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revision 1.5 by mlosch, Mon Jan 14 15:46:54 2008 UTC revision 1.9 by mlosch, Mon Jan 21 08:06:00 2008 UTC
# Line 1  Line 1 
1    % $Header$
2    % $Name$
3  \documentclass[12pt]{article}  \documentclass[12pt]{article}
4    
5  \usepackage[]{graphicx}  \usepackage[]{graphicx}
# Line 134  The maximum ice pressure $P_{\max}$, a m Line 136  The maximum ice pressure $P_{\max}$, a m
136  both thickness $h$ and compactness (concentration) $c$:  both thickness $h$ and compactness (concentration) $c$:
137  \begin{equation}  \begin{equation}
138    P_{\max} = P^{*}c\,h\,e^{[C^{*}\cdot(1-c)]},    P_{\max} = P^{*}c\,h\,e^{[C^{*}\cdot(1-c)]},
139  \label{icestrength}  \label{eq:icestrength}
140  \end{equation}  \end{equation}
141  with the constants $P^{*}$ and $C^{*}$. The nonlinear bulk and shear  with the constants $P^{*}$ and $C^{*}$. The nonlinear bulk and shear
142  viscosities $\eta$ and $\zeta$ are functions of ice strain rate  viscosities $\eta$ and $\zeta$ are functions of ice strain rate
# Line 160  tensor compuation the replacement pressu Line 162  tensor compuation the replacement pressu
162  \citep{hibler95} is used so that the stress state always lies on the  \citep{hibler95} is used so that the stress state always lies on the
163  elliptic yield curve by definition.  elliptic yield curve by definition.
164    
165    In the so-called truncated ellipse method the shear viscosity $\eta$
166    is capped to suppress any tensile stress \citep{hibler97, geiger98}:
167    \begin{equation}
168      \label{eq:etatem}
169      \eta = \min(\frac{\zeta}{e^2}
170      \frac{\frac{P}{2}-\zeta(\dot{\epsilon}_{11}+\dot{\epsilon}_{22})}
171      {\sqrt{(\dot{\epsilon}_{11}+\dot{\epsilon}_{22})^2
172          +4\dot{\epsilon}_{12}^2}}
173    \end{equation}
174    
175  In the current implementation, the VP-model is integrated with the  In the current implementation, the VP-model is integrated with the
176  semi-implicit line successive over relaxation (LSOR)-solver of  semi-implicit line successive over relaxation (LSOR)-solver of
177  \citet{zhang98}, which allows for long time steps that, in our case,  \citet{zhang98}, which allows for long time steps that, in our case,
# Line 376  boundaries. Line 388  boundaries.
388  The above argument can also be invoked to explain the small  The above argument can also be invoked to explain the small
389  differences between the free-slip and no-slip solutions on the C-grid.  differences between the free-slip and no-slip solutions on the C-grid.
390  Because of the non-linearities in the ice viscosities, in particular  Because of the non-linearities in the ice viscosities, in particular
391  along the boundaries, the no-slip boundary conditions has only a small  along the boundaries, the no-slip boundary conditions have only a small
392  impact on the solution.  impact on the solution.
393    
394  The difference between LSR and EVP solutions is largest in the  The difference between LSR and EVP solutions is largest in the
395  effective thickness and meridional velocity fields. The velocity field  effective thickness and meridional velocity fields. The EVP velocity
396  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
397  \citet{hunke01}. It can be dealt with by reducing EVP's internal time  \citet{hunke01}. It can be dealt with by reducing EVP's internal time
398  step (increasing the number of iterations along with the computational  step (increasing the number of iterations along with the computational
399  cost) or by regularizing the bulk and shear viscosities. We revisit  cost) or by regularizing the bulk and shear viscosities. We revisit
# Line 390  the latter option by reproducing some of Line 402  the latter option by reproducing some of
402  our C-grid no-slip cases: in a square domain with a few islands the  our C-grid no-slip cases: in a square domain with a few islands the
403  ice model is initialized with constant ice thickness and linearly  ice model is initialized with constant ice thickness and linearly
404  increasing ice concentration to the east. The model dynamics are  increasing ice concentration to the east. The model dynamics are
405  forced with a constant anticyclonic ocean gyre and variable  forced with a constant anticyclonic ocean gyre and by variable
406  atmospheric wind whose mean directed diagnonally to the north-east  atmospheric wind whose mean direction is diagnonal to the north-east
407  corner of the domain; ice volume and concentration are held constant  corner of the domain; ice volume and concentration are held constant
408  (no advection by ice velocity).  \reffig{hunke01} shows the ice  (no thermodynamics and no advection by ice velocity).
409  velocity field, its divergence, and the bulk viscosity $\zeta$ for the  \reffig{hunke01} shows the ice velocity field, its divergence, and the
410  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
411  \citet{hunke01}'s regularization has been implemented; compare to  C-EVPns case, where \citet{hunke01}'s regularization has been
412  Fig.\,4 in \citet{hunke01}. The regularization contraint limits ice  implemented; compare to Fig.\,4 in \citet{hunke01}. The regularization
413  strength and viscosities as a function of damping time scale,  contraint limits ice strength and viscosities as a function of damping
414  resolution and EVP-time step, effectively allowing the elastic waves to  time scale, resolution and EVP-time step, effectively allowing the
415  damp out more quickly \citep{hunke01}.  elastic waves to damp out more quickly \citep{hunke01}.
416  \begin{figure*}[htbp]  \begin{figure*}[htbp]
417    \includegraphics[width=\widefigwidth]{\fpath/hun12days}    \includegraphics[width=\widefigwidth]{\fpath/hun12days}
418    \caption{Hunke's test case.}    \caption{Ice flow, divergence and bulk viscosities of three
419        experiments with \citet{hunke01}'s test case: C-LSRns (top),
420        C-EVPns (middle), and C-EVPns with damping described in
421        \citet{hunke01} (bottom).}
422    \label{fig:hunke01}    \label{fig:hunke01}
423  \end{figure*}  \end{figure*}
424    
425  In the far right (``east'') side of the domain the ice concentration  In the far right (``east'') side of the domain the ice concentration
426  is close to one and the ice should be nearly rigid. The applied wind  is close to one and the ice should be nearly rigid. The applied wind
427  tends to push ice toward the upper right corner. Because the highly  tends to push ice toward the upper right corner. Because the highly
428  compact ice is confinded by the boundary, it resists any further  compact ice is confined by the boundary, it resists any further
429  compression and exhibits little motion in the rigid region on the  compression and exhibits little motion in the rigid region on the
430  right hand side. The C-LSRns solution (top row) allows high  right hand side. The C-LSRns solution (top row) allows high
431  viscosities in the rigid region suppressing nearly all flow.  viscosities in the rigid region suppressing nearly all flow.
432  \citet{hunke01}'s regularization for the C-EVPns solution (bottom row)  \citet{hunke01}'s regularization for the C-EVPns solution (bottom row)
433  clearly suppresses the noise present in $\nabla\cdot\vek{u}$ in the  clearly suppresses the noise present in $\nabla\cdot\vek{u}$ and
434  unregularized case (middle row), at the cost of reduced viscosities  $\log_{10}\zeta$ in the
435    unregularized case (middle row), at the cost of reduced viscosities.
436  These reduced viscosities lead to small but finite ice velocities  These reduced viscosities lead to small but finite ice velocities
437  which in turn can have a strong effect on solutions in the limit of  which in turn can have a strong effect on solutions in the limit of
438  nearly rigid regimes (arching and blocking, not shown).  nearly rigid regimes (arching and blocking, not shown).
439    
440    \ml{[Say something about performance? This is tricky, as the
441  %\begin{itemize}    perfomance depends strongly on the configuration. A run with slowly
442  %\item B-grid LSR no-slip    changing forcing is favorable for LSR, because then only very few
443  %\item C-grid LSR no-slip    iterations are required for convergences while EVP uses its fixed
444  %\item C-grid LSR slip    number of internal timesteps. If the forcing in changing fast, LSR
445  %\item C-grid EVP no-slip    needs far more iterations while EVP still uses the fixed number of
446  %\item C-grid EVP slip    internal timesteps. I have produces runs where for slow forcing LSR
447  %\end{itemize}    is much faster than EVP and for fast forcing, LSR is much slower
448      than EVP. EVP is certainly more efficient in terms of vectorization
449  %\subsection{B-grid vs.\ C-grid}    and MFLOPS on our SX8, but is that a criterion?]}
 %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.  
450    
451  \subsection{C-grid}  \subsection{C-grid}
452  \begin{itemize}  \begin{itemize}
# Line 485  differences between the two main options Line 493  differences between the two main options
493  \subsection{Arctic Domain with Open Boundaries}  \subsection{Arctic Domain with Open Boundaries}
494  \label{sec:arctic}  \label{sec:arctic}
495    
496  The Arctic domain of integration is illustrated in Fig.~\ref{???}.  It is  The Arctic domain of integration is illustrated in Fig.~\ref{???}.  It
497  carved out from, and obtains open boundary conditions from, the global  is carved out from, and obtains open boundary conditions from, the
498  cubed-sphere configuration of the Estimating the Circulation and Climate of  global cubed-sphere configuration of the Estimating the Circulation
499  the Ocean, Phase II (ECCO2) project \cite{men05a}.  The domain size is 420 by  and Climate of the Ocean, Phase II (ECCO2) project
500  384 grid boxes horizontally with mean horizontal grid spacing of 18 km.    \citet{menemenlis05}.  The domain size is 420 by 384 grid boxes
501    horizontally with mean horizontal grid spacing of 18 km.
502    
503  There are 50 vertical levels ranging in thickness from 10 m near the surface  There are 50 vertical levels ranging in thickness from 10 m near the surface
504  to approximately 450 m at a maximum model depth of 6150 m. Bathymetry is from  to approximately 450 m at a maximum model depth of 6150 m. Bathymetry is from
505  the National Geophysical Data Center (NGDC) 2-minute gridded global relief  the National Geophysical Data Center (NGDC) 2-minute gridded global relief
506  data (ETOPO2) and the model employs the partial-cell formulation of  data (ETOPO2) and the model employs the partial-cell formulation of
507  \cite{adc97}, which permits accurate representation of the bathymetry. The  \citet{adcroft97:_shaved_cells}, which permits accurate representation of the bathymetry. The
508  model is integrated in a volume-conserving configuration using a finite volume  model is integrated in a volume-conserving configuration using a finite volume
509  discretization with C-grid staggering of the prognostic variables. In the  discretization with C-grid staggering of the prognostic variables. In the
510  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
511  coupled to a sea-ice model described hereinabove.    coupled to a sea-ice model described hereinabove.  
512    
513  This particular ECCO2 simulation is initialized from rest using the January  This particular ECCO2 simulation is initialized from rest using the
514  temperature and salinity distribution from the World Ocean Atlas 2001 (WOA01)  January temperature and salinity distribution from the World Ocean
515  [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
516  1996-2001 period discussed in the study. Surface boundary conditions are from  32 years prior to the 1996--2001 period discussed in the study. Surface
517  the National Centers for Environmental Prediction and the National Center for  boundary conditions are from the National Centers for Environmental
518  Atmospheric Research (NCEP/NCAR) atmospheric reanalysis [Kistler et al.,  Prediction and the National Center for Atmospheric Research
519  2001]. Six-hourly surface winds, temperature, humidity, downward short- and  (NCEP/NCAR) atmospheric reanalysis [Kistler et al., 2001]. Six-hourly
520  long-wave radiations, and precipitation are converted to heat, freshwater, and  surface winds, temperature, humidity, downward short- and long-wave
521  wind stress fluxes using the Large and Pond [1981, 1982] bulk  radiations, and precipitation are converted to heat, freshwater, and
522  formulae. Shortwave radiation decays exponentially as per Paulson and Simpson  wind stress fluxes using the \citet{large81, large82} bulk formulae.
523  [1977]. Additionally the time-mean river run-off from Large and Nurser [2001]  Shortwave radiation decays exponentially as per Paulson and Simpson
524  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
525  surface salinity values from WOA01 with a relaxation time scale of 3  [2001] is applied and there is a relaxation to the monthly-mean
526  months. Vertical mixing follows Large et al. [1994] with background vertical  climatological sea surface salinity values from WOA01 with a
527  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
528  direct-space-time advection scheme with flux limiter is employed and there is  \citet{large94} with background vertical diffusivity of
529  no explicit horizontal diffusivity. Horizontal viscosity follows Leith [1996]  $1.5\times10^{-5}\text{\,m$^{2}$\,s$^{-1}$}$ and viscosity of
530  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
531  press].  Shortwave radiation decays exponentially as per Paulson and Simpson  advection scheme with flux limiter is employed \citep{hundsdorfer94}
532  [1977].  Additionally, the time-mean runoff of Large and Nurser [2001] is  and there is no explicit horizontal diffusivity. Horizontal viscosity
533  applied near the coastline and, where there is open water, there is a  follows \citet{lei96} but
534  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
535  45 days.  [in press].  Shortwave radiation decays exponentially as per Paulson
536    and Simpson [1977].  Additionally, the time-mean runoff of Large and
537    Nurser [2001] is applied near the coastline and, where there is open
538    water, there is a relaxation to monthly-mean WOA01 sea surface
539    salinity with a time constant of 45 days.
540    
541  Open water, dry  Open water, dry
542  ice, wet ice, dry snow, and wet snow albedo are, respectively, 0.15, 0.85,  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 Line 565  ice, wet ice, dry snow, and wet snow alb
565  \item C-grid LSR slip  \item C-grid LSR slip
566  \item C-grid EVP no-slip  \item C-grid EVP no-slip
567  \item C-grid EVP slip  \item C-grid EVP slip
568    \item C-grid LSR + TEM (truncated ellipse method, no tensile stress, new flag)
569  \item C-grid LSR no-slip + Winton  \item C-grid LSR no-slip + Winton
570  \item  speed-performance-accuracy (small)  \item  speed-performance-accuracy (small)
571    ice transport through Canadian Archipelago differences    ice transport through Canadian Archipelago differences
# Line 563  We anticipate small differences between Line 577  We anticipate small differences between
577  \begin{itemize}  \begin{itemize}
578  \item advection schemes: along the ice-edge and regions with large  \item advection schemes: along the ice-edge and regions with large
579    gradients    gradients
580  \item C-grid: more transport through narrow straits for no slip  \item C-grid: less transport through narrow straits for no slip
581    conditons, less for free slip    conditons, more for free slip
582  \item VP vs.\ EVP: speed performance, accuracy?  \item VP vs.\ EVP: speed performance, accuracy?
583  \item ocean stress: different water mass properties beneath the ice  \item ocean stress: different water mass properties beneath the ice
584  \end{itemize}  \end{itemize}
# Line 645  storing vs. recomputation of the model s Line 659  storing vs. recomputation of the model s
659  checkpointing loop.  checkpointing loop.
660  Again, an initial code adjustment is required to support TAFs  Again, an initial code adjustment is required to support TAFs
661  checkpointing capability.  checkpointing capability.
662  The code adjustments are sufficiently simply so as not to cause  The code adjustments are sufficiently simple so as not to cause
663  major limitations to the full nonlinear parent model.  major limitations to the full nonlinear parent model.
664  Once in place, an adjoint model of a new model configuration  Once in place, an adjoint model of a new model configuration
665  may be derived in about 10 minutes.  may be derived in about 10 minutes.
# Line 668  may be derived in about 10 minutes. Line 682  may be derived in about 10 minutes.
682  We demonstrate the power of the adjoint method  We demonstrate the power of the adjoint method
683  in the context of investigating sea-ice export sensitivities through Fram Strait  in the context of investigating sea-ice export sensitivities through Fram Strait
684  (for details of this study see Heimbach et al., 2007).  (for details of this study see Heimbach et al., 2007).
685    %\citep[for details of this study see][]{heimbach07}. %Heimbach et al., 2007).
686  The domain chosen is a coarsened version of the Arctic face of the  The domain chosen is a coarsened version of the Arctic face of the
687  high-resolution cubed-sphere configuration of the ECCO2 project  high-resolution cubed-sphere configuration of the ECCO2 project
688  (see Menemenlis et al. 2005). It covers the entire Arctic,  \citep[see][]{menemenlis05}. It covers the entire Arctic,
689  extends into the North Pacific such as to cover the entire  extends into the North Pacific such as to cover the entire
690  ice-covered regions, and comprises parts of the North Atlantic  ice-covered regions, and comprises parts of the North Atlantic
691  down to XXN to enable analysis of remote influences of the  down to XXN to enable analysis of remote influences of the
# Line 681  The adjoint models run efficiently on 80 Line 696  The adjoint models run efficiently on 80
696  (benchmarks have been performed both on an SGI Altix as well as an  (benchmarks have been performed both on an SGI Altix as well as an
697  IBM SP5 at NASA/ARC).  IBM SP5 at NASA/ARC).
698    
699  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
700  between 1992 and 1995.  years between 1992 and 1995. It is forced using realistic 6-hourly
701  It is forced using realistic 6-hourly NCEP/NCAR atmospheric state variables.  NCEP/NCAR atmospheric state variables. Over the open ocean these are
702  Over the open ocean these are converted into  converted into air-sea fluxes via the bulk formulae of
703  air-sea fluxes via the bulk formulae of Large and Yeager (2004).  \citet{large04}.  Derivation of air-sea fluxes in the presence of
704  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.  
705  The objective function chosen is sea-ice export through Fram Strait  The objective function chosen is sea-ice export through Fram Strait
706  computed for December 1995  computed for December 1995.  The adjoint model computes sensitivities
707  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
708  from 1995 to 1992 along this trajectory.  trajectory.  In principle all adjoint model variable (i.e., Lagrange
709  In principle all adjoint model variable (i.e. Lagrange multipliers)  multipliers) of the coupled ocean/sea-ice model are available to
710  of the coupled ocean/sea-ice model  analyze the transient sensitivity behaviour of the ocean and sea-ice
711  are available to analyze the transient sensitivity behaviour  state.  Over the open ocean, the adjoint of the bulk formula scheme
712  of the ocean and sea-ice state.  computes sensitivities to the time-varying atmospheric state.  Over
713  Over the open ocean, the adjoint of the bulk formula scheme  ice-covered parts, the sea-ice adjoint converts surface ocean
714  computes sensitivities to the time-varying atmospheric state.  sensitivities to atmospheric sensitivities.
715  Over ice-covered parts, the sea-ice adjoint converts  
716  surface ocean sensitivities to atmospheric sensitivities.  \reffig{4yradjheff}(a--d) depict sensitivities of sea-ice export
717    through Fram Strait in December 1995 to changes in sea-ice thickness
718  Fig. XXX(a--d) depict sensitivities of sea-ice export through Fram Strait  12, 24, 36, 48 months back in time. Corresponding sensitivities to
719  in December 1995 to changes in sea-ice thickness  ocean surface temperature are depicted in
720  12, 24, 36, 48 months back in time.  \reffig{4yradjthetalev1}(a--d).  The main characteristics is
721  Corresponding sensitivities to ocean surface temperature are  consistency with expected advection of sea-ice over the relevant time
722  depicted in Fig. XXX(a--d).  scales considered.  The general positive pattern means that an
723  The main characteristics is consistency with expected advection  increase in sea-ice thickness at location $(x,y)$ and time $t$ will
724  of sea-ice over the relevant time scales considered.  increase sea-ice export through Fram Strait at time $T_e$.  Largest
725  The general positive pattern means that an increase in  distances from Fram Strait indicate fastest sea-ice advection over the
726  sea-ice thickness at location $(x,y)$ and time $t$ will increase  time span considered.  The ice thickness sensitivities are in close
727  sea-ice export through Fram Strait at time $T_e$.  correspondence to ocean surface sentivitites, but of opposite sign.
728  Largest distances from Fram Strait indicate fastest sea-ice advection  An increase in temperature will incur ice melting, decrease in ice
729  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 $T_e$.  
730    
731  The picture is fundamentally different and much more complex  The picture is fundamentally different and much more complex
732  for sensitivities to ocean temperatures away from the surface.  for sensitivities to ocean temperatures away from the surface.
733  Fig. XXX (a--d) depicts ice export sensitivities to  \reffig{4yradjthetalev10??}(a--d) depicts ice export sensitivities to
734  temperatures at roughly 400 m depth.  temperatures at roughly 400 m depth.
735  Primary features are the effect of the heat transport of the North  Primary features are the effect of the heat transport of the North
736  Atlantic current which feeds into the West Spitsbergen current,  Atlantic current which feeds into the West Spitsbergen current,
# Line 770  sea-ice thickness at various prior times Line 780  sea-ice thickness at various prior times
780  -48 months}]  -48 months}]
781  {\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJtheta_arc_lev1_tim292_cmax5.0E+01.eps}}  {\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJtheta_arc_lev1_tim292_cmax5.0E+01.eps}}
782  }  }
783  \caption{Same as Fig. XXX but for sea surface temperature  \caption{Same as \reffig{4yradjheff} but for sea surface temperature
784  \label{fig:4yradjthetalev1}}  \label{fig:4yradjthetalev1}}
785  \end{figure}  \end{figure}
786    
# Line 795  parameters that we use here. What about Line 805  parameters that we use here. What about
805    
806  \paragraph{Acknowledgements}  \paragraph{Acknowledgements}
807  We thank Jinlun Zhang for providing the original B-grid code and many  We thank Jinlun Zhang for providing the original B-grid code and many
808  helpful discussions.  helpful discussions. ML thanks Elizabeth Hunke for multiple explanations.
809    
810  %\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}
 \bibliography{journal_abrvs,seaice,genocean,maths,mitgcmuv,bib/fram}  
811    
812  \end{document}  \end{document}
813    
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