## 1. Introduction

A sea ice model, in general, may contain subcomponents treating 1) dynamics (ice motion), 2) ice transport, 3) multiple ice thickness categories (including leads), 4) surface albedo, and 5) vertical thermodynamics. This paper is concerned with a scheme for the last of these processes. Hibler and Flato (1992) introduce many aspects of sea ice modeling and Ebert and Curry (1993) discuss the details of surface albedo representation.

At issue in the vertical thermodynamics part of an ice model are the vertical resolution in the ice and snow, and the representation of their conductivities, light transmission, and heat capacities including the latent heat of brine inclusions. Bitz et al. (1996) and Battisti et al. (1997) discuss the impact of snow and vertical resolution in the ice upon the representation of natural thickness variability. Thermodynamic models with a range of sophistications have been devised to represent sea ice in climate models. The simplest are fixed- or zero-heat-capacity slab models and the most sophisticated are multilayer models with ice properties dependent upon temperature and salinity. Semtner (1976) found that an intermediate model with one snow layer and two ice layers using constant heat conductivities and a simple parameterization of the brine content was able to mimic the seasonal cycle of the sophisticated multilayer model of Maykut and Untersteiner (1971).

*E,*per unit mass having the form (Bitz and Lipscomb 1999; Ono 1967)

*E*

*T, S*

*C*

*T*

*μS*

*L*

*μS*

*T*

*C, L, S,*and

*T*are the heat capacity, latent heat of fusion, salinity and temperature of the ice, respectively, and

*μ*is a constant equal to minus the freezing temperature (in °C) divided by the salinity. From (1) conservation of enthalpy for a mass,

*m,*of ice can be written in the variable heat capacity form: where

*F*is the heating of the ice.

Equation (2) poses a numerical difficulty—for while it is desirable to solve the ice temperature equation implicitly in order to avoid small time steps when the ice is thin, the variable heat capacity form of the equation, in general, requires that an iterative technique be used for multiple ice layers coupled by diffusion. Semtner sought to represent the brine content of the ice by a parameterization that avoided time stepping (2). A certain fraction of the solar radiation impinging upon the ice was stored in a brine pocket variable that released energy to the upper half of the ice under cooling conditions, maintaining its temperature at 0°C until the brine energy was exhausted. This parameterization achieves computational economy at the expense of decoupling the ice temperature from its brine content.

This paper presents a model that retains the layer structure of the Semtner model but treats the upper half of the ice as a variable heat capacity layer as in the more sophisticated models of Bitz and Lipscomb (1999), Ebert and Curry (1993), and Maykut and Untersteiner (1971). In this way, the brine reservoir is treated more physically and naturally than in Semtner’s model without an increase in computational expense. In particular, the new model follows the practice, advocated by Bitz and Lipscomb, of using −*E* as the energy needed for melting surface ice. They point out that this avoids double counting the melting of brine contained in the ice when ice melts at the surface. Thus, the reduction in brine that accompanies surface melting is automatically handled in the new model while it is ignored in the Semtner model. By accounting for sensible heat in the melting energy for both layers, the new model also avoids a small source of energy nonconservation found in the Semtner model (Bitz and Lipscomb 1999). A further advantage of the new model is that it is fully implicit, eliminating the need to switch to a zero-heat-capacity model when the ice becomes thin, as is typically done with the Semtner model. The next section presents the physical and numerical formulation of the new model. In the following section, results of the new model are compared with those of Semtner’s model. Conclusions are presented in the final section.

## 2. The model

The model consists of a zero-heat-capacity snow layer overlying two equally thick sea ice layers (Fig. 1). The upper ice layer has a variable heat capacity to represent brine pockets. The lower ice layer has a fixed heat capacity. The prognostic variables are *h*_{s}, the snow layer thickness; *h*_{i}, the ice layer thickness; and *T*_{1} and *T*_{2}, the upper and lower ice layer temperatures located at the midpoints of the layers *h*_{i}/4 and 3*h*_{i}/4 below the ice surface, respectively.

Two of the prognostic variables of Semtner’s model have been eliminated: the brine content of the upper ice and the snow temperature. A separate brine variable is no longer needed because the brine content in the new model is completely determined by the upper ice temperature and the (predetermined) ice salinity. The heat capacity of the snow layer is typically small relative that of the ice. In regions where snowfall is large, the depth of the snow layer relative to the ice is limited by snow-to-ice conversion taking place as seawater floods snow below the waterline. It has been estimated that 8% of sea ice in the Weddel Sea is formed through this process (Eicken et al. 1995). If all snow below the waterline is converted to sea ice, snow can represent at most 1 − *ρ*_{i}/*ρ*_{w} ≈ 0.1 of the sensible heat capacity of the snow–ice system. When latent heat capacity is also considered this ratio becomes even smaller since snow is fresh and does not form brine inclusions. For these reasons snow layer heat capacity has been neglected in the new model.

The ice model performs two functions. The first is to calculate the ice temperature and the second is to calculate changes in the thickness of ice and snow. In an atmosphere–ice model, the temperature calculation is typically done at each atmospheric physics time step, so that the surface temperature can be used as a boundary condition for vertical mixing. It is therefore desirable that this computation be as efficient as possible.

### a. Ice temperature calculation

*F*

_{s}. For this purpose we employ an expression for

*F*

_{s}that is linearized about the surface temperature at the last time step,

*T̂*

_{s}: Here

*F*

_{s}(

*T̂*

_{s}) and ∂[

*F*

_{s}(

*T̂*

_{s})]/∂

*T*

_{s}would typically be calculated using downward short- and longwave radiative fluxes, bulk formulas for latent and sensible heat fluxes, the Stefan–Boltzmann formula for the upward longwave, and the computed surface albedo for the upward shortwave. This part of the calculation is unchanged from Semtner (1976).

*K*

_{1/2}

*T*

_{1}

*T*

_{s}

*h*

_{i}/4 beneath the snow–ice interface. See Table 1 for other symbol definitions. Here

*K*

_{1/2}is determined by balancing the conductive flux of heat through the ice to the snow–ice interface with the conductive flux of heat through the snow away from the interface. Notice, that it also applies when

*h*

_{s}= 0. As in the Semtner model, the ice conductivity is assumed to be constant rather than a function of temperature and salinity, as it is in more sophisticated models (Maykut and Untersteiner 1971; Bitz and Lipscomb 1999).

*I*is the surface penetrating solar radiation absorbed by the ice

^{1}and

*K*

_{3/2}

*K*

_{i}

*h*

_{i}

*T*

_{2}gives Substituting (6) and (15) into (12) gives

*A*

_{1}

*T*

^{2}

_{1}

*B*

_{1}

*T*

_{1}+

*C*

_{1}= 0, where This is solved for

*T*

_{1}, which is subsequently used in (15) to find

*T*

_{2}and in (6) to find

*T*

_{s}. If this calculation gives a temperature greater than the melting temperature of snow (if there is snow cover) or sea ice (if there is none), the temperatures are recalculated using quadratic coefficients appropriate for the problem with the surface temperature fixed at freezing (

*T*

_{s}= 0°C when there is snow;

*T*

_{s}= −

*μS,*the melting temperature of sea ice, when there is no snow): and

*C*

_{1}is unchanged.

*A*

_{1},

*B*

_{1}, and

*C*

_{1}are given by (16)–(18). The new lower ice temperature is then given by (15) and the surface temperature by (6). If the surface temperature is greater than the freezing temperature of snow (when there is snow cover) or sea ice (when there is none), the surface temperature is fixed at the melting temperature of snow or sea ice, respectively, and the upper ice temperature is recomputed from (21) using the coefficients given by (19), (20), and (18). Then,

*T*

_{2}is recomputed from (15) and an energy flux,

*M*

_{s}

*K*

_{1/2}

*T*

_{1}

*T*

_{s}

*A*

*BT*

_{s}*M*

_{b}

*F*

_{b}

*K*

_{i}

*T*

_{f}

*T*

_{2}

*h*

_{i}

*F*

_{b}and the conductive flux of heat upward from the bottom.

### b. Calculation of ice and snow mass changes

*M*

_{b}is negative. In this case, the bottom layer thickness

*h*

_{2}(initially,

*h*

_{2}=

*h*

_{1}=

*h*

_{i}/2) is increased by

*h*

_{2}

*M*

_{b}

*t*

*E*

_{2}

*T*

_{f}

*S*

*E*

_{2}

*T, S*

*C*

*T*

*μS*

*L*

*M*

_{s}

*t*

*Lh*

_{s}

*E*

*T*

_{1}

*S*

*h*

_{1}

*E*

_{2}

*T*

_{2}

*S*

*h*

_{2}

*M*

_{b}is positive (bottom melting), the bottom ice thickness is reduced by the upper ice thickness by and the snow thickness by with excess melt energy of

*M*

_{b}

*t*

*Lh*

_{s}

*E*

*T*

_{1}

*S*

*h*

_{1}

*E*

_{2}

*T*

_{2}

*S*

*h*

_{2}

*ρ*

_{w}is the density of seawater. Now the temperature of the upper ice layer must be adjusted to account for the incorporation of zero-heat-capacity snow. This is done based upon enthalphy conservation, that is, by determining the temperature of the new upper layer that gives an enthalpy equal to the average enthalpies of the old upper layer and the added snow. For this purpose we note that the snow enthalpy, −

*L*=

*E*

_{2}(−

*μS, S*). Hence an expression for new upper ice temperature when adding lower-layer ice can also be used for adding snow simply by replacing

*T*

_{2}with −

*μS.*We now derive this expression that will also be used below for evening the ice layers.

*f*

_{1}of upper ice and 1 −

*f*

_{1}of lower ice,

*E*

*T*

_{1new}

*f*

_{1}

*E*

*T*

_{1}

*f*

_{1}

*E*

_{2}

*T*

_{2}

*T*

_{1new}gives where

*f*

_{1}of upper-layer ice and 1 −

*f*

_{1}of lower-layer ice gives a new lower layer temperature,

*T*

_{2new}

*T*

*T*

When upper-layer ice is converted to lower-layer ice it is possible for the new lower layer temperature to exceed the ice melting temperature, −*μS.* A number of adjustments might be made to avoid this condition. In a coupled atmosphere–ice model at the Geophysical Fluid Dynamics Laboratory (GFDL), this extra sensible energy has been used to melt equal thicknesses from the upper and lower layers with good results.

## 3. Comparison to Semtner’s model

To compare the new model with Semtner’s model both were forced with climatological seasonal fluxes of downward short- and longwave radiation, and sensible and latent heat fluxes. The values used were the same used by Semtner to compare his model with the Maykut and Untersteiner model. As in Semtner (1976), the upward longwave was calculated using the Stefan–Bolzmann relation, and upward shortwave by applying a surface albedo. Rather than using the observed albedos, however, snow was given an albedo of 0.8—reduced to 0.75 under melting conditions—and ice an albedo of 0.65. The latter value was tuned to give an annual mean ice thickness of 3 m. The ocean heat flux to the ice was neglected.

Figure 2 shows the seasonal variation of snow, ice, brine, and temperatures for the two models. For 30% penetrating radiation and a salinity of 1 per mil in the new model, there is close agreement between the two models in the simulation of all properties. The differences are not significant since the ice salinity for new model was chosen to produce agreement between the models with a round number. The surface temperatures predicted by the models (not shown) were virtually identical. When the salinity is increased in the new model, the agreement with the Semtner model ice thickness and surface temperature remain, but the internal temperatures show a relative seasonal lag and decrease in seasonal amplitude. Figure 3 shows the annual mean ice thickness of the two models with a range of settings for the fraction of solar radiation allowed to penetrate the ice. The sensitivity to this parameter is very similar for the two models. The new model has slightly less sensitivity due to having some storage of energy in brine pockets independent of the penetrating solar radiation (storage in brine pockets generally thickens the ice).

To compare the computational expense of the models, 1000 seasonal cycles of the above experiment were run on an SGI Indigo workstation and a Cray T90 supercomputer. The version of the Semtner model employed was that used as part of the National Center of Atmospheric Research CSM sea ice model (Bettge et al. 1996) available online at http://www.cgd.ucar.edu:80/ccr/bettge/ice/. The new model used 17% less CPU time on the SGI and 37% less CPU time on the Cray.

## 4. Conclusions

A reformulated three-layer sea ice model has been presented in this paper. This model offers several modest improvements over the widely used Semtner three-layer model. The upper ice layer is given a variable heat capacity putting the treatment of brine pockets on a firmer physical footing than in the Semtner model—although the Semtner and new models simulate similar seasonal cycles for a particular choice of ice salinity in the new model. The new model is fully implicit, allowing longer time steps and eliminating the need to use a zero-heat-capacity model when the ice becomes thin. The new ice model has been coupled to an atmospheric model at GFDL and runs stably without any limitation upon ice thickness. Finally, the new model is somewhat more computationally efficient than Semtner’s original model and carries fewer prognostic variables.

A FORTRAN code for the model described in this paper is available online at http://www.gfdl.gov/∼mw/.

## Acknowledgments

The author thanks Steve Griffies and Bob Hallberg for reviews of an early draft. Cecilia Bitz, Bill Lipscomb, and two anonymous reviewers made helpful comments leading to significant improvement of the paper.

## REFERENCES

Battisti, D. S., C. M. Bitz, and R. E. Moritz, 1997: Do general cir-culation models underestimate the natural variability in the Arctic climate?

*J. Climate,***10,**1909–1920.Bettge, T. W., J. W. Weatherly, W. M. Washington, D. Pollard, B. P. Briegleb, and W. G. Strand Jr., 1996: The NCAR CSM Sea Ice Model. NCAR Tech. Note TN-425+STR, 25 pp. [Available from NCAR, P.O. Box 3000, Boulder, CO 80307-3000.].

Bitz, C. M., and W. H. Lipscomb, 1999: A new energy-conserving sea ice model for climate study.

*J. Geophys. Res.,***104,**15 669–15 677.——, D. S. Battisti, R. E. Moritz, and J. A. Beesley, 1996: Low-frequency variability in the arctic atmosphere, sea ice, and upper-ocean climate system.

*J. Climate,***9,**394–408.Ebert, E. E, and J. A. Curry, 1993: An intermediate one-dimensional thermodynamic sea ice model for investigating ice–atmosphere interactions.

*J. Geophys. Res.,***98,**10 085–10 109.Eicken, H., H. Fischer, and P. Lemke, 1995: Effects of the snow cover on Antarctic sea ice and potential modulaton of its response to climate change.

*Ann. Glaciol.,***21,**369–376.Hibler, W. D, and G. M. Flato, 1992: Sea ice models.

*Climate Sytem Modeling,*K. Trenberth, Ed., Cambridge University Press, 413–436.Maykut, G. A., and N. Untersteiner, 1971: Some results from a time-dependent thermodynamic model of sea ice.

*J. Geophys. Res.,***76,**1550–1575.Ono, N., 1967: Specific heat and heat of fusion of sea ice.

*Physics of Snow and Ice,*H. Oura, Ed., Vol. I, Institute of Low Temperature Science, 599–610.Semtner, A. J., 1976: A model for the thermodynamic growth of sea ice in numerical investigations of climate.

*J. Phys. Oceanogr.,***6,**27–37.

Notation.

^{1}

All of the solar radiation that is absorbed by the ice is absorbed in the upper layer. However, some of the penetrating solar may pass through the ice to the ocean below. Indeed, it is desirable that a fraction, exp[−(*h*_{i})/2*h*_{o}] where *h*_{o} is the ice optical depth, of the penetrating solar pass through the ice so that the thermal forcing of thin ice does not become so great that temperatures above the melting temperature are predicted. This approach is taken by Bettge et al. (1996) in their use of the Semtner model.