| 51 |
|
|
| 52 |
\maketitle |
\maketitle |
| 53 |
|
|
| 54 |
\begin{abstract} |
\input{ceaice_abstract.tex} |
|
Some blabla |
|
|
\end{abstract} |
|
| 55 |
|
|
| 56 |
\section{Introduction} |
\input{ceaice_intro.tex} |
|
\label{sec:intro} |
|
| 57 |
|
|
| 58 |
more blabla |
\input{ceaice_model.tex} |
| 59 |
|
|
| 60 |
\section{Model} |
\input{ceaice_forward.tex} |
|
\label{sec:model} |
|
| 61 |
|
|
| 62 |
Traditionally, probably for historical reasons and the ease of |
\input{ceaice_adjoint.tex} |
|
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. |
|
| 63 |
|
|
| 64 |
A further advantage of the C-grid formulation is apparent in narrow |
\input{ceaice_concl.tex} |
|
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 maximum ice pressure $P_{\max}$, a measure of ice strength, depends on |
|
|
both thickness $h$ and compactness (concentration) $c$: |
|
|
\begin{equation} |
|
|
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 |
|
|
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 =& \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) |
|
|
(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*} |
|
|
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, |
|
|
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} |
|
|
|
|
|
For a first/detailed comparison between the different variants of the |
|
|
MIT sea ice model an idealized geometry of a periodic channel, |
|
|
1000\,km long and 500\,m wide on a non-rotating plane, with converging |
|
|
walls forming a symmetric funnel and a narrow strait of 40\,km width |
|
|
is used. The horizontal resolution is 5\,km throughout the domain |
|
|
making the narrow strait 8 grid points wide. The ice model is |
|
|
initialized with a complete ice cover of 50\,cm uniform thickness. The |
|
|
ice model is driven by a constant along channel eastward ocean current |
|
|
of 25\,cm/s that does not see the walls in the domain. All other |
|
|
ice-ocean-atmosphere interactions are turned off, in particular there |
|
|
is no feedback of ice dynamics on the ocean current. All thermodynamic |
|
|
processes are turned off so that ice thickness variations are only |
|
|
caused by convergent or divergent ice flow. Ice volume (effective |
|
|
thickness) and concentration are advected with a third-order scheme |
|
|
with a flux limiter \citep{hundsdorfer94} to avoid undershoots. This |
|
|
scheme is unconditionally stable and does not require additional |
|
|
diffusion. The time step used here is 1\,h. |
|
|
|
|
|
\reffig{funnelf0} compares the dynamic fields ice concentration $c$, |
|
|
effective thickness $h_{eff} = h\cdot{c}$, and velocities $(u,v)$ for |
|
|
five different cases at steady state (after 10\,years of integration): |
|
|
\begin{description} |
|
|
\item[B-LSRns:] LSR solver with no-slip boundary conditions on a B-grid, |
|
|
\item[C-LSRns:] LSR solver with no-slip boundary conditions on a C-grid, |
|
|
\item[C-LSRfs:] LSR solver with free-slip boundary conditions on a C-grid, |
|
|
\item[C-EVPns:] EVP solver with no-slip boundary conditions on a C-grid, |
|
|
\item[C-EVPfs:] EVP solver with free-slip boundary conditions on a C-grid, |
|
|
\end{description} |
|
|
\ml{[We have not implemented the EVP solver on a B-grid.]} |
|
|
\begin{figure*}[htbp] |
|
|
\includegraphics[width=\widefigwidth]{\fpath/all_086280} |
|
|
\caption{Ice concentration, effective thickness [m], and ice |
|
|
velocities [m/s] |
|
|
for 5 different numerical solutions.} |
|
|
\label{fig:funnelf0} |
|
|
\end{figure*} |
|
|
At a first glance, the solutions look similar. This is encouraging as |
|
|
the details of discretization and numerics should not affect the |
|
|
solutions to first order. In all cases the ice-ocean stress pushes the |
|
|
ice cover eastwards, where it converges in the funnel. In the narrow |
|
|
channel the ice moves quickly (nearly free drift) and leaves the |
|
|
channel as narrow band. |
|
|
|
|
|
A close look reveals interesting differences between the B- and C-grid |
|
|
results. The zonal velocity in the narrow channel is nearly the free |
|
|
drift velocity ( = ocean velocity) of 25\,cm/s for the C-grid |
|
|
solutions, regardless of the boundary conditions, while it is just |
|
|
above 20\,cm/s for the B-grid solution. The ice accelerates to |
|
|
25\,cm/s after it exits the channel. Concentrating on the solutions |
|
|
B-LSRns and C-LSRns, the ice volume (effective thickness) along the |
|
|
boundaries in the narrow channel is larger in the B-grid case although |
|
|
the ice concentration is reduces in the C-grid case. The combined |
|
|
effect leads to a larger actual ice thickness at smaller |
|
|
concentrations in the C-grid case. However, since the effective |
|
|
thickness determines the ice strength $P$ in Eq\refeq{icestrength}, |
|
|
the ice strength and thus the bulk and shear viscosities are larger in |
|
|
the B-grid case leading to more horizontal friction. This circumstance |
|
|
might explain why the no-slip boundary conditions in the B-grid case |
|
|
appear to be more effective in reducing the flow within the narrow |
|
|
channel, than in the C-grid case. Further, the viscosities are also |
|
|
sensitive to details of the velocity gradients. Via $\Delta$, these |
|
|
gradients enter the viscosities in the denominator so that large |
|
|
gradients tend to reduce the viscosities. This again favors more flow |
|
|
along the boundaries in the C-grid case: larger velocities |
|
|
(\reffig{funnelf0}) on grid points that are closer to the boundary by |
|
|
a factor $\frac{1}{2}$ than in the B-grid case because of the stagger |
|
|
nature of the C-grid lead numerically to larger tangential gradients |
|
|
across the boundary; these in turn make the viscosities smaller for |
|
|
less tangential friction and allow more tangential flow along the |
|
|
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 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 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 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{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 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} |
|
|
\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 |
|
|
\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 |
|
|
\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 \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 \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, |
|
|
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} |
|
|
|
|
|
\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 + 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 |
|
|
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: 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} |
|
|
|
|
|
\section{Adjoint sensiivities of the MITsim} |
|
|
|
|
|
\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 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. |
|
|
|
|
|
[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). |
|
|
%\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 |
|
|
\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 |
|
|
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 |
|
|
\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. |
|
|
|
|
|
\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. |
|
|
\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, |
|
|
the circulation around Svalbard, and ... |
|
|
|
|
|
\begin{figure}[t!] |
|
|
\centerline{ |
|
|
\subfigure[{\footnotesize -12 months}] |
|
|
{\includegraphics*[width=0.44\linewidth]{\fpath/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]{\fpath/run_4yr_ADJheff_arc_lev1_tim145_cmax2.0E+02.eps}} |
|
|
} |
|
|
|
|
|
\centerline{ |
|
|
\subfigure[{\footnotesize |
|
|
-36 months}] |
|
|
{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim218_cmax2.0E+02.eps}} |
|
|
% |
|
|
\subfigure[{\footnotesize |
|
|
-48 months}] |
|
|
{\includegraphics*[width=0.44\linewidth]{\fpath/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]{\fpath/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]{\fpath/run_4yr_ADJtheta_arc_lev1_tim145_cmax5.0E+01.eps}} |
|
|
} |
|
|
|
|
|
\centerline{ |
|
|
\subfigure[{\footnotesize |
|
|
-36 months}] |
|
|
{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJtheta_arc_lev1_tim218_cmax5.0E+01.eps}} |
|
|
% |
|
|
\subfigure[{\footnotesize |
|
|
-48 months}] |
|
|
{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJtheta_arc_lev1_tim292_cmax5.0E+01.eps}} |
|
|
} |
|
|
\caption{Same as \reffig{4yradjheff} 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? |
|
| 65 |
|
|
| 66 |
\paragraph{Acknowledgements} |
\paragraph{Acknowledgements} |
| 67 |
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 |
Parent Directory
| 
Revision Log
| 
Revision Graph
| 
Patch