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).

The treatment of the brine content of the ice is one demarcating feature distinguishing simple and sophisticated models. Since sea ice contains salt, it also contains an amount of liquid water needed to dilute its salt to a brine with freezing temperature equal to the sea ice temperature. Thus, as the ice is heated, for example, two changes occur: 1) the temperature of the ice and its brine increases, and 2) the higher temperature allows for a more dilute brine and so some ice melts, increasing the brine content of the ice. Consequently, sea ice conserves an enthalpy, E, per unit mass having the form (Bitz and Lipscomb 1999; Ono 1967)

ET, SCTμSLμST(1)

where 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:

View Expanded

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 T₁ and T₂, the upper and lower ice layer temperatures located at the midpoints of the layers h_i/4 and 3h_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

The surface temperature is determined from the diagnostic balance between the upward conduction of heat through snow and/or ice and upward flux of heat from the surface 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:

View Expanded

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).

The conductive flux to the surface is

K_1/2T₁T_s(4)

where

View Expanded

is the effective conductive coupling of the snow–ice layer between the surface and the upper layer ice temperature 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).

Now equating (3) and (4) gives the surface temperature

View Expanded

where we define

View Expanded

For the upper layer of the ice, (2) is written:

View Expanded

where I is the surface penetrating solar radiation absorbed by the ice¹ and

K_3/2K_ih_i(10)

is the conductive coupling between the two ice temperature points. The prognostic ice temperature equation for the fixed heat capacity ice lower layer is

View Expanded

The discrete forms of (9) and (11) are

View Expanded

where a hat denotes a value at the previous time step. In deriving the left-hand side of (12) we have used the fact that

View Expanded

Solving (13) for T₂ gives

View Expanded

Substituting (6) and (15) into (12) gives A₁T²₁ + B₁T₁ + C₁ = 0, where

View Expanded

This is solved for T₁, which is subsequently used in (15) to find T₂ 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):

View Expanded

and C₁ is unchanged.

To summarize the ice temperature time-stepping procedure: The new upper ice temperature is given by

View Expanded

where A₁, B₁, and C₁ 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, T₂ is recomputed from (15) and an energy flux,

M_sK_1/2T₁T_sABT_s(22)

is applied toward surface melting (thereby balancing the surface energy budget). Likewise, the energy for bottom melting (or freezing, if negative),

M_bF_bK_iT_fT₂h_i(23)

serves to balance the difference between the oceanic heat flux to the ice bottom F_b and the conductive flux of heat upward from the bottom.

b. Calculation of ice and snow mass changes

In addition to calculating ice temperature changes, the ice model must also readjust the sizes of the snow and ice layers 1) to accommodate mass fluxes at the upper and lower surfaces, 2) to convert snow below the waterline to ice, and 3) to equalize the thickness of the two ice layers. It is convenient to apply mass-increasing changes first. Snow is added to the snow layer and ice from two sources is added to the bottom layer. The first source is freezing at the ice ocean interface from (23) when M_b is negative. In this case, the bottom layer thickness h₂ (initially, h₂ = h₁ = h_i/2) is increased by

h₂M_btE₂T_fS(24)

where

E₂T, SCTμSL(25)

is the enthalpy of the lower, fixed heat capacity, ice. Notice that, even though it does not contain brine, the lower ice has the melting temperature of saline ice. Frazil from the ocean mixed layer, possibly formed to balance heat fluxes through leads when the mixed layer temperature falls to seawater freezing temperature, is added to the bottom of the ice along with the portion frozen at the ice–ocean interface. Both are added at seawater freezing temperature, changing the lower layer temperature to

View Expanded

Following, positive mass changes to the snow and ice layers, surface and bottom melting is applied. The surface melting that occurs to balance the (upper) surface energy budget described above reduces the snow thickness by

View Expanded

the upper ice thickness by

View Expanded

and the lower ice thickness by

View Expanded

An excess melt energy of

M_stLh_sET₁Sh₁E₂T₂Sh₂(30)

is applied to the ocean mixed layer. Similarly, if M_b is positive (bottom melting), the bottom ice thickness is reduced by

View Expanded

the upper ice thickness by

View Expanded

and the snow thickness by

View Expanded

with excess melt energy of

M_btLh_sET₁Sh₁E₂T₂Sh₂(34)

applied to the ocean mixed layer.

After the above adjustments have been made to the thickness of the snow and ice layers, two internal adjustments are made. The first converts snow below the waterline

View Expanded

into upper layer ice

View Expanded

where 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 = E₂(−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 T₂ with −μS. We now derive this expression that will also be used below for evening the ice layers.

Using (1) and (25) we can write the enthalpy of a new upper layer formed from a fraction f₁ of upper ice and 1 − f₁ of lower ice,

ET_1newf₁ET₁f₁E₂T₂(37)

Solving for this for T_1new gives

View Expanded

where

View Expanded

The final adjustment evens the upper and lower layers of ice to maintain the layer structure depicted in Fig. 1. When shifting mass between the two layers, the temperature of the donating layer remains unchanged, while the temperature of the receiving layer is determined from averaging the enthalphies of the two ice portions. Equation (38) is used for determining the new temperature of an upper layer when lower-layer ice is added to it. Averaging enthalpies for a new lower layer formed from a fraction f₁ of upper-layer ice and 1 − f₁ of lower-layer ice gives a new lower layer temperature,

T_2newT(40)

where T is given by (39).

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/.