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revision 1.7 by mlosch, Thu Mar 6 21:54:55 2008 UTC revision 1.12 by dimitri, Fri Aug 15 14:16:35 2008 UTC
# Line 5  The MITgcm sea ice model (MITsim) is bas Line 5  The MITgcm sea ice model (MITsim) is bas
5  viscous-plastic (VP) dynamic-thermodynamic sea ice model  viscous-plastic (VP) dynamic-thermodynamic sea ice model
6  \citep{zhang97} first introduced by \citet{hibler79, hibler80}. In  \citep{zhang97} first introduced by \citet{hibler79, hibler80}. In
7  order to adapt this model to the requirements of coupled  order to adapt this model to the requirements of coupled
8  ice-ocean simulations, many important aspects of the original code have  ice-ocean state estimation, many important aspects of the original code have
9  been modified and improved:  been modified and improved:
10  \begin{itemize}  \begin{itemize}
11  \item the code has been rewritten for an Arakawa C-grid, both B- and  \item the code has been rewritten for an Arakawa C-grid, both B- and
12    C-grid variants are available; the C-grid code allows for no-slip    C-grid variants are available; the C-grid code allows for no-slip
13    and free-slip lateral boundary conditions;    and free-slip lateral boundary conditions;
14  \item two different solution methods for solving the nonlinear  \item two different solution methods for solving the nonlinear
15    momentum equations have been adopted: LSOR \citep{zhang97}, EVP    momentum equations have been adopted: LSOR \citep{zhang97}, and EVP
16    \citep{hunke97};    \citep{hunke97};
17  \item ice-ocean stress can be formulated as in \citet{hibler87};  \item ice-ocean stress can be formulated as in \citet{hibler87} or as in \citet{cam08};
18  \item ice variables are advected by sophisticated advection schemes;  \item ice variables \ml{can be} advected by sophisticated, \ml{conservative}
19  \item growth and melt parameterizaion have been refined and extended    advection schemes \ml{with flux limiting};
20    in order to allow for automatic differentiation of the code.  \item growth and melt parameterizations have been refined and extended
21      in order to allow for more stable automatic differentiation of the code.
22  \end{itemize}  \end{itemize}
23  The model equations and their numerical realization are summarized  
24  below.  The sea ice model is tightly coupled to the ocean compontent of the MITgcm
25    \citep{mar97a}.  Heat, fresh water fluxes and surface stresses are computed
26    from the atmospheric state and modified by the ice model at every time step.
27    
28    
29  \subsection{Dynamics}  \subsection{Dynamics}
30  \label{sec:dynamics}  \label{app:dynamics}
31    
32  The momentum equation of the sea-ice model is  The momentum equation of the sea-ice model is
33  \begin{equation}  \begin{equation}
# Line 40  $\vtau_{air}$ and $\vtau_{ocean}$ are th Line 44  $\vtau_{air}$ and $\vtau_{ocean}$ are th
44  respectively;  respectively;
45  $g$ is the gravity accelation;  $g$ is the gravity accelation;
46  $\nabla\phi(0)$ is the gradient (or tilt) of the sea surface height;  $\nabla\phi(0)$ is the gradient (or tilt) of the sea surface height;
47  $\phi(0) = g\eta + p_{a}/\rho_{0}$ is the sea surface height potential  $\phi(0) = g\eta + p_{a}/\rho_{0} + mg/\rho_{0}$ is the sea surface
48  in response to ocean dynamics ($g\eta$) and to atmospheric pressure  height potential in response to ocean dynamics ($g\eta$), to
49  loading ($p_{a}/\rho_{0}$, where $\rho_{0}$ is a reference density);  atmospheric pressure loading ($p_{a}/\rho_{0}$, where $\rho_{0}$ is a
50  and $\vek{F}=\nabla\cdot\sigma$ is the divergence of the internal ice stress  reference density) and a term due to snow and ice loading \citep{campin08};
51  tensor $\sigma_{ij}$.  and $\vek{F}=\nabla\cdot\sigma$ is the divergence of the internal ice
52  When using the rescaled vertical coordinate system, z$^\ast$, of  stress tensor $\sigma_{ij}$. %
 \citet{cam08}, $\phi(0)$ also includes a term due to snow and ice  
 loading, $mg/\rho_{0}$.  
53  Advection of sea-ice momentum is neglected. The wind and ice-ocean stress  Advection of sea-ice momentum is neglected. The wind and ice-ocean stress
54  terms are given by  terms are given by
55  \begin{align*}  \begin{align*}
# Line 98  minor axis $e$ equal to $2$; they are gi Line 100  minor axis $e$ equal to $2$; they are gi
100      \left(\dot{\epsilon}_{11}^2+\dot{\epsilon}_{22}^2\right)      \left(\dot{\epsilon}_{11}^2+\dot{\epsilon}_{22}^2\right)
101      (1+e^{-2}) +  4e^{-2}\dot{\epsilon}_{12}^2 +      (1+e^{-2}) +  4e^{-2}\dot{\epsilon}_{12}^2 +
102      2\dot{\epsilon}_{11}\dot{\epsilon}_{22} (1-e^{-2})      2\dot{\epsilon}_{11}\dot{\epsilon}_{22} (1-e^{-2})
103    \right]^{\frac{1}{2}}    \right]^{\frac{1}{2}}.
104  \end{align*}  \end{align*}
105  The bulk viscosities are bounded above by imposing both a minimum  The bulk viscosities are bounded above by imposing both a minimum
106  $\Delta_{\min}=10^{-11}\text{\,s}^{-1}$ (for numerical reasons) and a  $\Delta_{\min}=10^{-11}\text{\,s}^{-1}$ (for numerical reasons) and a
# Line 147  timestep. According to the recommendatio Line 149  timestep. According to the recommendatio
149  EVP-model is stepped forward in time 120 times within the physical  EVP-model is stepped forward in time 120 times within the physical
150  ocean model time step (although this parameter is under debate), to  ocean model time step (although this parameter is under debate), to
151  allow for elastic waves to disappear.  Because the scheme does not  allow for elastic waves to disappear.  Because the scheme does not
152  require a matrix inversion it is fast in spite of the small timestep  require a matrix inversion it is fast in spite of the small internal
153    timestep and simple to implement on parallel computers
154  \citep{hunke97}. For completeness, we repeat the equations for the  \citep{hunke97}. For completeness, we repeat the equations for the
155  components of the stress tensor $\sigma_{1} =  components of the stress tensor $\sigma_{1} =
156  \sigma_{11}+\sigma_{22}$, $\sigma_{2}= \sigma_{11}-\sigma_{22}$, and  \sigma_{11}+\sigma_{22}$, $\sigma_{2}= \sigma_{11}-\sigma_{22}$, and
# Line 155  $\sigma_{12}$. Introducing the divergenc Line 158  $\sigma_{12}$. Introducing the divergenc
158  \dot{\epsilon}_{11}+\dot{\epsilon}_{22}$, and the horizontal tension  \dot{\epsilon}_{11}+\dot{\epsilon}_{22}$, and the horizontal tension
159  and shearing strain rates, $D_T =  and shearing strain rates, $D_T =
160  \dot{\epsilon}_{11}-\dot{\epsilon}_{22}$ and $D_S =  \dot{\epsilon}_{11}-\dot{\epsilon}_{22}$ and $D_S =
161  2\dot{\epsilon}_{12}$, respectively, and using the above abbreviations,  2\dot{\epsilon}_{12}$, respectively, and using the above
162  the equations can be written as:  abbreviations, the equations\refeq{evpequation} can be written as:
163  \begin{align}  \begin{align}
164    \label{eq:evpstresstensor1}    \label{eq:evpstresstensor1}
165    \frac{\partial\sigma_{1}}{\partial{t}} + \frac{\sigma_{1}}{2T} +    \frac{\partial\sigma_{1}}{\partial{t}} + \frac{\sigma_{1}}{2T} +
# Line 174  $T=E_{0}\Delta{t}$ with the tunable para Line 177  $T=E_{0}\Delta{t}$ with the tunable para
177  the external (long) timestep $\Delta{t}$. \citet{hunke97} recommend  the external (long) timestep $\Delta{t}$. \citet{hunke97} recommend
178  $E_{0} = \frac{1}{3}$.  $E_{0} = \frac{1}{3}$.
179    
180  For details of the spatial discretization, the reader is referred to  Our discretization differs from \citet{zhang98, zhang03} in the
181  \citet{zhang98, zhang03}. Our discretization differs only (but  underlying grid, namely the Arakawa C-grid, but is otherwise
182  importantly) in the underlying grid, namely the Arakawa C-grid, but is  straightforward. The EVP model, in particular, is discretized
 otherwise straightforward. The EVP model, in particular, is discretized  
183  naturally on the C-grid with $\sigma_{1}$ and $\sigma_{2}$ on the  naturally on the C-grid with $\sigma_{1}$ and $\sigma_{2}$ on the
184  center points and $\sigma_{12}$ on the corner (or vorticity) points of  center points and $\sigma_{12}$ on the corner (or vorticity) points of
185  the grid. With this choice all derivatives are discretized as central  the grid. With this choice all derivatives are discretized as central
# Line 236  temperature and salinity are different f Line 238  temperature and salinity are different f
238  %\end{itemize}  %\end{itemize}
239    
240  \subsection{Thermodynamics}  \subsection{Thermodynamics}
241  \label{sec:thermodynamics}  \label{app:thermodynamics}
242    
243  In the original formulation the sea ice model \citep{menemenlis05}  In its original formulation the sea ice model \citep{menemenlis05}
244  uses simple thermodynamics following the appendix of  uses simple thermodynamics following the appendix of
245  \citet{semtner76}. This formulation does not allow storage of heat  \citet{semtner76}. This formulation does not allow storage of heat,
246  (heat capacity of ice is zero, and this type of model is often refered  that is, the heat capacity of ice is zero. Upward conductive heat flux
247  to as a ``zero-layer'' model). Upward conductive heat flux is parameterized  is parameterized assuming a linear temperature profile and together
248  assuming a linear temperature profile and together with a constant ice  with a constant ice conductivity. It is expressed as
249  conductivity. It is expressed as $(K/h)(T_{w}-T_{0})$, where $K$ is  $(K/h)(T_{w}-T_{0})$, where $K$ is the ice conductivity, $h$ the ice
250  the ice conductivity, $h$ the ice thickness, and $T_{w}-T_{0}$ the  thickness, and $T_{w}-T_{0}$ the difference between water and ice
251  difference between water and ice surface temperatures. The surface  surface temperatures. This type of model is often refered to as a
252  heat flux is computed in a similar way to that of \citet{parkinson79}  ``zero-layer'' model. The surface heat flux is computed in a similar
253  and \citet{manabe79}.  way to that of \citet{parkinson79} and \citet{manabe79}.
254    
255  The conductive heat flux depends strongly on the ice thickness $h$.  The conductive heat flux depends strongly on the ice thickness $h$.
256  However, the ice thickness in the model represents a mean over a  However, the ice thickness in the model represents a mean over a
257  potentially very heterogeneous thickness distribution.  In order to  potentially very heterogeneous thickness distribution.  In order to
258  parameterize this sub-grid scale distribution for heat flux  parameterize a sub-grid scale distribution for heat flux
259  computations, the mean ice thickness $h$ is split into seven thickness  computations, the mean ice thickness $h$ is split into seven thickness
260  categories $H_{n}$ that are equally distributed between $2h$ and  categories $H_{n}$ that are equally distributed between $2h$ and a
261  minimum imposed ice thickness of $5\text{\,cm}$ by $H_n=  minimum imposed ice thickness of $5\text{\,cm}$ by $H_n=
262  \frac{2n-1}{7}\,h$ for $n\in[1,7]$. The heat fluxes computed for each  \frac{2n-1}{7}\,h$ for $n\in[1,7]$. The heat fluxes computed for each
263  thickness category area averaged to give the total heat flux. \ml{[I  thickness category is area-averaged to give the total heat flux
264    don't have citation for that, anyone?]}  \citep{hibler84}.
265    
266    \ml{[This is Ian Fenty's work and we may want to remove this paragraph
267      from the paper]: %
268  The atmospheric heat flux is balanced by an oceanic heat flux from  The atmospheric heat flux is balanced by an oceanic heat flux from
269  below.  The oceanic flux is proportional to  below.  The oceanic flux is proportional to
270  $\rho\,c_{p}\left(T_{w}-T_{fr}\right)$ where $\rho$ and $c_{p}$ are  $\rho\,c_{p}\left(T_{w}-T_{fr}\right)$ where $\rho$ and $c_{p}$ are
# Line 268  the density and heat capacity of sea wat Line 272  the density and heat capacity of sea wat
272  freezing point temperature that is a function of salinity. Contrary to  freezing point temperature that is a function of salinity. Contrary to
273  \citet{menemenlis05}, this flux is not assumed to instantaneously melt  \citet{menemenlis05}, this flux is not assumed to instantaneously melt
274  or create ice, but a time scale of three days is used to relax $T_{w}$  or create ice, but a time scale of three days is used to relax $T_{w}$
275  to the freezing point.  to the freezing point.}
276    %
277  The parameterization of lateral and vertical growth of sea ice follows  The parameterization of lateral and vertical growth of sea ice follows
278  that of \citet{hibler79, hibler80}.  that of \citet{hibler79, hibler80}.
279    
280  On top of the ice there is a layer of snow that modifies the heat flux  On top of the ice there is a layer of snow that modifies the heat flux
281  and the albedo \citep{zhang98}. If enough snow accumulates so that its  and the albedo \citep{zhang98}. Snow modifies the effective
282  weight submerges the ice and the snow is flooded, a simple mass  conductivity according to
283  conserving parameterization of snowice formation (a flood-freeze  \[\frac{K}{h} \rightarrow \frac{1}{\frac{h_{s}}{K_{s}}+\frac{h}{K}},\]
284  algorithm following Archimedes' principle) turns snow into ice until  where $K_s$ is the conductivity of snow and $h_s$ the snow thickness.
285  the ice surface is back at $z=0$ \citep{leppaeranta83}.  If enough snow accumulates so that its weight submerges the ice and
286    the snow is flooded, a simple mass conserving parameterization of
287    snowice formation (a flood-freeze algorithm following Archimedes'
288    principle) turns snow into ice until the ice surface is back at $z=0$
289    \citep{leppaeranta83}.
290    
291  Effective ich thickness (ice volume per unit area,  Effective ice thickness (ice volume per unit area,
292  $c\cdot{h}$), concentration $c$ and effective snow thickness  $c\cdot{h}$), concentration $c$ and effective snow thickness
293  ($c\cdot{h}_{s}$) are advected by ice velocities:  ($c\cdot{h}_{s}$) are advected by ice velocities:
294  \begin{equation}  \begin{equation}
# Line 316  the two ice layers and the thickness of Line 324  the two ice layers and the thickness of
324    this can only be accomplished with a 2nd-order advection scheme with    this can only be accomplished with a 2nd-order advection scheme with
325    flux limiter \citep{roe85}.}    flux limiter \citep{roe85}.}
326    
327    Further documentation and model code can be found at
328    \url{http://mitgcm.org}.
329    
 \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}  
330    
331  \end{itemize}  %\subsection{C-grid}
332    %\begin{itemize}
333    %\item no-slip vs. free-slip for both lsr and evp;
334    %  "diagnostics" to look at and use for comparison
335    %  \begin{itemize}
336    %  \item ice transport through Fram Strait/Denmark Strait/Davis
337    %    Strait/Bering strait (these are general)
338    %  \item ice transport through narrow passages, e.g.\ Nares-Strait
339    %  \end{itemize}
340    %\item compare different advection schemes (if lsr turns out to be more
341    %  effective, then with lsr otherwise I prefer evp), eg.
342    %  \begin{itemize}
343    %  \item default 2nd-order with diff1=0.002
344    %  \item 1st-order upwind with diff1=0.
345    %  \item DST3FL (SEAICEadvScheme=33 with diff1=0., should work, works for me)
346    %  \item 2nd-order wit flux limiter (SEAICEadvScheme=77 with diff1=0.)
347    %  \end{itemize}
348    %  That should be enough. Here, total ice mass and location of ice edge
349    %  is interesting. However, this comparison can be done in an idealized
350    %  domain, may not require full Arctic Domain?
351    %\item
352    %Do a little study on the parameters of LSR and EVP
353    %\begin{enumerate}
354    %\item convergence of LSR, how many iterations do you need to get a
355    %  true elliptic yield curve
356    %\item vary deltaTevp and the relaxation parameter for EVP and see when
357    %  the EVP solution breaks down (relative to the forcing time scale).
358    %  For this, it is essential that the evp solver gives use "stripeless"
359    %  solutions, that is your dtevp = 1sec solutions/or 10sec solutions
360    %  with SEAICE\_evpDampC = 615.
361    %\end{enumerate}
362    
363    %\end{itemize}
364    
365  %%% Local Variables:  %%% Local Variables:
366  %%% mode: latex  %%% mode: latex
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