a. Model description

In order to examine the effects of enthalpy conservation on modeling sea ice, we have implemented the enthalpy-distribution equation, (5), into a one-dimensional thickness-distribution sea-ice model. Although the model is one-dimensional, it takes into account the effects of ice divergence and shear on ice transport and ridging. The sea-ice model has 12 thickness categories, partitioned following a Gaussian distribution to obtain a thickness mesh that varies smoothly in space (Fig. 2). The 12-category ice thickness-distribution equation is solved following the Hibler (1980) finite-differencing procedure. The parameters governing the ridging process, such as the frictional dissipation coefficient, the ridge participation constant, and shear ridging parameter, are given by Flato and Hibler (1995; see their Table 3 for the standard case). The ice enthalpy-distribution equation, based on the same 12 ice categories, is solved following the same numerical procedure. Accompanying the ice model is a snow model described in terms of snow thickness distribution, g_s(h), corresponding to the ice thickness distribution, g(h). The treatment of the snow thickness distribution, the solution of the snow conservation equation, and the parameterization of surface albedo, dependent on the surface conditions of snow, ice, open water, and surface temperature, are described in detail by Flato and Hibler (1995; in their appendix and Table 2). The heat equation, (2), of the snow–ice system is solved using the Winton (2000) three-layer thermodynamic model, which divides the ice in each category into two layers of equal thickness beneath a layer of snow (Fig. 3) and calculates the surface temperature T_s, the upper ice temperature T₁, the lower ice temperature T₂, and the ice growth rate f.

The solution of (2) provides a vertical profile of ice temperature that can be used to calculate the ice-enthalpy distribution based on (4). The enthalpy distribution is then updated by solving (5), which is in turn used to update ice temperatures T₁ and T₂. However, the variability of the upper ice temperature, T₁, is considerably different from that of the lower ice temperature, T₂ (see Figs. 11 and 14 in section 4c), and the updated enthalpy distribution defined by (3) and (4) may not give enough information to distinguish the temperature changes in the individual layers. To deal with this difficulty, we assume that changes in the ice enthalpy in each layer because of ice advection, growth or decay, and ridging are confined to the same layer. This assumption implies that, when ice transfers to a different thickness category owing to any of the processes, its upper and lower enthalpy would merge, respectively, with the upper and lower enthalpy of the ice in that category. Merging of enthalpy following this rule is certainly accurate with regard to ice advection. It may not be so accurate with regard to ice growth or decay and ridging, but it is unlikely to cause large errors. On the basis of this assumption, we can solve two enthalpy conservation equations similar to (5) in form, one for the upper enthalpy and the other for the lower enthalpy, and then update T₁ and T₂ separately. To simplify the solution, we allow the term ρc_p in (3) to be constant.

The solution procedure consists of the following steps:

1. Solve the sea-ice momentum equations (see, e.g., Hibler 1979) to obtain ice velocity, divergence, and shear. This step is not performed by the one-dimensional model used in this study; rather, it is performed by a three-dimensional coupled ice–ocean model of Zhang et al. (1998b, 2000), whose daily outputs for ice velocity, divergence, and shear are used to drive the one-dimensional model (see details in section 3.2).

2. Solve the heat equation (2) of the snow–ice system using the Winton (2000) three-layer thermodynamic model. The ice enthalpy distribution is

   

   

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   in the upper layer and

   

   

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   in the lower layer, where N = 2.

3. Solve the ice thickness-distribution equation (1) by computing the changes in thickness distribution due to ice transport, ice growth or decay, lateral melting, and ridging, and obtain an updated thickness distribution g′.

4. Solve the snow-thickness distribution equation following the procedure of Flato and Hibler (1995) to obtain the snow depth for each ice thickness category.

5. Insert e₁ and e₂ separately into the enthalpy-distribution equation (5) to obtain two conservation equations; solve these two equations by computing the changes in enthalpy due to ice transport, ice growth or decay, lateral melting, and ridging and obtain an updated upper enthalpy distribution e^′₁ and a lower enthalpy distribution e^′₂.

6. Update ice temperatures T^′₁ and T^′₂ according to

   

   

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   Note that this procedure is also useful for models with general multilayer thermodynamics.

b. Forcing data

The model is driven by daily forcing data derived in 1996 for the North Pole area from observations and model outputs (as described below). The forcing data consist of geostrophic winds, air temperature 2 m above the surface, specific humidity, shortwave and longwave radiations, snow fall, oceanic heat flux, and ice velocity, divergence, and shear. The geostrophic winds at the North Pole are calculated using gridded sea level pressure (SLP) fields provided by I. G. Rigor from the International Arctic Buoy Program (IABP) (see Thorndike et al. 1983). The daily 2-m temperature data at the North Pole (Fig. 4a) are derived from buoys, manned drift stations, and all available land stations (Rigor et al. 2000; also see Martin and Munoz 1997). The specific humidity and longwave and shortwave radiative fluxes are calculated following the method of Parkinson and Washington (1979) based on the SLP and air temperature fields. The downward longwave and shortwave radiative fluxes are shown in Figs. 4b and 4c. Monthly varying snow fall rates for the North Pole area are from the Vowinckel and Orvig (1970) climatology, which are also listed by Zhang et al. (1998a) and Flato and Hibler (1995). These snowfall rates (water equivalent), in millimeters per month, for January through December, respectively, are 12.3, 5.6, 8.5, 2.2, 3.5, 11.1, 16.6, 23.4, 16.1, 14.5, 7.5, and 6.3. Oceanic heat flux is set to a constant 2 W m⁻².

As mentioned earlier the daily ice velocity, divergence, and shear are calculated by a three-dimensional coupled ice–ocean model of the Arctic Ocean and the adjacent seas (Zhang et al. 1998b, 2000). The model has a 40-km horizontal resolution and has been continuously driven by the aforementioned daily forcing from 1979 to 1996. The model solves ice momentum equations associated with a viscous plastic ice rheology, using a line successive relaxation technique to obtain a better plastic solution for ice motion, deformation, and stress (Zhang and Hibler 1997). The 3D model's output of 1996 daily ice velocity, divergence, and shear at the North Pole (Figs. 4d and 4e) is used as forcing data to determine the ice transport, deformation, and ridging in the 1D model. Plotted in Figs. 4d and 4e, the modeled ice divergence and shear are generally small from mid-March through mid-June because of a thicker ice cover that is stronger and more resistant to deformation. In other seasons, the simulated divergence and shear have considerable fluctuations, which allow the 1D model to simulate ice opening or closing and ridging.

a. Ice thickness

Figure 5 shows the 50-yr evolution of mean ice thickness ∫^∞₀ g(h)h dh for the three models. All three model runs start at an initial condition of uniform 0.26-m-thick ice, which falls into ice category 2 (Fig. 2). After about 20 years of spinup, all the models reach a steady state. The zero-layer model does not allow heat storage in the ice and therefore approaches the steady state more rapidly than the other two models. The annual mean ice thickness at the 50th year is 2.79 m for the full model, 2.52 m for the nonconservative model, and 2.45 m for the zero-layer model. The difference in the annual mean ice thickness between the full model and the nonconservative model is 0.27 m, or 11%, which indicates that not conserving ice enthalpy is likely to cause a significant underestimation of ice thickness.

Although the nonconservative model and the zero-layer model do not differ significantly in the annual mean ice thickness (Fig. 5), they differ considerably in the seasonal variability of ice thickness (Fig. 6). The annual thickness range is 1.49 m for the zero-layer model, 1.00 m for the nonconservative model, and 0.96 m for the full model. Compared to the models that use the three-layer thermodynamic scheme, the zero-layer model substantially exaggerates the seasonal variation of ice thickness. The amplitude of the seasonal cycle is in error as a result of ignoring the storage of heat in ice. This is in line with the results of Semtner (1976). However, Semtner's results are based on a calculation of ice with a single thickness of around 3 m, whereas the results shown in Fig. 6 are based on simultaneous calculations of ice with a distribution of all thicknesses ranging from 0 to 28 m. This indicates that Semtner's conclusion about the errors induced by the zero-layer model is as applicable to a model with a thickness distribution as to a model with a single ice type.

Note that the seasonal cycles for the full and nonconservative models track each other very closely, with the nonconservative model thickness always being about 0.27 m thinner. The phase of the cycle—the timing of maximum and minimum—is very similar for the two models. As noted above, however, the annual mean computed by the full model is about 11% larger than that computed by the nonconservative model. This can be attributed to the models' different behaviors in simulating ice temperature and ice growth/melt and in responding to surface albedo feedback, which is to be presented in detail in the following sections.

The thickness distributions simulated by the full model are shown in Figs. 7a and 7b and compared to those simulated by the other two models in Figs. 7c and 7d. Since the models have 12 thickness categories, the distributions are plotted for 12 bins. As can be seen, the models simulate ice in a range of thicknesses owing to the ridging process driven by the daily divergence and shear forcing. Most of the simulated ice, however, is about 1–3 m thick (categories 4 and 5). In winter, all three models create about the same amount of open water (category 1 in Figs. 2, 7c, and 7d). The existence of open water, though small, is due to the existence of ice divergence and shear (Figs. 4d and 4e) that allow ice opening. The full model generally predicts more thick ice than the other two models, the nonconservative model predicts more 1–2 m ice (categories 3 and 4) than the full model, and the zero-layer model predicts more 2–3 m ice (categories 4 and 5) than either three-layer model. In summer, the 0.4–2 m ice decreases quickly, so the area of open water and very thin ice (0–0.4 m) grows rapidly, particularly in the zero-layer model. As a result, the models all show a summer thickness-distribution function with two peaks, one in the open water category and the other in the 1–3 m categories. Again, the full model predicts more thick ice than the other two models. It also predicts a smaller area of open water and thin ice (Fig. 7d).

b. Ice growth and melt

The exaggeration of seasonal cycle of ice thickness by the zero-layer model and the underestimation of mean ice thickness by the nonconservative model are expected to be linked to the models' behavior in estimating ice growth and melt. Figure 8 shows how the simulated ice growth rates vary with ice thickness. In winter, there is a huge ice growth over open water. The growth rates drop drastically for 0.4–1 m ice and then diminishes with increasing ice thickness (Fig. 8a). The growth rates estimated by the full model are about the same as those estimated by the nonconservative model, whereas the growth rates estimated by the zero-layer model are about 0.5 m yr⁻¹ larger than those estimated by the other two models over all thicknesses (Fig. 8b).

In summer, ice melts across the board (Fig. 8c). Magnified in Fig. 8d, the melt rates calculated by the full model and the nonconservative model change sharply for 0.4–2 m ice and gradually for thicker ice. The melt rates calculated by the zero-layer model, on the other hand, stay essentially the same over all the ice categories. This is because the model treats ice as if it has no heat capacity, that is, as if it has a linear temperature gradient. This treatment allows ice of all thicknesses to respond immediately and similarly to any change in thermal forcing at the surface. The lack of heat capacity in the ice also causes the zero-layer model to predict substantially more summer melting, which ranges depending on ice thickness from 0.5 to 4 m yr⁻¹ more than the other two models. This explains why the model tends to considerably exaggerate the seasonal cycle (Fig. 6).

The nonconservative model, in comparison with the enthalpy-conserving model, also predicts more summer melting but to a much lesser degree. The increase in the melt rate ranges from about 1 m yr⁻¹ for thin ice to about 0.5 m yr⁻¹ for very thick ice. Because of the increase in melting, the annual mean growth rate for the nonconservative model is slightly lower than that for the full model (Fig. 8f). Although the increase in melting only causes a slight increase in the seasonal variability of the ice thickness, over time it is responsible for an 11% underestimation of mean ice thickness (Fig. 5).

The seasonal variability in ice growth rate for different ice types is illustrated in Fig. 9. For open water and very thin ice (0–0.4 m), all three models simulate huge ice growth in winter, early spring, and fall (Fig. 9a). The melt season is limited to the summer three months (June–August). The difference among the three models is small relative to the large values of the growth rates. This is no surprise since the zero-layer thermodynamic model is expected to be almost equivalent to the three-layer models for 0–0.4 m ice, and the difference in ice growth rate between the two three-layer models with or without conserving ice enthalpy is generally within the range of 1 m yr⁻¹ (Fig. 8), which is not discernible in Fig. 9a.

For ice thicker than 0.4 m, the melt season increases to about 4 months (Figs. 9b–9f) because of the effect of heat conduction in the ice. The difference in growth rate among the models becomes conspicuous now that the growth rates are considerably smaller than those for very thin ice. Again, the melt rates calculated by the zero-layer model are much larger than those calculated by the other two models, while the melt rates calculated by the nonconservative model are slightly larger than those calculated by the full model. The exaggeration of seasonal variability by the zero-layer model (Fig. 6) is prominently reflected in the behavior of the predicted growth and melt rates. In addition, the timing of ice thaw and freeze-up estimated by the zero-layer model is considerably different from that estimated by the other two models except for 0.4–2 m ice (Fig. 9b). The nonconservative model, on the other hand, differs from the full model only in the timing of ice freeze-up for 0.4–5 m ice (Figs. 9b and c), delaying the fall ice freeze-up by about 6 days. For very thick ice (11–28 m), the two models that use the three-layer thermodynamic scheme hardly predict any ice growth all year round, whereas the zero-layer model estimates ice growth in winter, early spring, and fall regardless of ice thickness.

Another feature shown in Fig. 9 is that the nonconservative model generally estimates melt rates slightly elevated from the enthalpy-conserving model in several months after June. This is linked to the fact that the nonconservative model estimates a lower aggregate surface albedo than the full model for the same period (Figs. 10a and 10b). In fact, the higher ice melt and the lower surface albedo are closely related to the surface albedo feedback; that is, the increased ice melt reduces mean ice thickness by increasing 1–2 m ice and decreasing the ice thicker than 2 m in comparison with the full model (Fig. 7c). During summer melting, the 1–2 m ice is easily turned into open water or very thin ice (0–0.4 m). As a result, a larger area of open water and very thin ice is created by the nonconservative model (Fig. 7d), which lowers the summer surface albedo (Figs. 10a and 10b). The reduced albedo in turn allows more absorption of solar radiation at the surface (Figs. 10c and 10d) and therefore further enhances ice melt. Clearly, this is an effect of the positive surface albedo feedback.

The zero-layer model, on the other hand, underestimates the surface albedo more significantly during that period (Figs. 10a and 10b). This is because of the same surface albedo feedback that links the excessive ice melt to an excessive absorption of solar radiation at the surface (Figs. 10c and 10d). Note that the excessive solar absorption not only leads to an excessively rapid ice melt but also to an excessively rapid snow melt. In fact, the snow is melted so much that its build-up is delayed by almost one month (Fig. 10e), which further reduces the surface albedo. This shows that use of different thermodynamic models, whether allowing heat storage in ice or whether conserving ice enthalpy, not only affects the simulation of ice but also affects the simulation of other physical variables in the snow–ice system in which the well-established surface albedo feedback mechanism plays a prominent role.

c. Ice temperatures

We have described the likely linkage between the surface albedo feedback and the nonconservative model's overestimation of ice melt and underestimation of ice thickness. However, since the nonconservative model only differs from the full model in its inability to conserve ice enthalpy for various sea ice processes, we expect that this model deficiency is the original source of bias and is likely to manifest with the simulated ice interior temperatures. Seasonal snapshots of the temperatures within the two ice layers (Fig. 11) show that the temperatures in both the upper and lower layers vary substantially with ice thickness. They are also subject to considerable seasonal variability, particularly for 0–15 m ice. However, for the thickest ice (>20 m), the lower-layer temperature is more or less close to the freezing temperature of seawater (−1.96°C), and the upper-layer temperature is between the freezing temperature of seawater and the melting temperature of ice (specified as −0.11°C). They also exhibit less seasonal variability. This indicates that the surface conditions have less impact on the temperatures in the deep interior of very thick ice. For most ice thicknesses, the lower layer is warmer than the upper layer except for July through September. One notable feature is that, with every passing day, the coldest temperature in each layer moves, at a different pace, toward thicker ice. In November, the coldest upper and lower temperatures occur almost simultaneously in 2–3 m ice. They move to about 4-m ice in March. By July, the coldest temperature in the upper layer occurs in ice about 15 m thick, while the coldest temperature in the lower layer occurs in ice about 9 m thick. This represents the annual cold wave from the atmosphere, chilling and penetrating the ice.

The implementation of (5) and the conservation of ice thermal energy affect the prediction of the temperatures within the ice. As shown in Fig. 11, the temperatures estimated by the full model for both the upper and lower layers are generally lower than those estimated by the nonconservative model for 3–18 m ice. An interesting feature is that after summer the temperatures estimated by the full model for both the upper and lower layers become significantly lower for 0–5 m ice in September (Fig. 11f), for 1–6 m ice in October (Fig. 11g), and for 2–8 m ice in November (Fig. 11h). This suggests that the ice temperatures estimated by the enthalpy-conserving model become progressively lower for 0–8 m ice as time progresses after summer. This is because the implementation of (5) causes a redistribution of enthalpy from thin ice categories to thick ones in the event of active ice growth and ridging after a summer of ice decay. In September, the minimum ice temperatures, driven by atmospheric forcing, occur in the very thin ice, and the transfer of thin ice into thicker ice reduces the temperatures in 0–5 m ice. In October, the minimum temperatures occur in 1-m ice and the continuing transfer of thin ice into thicker ice reduces the temperatures in 1–6 m ice. In November, the minimum temperatures occur in 2–3 m, and the temperatures are lower in 2–8 m ice. This is why the temperature estimated by the full model is progressively lower for the 0–8 m ice, where the ice transfer is most active and therefore the enthalpy redistribution appears to be most noticeable.

The transfer of “cold” to thicker ice by thermodynamic growth and mechanical ridging in fall is further illustrated in Fig. 12, which plots the differences in the upper ice temperature between the full model and the nonconservative model over 0–8 m ice (the first 8 ice categories). The differences are shown over a one-year cycle starting from July. In July, the difference is small. In August, the difference trough at category 2 deepens, indicating an intensification of enthalpy redistribution driven by an increasing ice growth and ridging. The trough then moves to category 3 in September, to category 5 in October, and to category 6 in November. It reaches the maximum in depth in October at category 5 and then flattens, gradually spreading over a larger thickness range and moving toward ice thicker than 8 m in the following months. This shows a clear after-summer enthalpy redistribution given that the differences in the lower temperature between the enthalpy-conserving model and the nonconservative model are similar with smaller amplitude (not shown).

The enthalpy redistribution from thin to thicker ice in fall impacts ice growth, as shown in Fig. 13, which plots the difference of ice growth rates between the nonconservative model and the enthalpy-conserving model. In both July and August, the nonconservative model considerably overpredicts ice melt rates over all ice thicknesses, which, as mentioned before, is linked to a decrease in surface albedo (Figs. 10a and 10b). In these two months, the growth-rate differences have a trough at ice category 2. In the following months, from September to November, the trough of the growth-rate difference follows the peak of the temperature differences as the ice enthalpy redistributes. This is because the progressively elevated ice interior temperatures in fall, simulated by the nonconservative model, affect the heat convergence or divergence at the surface or the bottom of the ice by changing the heat conduction in the ice. This leads to lower ice growth rates in fall (Figs. 9 and 13). It also leads to a delay of fall ice freeze-up over 0.4–5 m ice (Figs. 9b and 9c).

Although the enthalpy transfer may continue in thicker ice after fall, Fig. 12 hardly shows a continuing enthalpy redistribution over 0–5 m ice, which may indicate that the enthalpy transfer from thin ice to thicker ice after fall is not significant in that thickness range. From December to March the temperature differences have a peak at ice category 2 and, at the same time, a trough at category 3. We do not exactly know what causes that. Meanwhile, the growth-rate differences have a trough at category 2 in November through January. Apparently, the linkage between the growth rates and the ice temperatures is not as strong in other seasons as in fall. Nevertheless, we may still attribute the nonconservative model's bias in predicting ice thickness ultimately to its inability to transfer and conserve ice enthalpy, based on the linkage between its elevated ice temperatures and suppressed ice growth in fall. That is, the model's failure to transfer and conserve ice enthalpy leads to warmer ice interior temperatures progressively over 0–8 m ice in comparison with the full model (Figs. 11 and 12); the warmer ice temperatures lead to a suppressed ice growth in fall (Figs. 9 and 13); the suppressed ice growth leads to a larger area of 1–2 m ice (Fig. 7c), which leads to a larger area of open water and very thin ice (0–0.4 m) during summer melting (Fig. 7d); a larger area of open water and very thin ice leads to a lower average surface albedo and stronger absorption of solar radiation at the surface, which enhances ice melt in summer. Such a sequence, we believe, causes the nonconservative model to underestimate ice thickness.

The seasonal evolution of the vertical temperature profiles for three selected ice categories is shown in Fig. 14. Unlike the linear profiles calculated by the zero-layer model, the profiles calculated by the two models with a three-layer thermodynamic scheme are characterized by a strong nonlinearity (i.e., considerable curvature in the profile), except in summer over relatively thin ice. The strong seasonal variability in the curvature of the profile illustrates how the heat or cold stored in ice affects ice growth and melt. The transfer of ice enthalpy in fall is again reflected in Figs. 14e and 14f, which show noticeably warmer temperatures for the nonconservative model at ice category 3 in September and at category 5 in November. This small difference in temperature profiles gives rise to a considerable difference in mean thickness (Fig. 5).

Figure 14 illustrates well why the zero-layer model exaggerates the seasonal variability of ice thickness with an overestimation of ice melt in summer and an overestimation of ice growth in other seasons. In fall, winter, and early spring, the zero-layer model's linear temperature profile has a larger temperature gradient at the ice bottom, which conducts more heat upward and therefore exaggerates ice growth (Figs. 14a, 14b, 14e, and 14f). In late spring and summer, the zero-layer model uses all the downward heat flux from the surface to melt ice, while the three-layer models use a portion of the downward heat flux to warm the ice (Figs. 14c and 14d).

d. Enthalpy distribution

Figure 15 shows the simulated enthalpy distribution, e(h)dh, and cumulative enthalpy distribution, E(h). As mentioned before, the enthalpy distribution function is not a normalized function, nor is the cumulative function. However, they describe how much enthalpy the ice possesses and how the enthalpy is distributed over different ice categories. Also note that the calculation of ice enthalpy is referenced to zero absolute temperature (T = 0 K) so that enthalpy is always positive. In either winter or summer, most of the ice enthalpy is stored in ice categories 5 and 6 (2–4 m thick) although most of the ice is concentrated in categories 4 and 5 (2–3 m thick; Fig. 7). This is because the amount of ice enthalpy is proportional to both ice thickness and concentration. The ice in category 4 has relatively large concentration, but it is thin and therefore contains less enthalpy. As the ice becomes thicker, its enthalpy per unit area becomes larger, but its concentration decreases more rapidly. As a result, the amount of enthalpy decreases for thick ice categories. Comparing Figs. 15a and 15b, we see that the winter ice has more enthalpy than the summer ice (this is shown more clearly by the cumulative distribution), although the summer ice is warmer than the winter ice (Figs. 11 and 14). This is because winter has more ice than summer has. Compared to the nonconservative model, the full model generally estimates more enthalpy for ice thicker than 2 m and less enthalpy for ice thinner than 2 m. This is because the former calculates more ice thicker than 2 m and, at the same time, less ice thinner than 2 m (Fig. 7). Overall, the total enthalpy simulated by the full model is larger than that simulated by the nonconservative model. This is attributed to the fact that the nonconservative model, although predicting warmer ice interior temperatures, underestimates ice thickness, which indicates that the changes in ice thickness distribution g are more influential on changes in enthalpy distribution e.

e. Sensitivity to input forcing

How do the models react to changes in the input forcing? To examine the behavior of the three models in response to changes in thermal and mechanical forcing, we conducted two series of sensitivity studies in which the model forcing was moderately changed. In the first series, the downward shortwave radiation (Fig. 4c) was proportionally increased or decreased by 5%, while the rest of the forcing remained the same. In the second series, the ice shear (Fig. 4e) was proportionally increased or decreased by 25%.

The models' responses to the perturbations of shortwave radiation are shown in Table 1, and responses to the perturbations of ice shear in Table 2; only the 50th-yr mean ice thicknesses are listed in the two tables. As can be seen, the models are more or less sensitive to the perturbations in input forcing. Within the range of thermal and mechanical perturbations used in the study, the ice thickness estimated by the full model is consistently about 10% larger than that estimated by the nonconservative model. Although the nonconservative model and zero-layer model do not differ much in annual mean ice thickness, they do in seasonal variability of thickness, as shown for the standard case in Fig. 6.

We have also tested these three models using the atmospheric forcing data for 1993. The model behaviors illustrated in Figs. 5 and 6 are essentially the same for 1993 (not shown). All these sensitivity runs lead us to believe that, under moderately different dynamic and thermodynamic conditions, the behavior of these models, relative to each other, would remain more or less the same.