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%%%--------------------------------------%%% |
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\subsection{THSICE: The Thermodynamic Sea Ice Package} |
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\label{sec:pkg:thsice} |
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\begin{rawhtml} |
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<!-- CMIREDIR:package_thsice: --> |
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|
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{\bf Important note:} |
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This document has been written by Stephanie Dutkiewicz |
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and describes an earlier implementation of the sea-ice package. |
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This needs to be updated to reflect the recent changes (JMC). |
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|
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\noindent |
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This thermodynamic ice model is based on the 3-layer model by Winton (2000). |
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and the energy-conserving LANL CICE model (Bitz and Lipscomb, 1999). |
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The model considers two equally thick ice layers; the upper layer has |
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a variable specific heat resulting from brine pockets, |
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the lower layer has a fixed heat capacity. A zero heat capacity snow |
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layer lies above the ice. Heat fluxes at the top and bottom |
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surfaces are used to calculate the change in ice and snow layer |
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thickness. Grid cells of the ocean model are |
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either fully covered in ice or are open water. There is |
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a provision to parametrize ice fraction (and leads) in this package. |
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Modifications are discussed in small font following the |
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subroutine descriptions. |
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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|
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\vspace{1cm} |
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|
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\noindent |
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The ice model is called from {\it thermodynamics.F}, subroutine |
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{\it ice\_forcing.F} is called in place of {\it external\_forcing\_surf.F}. |
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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|
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\vspace{1cm} |
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\noindent |
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{\bf \underline{subroutine ICE\_FORCING}} |
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|
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\noindent |
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In {\it ice\_forcing.F}, we calculate the freezing potential of the |
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ocean model surface layer of water: |
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\[ |
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{\bf frzmlt} = (T_f - SST) \frac{c_{sw} \rho_{sw} \Delta z}{\Delta t} |
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\] |
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where $c_{sw}$ is seawater heat capacity, |
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$\rho_{sw}$ is the seawater density, $\Delta z$ |
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is the ocean model upper layer thickness and $\Delta t$ is the model (tracer) |
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timestep. The freezing temperature, $T_f=\mu S$ is a function of the |
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salinity. |
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|
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|
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1) Provided there is no ice present and {\bf frzmlt} is less than 0, |
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the surface tendencies of wind, heat and freshwater are calculated |
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as usual (ie. as in {\it external\_forcing\_surf.F}). |
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|
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2) If there is ice present in the grid cell |
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we call the main ice model routine {\it ice\_therm.F} (see below). |
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Output from this routine gives net heat and freshwater flux |
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affecting the top of the ocean. |
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|
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Subroutine {\it ice\_forcing.F} uses these values to find the |
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sea surface tendencies |
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in grid cells. When there is ice present, |
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the surface stress tendencies are |
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set to zero; the ice model is purely thermodynamic and the |
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effect of ice motion on the sea-surface is not examined. |
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|
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Relaxation of surface $T$ and $S$ is only allowed equatorward |
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of {\bf relaxlat} (see {\bf DATA.ICE below}), and no relaxation is |
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allowed under the ice at any latitude. |
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|
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\noindent |
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{\tiny (Note that there is provision for allowing grid cells to have both |
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open water and seaice; if {\bf compact} is between 0 and 1)} |
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\vspace{1cm} |
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\noindent |
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{\bf {\underline{ subroutine ICE\_FREEZE}}} |
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|
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This routine is called from {\it thermodynamics.F} |
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after the new temperature calculation, {\it calc\_gt.F}, |
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but before {\it calc\_gs.F}. |
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In {\it ice\_freeze.F}, any ocean upper layer grid cell |
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with no ice cover, but with temperature below freezing, |
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$T_f=\mu S$ has ice initialized. |
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We calculate {\bf frzmlt} from all the grid cells in |
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the water column that have a temperature less than |
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freezing. In this routine, any water below the surface |
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that is below freezing is set to $T_f$. |
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A call to |
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{\it ice\_start.F} is made if {\bf frzmlt} $>0$, |
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and salinity tendancy is updated for brine release. |
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|
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\noindent |
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{\tiny (There is a provision for fractional ice: |
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In the case where the grid cell has less ice coverage than |
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{\bf icemaskmax} we allow {\it ice\_start.F} to be called).} |
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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|
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\vspace{1cm} |
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\noindent |
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{\bf {\underline{ subroutine ICE\_START}}} |
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|
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\noindent |
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The energy available from freezing |
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the sea surface is brought into this routine as {\bf esurp}. |
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The enthalpy of the 2 layers of any new ice is calculated as: |
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\begin{eqnarray} |
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q_1 & = & -c_{i}*T_f + L_i \nonumber \\ |
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q_2 & = & -c_{f}T_{mlt}+ c_{i}(T_{mlt}-T{f}) + L_i(1-\frac{T_{mlt}}{T_f} |
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\nonumber \\ |
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\end{eqnarray} |
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where $c_f$ is specific heat of liquid fresh water, $c_i$ is the |
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specific heat of fresh ice, $L_i$ is latent heat of freezing, |
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$\rho_i$ is density of ice and |
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$T_{mlt}$ is melting temperature of ice with salinity of 1. |
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The height of a new layer of ice is |
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\[ |
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h_{i new} = \frac{{\bf esurp} \Delta t}{qi_{0av}} |
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\] |
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where $qi_{0av}=-\frac{\rho_i}{2} (q_1+q_2)$. |
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|
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The surface skin temperature $T_s$ and ice temperatures |
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$T_1$, $T_2$ and the sea surface temperature are set at $T_f$. |
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|
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\noindent |
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{\tiny ( There is provision for fractional ice: |
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new ice is formed over open water; the first freezing in the cell |
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must have a height of {\bf himin0}; this determines the ice |
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fraction {\bf compact}. If there is already ice in the grid cell, |
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the new ice must have the same height and the new ice fraction |
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is |
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\[ |
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i_f=(1-\hat{i_f}) \frac{h_{i new}}{h_i} |
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\] |
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where $\hat{i_f}$ is ice fraction from previous timestep |
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and $h_i$ is current ice height. Snow is redistributed |
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over the new ice fraction. The ice fraction is |
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not allowed to become larger than {\bf iceMaskmax} and |
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if the ice height is above {\bf hihig} then freezing energy |
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comes from the full grid cell, ice growth does not occur |
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under orginal ice due to freezing water. |
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} |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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|
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\vspace{1cm} |
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\noindent |
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{\bf {\underline{subroutine ICE\_THERM}}} |
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|
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\noindent |
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The main subroutine of this package is {\it ice\_therm.F} where the |
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ice temperatures are calculated and the changes in ice and snow |
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thicknesses are determined. Output provides the net heat and fresh |
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water fluxes that force the top layer of the ocean model. |
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|
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If the current ice height is less than {\bf himin} then |
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the ice layer is set to zero and the ocean model upper layer temperature |
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is allowed to drop lower than its freezing temperature; and atmospheric |
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fluxes are allowed to effect the grid cell. |
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If the ice height is greater than {\bf himin} we proceed with |
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the ice model calculation. |
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|
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We follow the procedure |
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of Winton (1999) -- see equations 3 to 21 -- to calculate |
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the surface and internal ice temperatures. |
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The surface temperature is found from the balance of the |
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flux at the surface $F_s$, the shortwave heat flux absorbed by the ice, |
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{\bf fswint}, and |
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the upward conduction of heat through the snow and/or ice, $F_u$. |
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We linearize $F_s$ about the surface temperature, $\hat{T_s}$, |
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at the previous timestep (where \mbox{}$\hat{ }$ indicates the value at |
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the previous timestep): |
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\[ |
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F_s (T_s) = F_s(\hat{T_s}) + \frac{\partial F_s(\hat{T_s)}}{\partial T_s} |
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(T_s-\hat{T_s}) |
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\] |
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where, |
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\[ |
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F_s = F_{sensible}+F_{latent}+F_{longwave}^{down}+F_{longwave}^{up}+ (1- |
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\alpha) F_{shortwave} |
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\] |
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and |
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\[ |
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\frac{d F_s}{dT} = \frac{d F_{sensible}}{dT} + \frac{d F_{latent}}{dT} |
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+\frac{d F_{longwave}^{up}}{dT}. |
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\] |
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$F_s$ and $\frac{d F_s}{dT}$ are currently calculated from the {\bf BULKF} |
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package described separately, but could also be provided by an atmospheric |
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model. The surface albedo is calculated from the ice height and/or |
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surface temperature (see below, {\it srf\_albedo.F}) and the |
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shortwave flux absorbed in the ice is |
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\[ |
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{\bf fswint} = (1-e^{\kappa_i h_i})(1-\alpha) F_{shortwave} |
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\] |
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where $\kappa_i$ is bulk extinction coefficient. |
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|
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The conductive flux to the surface is |
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\[ |
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F_u=K_{1/2}(T_1-T_s) |
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\] |
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where $K_{1/2}$ is the effective conductive coupling of the snow-ice |
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layer between the surface and the mid-point of the upper layer of ice |
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$ |
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K_{1/2}=\frac{4 K_i K_s}{K_s h_i + 4 K_i h_s} |
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$. |
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$K_i$ and $K_s$ are constant thermal conductivities of seaice and snow. |
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|
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From the above equations we can develop a system of equations to |
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find the skin surface temperature, $T_s$ and the two ice layer |
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temperatures (see Winton, 1999, for details). We solve these |
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equations iteratively until the change in $T_s$ is small. |
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When the surface temperature is greater then |
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the melting temperature of the surface, the temperatures are |
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recalculated setting $T_s$ to 0. The enthalpy |
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of the ice layers are calculated in order to keep track of the energy in the |
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ice model. Enthalpy is defined, here, as the energy required to melt a |
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unit mass of seaice with temperature $T$. |
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For the upper layer (1) with brine pockets and |
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the lower fresh layer (2): |
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\begin{eqnarray} |
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q_1 & = & - c_f T_f + c_i (T_f-T)+ L_{i}(1-\frac{T_f}{T}) |
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\nonumber \\ |
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q_2 & = & -c_i T+L_i \nonumber |
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\end{eqnarray} |
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where $c_f$ is specific heat of liquid fresh water, $c_i$ is the |
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specific heat of fresh ice, and $L_i$ is latent heat of melting fresh ice. |
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|
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|
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|
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From the new ice temperatures, we can calculate |
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the energy flux at the surface available for melting (if $T_s$=0) |
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and the energy at the ocean-ice interface for either melting or freezing. |
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\begin{eqnarray} |
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E_{top} & = & (F_s- K_{1/2}(T_s-T_1) ) \Delta t |
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\nonumber \\ |
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E_{bot} &= & (\frac{4K_i(T_2-T_f)}{h_i}-F_b) \Delta t |
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\nonumber |
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\end{eqnarray} |
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where $F_b$ is the heat flux at the ice bottom due to the sea surface |
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temperature variations from freezing. |
261 |
If $T_{sst}$ is above freezing, $F_b=c_{sw} \rho_{sw} |
262 |
\gamma (T_{sst}-T_f)u^{*}$, $\gamma$ is the heat transfer coefficient |
263 |
and $u^{*}=QQ$ is frictional velocity between ice |
264 |
and water. If $T_{sst}$ is below freezing, |
265 |
$F_b=(T_f - T_{sst})c_f \rho_f \Delta z /\Delta t$ and set $T_{sst}$ |
266 |
to $T_f$. We also |
267 |
include the energy from lower layers that drop below freezing, |
268 |
and set those layers to $T_f$. |
269 |
|
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If $E_{top}>0$ we melt snow from the surface, if all the snow is melted |
271 |
and there is energy left, we melt the ice. If the ice is all gone |
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and there is still energy left, we apply the left over energy to |
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heating the ocean model upper layer (See Winton, 1999, equations 27-29). |
274 |
Similarly if $E_{bot}>0$ we melt ice from the bottom. If all the ice |
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is melted, the snow is melted (with energy from the ocean model upper layer |
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if necessary). If $E_{bot}<0$ we grow ice at the bottom |
277 |
\[ |
278 |
\Delta h_i = \frac{-E_{bot}}{(q_{bot} \rho_i)} |
279 |
\] |
280 |
where $q_{bot}=-c_{i} T_f + L_i$ is the enthalpy of the new ice, |
281 |
The enthalpy of the second ice layer, $q_2$ needs to be modified: |
282 |
\[ |
283 |
q_2 = \frac{ \hat{h_i}/2 \hat{q_2} + \Delta h_i q_{bot} } |
284 |
{\hat{h_i}/{2}+\Delta h_i} |
285 |
\] |
286 |
|
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If there is a ice layer and the overlying air temperature is |
288 |
below 0$^o$C then any precipitation, $P$ joins the snow layer: |
289 |
\[ |
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\Delta h_s = -P \frac{\rho_f}{\rho_s} \Delta t, |
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\] |
292 |
$\rho_f$ and $\rho_s$ are the fresh water and snow densities. |
293 |
Any evaporation, similarly, removes snow or ice from the surface. |
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We also calculate the snow age here, in case it is needed for |
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the surface albedo calculation (see {\it srf\_albedo.F} below). |
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|
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For practical reasons we limit the ice growth to {\bf hilim} |
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and snow is limited to {\bf hslim}. We converts any |
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ice and/or snow above these limits back to water, maintaining the salt |
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balance. Note however, that heat is not conserved in this |
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conversion; sea surface temperatures below the ice are not |
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recalculated. |
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|
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If the snow/ice interface is below the waterline, snow is converted |
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to ice (see Winton, 1999, equations 35 and 36). The subroutine |
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{\it new\_layers\_winton.F}, described below, repartitions the ice into |
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equal thickness layers while conserving energy. |
308 |
|
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The subroutine {\it ice\_therm.F} now calculates the heat and fresh |
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water fluxes affecting the ocean model surface layer. The heat flux: |
311 |
\[ |
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q_{net}= {\bf fswocn} - F_{b} - \frac{{\bf esurp}}{\Delta t} |
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\] |
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is composed of the shortwave flux that has passed through the |
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ice layer and is absorbed by the water, {\bf fswocn}$=QQ$, |
316 |
the ocean flux to the ice $F_b$, |
317 |
and the surplus energy left over from the melting, {\bf esurp}. |
318 |
The fresh water flux is determined from the amount of |
319 |
fresh water and salt in the ice/snow system before and after the |
320 |
timestep. |
321 |
|
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\noindent |
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{\tiny (There is a provision for fractional ice: |
324 |
If ice height is above {\bf hihig} then all energy from freezing at |
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sea surface is used only in the open water aparts of the cell (ie. |
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$F_b$ will only have the conduction term). |
327 |
The melt energy is partitioned by {\bf frac\_energy} between melting |
328 |
ice height and ice extent. However, once ice height drops below |
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{\bf himon0} then all energy melts ice extent.} |
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\vspace{1cm} |
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|
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\noindent |
335 |
{\bf {\underline{subroutine SFC\_ALBEDO} } } |
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|
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\noindent |
338 |
The routine {\it ice\_therm.F} calls this routine to determine |
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the surface albedo. There are two calculations provided here: |
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|
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\noindent |
342 |
{\bf 1)} from LANL CICE model |
343 |
\[ \alpha = f_s \alpha_s + (1-f_s) (\alpha_{i_{min}} |
344 |
+ (\alpha_{i_{max}}- \alpha_{i_{min}}) (1-e^{-h_i/h_{\alpha}})) |
345 |
\] |
346 |
where $f_s$ is 1 if there is snow, 0 if not; the snow albedo, |
347 |
$\alpha_s$ has two values |
348 |
depending on whether $T_s<0$ or not; $\alpha_{i_{min}}$ and |
349 |
$\alpha_{i_{max}}$ are ice albedos for thin melting ice, and |
350 |
thick bare ice respectively, and $h_{\alpha}$ is a scale |
351 |
height. |
352 |
|
353 |
\noindent |
354 |
{\bf 2)} From GISS model (Hansen et al 1983) |
355 |
\[ |
356 |
\alpha = \alpha_i e^{-h_s/h_a} + \alpha_s (1-e^{-h_s/h_a}) |
357 |
\] |
358 |
where $\alpha_i$ is a constant albedo for bare ice, $h_a$ |
359 |
is a scale height and $\alpha_s$ is a variable snow albedo. |
360 |
\[ |
361 |
\alpha_s = \alpha_1 + \alpha_2 e^{-\lambda_a a_s} |
362 |
\] |
363 |
where $\alpha_1$ is a constant, $\alpha_2$ depends on $T_s$, |
364 |
$a_s$ is the snow age, and $\lambda_a$ is a scale frequency. |
365 |
The snow age is calculated in {\it ice\_therm.F} and is given |
366 |
in equation 41 in Hansen et al (1983). |
367 |
|
368 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
369 |
|
370 |
\vspace{1cm} |
371 |
|
372 |
\noindent |
373 |
{\bf {\underline{subroutine NEW\_LAYERS\_WINTON}}} |
374 |
|
375 |
\noindent |
376 |
The subroutine |
377 |
{\it new\_layers\_winton.F} repartitions the ice into |
378 |
equal thickness layers while conserving energy. We pass |
379 |
to this subroutine, the ice layer enthalpies after |
380 |
melting/growth and the new height of the ice layers. |
381 |
The ending layer height should be half the sum of the |
382 |
new ice heights from {\it ice\_therm.F}. The enthalpies |
383 |
of the ice layers are adjusted accordingly to maintain |
384 |
total energy in the ice model. If layer 2 height is |
385 |
greater than layer 1 height then layer 2 gives ice to |
386 |
layer 1 and: |
387 |
\[ |
388 |
q_1=f_1 \hat{q_1} + (1-f1) \hat{q_2} |
389 |
\] |
390 |
where $f_1$ is the fraction of the new to old upper layer heights. |
391 |
$T_1$ will therefore also have changed. |
392 |
Similarly for when ice layer height 2 is less than |
393 |
layer 1 height, except here we need to to be careful |
394 |
that the new $T_2$ does not fall below the melting temperature. |
395 |
|
396 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
397 |
|
398 |
\vspace{1cm} |
399 |
|
400 |
\noindent |
401 |
{\bf {\underline{Initializing subroutines}}} |
402 |
|
403 |
\noindent |
404 |
{\it ice\_init.F}: |
405 |
Set ice variables to zero, or reads in pickup information |
406 |
from {\bf pickup.ic} (which was written out in {\it checkpoint.F}) |
407 |
|
408 |
\noindent |
409 |
{\it ice\_readparms.F}: |
410 |
Reads {\bf data.ice} |
411 |
|
412 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
413 |
|
414 |
\vspace{1cm} |
415 |
|
416 |
\noindent |
417 |
{\bf {\underline{Diagnostic subroutines}}} |
418 |
|
419 |
\noindent |
420 |
{\it ice\_ave.F}: |
421 |
Keeps track of means of the ice variables |
422 |
|
423 |
\noindent |
424 |
{\it ice\_diags.F}: |
425 |
Finds averages and writes out diagnostics |
426 |
|
427 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
428 |
\vspace{1cm} |
429 |
|
430 |
\noindent |
431 |
{\bf {\underline{Common Blocks}}} |
432 |
|
433 |
\noindent |
434 |
{\it ICE.h}: Ice Varibles, also |
435 |
{\bf relaxlat} and {\bf startIceModel} |
436 |
|
437 |
\noindent |
438 |
{\it ICE\_DIAGS.h}: matrices for diagnostics: averages of fields |
439 |
from {\it ice\_diags.F} |
440 |
|
441 |
\noindent |
442 |
{\it BULKF\_ICE\_CONSTANTS.h} (in {\bf BULKF} package): |
443 |
all the parameters need by the ice model |
444 |
|
445 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
446 |
\vspace{1cm} |
447 |
|
448 |
\noindent |
449 |
{\bf {\underline{Input file DATA.ICE}}} |
450 |
|
451 |
\noindent |
452 |
Here we need to set {\bf StartIceModel}: which is 1 if the |
453 |
model starts from no ice; and 0 if there is a pickup file |
454 |
with the ice matrices ({\bf pickup.ic}) which is read |
455 |
in {\it ice\_init.F} and written out in {\it checkpoint.F}. |
456 |
The parameter {\bf relaxlat} defines the latitude poleward |
457 |
of which there is no relaxing of surface $T$ or $S$ to |
458 |
observations. This avoids the relaxation forcing the ice |
459 |
model at these high latitudes. |
460 |
|
461 |
\noindent |
462 |
({\tiny Note: {\bf hicemin} is set to 0 here. If the |
463 |
provision for allowing grid cells to have both |
464 |
open water and seaice is ever implemented, this would |
465 |
be greater than 0}) |
466 |
|
467 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
468 |
\vspace{1cm} |
469 |
|
470 |
\noindent |
471 |
{\bf {\underline{Important Notes}}} |
472 |
|
473 |
\noindent |
474 |
{\bf 1)} heat fluxes have different signs in the ocean and ice |
475 |
models. |
476 |
|
477 |
\noindent |
478 |
{\bf 2)} {\bf StartIceModel} must be changed in {\bf data.ice}: |
479 |
1 (if starting from no ice), 0 (if using pickup.ic file). |
480 |
|
481 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
482 |
|
483 |
\vspace{1cm} |
484 |
|
485 |
\noindent |
486 |
{\bf {\underline{References}}} |
487 |
|
488 |
\noindent |
489 |
Bitz, C.M. and W.H. Lipscombe, 1999: An Energy-Conserving |
490 |
Thermodynamic Model of Sea Ice. |
491 |
{\it Journal of Geophysical Research}, 104, 15,669 -- 15,677. |
492 |
|
493 |
\vspace{.2cm} |
494 |
|
495 |
\noindent |
496 |
Hansen, J., G. Russell, D. Rind, P. Stone, A. Lacis, S. Lebedeff, |
497 |
R. Ruedy and L.Travis, 1983: Efficient Three-Dimensional |
498 |
Global Models for Climate Studies: Models I and II. |
499 |
{\it Monthly Weather Review}, 111, 609 -- 662. |
500 |
|
501 |
\vspace{.2cm} |
502 |
|
503 |
\noindent |
504 |
Hunke, E.C and W.H. Lipscomb, circa 2001: CICE: the Los Alamos |
505 |
Sea Ice Model Documentation and Software User's Manual. |
506 |
LACC-98-16v.2.\\ |
507 |
(note: this documentation is no longer available as CICE has progressed |
508 |
to a very different version 3) |
509 |
|
510 |
|
511 |
\vspace{.2cm} |
512 |
|
513 |
\noindent |
514 |
Winton, M, 2000: A reformulated Three-layer Sea Ice Model. |
515 |
{\it Journal of Atmospheric and Ocean Technology}, 17, 525 -- 531. |
516 |
|
517 |
|
518 |
|
519 |
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% \end{document} |