| 1 |
\section{Sea Ice Model Formulation} |
| 2 |
\label{app:model} |
| 3 |
|
| 4 |
\ml{[All of this (and more) should go into the documentation, but we |
| 5 |
can remove a large part of the text because it is completely |
| 6 |
redundant. I leave it in for now \ldots]} |
| 7 |
|
| 8 |
\subsection{Dynamics} |
| 9 |
\label{app:dynamics} |
| 10 |
|
| 11 |
The momentum equation of the sea-ice model is |
| 12 |
\begin{equation} |
| 13 |
\label{eq:momseaice} |
| 14 |
m \frac{D\vek{u}}{Dt} = -mf\vek{k}\times\vek{u} + \vtau_{air} + |
| 15 |
\vtau_{ocean} - m \nabla{\phi(0)} + \vek{F}, |
| 16 |
\end{equation} |
| 17 |
where $m=m_{i}+m_{s}$ is the ice and snow mass per unit area; |
| 18 |
$\vek{u}=u\vek{i}+v\vek{j}$ is the ice velocity vector; |
| 19 |
$\vek{i}$, $\vek{j}$, and $\vek{k}$ are unit vectors in the $x$, $y$, and $z$ |
| 20 |
directions, respectively; |
| 21 |
$f$ is the Coriolis parameter; |
| 22 |
$\vtau_{air}$ and $\vtau_{ocean}$ are the wind-ice and ocean-ice stresses, |
| 23 |
respectively; |
| 24 |
$g$ is the gravity accelation; |
| 25 |
$\nabla\phi(0)$ is the gradient (or tilt) of the sea surface height; |
| 26 |
$\phi(0) = g\eta + p_{a}/\rho_{0} + mg/\rho_{0}$ is the sea surface |
| 27 |
height potential in response to ocean dynamics ($g\eta$), to |
| 28 |
atmospheric pressure loading ($p_{a}/\rho_{0}$, where $\rho_{0}$ is a |
| 29 |
reference density) and a term due to snow and ice loading \citep{campin08}; |
| 30 |
and $\vek{F}=\nabla\cdot\sigma$ is the divergence of the internal ice |
| 31 |
stress tensor $\sigma_{ij}$. % |
| 32 |
Advection of sea-ice momentum is neglected. The wind and ice-ocean stress |
| 33 |
terms are given by |
| 34 |
\begin{align*} |
| 35 |
\vtau_{air} = & \rho_{air} C_{air} |\vek{U}_{air} -\vek{u}| |
| 36 |
R_{air} (\vek{U}_{air} -\vek{u}), \\ |
| 37 |
\vtau_{ocean} = & \rho_{ocean}C_{ocean} |\vek{U}_{ocean}-\vek{u}| |
| 38 |
R_{ocean}(\vek{U}_{ocean}-\vek{u}), \\ |
| 39 |
\end{align*} |
| 40 |
where $\vek{U}_{air/ocean}$ are the surface winds of the atmosphere |
| 41 |
and surface currents of the ocean, respectively; $C_{air/ocean}$ are |
| 42 |
air and ocean drag coefficients; $\rho_{air/ocean}$ are reference |
| 43 |
densities; and $R_{air/ocean}$ are rotation matrices that act on the |
| 44 |
wind/current vectors. |
| 45 |
|
| 46 |
For an isotropic system the stress tensor $\sigma_{ij}$ ($i,j=1,2$) can |
| 47 |
be related to the ice strain rate and strength by a nonlinear |
| 48 |
viscous-plastic (VP) constitutive law \citep{hibler79, zhang97}: |
| 49 |
\begin{equation} |
| 50 |
\label{eq:vpequation} |
| 51 |
\sigma_{ij}=2\eta(\dot{\epsilon}_{ij},P)\dot{\epsilon}_{ij} |
| 52 |
+ \left[\zeta(\dot{\epsilon}_{ij},P) - |
| 53 |
\eta(\dot{\epsilon}_{ij},P)\right]\dot{\epsilon}_{kk}\delta_{ij} |
| 54 |
- \frac{P}{2}\delta_{ij}. |
| 55 |
\end{equation} |
| 56 |
The ice strain rate is given by |
| 57 |
\begin{equation*} |
| 58 |
\dot{\epsilon}_{ij} = \frac{1}{2}\left( |
| 59 |
\frac{\partial{u_{i}}}{\partial{x_{j}}} + |
| 60 |
\frac{\partial{u_{j}}}{\partial{x_{i}}}\right). |
| 61 |
\end{equation*} |
| 62 |
The maximum ice pressure $P_{\max}$, a measure of ice strength, depends on |
| 63 |
both thickness $h$ and compactness (concentration) $c$: |
| 64 |
\begin{equation} |
| 65 |
P_{\max} = P^{*}c\,h\,e^{[C^{*}\cdot(1-c)]}, |
| 66 |
\label{eq:icestrength} |
| 67 |
\end{equation} |
| 68 |
with the constants $P^{*}$ and $C^{*}$. The nonlinear bulk and shear |
| 69 |
viscosities $\eta$ and $\zeta$ are functions of ice strain rate |
| 70 |
invariants and ice strength such that the principal components of the |
| 71 |
stress lie on an elliptical yield curve with the ratio of major to |
| 72 |
minor axis $e$ equal to $2$; they are given by: |
| 73 |
\begin{align*} |
| 74 |
\zeta =& \min\left(\frac{P_{\max}}{2\max(\Delta,\Delta_{\min})}, |
| 75 |
\zeta_{\max}\right) \\ |
| 76 |
\eta =& \frac{\zeta}{e^2} \\ |
| 77 |
\intertext{with the abbreviation} |
| 78 |
\Delta = & \left[ |
| 79 |
\left(\dot{\epsilon}_{11}^2+\dot{\epsilon}_{22}^2\right) |
| 80 |
(1+e^{-2}) + 4e^{-2}\dot{\epsilon}_{12}^2 + |
| 81 |
2\dot{\epsilon}_{11}\dot{\epsilon}_{22} (1-e^{-2}) |
| 82 |
\right]^{\frac{1}{2}}. |
| 83 |
\end{align*} |
| 84 |
The bulk viscosities are bounded above by imposing both a minimum |
| 85 |
$\Delta_{\min}=10^{-11}\text{\,s}^{-1}$ (for numerical reasons) and a |
| 86 |
maximum $\zeta_{\max} = P_{\max}/\Delta^*$, where |
| 87 |
$\Delta^*=(5\times10^{12}/2\times10^4)\text{\,s}^{-1}$. For stress |
| 88 |
tensor computation the replacement pressure $P = 2\,\Delta\zeta$ |
| 89 |
\citep{hibler95} is used so that the stress state always lies on the |
| 90 |
elliptic yield curve by definition. |
| 91 |
|
| 92 |
In the so-called truncated ellipse method the shear viscosity $\eta$ |
| 93 |
is capped to suppress any tensile stress \citep{hibler97, geiger98}: |
| 94 |
\begin{equation} |
| 95 |
\label{eq:etatem} |
| 96 |
\eta = \min\left(\frac{\zeta}{e^2}, |
| 97 |
\frac{\frac{P}{2}-\zeta(\dot{\epsilon}_{11}+\dot{\epsilon}_{22})} |
| 98 |
{\sqrt{(\dot{\epsilon}_{11}+\dot{\epsilon}_{22})^2 |
| 99 |
+4\dot{\epsilon}_{12}^2}}\right). |
| 100 |
\end{equation} |
| 101 |
|
| 102 |
In the current implementation, the VP-model is integrated with the |
| 103 |
semi-implicit line successive over relaxation (LSOR)-solver of |
| 104 |
\citet{zhang98}, which allows for long time steps that, in our case, |
| 105 |
are limited by the explicit treatment of the Coriolis term. The |
| 106 |
explicit treatment of the Coriolis term does not represent a severe |
| 107 |
limitation because it restricts the time step to approximately the |
| 108 |
same length as in the ocean model where the Coriolis term is also |
| 109 |
treated explicitly. |
| 110 |
|
| 111 |
\citet{hunke97}'s introduced an elastic contribution to the strain |
| 112 |
rate in order to regularize Eq.\refeq{vpequation} in such a way that |
| 113 |
the resulting elastic-viscous-plastic (EVP) and VP models are |
| 114 |
identical at steady state, |
| 115 |
\begin{equation} |
| 116 |
\label{eq:evpequation} |
| 117 |
\frac{1}{E}\frac{\partial\sigma_{ij}}{\partial{t}} + |
| 118 |
\frac{1}{2\eta}\sigma_{ij} |
| 119 |
+ \frac{\eta - \zeta}{4\zeta\eta}\sigma_{kk}\delta_{ij} |
| 120 |
+ \frac{P}{4\zeta}\delta_{ij} |
| 121 |
= \dot{\epsilon}_{ij}. |
| 122 |
\end{equation} |
| 123 |
%In the EVP model, equations for the components of the stress tensor |
| 124 |
%$\sigma_{ij}$ are solved explicitly. Both model formulations will be |
| 125 |
%used and compared the present sea-ice model study. |
| 126 |
The EVP-model uses an explicit time stepping scheme with a short |
| 127 |
timestep. According to the recommendation of \citet{hunke97}, the |
| 128 |
EVP-model is stepped forward in time 120 times within the physical |
| 129 |
ocean model time step (although this parameter is under debate), to |
| 130 |
allow for elastic waves to disappear. Because the scheme does not |
| 131 |
require a matrix inversion it is fast in spite of the small internal |
| 132 |
timestep and simple to implement on parallel computers |
| 133 |
\citep{hunke97}. For completeness, we repeat the equations for the |
| 134 |
components of the stress tensor $\sigma_{1} = |
| 135 |
\sigma_{11}+\sigma_{22}$, $\sigma_{2}= \sigma_{11}-\sigma_{22}$, and |
| 136 |
$\sigma_{12}$. Introducing the divergence $D_D = |
| 137 |
\dot{\epsilon}_{11}+\dot{\epsilon}_{22}$, and the horizontal tension |
| 138 |
and shearing strain rates, $D_T = |
| 139 |
\dot{\epsilon}_{11}-\dot{\epsilon}_{22}$ and $D_S = |
| 140 |
2\dot{\epsilon}_{12}$, respectively, and using the above |
| 141 |
abbreviations, the equations\refeq{evpequation} can be written as: |
| 142 |
\begin{align} |
| 143 |
\label{eq:evpstresstensor1} |
| 144 |
\frac{\partial\sigma_{1}}{\partial{t}} + \frac{\sigma_{1}}{2T} + |
| 145 |
\frac{P}{2T} &= \frac{P}{2T\Delta} D_D \\ |
| 146 |
\label{eq:evpstresstensor2} |
| 147 |
\frac{\partial\sigma_{2}}{\partial{t}} + \frac{\sigma_{2} e^{2}}{2T} |
| 148 |
&= \frac{P}{2T\Delta} D_T \\ |
| 149 |
\label{eq:evpstresstensor12} |
| 150 |
\frac{\partial\sigma_{12}}{\partial{t}} + \frac{\sigma_{12} e^{2}}{2T} |
| 151 |
&= \frac{P}{4T\Delta} D_S |
| 152 |
\end{align} |
| 153 |
Here, the elastic parameter $E$ is redefined in terms of a damping timescale |
| 154 |
$T$ for elastic waves \[E=\frac{\zeta}{T}.\] |
| 155 |
$T=E_{0}\Delta{t}$ with the tunable parameter $E_0<1$ and |
| 156 |
the external (long) timestep $\Delta{t}$. \citet{hunke97} recommend |
| 157 |
$E_{0} = \frac{1}{3}$. |
| 158 |
|
| 159 |
Our discretization differs from \citet{zhang98, zhang03} in the |
| 160 |
underlying grid, namely the Arakawa C-grid, but is otherwise |
| 161 |
straightforward. The EVP model, in particular, is discretized |
| 162 |
naturally on the C-grid with $\sigma_{1}$ and $\sigma_{2}$ on the |
| 163 |
center points and $\sigma_{12}$ on the corner (or vorticity) points of |
| 164 |
the grid. With this choice all derivatives are discretized as central |
| 165 |
differences and averaging is only involved in computing $\Delta$ and |
| 166 |
$P$ at vorticity points. |
| 167 |
|
| 168 |
For a general curvilinear grid, one needs in principle to take metric |
| 169 |
terms into account that arise in the transformation of a curvilinear |
| 170 |
grid on the sphere. For now, however, we can neglect these metric |
| 171 |
terms because they are very small on the \ml{[modify following |
| 172 |
section3:] cubed sphere grids used in this paper; in particular, |
| 173 |
only near the edges of the cubed sphere grid, we expect them to be |
| 174 |
non-zero, but these edges are at approximately 35\degS\ or 35\degN\ |
| 175 |
which are never covered by sea-ice in our simulations. Everywhere |
| 176 |
else the coordinate system is locally nearly cartesian.} However, for |
| 177 |
last-glacial-maximum or snowball-earth-like simulations the question |
| 178 |
of metric terms needs to be reconsidered. Either, one includes these |
| 179 |
terms as in \citet{zhang03}, or one finds a vector-invariant |
| 180 |
formulation for the sea-ice internal stress term that does not require |
| 181 |
any metric terms, as it is done in the ocean dynamics of the MITgcm |
| 182 |
\citep{adcroft04:_cubed_sphere}. |
| 183 |
|
| 184 |
Lateral boundary conditions are naturally ``no-slip'' for B-grids, as |
| 185 |
the tangential velocities points lie on the boundary. For C-grids, the |
| 186 |
lateral boundary condition for tangential velocities is realized via |
| 187 |
``ghost points'', allowing alternatively no-slip or free-slip |
| 188 |
conditions. In ocean models free-slip boundary conditions in |
| 189 |
conjunction with piecewise-constant (``castellated'') coastlines have |
| 190 |
been shown to reduce in effect to no-slip boundary conditions |
| 191 |
\citep{adcroft98:_slippery_coast}; for sea-ice models the effects of |
| 192 |
lateral boundary conditions have not yet been studied. |
| 193 |
|
| 194 |
Moving sea ice exerts a stress on the ocean which is the opposite of |
| 195 |
the stress $\vtau_{ocean}$ in Eq.\refeq{momseaice}. This stess is |
| 196 |
applied directly to the surface layer of the ocean model. An |
| 197 |
alternative ocean stress formulation is given by \citet{hibler87}. |
| 198 |
Rather than applying $\vtau_{ocean}$ directly, the stress is derived |
| 199 |
from integrating over the ice thickness to the bottom of the oceanic |
| 200 |
surface layer. In the resulting equation for the \emph{combined} |
| 201 |
ocean-ice momentum, the interfacial stress cancels and the total |
| 202 |
stress appears as the sum of windstress and divergence of internal ice |
| 203 |
stresses: $\delta(z) (\vtau_{air} + \vek{F})/\rho_0$, \citep[see also |
| 204 |
Eq.\,2 of][]{hibler87}. The disadvantage of this formulation is that |
| 205 |
now the velocity in the surface layer of the ocean that is used to |
| 206 |
advect tracers, is really an average over the ocean surface |
| 207 |
velocity and the ice velocity leading to an inconsistency as the ice |
| 208 |
temperature and salinity are different from the oceanic variables. |
| 209 |
|
| 210 |
%\subparagraph{boundary conditions: no-slip, free-slip, half-slip} |
| 211 |
%\begin{itemize} |
| 212 |
%\item transition from existing B-Grid to C-Grid |
| 213 |
%\item boundary conditions: no-slip, free-slip, half-slip |
| 214 |
%\item fancy (multi dimensional) advection schemes |
| 215 |
%\item VP vs.\ EVP \citep{hunke97} |
| 216 |
%\item ocean stress formulation \citep{hibler87} |
| 217 |
%\end{itemize} |
| 218 |
|
| 219 |
\subsection{Thermodynamics} |
| 220 |
\label{app:thermodynamics} |
| 221 |
|
| 222 |
In its original formulation the sea ice model \citep{menemenlis05} |
| 223 |
uses simple thermodynamics following the appendix of |
| 224 |
\citet{semtner76}. This formulation does not allow storage of heat, |
| 225 |
that is, the heat capacity of ice is zero. Upward conductive heat flux |
| 226 |
is parameterized assuming a linear temperature profile and together |
| 227 |
with a constant ice conductivity. It is expressed as |
| 228 |
$(K/h)(T_{w}-T_{0})$, where $K$ is the ice conductivity, $h$ the ice |
| 229 |
thickness, and $T_{w}-T_{0}$ the difference between water and ice |
| 230 |
surface temperatures. This type of model is often refered to as a |
| 231 |
``zero-layer'' model. The surface heat flux is computed in a similar |
| 232 |
way to that of \citet{parkinson79} and \citet{manabe79}. |
| 233 |
|
| 234 |
The conductive heat flux depends strongly on the ice thickness $h$. |
| 235 |
However, the ice thickness in the model represents a mean over a |
| 236 |
potentially very heterogeneous thickness distribution. In order to |
| 237 |
parameterize a sub-grid scale distribution for heat flux |
| 238 |
computations, the mean ice thickness $h$ is split into seven thickness |
| 239 |
categories $H_{n}$ that are equally distributed between $2h$ and a |
| 240 |
minimum imposed ice thickness of $5\text{\,cm}$ by $H_n= |
| 241 |
\frac{2n-1}{7}\,h$ for $n\in[1,7]$. The heat fluxes computed for each |
| 242 |
thickness category is area-averaged to give the total heat flux |
| 243 |
\citep{hibler84}. |
| 244 |
|
| 245 |
\ml{[This is Ian Fenty's work and we may want to remove this paragraph |
| 246 |
from the paper]: % |
| 247 |
The atmospheric heat flux is balanced by an oceanic heat flux from |
| 248 |
below. The oceanic flux is proportional to |
| 249 |
$\rho\,c_{p}\left(T_{w}-T_{fr}\right)$ where $\rho$ and $c_{p}$ are |
| 250 |
the density and heat capacity of sea water and $T_{fr}$ is the local |
| 251 |
freezing point temperature that is a function of salinity. Contrary to |
| 252 |
\citet{menemenlis05}, this flux is not assumed to instantaneously melt |
| 253 |
or create ice, but a time scale of three days is used to relax $T_{w}$ |
| 254 |
to the freezing point.} |
| 255 |
% |
| 256 |
The parameterization of lateral and vertical growth of sea ice follows |
| 257 |
that of \citet{hibler79, hibler80}. |
| 258 |
|
| 259 |
On top of the ice there is a layer of snow that modifies the heat flux |
| 260 |
and the albedo \citep{zhang98}. Snow modifies the effective |
| 261 |
conductivity according to |
| 262 |
\[\frac{K}{h} \rightarrow \frac{1}{\frac{h_{s}}{K_{s}}+\frac{h}{K}},\] |
| 263 |
where $K_s$ is the conductivity of snow and $h_s$ the snow thickness. |
| 264 |
If enough snow accumulates so that its weight submerges the ice and |
| 265 |
the snow is flooded, a simple mass conserving parameterization of |
| 266 |
snowice formation (a flood-freeze algorithm following Archimedes' |
| 267 |
principle) turns snow into ice until the ice surface is back at $z=0$ |
| 268 |
\citep{leppaeranta83}. |
| 269 |
|
| 270 |
Effective ice thickness (ice volume per unit area, |
| 271 |
$c\cdot{h}$), concentration $c$ and effective snow thickness |
| 272 |
($c\cdot{h}_{s}$) are advected by ice velocities: |
| 273 |
\begin{equation} |
| 274 |
\label{eq:advection} |
| 275 |
\frac{\partial{X}}{\partial{t}} = - \nabla\cdot\left(\vek{u}\,X\right) + |
| 276 |
\Gamma_{X} + D_{X} |
| 277 |
\end{equation} |
| 278 |
where $\Gamma_X$ are the thermodynamic source terms and $D_{X}$ the |
| 279 |
diffusive terms for quantities $X=(c\cdot{h}), c, (c\cdot{h}_{s})$. |
| 280 |
% |
| 281 |
From the various advection scheme that are available in the MITgcm |
| 282 |
\citep{mitgcm02}, we choose flux-limited schemes |
| 283 |
\citep[multidimensional 2nd and 3rd-order advection scheme with flux |
| 284 |
limiter][]{roe85, hundsdorfer94} to preserve sharp gradients and edges |
| 285 |
that are typical of sea ice distributions and to rule out unphysical |
| 286 |
over- and undershoots (negative thickness or concentration). These |
| 287 |
scheme conserve volume and horizontal area and are unconditionally |
| 288 |
stable, so that we can set $D_{X}=0$. \ml{[do we need to proove that? |
| 289 |
can we proove that? citation?]} |
| 290 |
|
| 291 |
There is considerable doubt about the reliability of such a simple |
| 292 |
thermodynamic model---\citet{semtner84} found significant errors in |
| 293 |
phase (one month lead) and amplitude ($\approx$50\%\,overestimate) in |
| 294 |
such models---, so that today many sea ice models employ more complex |
| 295 |
thermodynamics. A popular thermodynamics model of \citet{winton00} is |
| 296 |
based on the 3-layer model of \citet{semtner76} and treats brine |
| 297 |
content by means of enthalphy conservation. This model requires in |
| 298 |
addition to ice-thickness and compactness (fractional area) additional |
| 299 |
state variables to be advected by ice velocities, namely enthalphy of |
| 300 |
the two ice layers and the thickness of the overlying snow layer. |
| 301 |
\ml{[Jean-Michel, your turf: ]Care must be taken in advecting these |
| 302 |
quantities in order to ensure conservation of enthalphy. Currently |
| 303 |
this can only be accomplished with a 2nd-order advection scheme with |
| 304 |
flux limiter \citep{roe85}.} |
| 305 |
|
| 306 |
%%% Local Variables: |
| 307 |
%%% mode: latex |
| 308 |
%%% TeX-master: "ceaice" |
| 309 |
%%% End: |
Parent Directory
| 
Revision Log
| 
Revision Graph