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-revision 1.5 by mlosch,
Mon Jan 14 15:46:54 2008 UTC
+revision 1.6 by mlosch,
Tue Jan 15 16:04:50 2008 UTC
 Line 160 
 tensor compuation the replacement pressu
  \citep{hibler95} is used so that the stress state always lies on the
  elliptic yield curve by definition.
+ In the so-called truncated ellipse method the shear viscosity $\eta$
+ is capped to suppress any tensile stress \citep{hibler97, geiger98}:
+ \begin{equation}
+   \label{eq:etatem}
+   \eta = \min(\frac{\zeta}{e^2}
+   \frac{\frac{P}{2}-\zeta(\dot{\epsilon}_{11}+\dot{\epsilon}_{22})}
+   {\sqrt{(\dot{\epsilon}_{11}+\dot{\epsilon}_{22})^2
+       +4\dot{\epsilon}_{12}^2}}
+ \end{equation}
  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,
-Line 376 
 boundaries.
+Line 386 
 boundaries.
  The above argument can also be invoked to explain the small
  differences between the free-slip and no-slip solutions on the C-grid.
  Because of the non-linearities in the ice viscosities, in particular
- along the boundaries, the no-slip boundary conditions has only a small
+ along the boundaries, the no-slip boundary conditions have only a small
  impact on the solution.
  The difference between LSR and EVP solutions is largest in the
- effective thickness and meridional velocity fields. The velocity field
+ effective thickness and meridional velocity fields. The EVP velocity
- appears to be a little noisy. This noise has been address by
+ fields appears to be a little noisy. This noise has been address by
  \citet{hunke01}. It can be dealt with by reducing EVP's internal time
  step (increasing the number of iterations along with the computational
  cost) or by regularizing the bulk and shear viscosities. We revisit
-Line 390 
 the latter option by reproducing some of
+Line 400 
 the latter option by reproducing some of
  our C-grid no-slip cases: in a square domain with a few islands the
  ice model is initialized with constant ice thickness and linearly
  increasing ice concentration to the east. The model dynamics are
- forced with a constant anticyclonic ocean gyre and variable
+ forced with a constant anticyclonic ocean gyre and by variable
- atmospheric wind whose mean directed diagnonally to the north-east
+ atmospheric wind whose mean direction is diagnonal to the north-east
  corner of the domain; ice volume and concentration are held constant
- (no advection by ice velocity).  \reffig{hunke01} shows the ice
+ (no thermodynamics and no advection by ice velocity).
- velocity field, its divergence, and the bulk viscosity $\zeta$ for the
+ \reffig{hunke01} shows the ice velocity field, its divergence, and the
- cases C-LRSns and C-EVPns, and for a C-EVPns case, where
+ bulk viscosity $\zeta$ for the cases C-LRSns and C-EVPns, and for a
- \citet{hunke01}'s regularization has been implemented; compare to
+ C-EVPns case, where \citet{hunke01}'s regularization has been
- Fig.\,4 in \citet{hunke01}. The regularization contraint limits ice
+ implemented; compare to Fig.\,4 in \citet{hunke01}. The regularization
- strength and viscosities as a function of damping time scale,
+ contraint limits ice strength and viscosities as a function of damping
- resolution and EVP-time step, effectively allowing the elastic waves to
+ time scale, resolution and EVP-time step, effectively allowing the
- damp out more quickly \citep{hunke01}.
+ elastic waves to damp out more quickly \citep{hunke01}.
  \begin{figure*}[htbp]
    \includegraphics[width=\widefigwidth]{\fpath/hun12days}
-   \caption{Hunke's test case.}
+   \caption{Ice flow, divergence and bulk viscosities of three
+     experiments with \citet{hunke01}'s test case: C-LSRns (top),
+     C-EVPns (middle), and C-EVPns with damping described in
+     \citet{hunke01} (bottom).}
    \label{fig:hunke01}
  \end{figure*}
  In the far right (``east'') side of the domain the ice concentration
  is close to one and the ice should be nearly rigid. The applied wind
  tends to push ice toward the upper right corner. Because the highly
- compact ice is confinded by the boundary, it resists any further
+ compact ice is confined by the boundary, it resists any further
  compression and exhibits little motion in the rigid region on the
  right hand side. The C-LSRns solution (top row) allows high
  viscosities in the rigid region suppressing nearly all flow.
  \citet{hunke01}'s regularization for the C-EVPns solution (bottom row)
- clearly suppresses the noise present in $\nabla\cdot\vek{u}$ in the
+ clearly suppresses the noise present in $\nabla\cdot\vek{u}$ and
- unregularized case (middle row), at the cost of reduced viscosities
+ $\log_{10}\zeta$ in the
+ unregularized case (middle row), at the cost of reduced viscosities.
  These reduced viscosities lead to small but finite ice velocities
  which in turn can have a strong effect on solutions in the limit of
  nearly rigid regimes (arching and blocking, not shown).
- %\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;
-Line 485 
 differences between the two main options
+Line 480 
 differences between the two main options
  \subsection{Arctic Domain with Open Boundaries}
  \label{sec:arctic}
- The Arctic domain of integration is illustrated in Fig.~\ref{???}.  It is
+ The Arctic domain of integration is illustrated in Fig.~\ref{???}.  It
- carved out from, and obtains open boundary conditions from, the global
+ is carved out from, and obtains open boundary conditions from, the
- cubed-sphere configuration of the Estimating the Circulation and Climate of
+ global cubed-sphere configuration of the Estimating the Circulation
- the Ocean, Phase II (ECCO2) project \cite{men05a}.  The domain size is 420 by
+ and Climate of the Ocean, Phase II (ECCO2) project
-grid boxes horizontally with mean horizontal grid spacing of 18 km.
+ \citet{menemenlis05}.  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
+ \citet{adcroft97:_shaved_cells}, 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
+ ocean, the non-linear equation of state of \citet{jackett95}.  The ocean model is
  coupled to a sea-ice model described hereinabove.
- This particular ECCO2 simulation is initialized from rest using the January
+ This particular ECCO2 simulation is initialized from rest using the
- temperature and salinity distribution from the World Ocean Atlas 2001 (WOA01)
+ January temperature and salinity distribution from the World Ocean
- [Conkright et al., 2002] and it is integrated for 32 years prior to the
+ Atlas 2001 (WOA01) [Conkright et al., 2002] and it is integrated for
--2001 period discussed in the study. Surface boundary conditions are from
+years prior to the 1996--2001 period discussed in the study. Surface
- the National Centers for Environmental Prediction and the National Center for
+ boundary conditions are from the National Centers for Environmental
- Atmospheric Research (NCEP/NCAR) atmospheric reanalysis [Kistler et al.,
+ Prediction and the National Center for Atmospheric Research
-]. Six-hourly surface winds, temperature, humidity, downward short- and
+ (NCEP/NCAR) atmospheric reanalysis [Kistler et al., 2001]. Six-hourly
- long-wave radiations, and precipitation are converted to heat, freshwater, and
+ surface winds, temperature, humidity, downward short- and long-wave
- wind stress fluxes using the Large and Pond [1981, 1982] bulk
+ radiations, and precipitation are converted to heat, freshwater, and
- formulae. Shortwave radiation decays exponentially as per Paulson and Simpson
+ wind stress fluxes using the \citet{large81, large82} bulk formulae.
- [1977]. Additionally the time-mean river run-off from Large and Nurser [2001]
+ Shortwave radiation decays exponentially as per Paulson and Simpson
- is applied and there is a relaxation to the monthly-mean climatological sea
+ [1977]. Additionally the time-mean river run-off from Large and Nurser
- surface salinity values from WOA01 with a relaxation time scale of 3
+ [2001] is applied and there is a relaxation to the monthly-mean
- months. Vertical mixing follows Large et al. [1994] with background vertical
+ climatological sea surface salinity values from WOA01 with a
- diffusivity of 1.5 � 10-5 m2 s-1 and viscosity of 10-3 m2 s-1. A third order,
+ relaxation time scale of 3 months. Vertical mixing follows
- direct-space-time advection scheme with flux limiter is employed and there is
+ \citet{large94} with background vertical diffusivity of
- no explicit horizontal diffusivity. Horizontal viscosity follows Leith [1996]
+ $1.5\times10^{-5}\text{\,m$^{2}$\,s$^{-1}$}$ and viscosity of
- but modified to sense the divergent flow as per Fox-Kemper and Menemenlis [in
+ $10^{-3}\text{\,m$^{2}$\,s$^{-1}$}$. A third order, direct-space-time
- press].  Shortwave radiation decays exponentially as per Paulson and Simpson
+ advection scheme with flux limiter is employed \citep{hundsdorfer94}
- [1977].  Additionally, the time-mean runoff of Large and Nurser [2001] is
+ and there is no explicit horizontal diffusivity. Horizontal viscosity
- applied near the coastline and, where there is open water, there is a
+ follows \citet{lei96} but
- relaxation to monthly-mean WOA01 sea surface salinity with a time constant of
+ modified to sense the divergent flow as per Fox-Kemper and Menemenlis
-days.
+ [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,
 Line 552 
 ice, wet ice, dry snow, and wet snow alb
  \item C-grid LSR slip
  \item C-grid EVP no-slip
  \item C-grid EVP slip
+ \item C-grid LSR + TEM (truncated ellipse method, no tensile stress, new flag)
  \item C-grid LSR no-slip + Winton
  \item  speed-performance-accuracy (small)
    ice transport through Canadian Archipelago differences
-Line 563 
 We anticipate small differences between
+Line 564 
 We anticipate small differences between
  \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
+ \item C-grid: less transport through narrow straits for no slip
-   conditons, less for free slip
+   conditons, more for free slip
  \item VP vs.\ EVP: speed performance, accuracy?
  \item ocean stress: different water mass properties beneath the ice
  \end{itemize}
-Line 645 
 storing vs. recomputation of the model s
+Line 646 
 storing vs. recomputation of the model s
  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
+ The code adjustments are sufficiently simple 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.
-Line 668 
 may be derived in about 10 minutes.
+Line 669 
 may be derived in about 10 minutes.
  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
- (see Menemenlis et al. 2005). It covers the entire Arctic,
+ \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
-Line 681 
 The adjoint models run efficiently on 80
+Line 683 
 The adjoint models run efficiently on 80
  (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
+ Following a 1-year spinup, the model has been integrated for four
- between 1992 and 1995.
+ years between 1992 and 1995. It is forced using realistic 6-hourly
- It is forced using realistic 6-hourly NCEP/NCAR atmospheric state variables.
+ NCEP/NCAR atmospheric state variables. Over the open ocean these are
- Over the open ocean these are converted into
+ converted into air-sea fluxes via the bulk formulae of
- air-sea fluxes via the bulk formulae of Large and Yeager (2004).
+ \citet{large04}.  Derivation of air-sea fluxes in the presence of
- Derivation of air-sea fluxes in the presence of sea-ice is handled
+ sea-ice is handled by the ice model as described in \refsec{model}.
- by the ice model as described in Section XXX.
  The objective function chosen is sea-ice export through Fram Strait
- computed for December 1995
+ computed for December 1995.  The adjoint model computes sensitivities
- The adjoint model computes sensitivities to sea-ice export back in time
+ to sea-ice export back in time from 1995 to 1992 along this
- from 1995 to 1992 along this trajectory.
+ trajectory.  In principle all adjoint model variable (i.e., Lagrange
- In principle all adjoint model variable (i.e. Lagrange multipliers)
+ multipliers) of the coupled ocean/sea-ice model are available to
- of the coupled ocean/sea-ice model
+ analyze the transient sensitivity behaviour of the ocean and sea-ice
- are available to analyze the transient sensitivity behaviour
+ state.  Over the open ocean, the adjoint of the bulk formula scheme
- of the ocean and sea-ice state.
+ computes sensitivities to the time-varying atmospheric state.  Over
- Over the open ocean, the adjoint of the bulk formula scheme
+ ice-covered parts, the sea-ice adjoint converts surface ocean
- computes sensitivities to the time-varying atmospheric state.
+ sensitivities to atmospheric sensitivities.
- 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
- Fig. XXX(a--d) depict sensitivities of sea-ice export through Fram Strait
+, 24, 36, 48 months back in time. Corresponding sensitivities to
- in December 1995 to changes in sea-ice thickness
+ ocean surface temperature are depicted in
-, 24, 36, 48 months back in time.
+ \reffig{4yradjthetalev1}(a--d).  The main characteristics is
- Corresponding sensitivities to ocean surface temperature are
+ consistency with expected advection of sea-ice over the relevant time
- depicted in Fig. XXX(a--d).
+ scales considered.  The general positive pattern means that an
- The main characteristics is consistency with expected advection
+ increase in sea-ice thickness at location $(x,y)$ and time $t$ will
- of sea-ice over the relevant time scales considered.
+ increase sea-ice export through Fram Strait at time $T_e$.  Largest
- The general positive pattern means that an increase in
+ distances from Fram Strait indicate fastest sea-ice advection over the
- sea-ice thickness at location $(x,y)$ and time $t$ will increase
+ time span considered.  The ice thickness sensitivities are in close
- sea-ice export through Fram Strait at time $T_e$.
+ correspondence to ocean surface sentivitites, but of opposite sign.
- Largest distances from Fram Strait indicate fastest sea-ice advection
+ An increase in temperature will incur ice melting, decrease in ice
- over the time span considered.
+ thickness, and therefore decrease in sea-ice export at time $T_e$.
- 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
+ \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,
-Line 770 
 sea-ice thickness at various prior times
+Line 767 
 sea-ice thickness at various prior times
  -48 months}]
  {\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJtheta_arc_lev1_tim292_cmax5.0E+01.eps}}
  }
- \caption{Same as Fig. XXX but for sea surface temperature
+ \caption{Same as \reffig{4yradjheff} but for sea surface temperature
  \label{fig:4yradjthetalev1}}
  \end{figure}
-Line 795 
 parameters that we use here. What about
+Line 792 
 parameters that we use here. What about
  \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.
  %\bibliography{bib/journal_abrvs,bib/seaice,bib/genocean,bib/maths,bib/mitgcmuv,bib/fram}
- \bibliography{journal_abrvs,seaice,genocean,maths,mitgcmuv,bib/fram}
+ \bibliography{journal_abrvs,seaice,genocean,maths,mixing,mitgcmuv,bib/fram}
  \end{document}

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