--- MITgcm_contrib/articles/ceaice/ceaice.tex	2008/01/10 15:47:32	1.4
+++ MITgcm_contrib/articles/ceaice/ceaice.tex	2008/01/21 08:06:00	1.9
@@ -1,6 +1,9 @@
+% $Header: /home/ubuntu/mnt/e9_copy/MITgcm_contrib/articles/ceaice/ceaice.tex,v 1.9 2008/01/21 08:06:00 mlosch Exp $
+% $Name:  $
 \documentclass[12pt]{article}
 
-\usepackage{graphicx,subfigure}
+\usepackage[]{graphicx}
+\usepackage{subfigure}
 
 \usepackage[round,comma]{natbib}
 \bibliographystyle{bib/agu04}
@@ -129,11 +132,11 @@
     \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) $c$: 
+The maximum ice pressure $P_{\max}$, a measure of ice strength, depends on
+both thickness $h$ and compactness (concentration) $c$:
 \begin{equation}
-  P = P^{*}c\,h\,e^{[C^{*}\cdot(1-c)]},
-\label{icestrength}
+  P_{\max} = P^{*}c\,h\,e^{[C^{*}\cdot(1-c)]},
+\label{eq:icestrength}
 \end{equation}
 with the constants $P^{*}$ and $C^{*}$. The nonlinear bulk and shear
 viscosities $\eta$ and $\zeta$ are functions of ice strain rate
@@ -141,8 +144,9 @@
 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} \\
+  \zeta =& \min\left(\frac{P_{\max}}{2\max(\Delta,\Delta_{\min})},
+   \zeta_{\max}\right) \\
+  \eta =& \frac{\zeta}{e^2} \\
   \intertext{with the abbreviation}
   \Delta = & \left[
     \left(\dot{\epsilon}_{11}^2+\dot{\epsilon}_{22}^2\right)
@@ -150,6 +154,24 @@
     2\dot{\epsilon}_{11}\dot{\epsilon}_{22} (1-e^{-2})
   \right]^{-\frac{1}{2}}
 \end{align*}
+The bulk viscosities are bounded above by imposing both a minimum
+$\Delta_{\min}=10^{-11}\text{\,s}^{-1}$ (for numerical reasons) and a
+maximum $\zeta_{\max} = P_{\max}/\Delta^*$, where
+$\Delta^*=(5\times10^{12}/2\times10^4)\text{\,s}^{-1}$. For stress
+tensor compuation the replacement pressure $P = 2\,\Delta\zeta$
+\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,
@@ -366,39 +388,65 @@
 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
-appears to be a little noisy. This noise has been address by
+effective thickness and meridional velocity fields. The EVP velocity
+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) or by regularizing the bulk
-and shear viscosities. We revisit the latter option by reproducing the
-results of \citet{hunke01} for the C-grid no-slip cases.
+step (increasing the number of iterations along with the computational
+cost) or by regularizing the bulk and shear viscosities. We revisit
+the latter option by reproducing some of the results of
+\citet{hunke01}, namely the experiment described in her section~4, for
+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 by variable
+atmospheric wind whose mean direction is diagnonal to the north-east
+corner of the domain; ice volume and concentration are held constant
+(no thermodynamics and no advection by ice velocity).
+\reffig{hunke01} shows the ice velocity field, its divergence, and the
+bulk viscosity $\zeta$ for the cases C-LRSns and C-EVPns, and for a
+C-EVPns case, where \citet{hunke01}'s regularization has been
+implemented; compare to Fig.\,4 in \citet{hunke01}. The regularization
+contraint limits ice strength and viscosities as a function of damping
+time scale, resolution and EVP-time step, effectively allowing the
+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*}
 
-\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.
+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 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 $\nabla\cdot\vek{u}$ and
+$\log_{10}\zeta$ 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).
+
+\ml{[Say something about performance? This is tricky, as the
+  perfomance depends strongly on the configuration. A run with slowly
+  changing forcing is favorable for LSR, because then only very few
+  iterations are required for convergences while EVP uses its fixed
+  number of internal timesteps. If the forcing in changing fast, LSR
+  needs far more iterations while EVP still uses the fixed number of
+  internal timesteps. I have produces runs where for slow forcing LSR
+  is much faster than EVP and for fast forcing, LSR is much slower
+  than EVP. EVP is certainly more efficient in terms of vectorization
+  and MFLOPS on our SX8, but is that a criterion?]}
 
 \subsection{C-grid}
 \begin{itemize}
@@ -445,45 +493,50 @@
 \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.  
+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
+\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
-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.
+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 \citet{large81, large82} 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
+\citet{large94} with background vertical diffusivity of
+$1.5\times10^{-5}\text{\,m$^{2}$\,s$^{-1}$}$ and viscosity of
+$10^{-3}\text{\,m$^{2}$\,s$^{-1}$}$. A third order, direct-space-time
+advection scheme with flux limiter is employed \citep{hundsdorfer94}
+and there is no explicit horizontal diffusivity. Horizontal viscosity
+follows \citet{lei96} 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,
@@ -512,6 +565,7 @@
 \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
@@ -523,8 +577,8 @@
 \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 C-grid: less transport through narrow straits for no 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}
@@ -605,7 +659,7 @@
 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.
@@ -628,9 +682,10 @@
 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 
@@ -641,46 +696,41 @@
 (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.
+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.
-
-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 $T_e$.
+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.
-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,
@@ -730,7 +780,7 @@
 -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}
 
@@ -755,7 +805,7 @@
 
 \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}
 
