaxation parameter for EVP and see when - the EVP solution breaks down (relative to the forcing time scale). - For this, it is essential that the evp solver gives use "stripeless" - solutions, that is your dtevp = 1sec solutions/or 10sec solutions - with SEAICE\_evpDampC = 615. -\end{enumerate} -\end{itemize} - -\section{Forward sensitivity experiments} -\label{sec:forward} - -A second series of forward sensitivity experiments have been carried out on an -Arctic Ocean domain with open boundaries. Once again the objective is to -compare the old B-grid LSR dynamic solver with the new C-grid LSR and EVP -solvers. One additional experiment is carried out to illustrate the -differences between the two main options for sea ice thermodynamics in the MITgcm. - -\subsection{Arctic Domain with Open Boundaries} -\label{sec:arctic} - -The Arctic domain of integration is illustrated in Fig.~\ref{???}. It is -carved out from, and obtains open boundary conditions from, the global -cubed-sphere configuration of the Estimating the Circulation and Climate of -the Ocean, Phase II (ECCO2) project \cite{men05a}. The domain size is 420 by -384 grid boxes horizontally with mean horizontal grid spacing of 18 km. - -There are 50 vertical levels ranging in thickness from 10 m near the surface -to approximately 450 m at a maximum model depth of 6150 m. Bathymetry is from -the National Geophysical Data Center (NGDC) 2-minute gridded global relief -data (ETOPO2) and the model employs the partial-cell formulation of -\cite{adc97}, which permits accurate representation of the bathymetry. The -model is integrated in a volume-conserving configuration using a finite volume -discretization with C-grid staggering of the prognostic variables. In the -ocean, the non-linear equation of state of \cite{jac95}. The ocean model is -coupled to a sea-ice model described hereinabove. - -This particular ECCO2 simulation is initialized from rest using the January -temperature and salinity distribution from the World Ocean Atlas 2001 (WOA01) -[Conkright et al., 2002] and it is integrated for 32 years prior to the -1996-2001 period discussed in the study. Surface boundary conditions are from -the National Centers for Environmental Prediction and the National Center for -Atmospheric Research (NCEP/NCAR) atmospheric reanalysis [Kistler et al., -2001]. Six-hourly surface winds, temperature, humidity, downward short- and -long-wave radiations, and precipitation are converted to heat, freshwater, and -wind stress fluxes using the Large and Pond [1981, 1982] bulk -formulae. Shortwave radiation decays exponentially as per Paulson and Simpson -[1977]. Additionally the time-mean river run-off from Large and Nurser [2001] -is applied and there is a relaxation to the monthly-mean climatological sea -surface salinity values from WOA01 with a relaxation time scale of 3 -months. Vertical mixing follows Large et al. [1994] with background vertical -diffusivity of 1.5 × 10-5 m2 s-1 and viscosity of 10-3 m2 s-1. A third order, -direct-space-time advection scheme with flux limiter is employed and there is -no explicit horizontal diffusivity. Horizontal viscosity follows Leith [1996] -but modified to sense the divergent flow as per Fox-Kemper and Menemenlis [in -press]. Shortwave radiation decays exponentially as per Paulson and Simpson -[1977]. Additionally, the time-mean runoff of Large and Nurser [2001] is -applied near the coastline and, where there is open water, there is a -relaxation to monthly-mean WOA01 sea surface salinity with a time constant of -45 days. - -Open water, dry -ice, wet ice, dry snow, and wet snow albedo are, respectively, 0.15, 0.85, -0.76, 0.94, and 0.8. - -\begin{itemize} -\item Configuration -\item OBCS from cube -\item forcing -\item 1/2 and full resolution -\item with a few JFM figs from C-grid LSR no slip - ice transport through Canadian Archipelago - thickness distribution - ice velocity and transport -\end{itemize} +\input{ceaice_abstract.tex} -\begin{itemize} -\item Arctic configuration -\item ice transport through straits and near boundaries -\item focus on narrow straits in the Canadian Archipelago -\end{itemize} - -\begin{itemize} -\item B-grid LSR no-slip -\item C-grid LSR no-slip -\item C-grid LSR slip -\item C-grid EVP no-slip -\item C-grid EVP slip -\item C-grid LSR no-slip + Winton -\item speed-performance-accuracy (small) - ice transport through Canadian Archipelago differences - thickness distribution differences - ice velocity and transport differences -\end{itemize} - -We anticipate small differences between the different models due to: -\begin{itemize} -\item advection schemes: along the ice-edge and regions with large - gradients -\item C-grid: more transport through narrow straits for no slip - conditons, less for free slip -\item VP vs.\ EVP: speed performance, accuracy? -\item ocean stress: different water mass properties beneath the ice -\end{itemize} +\input{ceaice_intro.tex} -\section{Adjoint sensitivity experiment} -\label{sec:adjoint} +\input{ceaice_model.tex} -Adjoint sensitivity experiment on 1/2-res setup - Sensitivity of sea ice volume flow through Fram Strait +\input{ceaice_forward.tex} -\section{Adjoint sensiivities of the MITsim} +\input{ceaice_adjoint.tex} -\subsection{The adjoint of MITsim} - -The ability to generate tangent linear and adjoint model components -of the MITsim has been a main design task. -For the ocean the adjoint capability has proven to be an -invaluable tool for sensitivity analysis as well as state estimation. -In short, the adjoint enables very efficient computation of the gradient -of scalar-valued model diagnostics (called cost function or objective function) -with respect to many model "variables". -These variables can be two- or three-dimensional fields of initial -conditions, 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, (time-varying) sensitivities of the objective function -to any element of the control space can be computed very efficiently in -one single adjoint -model integration, provided an efficient adjoint model is available. - -[REFERENCES] - - -The adjoint operator (ADM) is the transpose of the tangent linear operator (TLM) -of the full (in general nonlinear) forward model, i.e. the MITsim. -The TLM maps perturbations of elements of the control space -(e.g. initial ice thickness distribution) -via the model Jacobian -to a perturbation in the objective function -(e.g. sea-ice export at the end of the integration interval). -\textit{Tangent} linearity ensures that the derivatives are evaluated -with respect to the underlying model trajectory at each point in time. -This is crucial for nonlinear trajectories and the presence of different -regimes (e.g. effect of the seaice growth term at or away from the -freezing point of the ocean surface). -Ensuring tangent linearity can be easily achieved by integrating -the full model in sync with the TLM to provide the underlying model state. -Ensuring \textit{tangent} adjoints is equally crucial, but much more -difficult to achieve because of the reverse nature of the integration: -the adjoint accumulates sensitivities backward in time, -starting from a unit perturbation of the objective function. -The adjoint model requires the model state in reverse order. -This presents one of the major complications in deriving an -exact, i.e. \textit{tangent} adjoint model. - -Following closely the development and maintenance of 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 (Giering and Kaminski, 1998) -to derive TLM and ADM code of the MITsim. -Briefly, the nonlinear parent model is fed to the AD tool which produces -derivative code for the specified control space and objective function. -Following this approach has (apart from its evident success) -several advantages: -(1) the adjoint model is the exact adjoint operator of the parent model, -(2) the adjoint model can be kept up to date with respect to ongoing -development of the parent model, and adjustments to the parent model -to extend the automatically generated adjoint are incremental changes -only, rather than extensive re-developments, -(3) the parallel structure of the parent model is preserved -by the adjoint model, ensuring efficient use in high performance -computing environments. - -Some initial code adjustments are required to support dependency analysis -of the flow reversal and certain language limitations which may lead -to irreducible flow graphs (e.g. GOTO statements). -The problem of providing the required model state in reverse order -at the time of evaluating nonlinear or conditional -derivatives is solved via balancing -storing vs. recomputation of the model state in a multi-level -checkpointing loop. -Again, an initial code adjustment is required to support TAFs -checkpointing capability. -The code adjustments are sufficiently simply so as not to cause -major limitations to the full nonlinear parent model. -Once in place, an adjoint model of a new model configuration -may be derived in 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 Fram Strait} - -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). -The domain chosen is a coarsened version of the Arctic face of the -high-resolution cubed-sphere configuration of the ECCO2 project -(see Menemenlis et al. 2005). 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 Large and Yeager (2004). -Derivation of air-sea fluxes in the presence of sea-ice is handled -by the ice model as described in Section XXX. -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. - -Fig. XXX(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 Fig. XXX(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. -Fig. XXX (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]{figs/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]{figs/run_4yr_ADJheff_arc_lev1_tim145_cmax2.0E+02.eps}} -} - -\centerline{ -\subfigure[{\footnotesize --36 months}] -{\includegraphics*[width=0.44\linewidth]{figs/run_4yr_ADJheff_arc_lev1_tim218_cmax2.0E+02.eps}} -% -\subfigure[{\footnotesize --48 months}] -{\includegraphics*[width=0.44\linewidth]{figs/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]{figs/run_4yr_ADJtheta_arc_lev1_tim072_cmax5.0E+01.eps}} -%\includegraphics*[width=.3\textwidth]{H_c.bin_res_100_lev1.pdf} -% -\subfigure[{\footnotesize -24 months}] -{\includegraphics*[width=0.44\linewidth]{figs/run_4yr_ADJtheta_arc_lev1_tim145_cmax5.0E+01.eps}} -} - -\centerline{ -\subfigure[{\footnotesize --36 months}] -{\includegraphics*[width=0.44\linewidth]{figs/run_4yr_ADJtheta_arc_lev1_tim218_cmax5.0E+01.eps}} -% -\subfigure[{\footnotesize --48 months}] -{\includegraphics*[width=0.44\linewidth]{figs/run_4yr_ADJtheta_arc_lev1_tim292_cmax5.0E+01.eps}} -} -\caption{Same as Fig. XXX but for sea surface temperature -\label{fig:4yradjthetalev1}} -\end{figure} - - - -\section{Discussion and conclusion} -\label{sec:concl} - -The story of the paper could be: -B-grid ice model + C-grid ocean model does not work properly for us, -therefore C-grid ice model with advantages: -\begin{enumerate} -\item stress coupling simpler (no interpolation required) -\item different boundary conditions -\item advection schemes carry over trivially from main code -\end{enumerate} -LSR/EVP solutions are similar with appropriate bcs, evp parameters as -a function of forcing time scale (when does VP solution break -down). Same for LSR solver, provided that it works (o: -Which scheme is more efficient for the resolution/time stepping -parameters that we use here. What about other resolutions? +\input{ceaice_concl.tex} \paragraph{Acknowledgements} We thank Jinlun Zhang for providing the original B-grid code and many -helpful discussions. +helpful discussions. ML thanks Elizabeth Hunke for multiple explanations. + +This work is a contribution to Estimating the Circulation and Climate of the +Ocean, Phase II (ECCO2). The ECCO2 project (http://ecco2.org/) is sponsored +by the NASA Modeling Analysis and Prediction (MAP) program. D. Menemenlis +carried out this work at the Jet Propulsion Laboratory, California Institute +of Technology under contract with the National Aeronautics and Space +Administration. -\bibliography{bib/journal_abrvs,bib/seaice,bib/genocean,bib/maths,bib/mitgcmuv,bib/fram} +\bibliography{bib/journal_abrvs,bib/seaice,bib/genocean,bib/maths,bib/mitgcmuv,bib/fram,bib/mit_biblio} \end{document} @@ -676,3 +84,54 @@ %%% mode: latex %%% TeX-master: t %%% End: + + +A Dynamic-Thermodynamic Sea ice Model for Ocean Climate + Estimation on an Arakawa C-Grid + +Introduction + +Ice Model: + Dynamics formulation. + B-C, LSR, EVP, no-slip, slip + parallellization + Thermodynamics formulation. + 0-layer Hibler salinity + snow + 3-layer Winton + +Idealized tests + Funnel Experiments + Downstream Island tests + B-grid LSR no-slip + C-grid LSR no-slip + C-grid LSR slip + C-grid EVP no-slip + C-grid EVP slip + +Arctic Setup + Configuration + OBCS from cube + forcing + 1/2 and full resolution + with a few JFM figs from C-grid LSR no slip + ice transport through Canadian Archipelago + thickness distribution + ice velocity and transport + +Arctic forward sensitivity experiments + B-grid LSR no-slip + C-grid LSR no-slip + C-grid LSR slip + C-grid EVP no-slip + C-grid EVP slip + C-grid LSR no-slip + Winton + speed-performance-accuracy (small) + ice transport through Canadian Archipelago differences + thickness distribution differences + ice velocity and transport differences + +Adjoint sensitivity experiment on 1/2-res setup + Sensitivity of sea ice volume flow through Fram Strait +*** Sensitivity of sea ice volume flow through Canadian Archipelago + +Summary and conluding remarks