The sensitivity of the Arctic Ocean ice cover on the atmospheric poleward energy flux, D, is studied using a coupled column model of the ocean, ice, and atmosphere. In the model, the ice cover is described by a thickness distribution and the atmosphere is a simple two-stream gray body in radiative equilibrium. It is shown that the thickness distribution, in combination with the albedo function, gives a strong nonlinear response to positive perturbations of D. The response on D is sensitive to the albedo parameterization and the shape of the thickness distribution, controlled by ridging and divergence. An increase of about 9 W m−2 from a standard value of D = 103 W m−2 has a dramatic effect, reducing the ice thickness by more than 2 m, and generates a large open-water fraction during summer. The reduction of ice thickness is characterized by a clear transition between two regimes, going from a regime where first-year ice survives the next summer melt period to a seasonal ice regime where the first-year ice melts completely. It is shown that the existence of seasonal ice regime is dependent on the surface mixed layer thickness. The model enters a completely ice-free state if the thickness of the mixed layer is increased above a threshold value. The adjustment timescale for the ice cover is 6 years for small positive and negative perturbations in D. For larger positive perturbations of about 10 W m−2, the adjustment timescale is up to 20 years.
 The Arctic Ocean ice cover together with the overlying atmosphere makes up a complicated dynamic and thermodynamic system. This system includes numerous subprocesses, which are certainly important but hard to describe in detail. Examples of such processes are formation of pressure ridges, melt ponds on the ice floes, cloud formation and precipitation. Although internal subprocesses can play an important role there should also exist a strong and fundamental coupling between the basic state of the system and the energy supply from external sources. The main energy sources for the Arctic atmosphere and ice cover are atmospheric transport of warm and moist air from lower latitudes, solar radiation, and oceanic heat conducted through the ice. When the ice cover is at equilibrium, that is, with no net growth or melt over the year, there is a balance between the total heat supply and the thermal radiation to space. Using a very simplistic approach Thorndike , hereafter TH92, identified these processes and linked them together in a time-dependent and fully coupled ocean-ice-atmosphere model. The model forcing was step functions with winter and summer values but otherwise constant in time during each season, and the processes in the atmosphere and ice were simplified down to the “bare bones physics.” An expression, based on the model, for the equilibrium ice thickness in terms of the poleward energy flux in the atmosphere, solar radiation, total optical thickness of the atmosphere, oceanic heat flux and thermal conductivity was one of the major results. In particular, it was shown that the ice thickness is highly dependent on the poleward energy transport in the atmosphere. This model allows for two steady states. A perennial ice cover and a totally ice-free state, whereas a possible solution with a seasonal ice cover is unsteady and therefore not realized.
 The coupled ocean-ice-atmosphere system has later been studied by more sophisticated models and with interannual variability of the forcing. Bitz et al.  used an ocean-ice-atmosphere column where the atmospheric part consisted of 18 layers including a full radiative and convection scheme. The ocean was a fixed 50-m mixed layer and the ice cover was modeled as a uniform slab. The natural variability of the ice cover was then addressed by applying stochastic variations to the poleward energy transport and cloudiness. One important result was that high-frequency variations in the forcing may generate variability of the ice thickness at much lower frequencies. In the model study by Holland and Curry , the ice cover was described by a thickness distribution and the atmospheric part included a radiative transfer scheme. They investigated the variability of the Arctic ice volume, applying stochastic variations to both surface air temperature and ice divergence and found that inclusion of a thickness distribution significantly affects the variability.
 The main objective with the present investigation is to explore the response of the ice thickness distribution when changing the poleward energy transport in the atmosphere. It may been seen as an extension of TH92 with a more sophisticated model tool. Compared to TH92 the present model includes additional characteristics and processes which we believe are in the next order of importance. These are an ice thickness distribution, a snow layer on the ice, ice divergence, ridging processes, turbulent heat fluxes at the ice/ocean surface, and an active oceanic mixed layer. The forcing consists of monthly mean values instead of step functions over a cold and warm season. The main difference in our model approach to the earlier studies by Holland and Curry  and Bitz et al.  is that we use an ice sheet with a thickness distribution that is fully coupled to the atmosphere and ocean. Another aspect of the model is that it is relatively simple since the aim is not to model the system in detail but instead to focus on the overall function.
 Many model studies of the Arctic ice-ocean-atmosphere system include much more sophisticated descriptions of subprocesses than is the case here. The dynamic response of the ice cover to wind forcing in combination with the thermodynamic growth/melt has been studied using dynamic-thermodynamic models [e.g., Hibler, 1979], which generates horizontal fields of ice speed and ice thickness as resolved by the grid. The representation of the ice cover can be refined further using a thickness distribution within each grid square [Flato and Hibler, 1995]. Models for thermodynamic processes in the ice may include the effect of melt ponds and have several temperature points in the ice and snow [Ebert and Curry, 1993]. There are detailed models for transmission of the solar radiation within the ice [Ebert et al., 1995], and for the heat fluxes at the ice/ocean interface [Holland et al., 1997]. Models for Arctic atmosphere normally include radiative transfer schemes for shortwave and longwave radiation with separate wavelength bands and the effect of clouds at different levels [e.g., Beesley, 2000]. A full treatment of moisture including a cloud formation scheme was implemented in a specific model study in connection with the SHEBA experiment [Pinto et al., 1999].
 A truly quantitative ranking of the impact of different processes and to which level of details these need to be described is of course hard to find but we motivate our level of the modeling effort by comparing with observations and results from other investigations where a quantitative judgement can be made. We also make a direct comparison of the result from our standard simplistic atmospheric model with a more detailed model formulation of the atmospheric radiative transfer. The model is described in section 2 and results in section 3, including verification of the standard case annual cycle (3.1), the steady state sensitivity of the ice cover to changes of different forcing quantities (3.2), and some aspects of the time-dependent response (3.3). Some of the results are discussed and put in perspective in section 4. Conclusions are given in section 5.
2. Model Description and Forcing
 The most serious simplification made in the model by TH92 is probably that the ice cover was described as a slab with only one ice thickness. No leads exist then where solar radiation can penetrate, heat up the water column, and generate basal melting. The single thickness does not permit a description of the large turbulent heat fluxes and ice growth during winter in areas with open water or thin ice. Another important characteristic, omitted in TH92, is that the Arctic Ocean ice cover is not in a thermodynamic steady state since large amounts of ice are exported out of the area each year resulting in a net ice growth. The export occurs at a rate of about 12% of the basin area each year [Kwok and Rothrock, 1999] and represents an average heat flux of about 4 W m−2. An active mixed layer having a thickness dependent on the buoyancy flux and mechanical forcing, instead of a layer with fixed thickness as in TH92, is likely needed in order to distribute and store the oceanic heat in a realistic way.
 The model we use here is a column model representing the horizontally average properties across the Arctic Ocean, excluding the Barents Sea. The ice cover is described by a thickness distribution and the model is fully coupled between ocean, ice, and atmosphere regarding the heat fluxes but there is no explicit interaction concerning moisture fluxes and precipitation (see Figure 1 for a schematic sketch).
 The ice and ocean models are nearly identical with that of Björk  with some modifications to handle the differences in surface stress between ice and open water and also some slight changes of parameter values (see Table 2). The ocean model is a column with a dynamic mixed layer on top, which deepens during winter by entrainment forced by the ice/ocean stress and negative buoyancy flux. During summer when the buoyancy flux is positive, due to melting and river water input, it becomes thinner again. The column below the mixed layer is maintained by inflow from Bering Strait and a geostrophically controlled outflow. The ocean model also includes a procedure that redistributes the properties in the column in order to mimic the circulation due to the large ice production and salinity increase in coastal polynyas. This process is handled by circulating mixed layer water through a hypothetical shelf, where the salinity is enhanced, from where it flows back in the column and interleaves at the level of matching density. The properties at the lowest level in the model are prescribed to have a typical temperature and salinity of the Atlantic water. The ocean model has shown to be able to reproduce the T-S structure in the upper layers [Björk, 1989] as well as time-dependent tracers such as bomb tritium and Freons [Becker and Björk, 1996].
 The model ice cover is partitioned in thickness categories, which develop independently in a Lagrangian fashion in time. New categories are created when new ice is formed in the leads during winter and by ridging which transform thin ice to piles of thicker ice. Categories disappear when thin ice melts completely during summer. The divergence and ridging are the processes that create open water during winter where new ice can form. The rate of open-water formation due to divergence, e, is connected to the ice export while the rate of open-water formation due to ridging, r, is prescribed directly. The redistribution of ice by the ridging process is parameterized using a participation parameter, go, and a thickness multiplier, M. Categories within the thinnest go fraction of the ice distribution are ridged to M times the original ice thickness. Each ice category has one internal temperature point and a snow layer on top. The snow albedo is a prescribed annual cycle based on observations (Table 1), and the surface albedo parameterization for bare ice, which play a central role in this analyze, is given by
according to Maykut , where H is the ice thickness (see also Figure 2). The maximum value for αice is reached when H = 2.3 m. The reason to use this albedo parameterization is that it is simple but still includes the well established characteristics of rapidly decreasing albedo with thickness for thin ice [Ebert and Curry, 1993].
Table 1. Model Forcing and Seasonal Dependent Parametersa
The first parameter, the poleward energy flux, is based on rawinsonde data and the GFDL model [Overland and Turet, 1994]; the incoming solar radiation is from Russian North Pole drift stations, obtained from the National Snow and Ice Data Center (NSIDC). The divergence is based on an 18-year-long time series, 1978 to 1996, of ice motion in the Fram Strait from satellite passive microwave data [Kwok and Rothrock, 1999]. Some ice is also exported into the Barents Sea and through the Canadian Archipelago. We estimate this export to about 20% of the Fram Strait export, and the total export is then 1.2 times the Fram Strait value. The divergence is computed by dividing the ice export with the ice-covered area, assumed equal to the total area, except during the summer months, when it is decreased by 12% in order to account for the open-water fraction. The annual cycle of the river runoff is from Becker and Björk . Volume flow, salinity, and temperature in the Bering Strait are based on 4-year-long time record [Roach et al., 1995]. Here the flow is increased by 10%. Snow albedo and precipitation are from Maykut . Wind speed, wind standard deviation (SDA), and ice/wind speed ratio are from Björk .
Energy flux, W m−2
Solar radiation, W m−2
Precipitation, 10−9 m s−1
Runoff, 106 m3 s−1
Flow, 106 m3 s−1
Wind velocity, m s−1
Wind SDA, m s−1
Ice/wind speed ratio
 The atmospheric model is similar to TH92 and is a two stream, gray body, radiative equilibrium, atmosphere which is transparent to solar radiation. The leading idea behind the use of this, admittedly very simplified, atmospheric model is to get a reasonable downward longwave radiation to the ice surface for a given poleward heat flux and cloudiness in the atmosphere. An attractive property of this model is that the rather complicated state of water in the atmosphere, as determined by the water vapor distribution and different kinds of clouds at different levels, is collectively described by one single parameter; the total optical thickness, N. The full equations for the atmospheric model and the coupling procedure with the heterogeneous ice/ocean surface are given in the Appendix'Coupling at the Surface'. There are of course also drawbacks with this simplification. It has been shown that the thermal equilibrium of the atmosphere and ice is sensitive not only to the amount of clouds but also to the level occupied by clouds [Beesley, 2000]. Increasing the amount of low-level clouds give an equilibrium with thicker ice while more high level clouds give thinner ice. The atmospheric model differs from TH92 by including effects of the turbulent heat flux which is needed in connection with the thickness distribution since the turbulent heat fluxes can be very large over thin ice and open water. The heat supply/removal to/from the atmosphere by the turbulent flux is treated as a heat source/sink, which is distributed close to the surface using an exponentially decaying function with optical height. The decay scale is set by a parameter, ζ, with a value much smaller than N.
 The model is forced by monthly means of poleward energy flux at the vertical boundary, solar radiation at the surface, snow precipitation, river runoff, Bering Strait inflow, ice export, and wind. The total optical thickness is used to tune the model ice thickness and is given as a prescribed time-dependent parameter with an annual cycle. Since N is an integrated measure of all absorbers of thermal radiation in the atmosphere, of which cloud particles play a dominate role, it is prescribed to follow the annual cycle of cloudiness (Figure 3). It is hard to evaluate directly how well the tuned optical thickness in Figure 3 describes the actual conditions, since no data of total optical thickness exists, to our knowledge. Using model result from a more sophisticated radiative convective column model, NCAR Community Climate Model (CCM2, CRM) [Hack et al., 1993] it is possible to estimate the total optical thickness for different cloud conditions. The clear sky optical thickness from this model, using the 1980–1987 January mean temperature and humidity profile from the Historical Arctic Rawinsonde Archive (HARA) data set, obtained National Snow and Ice Data Center (NSIDC), Boulder, Colorado, is about 0.3. A realistic summer Arctic cloud coverage with 70% of low clouds, 25% of middle and high clouds, respectively, corresponds to an optical thickness of about 4.9. For the winter profile with low and middle cloud fraction at 50%, and 25% high cloud coverage gives an optical thickness of about 2.3. The low-level cloud fraction is about a factor 2 above climatological mean, taken from the Russian North Pole drift stations (NP stations). We motivate this high value by the fact that the CRM model does not include ice crystals, which would increase the optical thickness. The values of the total optical thickness computed from the CRM model in this way are relatively close to the values used in the present model (Figure 3).
 The standard forcing is shown in Table 1 and the standard values of some selected model parameters are presented in Table 2.
Table 2. Values of Constants and Standard Case Parameters
 In the light of observed thinning of the ice cover from about 3 m in the 1970s to about 2 m in the 1990s [Rothrock et al., 1999] there is actually some ambiguity in specifying an observed ice thickness. We have conservatively chosen the commonly used 3 m [McLaren et al., 1994] as the observed annual mean thickness and tuned the model towards that by adjusting the total optical thickness. The model is run using interpolated values between the monthly means of the forcing quantities and with a time step of one day.
3.1. Standard Case, Annual Cycle
 In this section we show that the model can reproduce the present day characteristics of the upper ocean, ice and lower atmosphere. The standard case gives an ice cover with an annual mean area averaged thickness of 3.3 m and with a seasonal thickness variation of 0.5 m, similar to that of Björk . In late winter, the modeled ice thickness distribution agrees well with ice draft data from submarine cruises from 1991, 1992, and 1994 (Figure 4). The melting season is about 90 days with a start in the beginning of June and ends in late August which agrees well with data [Rigor et al., 2000]. At the end of the melting season, about 12% of the area is ice free, less than 1% is covered by ice thinner than 1 m, 65% is in the interval 1 m to 4 m, and 25% of the total area is covered by ice thicker than 4 m (Figure 9b). The annual cycle of the surface air temperature is close to the observations at the NP stations (Figure 5). The temperature profile aloft (not shown) is hard to compare directly with observations, by the use of optical height as vertical coordinate. Using the CRM model with the winter profile of humidity and clouds described in section 2, it is possible to roughly estimate how the optical thickness is related to height. The present model does not capture the large scale inversion of the lowest 3000 m in the 80–87 January mean profile from the HARA data set and is about 10°C colder than data at 2000 m, but is closer to the observed profile at higher levels. The main reason for the discrepancy is probably that the model distributes D evenly throughout the atmosphere. The lowest 200 m agree with the data within 2°C. The annual cycle of the downward longwave radiation at the surface follows close to the observed at the NP stations (Figure 6). Modeled upward longwave radiation at the top of the atmosphere during winter is 140 W m−2, which is about 25 W m−2 below the observed mean value according to the Earth Radiation Budget Experiment (ERBE) [Briegleb and Bromwich, 1998]. There are several possible reasons for this discrepancy. The model does not consider the vertical distribution of clouds or D which means that a larger/smaller amount of heat will radiate to space if the atmosphere is covered by low/high clouds [Beesley, 2000]. A second possibility is that the model domain does not cover the entire area north of 70°N. Processes in the Barents Sea and on the adjacent continents may act as a heat source/sink for the poleward energy flux. Third, the ERBE mean value is based on a 4 year long time record, which may be too short to be considered as a climatic winter mean value. During summer the model gives an outgoing longwave radiation at top of the atmosphere of 140 W m−2 compared to the observed 220 W m−2. This larger difference (80 W m−2), is a direct consequence of the simplified atmospheric model which does not allow for absorption of solar radiation. Jin et al.  estimated the absorbed solar radiation under clear sky conditions in the atmosphere to be about 100 W m−2. The net radiation at the surface agrees well with the measured at the NP stations (not shown) with a net heat loss at the surface of about 20 W m−2 in the winter and a net heat gain of about 40 W m−2 in summer. In summary, the atmospheric model is able to reproduce the present state of the lower atmosphere, while processes in the upper atmosphere may not be properly modeled. In the present study we focus on the coupling of the ice/water surface, so we do not consider the difference in the upper atmosphere to be a serious error.
 The ice/water surface temperature is strongly coupled to the surface air temperature through the turbulent heat fluxes, resulting in a rather small temperature difference, of less than 1°C between the atmospheric ground temperature and the horizontal average surface temperature. Note that the temperature difference for thin ice categories is much larger in general. The tight temperature coupling between the lowest atmosphere and the ice surface is a well-known fact from observations, except for very calm conditions [Overland and Guest, 1991]. During the winter, the horizontally averaged turbulent heat flux is directed from the atmosphere to the surface, here defined as a positive value, and this acts as a net heat sink for the lower atmosphere of about 1 W m−2 (Figure 7a). The strong turbulent heat loss from open water and thin ice during winter is slightly exceeded by heat gain over the thick and therefore cold ice categories (Figure 7b), which is about 10 W m−2 and agrees well with values for perennial ice in the study by Maykut . During summer, the averaged turbulent heat flux is directed from the ice to the atmosphere and is about 1 W m−2 (Figure 7a). The surface temperature is 0°C over all ice categories during the melt season and the weak upward heat flux dominates over the downward directed heat flux over the colder open-water fraction (Figure 7b). It should be noticed that the present coupled system is not very sensitive to the choice of turbulent heat flux parameterization since the atmospheric ground temperature will adjust toward the ice surface temperature (see also section 3.2).
 The ocean mixed layer is about 16 m during summer with a salinity of about 30, and the temperature is 0.8°C above the freezing point. During autumn and winter, the mixed layer deepens due to convection/entrainment when the buoyancy flux is negative. The maximum mixed layer thickness is about 40 m. The winter salinity and temperature profile for the upper ocean produced by the model follows data from the Environmental Working Group (EWG) Atlas of the Arctic Ocean  (Figure 8). The solar radiation warms the upper ocean above the freezing point, which results in an oceanic heat flux of about 15 W m−2 during summer, and 0.8–1.3 W m−2 during winter. This winter heat flux is dominated by heat stored from the previous summer by absorption of solar radiation below the shallow summer mixed layer.
3.2. Steady State Sensitivity
 This section describes the steady state (meaning a cyclic repetitive state) response to changes in different quantities. We mainly analyze sensitivities to one quantity at a time but some selected combined sensitivities are given at the end of the section. A frequently used characteristic of the ice cover is the annual mean of the area averaged ice thickness defined as
where Hi and ai are the ice thickness and area fraction for ice category i, respectively, n is the number of ice categories, including open-water fraction, and τ is the number of time steps during one year. It should be noted that is the same as the annual mean ice volume divided by the area. Another important characteristic is the thermodynamic equilibrium ice thickness, Heq, which is the thickness an individual ice floe will reach if it remains long enough in the basin to reach thermodynamic equilibrium when the growth during winter equals the summer melting. Heq is generally different and larger than since the ice cover as a whole is not in thermodynamic equilibrium due to ice export. There are two important times scales that determines the shape of the ice thickness distribution. The thermodynamic adjustment timescale, Teq, for a flow to reach Heq, starting from zero thickness, and the residence timescale, Tres, for a typical flow in the basin. Tres can be defined as Ab/Aexp, where Ab is the total basin area and Aexp is the exported ice area per time unit. In general, there will be dominating ice categories occupying a large area fraction with thicknesses close to Heq when Teq ≪ Tres.
3.2.1. Sensitivity to Poleward Energy Flux in the Atmosphere
 The relation between and the annual mean of the poleward energy flux in the atmosphere, D, is shown in Figure 9a. The response is nearly linear for negative perturbations from the standard value (103 W m−2), with about 1 m decrease of ice thickness when decreasing D with 10 W m−2. The result in this range is similar to the model by TH92 (see Figure 10). For positive perturbations there is a dramatic increase in sensitivity starting at about 109 W m−2 and gives a knee like structure of the response curve. In this regime, the ice thickness decreases by about 1.5 m when D is increased by 3 W m−2. This behavior is fundamentally different from TH92 and is a result of the thickness distribution in combination with the nonlinear ice albedo function (1). The ice cover becomes generally thinner with increasing D but the shape of the distribution changes also and becomes more steep (see Figure 9b), comparing e.g., D = 90 W m−2 and D = 103 W m−2. This steepening results from the decrease of Heq with D and also shorter Teq, which at some point becomes less than Tres. When the steep part of the distribution is thinner than 2.3 m, where the bare ice albedo is a function of thickness, the albedo effect is magnified which results in a much stronger response. For D larger than 109 W m−2 more than 60% of the ice cover will be in the nonlinear regime of the albedo curve (Figure 9b) which results in a high sensitivity. At some point near D = 112 W m−2 the ice cover goes from a regime where first-year-level ice survives the summer and continues to grow during the following winter, to a regime with a seasonal ice cover when all ice at the end of the melting season is ridged ice and no level ice remains. The open-water fraction during the summer increases dramatically when passing this transition (Figure 9b) which is also manifested by a sharp bend of the response curve (Figure 9a). The curve becomes more linear again in the regime with no multiyear level ice (113 < D < 130) since is then mainly controlled by ice growth during one single winter season starting from zero thickness. A seasonal regime is not possible with the TH92 model. The mechanisms needed to obtain seasonal ice are discussed further below. Both in the present study and in TH92 the ocean becomes ice free during the whole year for D larger than 130 W m−2.
 As given by Bitz et al. , using a single ice category ice-atmosphere model, the ice thickness is even more sensitive to perturbations of D if these are concentrated to the summer period. Increasing D by 16% during June, July and August, which represents a 4% increase for the entire year, or an annual mean of 107 W m−2, gives of about 1 m and the open-water fraction at the end of the melting season is 85%.
 An extreme case with a low poleward energy flux of 75 W m−2 is characterized by net accumulation of snow, an ice thickness of about 6 m with an annual variation of less than 0.3 m (Figure 9a). The annual mean of downward longwave radiation is 30 W m−2 below the standard case value (208 W m−2) and the winter surface temperature is −40°C, compared to −30°C in the standard case. When the ice is thicker, a larger fraction of the imported fresh water from rivers and Bering Strait is exported as ice and less fresh water is stored in the water column, where the surface salinity is about 33.5 psu.
 A case with high D of 115 W m−2 is characterized by a mostly seasonal ice cover with a maximum area averaged ice thickness of about 2.0 m at the end of winter and the open-water fraction is about 90% at the end of melting season (Figure 9b). The ice melt season is increased by 1 month, to the end of September, since the large open-water fraction absorbs much more solar radiation than in the standard case which gives a rapid warming of the mixed layer. This is followed by 1-month cooling period and the ice production does not start until late October. The winter surface temperature is about −20°C and the highest summer mixed layer temperature is −0.4°C.
Figure 10 shows in addition to the response curve by TH92 some discrete points from the study by Bitz et al.  and the results when the gray body atmospheric part of our model is replaced with a more detailed model of the atmospheric radiative transfer, including clouds at three different levels with cloud statistics according to Table 3, and also a more detailed description of the surface albedo. The ice thickness from Bitz et al.  is generally smaller compared to our standard case and the slope inferred from these points is larger. The larger sensitivity is mainly explained by that a slab ice cover is more sensitive than the inhomogeneous ice cover with thin and thick ice, see also Figure 14a. The more detailed atmospheric model gives, however, a similar response curve as in the standard case model except for the seasonal ice regime where the response is more flat. This implies that the overall response of the ice thickness to the poleward energy flux in the atmosphere do not to rely on a detailed radiative transfer scheme in the atmosphere. A similar conclusion was also made by Bitz et al. . It should be noticed that both the gray body and the detailed model works under the restriction of an prescribed annual cycle of cloudiness, which is used as a tuning quantity. The ice cover is extremely sensitive to the choice of cloud cover and cloud height using the detailed radiative transfer scheme and we consider this model less useful in the present context since one ends up with a larger number of tunable quantities than in the simpler model. Another aspect is that the detailed model is very impractical to use in the present type of sensitivity studies since it is very time consuming.
Table 3. Description of Clouds in the CRM Simulation in Figure 10
Frequency of Occurrence, %
Amount When Present, %
Water Path, g m−2
Probability of low clouds is 68% from 20 May to 15 September, and 19% from 1 November to 1 May: transitions between these two periods are linear with time.
Water paths of low clouds are 30 from 20 May to 15 September, and 15 from 1 November to 1 May: transitions between these two periods are linear with time.
 Changing the total optical thickness, N, gives a response very much similar to changing D, expressed in terms of percent. The stepwise behavior in occurs when both D and N have increased by 6%, and a decrease of about 20% results in net accumulation of snow. The similarity of the response for N and D is explained by the fact that the relative change in N or D has similar effect on the surface temperature in the atmosphere model, which can be shown from equation (B9). It is possible to find the same result as in the standard case by both perturbating the annual cycles of D and N, e.g., increase D with 5% and decrease N with 4%. One should, however, be cautious when interpreting the response on N in terms of cloudiness since not only the amount of clouds but also cloud level is important. Beesley  shows that increasing the amount of low level clouds increases the equilibrium thickness while increasing mid level and high clouds have the opposite effect.
3.2.3. Sensitivity to Divergence and Ridging
 The divergence of ice, e, which is closely related to the ice export, has a large influence on the conditions in the Arctic Ocean. It represents a direct sink of ice, which must be compensated by a net ice growth in the basin each year and the ice cover as a whole is therefore not in thermodynamic equilibrium with the atmosphere. It also represents an outflow of equivalent freshwater, which is of the same magnitude as the river inflow. The shape of the ice thickness distribution is largely dependent on the divergence since it promotes opening of leads where new ice can form during winter. Excluding divergence entirely (e = 0) gives m and e = 0.05 yr−1 gives For divergence above e = 0.05 yr−1, decreases in an exponential fashion and is about 2 m when e = 0.3 yr−1 (Figure 11). A strong response on ice divergence has also been found by several other studies [e.g., Holland and Curry, 1999]. The exported ice volume is more constant, however, since increased areal export is nearly compensated by decreasing ice thickness (Figure 11). The freshwater budget is thus not very much affected by the areal export. The amount of ridging has little effect on (not shown) which is a result of a somewhat subtle effect on the thickness distribution in combination with absorption of solar radiation. Increasing ridging results not only in more thick ice but also larger area fraction of thin ice. When the thin ice melts completely during the summer, it results in more absorption of solar radiation and in turn a larger overall melting. The increase in ice production due to formation of open-water areas by ridging in winter is nearly balanced by increased ice melt during the summer (see Björk  for a comprehensive discussion). A two-fold increase from the standard case of ridging activity (r = 0.4 yr−1 to r = 0.8 yr−1) increases from 3.3 to 3.5 m, and if the ridging process is removed entirely (r = 0) it has even less effect. The shape of the thickness distribution differs significantly between the different cases. With no ridging activity the ice thickness distribution is nearly linear and the thickest category is about 6 m, see also Figure 13b. For the case with large ridging activity, the thickness distribution is dominated by ice categories about 3 m thick and the area fraction of thick ice (H > 4 m) increases with about 30% compared to the standard case. Increasing the thickness multiplier from 6 in the standard case to 15, which is the standard case value as given by Holland and Curry , increases from 3.3 to 3.5 m. A rather small change for such large perturbation.
3.2.4. Sensitivity to the Turbulent Heat Flux Parameterization
 Different types of parameterizations affect mainly the adjustment timescale and extent of the boundary layer. As long as the timescale is shorter than the seasonal changes of forcing and the boundary layer is much thinner than the whole atmosphere it really does not matter. Two tests have been made to investigate the sensitivity to the formulation of the turbulent heat fluxes. Increasing the vertical e-folding scale, ζ, by a factor six, increases by only 1% and the net surface heat flux changes by less than 2%. The small difference of ζ can also be deduced directly from the equation for the downward longwave radiation at the surface (equation (A16)). When the latent heat flux is increased from 20% of the sensible heat flux in the standard case parameterization (see equation (A3)) to 200%, increases from 3.3 to 3.4 m. These tests show that the model is not sensitive to the parameterization of the turbulent heat fluxes.
3.2.5. Combined Sensitivity Study
 Here the knee-like behavior near D = 109 W m−2 in Figure 9a is studied, with special emphasis to find how robust this behavior is to changes of different parameters and forcing in the model. We have focused on the albedo parameterization, ridging activity, and ice export since these are the elements of the model most heavily involved in shaping the ice thickness distribution and thus interacting with the absorption of solar radiation. The analysis comprises seven different cases, which are presented in Table 4. In case 1, the albedo increases linearly with the ice thickness for ice thinner than 1.2 m instead of a nonlinear increase up to 2.3 m. Case 2 is for a constant ice albedo, but the snow and open-water albedo are held at the standard case values. The ridging activity is changed in case 3 and 4. Case 5 and 6 shows the response for different values of the annual mean divergence. Case 7 is a slab version of the model where only one ice category is allowed. This case is included in order to show how the behavior depends upon having a resolved ice thickness distribution or not. The model has been tuned towards the standard ice thickness of 3.3 m for D = 103 W m−2 using N as a tuning quantity. The required tuning is small for cases 1–4. Changing the divergence (case 5 and 6), and the slab version (case 7) makes it necessary with somewhat larger tuning in order to keep the standard case ice thickness.
Table 4. Different Cases in the Combined Sensitivity Studya
Change From Standard
Tuning N, %
The last column shows the required tuning of the total optical thickness, N, in order to keep the ice thickness, , at the standard case value. See Table 1, Table 2 and text for definitions of quantities and standard case values. In the slab ice version an artificial heat source of 2 W m−2 is added to the mixed layer, simulating absorbed solar radiation in leads.
αice = min(0.08 + 0.5H, 0.64)
αice = 0.64
r = 0.8 yr−1
r = 0.0 yr−1
e = 0.22 yr−1
e = 0.07 yr−1
 The albedo parameterization has a large impact on the relationship between and D. In the linear albedo case, the ice thickness decreases linearly with D up to 115 W m−2, compared to 109 W m−2 in the standard case (Figure 12), which means that the ice cover can sustain significantly larger poleward energy flux before the transition to seasonal ice occurs. The major change in ice thickness occurs when a large area fraction (about 70%) enters the interval H < 1.2 m where the albedo increases with thickness. In the constant albedo case, the thickness decreases linearly with D for the entire range and there is no distinct transition at all.
 Changing quantities associated with deformation processes, cases 3–7, result in different shapes of the ice thickness distribution and therefore a different dependence on D. The ridging activity affects the thickness distribution by producing thick ice but also by increasing the amount of thin ice which in turn interacts with the ice albedo. A large ridging activity (case 3) results in a slightly larger area fraction of thin ice (H < 2.3 m) and the knee-like structure is therefore reached at a somewhat lower D-value (Figure 13a). Excluding the ridging (case 4) decreases the amount of thin ice and open water during summer which also reduce the absorption of solar radiation. The oceanic heat flux is then generally smaller which implies a larger Heq and Teq. The ice thickness distribution is then nearly linear (Figure 13b). D has to be above about 115 W m−2 before the larger part of the ice cover is within the nonlinear regime, compared to 109 W m−2 in the standard case. This shows that the ridging process in general has a rather weak effect in this context, but it acts to make the ice cover more sensitive to positive perturbations in D.
 The residence time of the ice, Tres, changes with the divergence, which in turn affects the ice thickness distribution. When Tres is much longer than Teq, the ice cover behaves more like in a closed basin. Ice both thinner and thicker than Heq will have enough time to adjust towards Heq and form dominating ice categories near Heq. Conversely when Tres < Teq, the distribution will be more linear with no dominant ice categories around Heq. These features are evident in cases 5 and 6. In case 5 (with high divergence), D can be increased to higher values compared to the standard case, about 118 W m−2, before a transition to seasonal ice occurs (Figure 14a). The reason for this is that Tres is shorter than Teq over a large range in D, which gives a linear shape of the thickness distribution (Figure 14b). Tres and Teq are about 5 years and 7 years, respectively, for D = 103 W m−2. In the vicinity of D = 115 W m−2, Teq becomes shorter than Tres and ice categories near Heq (=2.6 m) cover a large area fraction. In case 6 (with low divergence), Tres is 15 years for D = 103 W m−2, which is twice as long as Teq. A large area fraction is close to Heq and just outside the nonlinear albedo regime (Figure 14c). A small positive perturbation in D decreases the albedo for this dominating ice category, and the ice cover enters the seasonal regime for D = 110 W m−2. Using a single ice category (case 7) instead of a thickness distribution makes the model relatively similar to TH92. Since no open water is present, the oceanic heat flux will be zero and has to be introduced artificially in order to achieve a realistic ice thickness. This case shows a stepwise transition to a seasonal ice cover when D becomes larger than a critical value when the ice becomes so thin that it enters the nonlinear albedo regime. The response curve for case 7 is similar to case 6 (with small divergence) also having a stepwise transition to the seasonal regime. Using a single category may be seen as an extreme case of the full thickness distribution with very low divergence and ridging and has therefore a similar response as in case 6. A seasonal ice cover is realized in case 7 in contrast to the TH92 model in which no seasonal ice was possible. The reason for this is that it takes longer time to cool the mixed layer to the freezing point from an initially colder and deeper mixed layer compared to a warmer and shallower, even if both cases have the same heat content initially. No ice will form if the time to cool the mixed layer to the freezing point is longer than the winter period with net negative heat flux. The seasonal ice cover in case 7 diminishes completely if the mixed layer is increased artificially by doubling the wind speed. The existence of a seasonal ice cover is thus dependent on a relatively shallow mixed layer, which in our case is about 20 m as a yearly average, compared to 50 m in TH92.
3.2.6. Multiple Steady States
 In the present model and also in TH92 it is possible to achieve two different steady states, one completely ice free and one with perennial ice, for identical forcing. The existence of such multiple steady states are related to the cooling timescale as described in the previous section. If all ice is removed at the beginning of summer in the standard case, the ocean becomes so warm that it won't freeze during the following winter. A large reduction of the atmospheric poleward energy flux, D, to about 80 W m−2 from the standard case value 103 W m−2 is then needed in order to come back to a state with perennial ice. The transition from the state with no ice to a state with perennial ice is wind dependent and it occurs when D = 95 W m−2 for a 50% wind reduction and when D = 70 W m−2 for 50% wind increase compared to the standard case.
3.3. Time-Dependent Response
 A full treatment of the time-dependent response for different frequencies and type of forcing is beyond the scope of the present work but we show some aspects here and make also comparisons with the results from other model studies of the time-dependent characteristics.
3.3.1. Adjustment Timescale
 The adjustment timescale (e-folding) to reach a new cyclic repetitive steady state when D is perturbated from the standard case value is dependent on the amplitude and sign of the perturbation (Figure 15). The adjustment timescale for negative and small positive perturbations is 6 years. For larger positive perturbations of about 6–10 W m−2 (in the knee like region in Figure 9a), the timescale is up to 20 years. The ice must then change to the seasonal regime and become much thinner in order to reach a new equilibrium, which takes longer time. Larger positive perturbations (D > 113 W m−2) melt the perennial ice cover more rapidly and the adjustment time decreases.
 The timescale for negative and small positive perturbations agrees well with the 6-year timescale reported by Bitz et al. , for the case with ice export, using a coupled ice-atmosphere model with a highly resolved temperature profile in the ice but no thickness distribution. In the study by Holland and Curry , using an ice thickness distribution model and several layers in the ice, but forced with fluctuations in the air temperature instead of D, the timescale is 6.5 years.
 The thickness distribution has a large effect on the response time. The adjustment timescale is generally longer in the single category case (case 7). It is about 13 years for perturbations of D near the standard value and has a very large peak value for positive perturbations just large enough to force the ice into the seasonal regime. The longer timescale is mainly due to the nonexisting thin ice categories that are an important part of the thickness distribution. Thin ice categories are able to adjust rapidly to new forcing conditions and thereby change the open-water fraction and also the absorption of solar radiation.
 For the standard case values in TH92 the adjustment timescale is 3 years. It is hard to trace exactly why this model gives shorter timescale compared to the more sophisticated models since the approaches are very different. One critical difference might be that the TH92 model does not include any effects of brine pockets in the ice. The ice will generally have more rapid temperature adjustment without internal phase changes. The adjustment timescale is reduced to about 4 years in the present model if the brine effect is removed (with full thickness distribution).
3.3.2. Stochastical Forcing of Poleward Energy Flux
 A stochastic variation, based on the observed statistics, is now added to the annual cycle of the poleward energy flux. A characteristic timescale of duration of 5 days is used, simulating the low-pressure systems entering the Arctic. The standard deviation is 33 W m−2 and 18 W m−2 during winter and summer, respectively. The standard deviation is linearly interpolated between the winter and summer values during spring and autumn (see Bitz et al.  for details). In a 1000-year simulation the variance of the ice thickness is 0.068 m2, which agrees well with the 0.08-m2 variance in the study by Holland and Curry  when forcing a model, including a thickness distribution, with stochastic fluctuations of the air temperature. It is interesting to compare with Holland and Curry  since the forcing by air temperature fluctuations represents an internal quantity of the system while the forcing by D in this study is an external quantity. Figure 16 shows the observed variance of temperature fluctuations from the NP stations compared with the model computed variance using the stochastic D-forcing. Although there are some discrepancies between the model and observations, most pronounced is the peak in variance that occurs in November from the model but in January from data, it shows that the model is able to generate realistic temperature fluctuations at the surface in the central parts of the Arctic Ocean from fluctuations of D at the border. The similarity between the observed temperature fluctuations (which form the basis of the stochastic forcing by Holland and Curry  and the temperature fluctuations in our study implies that the ice cover in both studies has been exposed to similar forcing. It is not then surprising that the variance of the ice thickness is about equal since the description of the ice cover is similar in both models.
 The standard deviation is much larger (0.75 m2) in the study by Bitz et al. , using the same stochastic D forcing but with a single category ice model. The variance of our model using a single ice category (0.066 m2) is almost equal to the case with full thickness distribution (0.068 m2). An explanation to this similarity is that the longer adjustment timescale in the slab version is compensated by the strong stepwise steady state response on D (Figure 14a). Using a single ice category but changing the albedo parameterization to the one by Bitz et al. , where the main difference is that the snow albedo is dependent on snow thickness, results in a much larger variance (1.1 m2) of which a large part is caused by snow accumulation during some periods.
 We have shown some of the time-dependent properties of a coupled ocean-ice-atmosphere model, including a thickness distribution. A full treatment of the time-dependent behavior is, however, a complex subject and needs a much more extensive analyze in order to obtain a full understanding. One of the things adding to the complexity is that the problem includes at least three important timescales: the equilibrium timescale for individual floes, Teq, the residence time, Tres, and the adjustment timescale of the ice cover as a whole. The residence time, for example, controls the memory of the ice cover such that most ice that has been exposed to extreme forcing during a particular year and become much thinner or thicker will leave the area within the residence time.
4. Discussion and Future Outlook
 As we have shown here it is possible to obtain a realistic description of the upper ocean, ice cover, and lower atmosphere using a rather simplified ocean-ice-atmosphere model which is tuned by an optical thickness parameter. The model gives similar results as the more simplified TH92 model in some aspects but there are also important differences. The main finding is that the ice cover is much more sensitive to the poleward energy flux, D, with a drastic decrease of the ice thickness when increasing D above a critical value. This decrease is explained by the thickness-dependent ice albedo in combination with the slope of the thickness distribution. The result is thus sensitive to different albedo parameterizations. An albedo formulation given by physical processes is preferred instead of an empirical formulation. The albedo is one of the more complicated quantities in the ice-atmosphere system and involves numerous surface characteristics such as snow temperature, snow age, melt pond fraction, and frost flowers. The surface albedo is also highly dependent on the amount and timing of snowfall, which in turn is generated by complicated atmospheric processes. It is beyond the scope of the present work to investigate all possible aspects of the surface albedo and we leave this to future work. The standard case albedo parameterization should, however, capture the basic and well known characteristics such as high snow albedo, and a decrease with thickness for bare ice [Ebert and Curry, 1993].
 It is shown that the ice thickness decreases with the optical thickness, N, in the same way as for D. These two quantities are in reality likely to be connected such that larger total energy flux will include more latent heat flux, which in turn will increase the moisture content and generate more clouds in the Arctic. Such effects are not included in the present study. An interesting development might be to include the moisture budget in the atmospheric model and a parameterization of cloud processes. It will then be possible to compute the precipitation instead of prescribing it. Such processes might need a more sophisticated atmospheric model. However, the present very simplified atmospheric model seems appropriate at this stage.
 Although it might be necessary to have very detailed formulations of many processes in the system and use GCM:s in order to achieve realistic climate prediction, we think that the present results are encouraging for future developments of realistic simplified models. Such models may contribute to a better understanding of the overall functioning of the system but not necessarily be able to reproduce all details. One such simplification that can be explored further is that the parameterization of the turbulent heat fluxes is not very critical in a fully coupled system if the thermal adjustment timescale in the turbulent boundary layer is short compared with the forcing timescale.
 The Arctic Ocean ice cover seems, according to this model study, very sensitive to variations in the poleward energy flux in the atmosphere, D. A positive perturbation of only 9 W m−2 from the standard case value of 103 W m−2 results in a decrease of the mean ice thickness of more than 50%. If this sensitivity is real and noting that the poleward energy flux is a naturally fluctuating quantity it implies that a strong feed-back mechanism is at work since we otherwise would have some indications of periods with a very much reduced ice cover, which is not the case. The Arctic Ocean ice cover seems to be a rather stable feature judging from geological records. Kneis et al. , for example, used a multi proxy approach to show that there has been a permanent ice cover during the last 150 000 years in the eastern part of the Eurasian basin.
 A very potent stabilizing mechanism is that D seems to be tightly coupled to the meridional temperature gradient in the atmosphere such that generally thinner ice cover, which is associated with higher surface temperatures, decreases the temperature contrast between the Arctic and mid latitudes and reduce the poleward energy flux and vice versa. This mechanism was already introduced by TH92 and has been studied further with connection with cloud formation by Beesley , but we think it merits further investigation. The dependence between the meridional temperature gradient and the poleward energy flux is a very robust feature and empirical relations with small scatter can be established [Stone and Miller, 1980]. These relations also show a clear annual cycle with less transport during summer when the temperature gradient is reduced and larger transports during winter. Such an annual cycle is also evident in the data of the poleward energy flux at 70°N (Table 1).
 In the light of the high sensitivity to D it is not surprising that changes of the ice thickness as reported lately by Rothrock et al.  indeed occur. The ice cover should have a rather large natural variation since the poleward energy flux is a naturally fluctuating quantity with an interannual variability of about 10 W m−2 [Overland and Turet, 1994]. The quite long adjustment timescale of about 6 years due to the thermal inertia in the ice suggest that some filtering occurs such that changes with timescales of just 1 or 2 years will be small. This is, however, not the only possible response in a system with nonlinear effects. Bitz et al.  showed that a significant low-frequency variability in the ice thickness remains after removing all low-frequency forcing of the stochastically produced time series of D.
 Another interesting aspect is how ice thickness changes are related to the observed dominant modes of variability in the atmospheric circulation pattern such as the Arctic Oscillation (AO) [Thompson and Wallace, 1998]. There seems not to be any strong relation between the AO and the atmospheric poleward energy flux [Adams et al., 2000] but the ice drift pattern is markedly different for different phases of the AO. During the latest positive AO phase, there has generally been a large ice drift from the Siberian side of the Basin towards Canada [Polyakov and Johnson, 2000]. This ice drift pattern seems also associated with an increase of the surface air temperature [Rigor et al., 2002] and a decrease of the ice thickness [Rothrock et al., 1999]. Such effects cannot be reproduced with the present basin scale model domain, but may be studied with a more regional column model in which the ice divergence is increased locally.
 We have demonstrated the following:
The model ice cover is very sensitive to positive perturbations of the poleward atmospheric energy flux, D. Increasing D by about 9 W m−2 from a standard value of 103 W m−2 reduces the ice thickness by more than 2 m and generates a large open-water fraction during summer.
The steady state and time-dependent response of the ice thickness on D is very much dependent on the albedo parameterization. The shape of the thickness distribution as controlled by divergence and ridging is also important.
A seasonal ice cover is realized for moderately large values of D in contrast to the model study by Thorndike  where this type of regime was unstable. The existence of the seasonal regime is dependent on the thickness of the ocean mixed layer.
The timescale to adjust the ice thickness to a sudden change of D from the standard case value is highly dependent on the sign and magnitude of the perturbation. Negative and small positive perturbations (<5 W m−2) give an adjustment timescale of about 6 years while the adjustment to larger positive perturbations, around 10 W m−2, may take up to 20 years.
The relatively simplistic approach for the atmospheric part of the present model seems not to be critical for the general results of the study. Using a more detailed atmospheric model gives similar results.
Appendix A:: The Atmosphere Model
 The atmosphere is described with a gray atmospheric model that is transparent to solar radiation [Thorndike, 1992]. The vertical coordinate is given in optical height, η, which describes the amount of absorbers in the longwave spectrum over a certain distance
The total optical depth, N, is defined as the optical height at top of the atmosphere, η(∞). In the present model formulation the turbulent heat flux to the surface is included.
where Ts is the surface temperature, Ta(0) is the ground air temperature, and CT is a parameter which depends on wind speed, W, the turbulent transfer coefficient for sensible heat, Cs, the density of air, ρa, and heat capacity of air, cpa.
The factor 1.2 accounts for the latent heat flux such that the turbulent heat flux is 1.2 times the sensible. This simplification is motivated by that the latent heat flux is expected to be weak throughout the year. During the Arctic winter, the cold atmosphere absorbs a small amount of water vapor and during the summer both the surface and ground air temperature are close to the melting point which results in a small difference in the specific humidity, thus weak evaporation/sublimation. The effect of the turbulent heat flux is distributed exponentially in the lowest atmosphere at the scale ζ. The heat supply or removal of heat in the atmosphere, FT, due to the surface turbulent heat flux is then
The poleward energy flux from the lower latitudes, D, is spread evenly throughout the atmosphere. The atmosphere is assumed to be in thermal equilibrium, which means that the absorbed longwave radiation, heat supply by D and FT is balanced by up- and downward thermal radiation in an atmospheric layer, dη.
where, FUP and FDN, are the longwave upward and downward radiation, respectively. These fluxes vary with height due to absorption and reradiation, R.
Two boundary conditions for the differential equations above are needed. The upward longwave radiation at the surface is given by a linearization of the Stefan-Bolzman law near the freezing point. The downward radiation is zero at the top of atmosphere
The solutions for the longwave radiation and reradiation in the atmosphere are
which is equivalent to the solution by Thorndike  when LFD = LFU = LR = 0. The turbulent heat flux at the surface is expected to affect the lower part of the atmosphere, thus N/ζ ≫ 1, which means that the terms e−N/ζ can be neglected. The turbulent coefficients LFD(η), LFU(η), and LR(η) are then
At the bottom of the atmosphere the downward longwave radiation is
An expression for Ta(0) as a function of Ts can be found by using the relation
Appendix B: Coupling at the Surface
 The surface is coupled to the atmosphere by using the radiation balance at the surface. The upward conductive heat flux from the ice to the surface is balanced by the net heat flux at the surface.
where T is the internal ice temperature or ocean temperature depending on ice thickness, Fsw is the incoming solar radiation, α is the albedo, the third term is the upward longwave radiation (A8), and the last term is the turbulent heat flux from the surface (A2). For thin ice (H < 0.25 m) the conductive heat flux is assumed to be constant through the ice and T is therefore the mixed layer temperature. For thicker ice, the temperature gradient is different between the upper and lower part of the ice and T is the internal temperature. CC is a coefficient given by conductivity, ice and snow thickness.
where ksnow and kt are the thermal conductivity for snow and sea ice, respectively, H and h are the ice and snow thickness, respectively, and Γ is a step function defined as
The conductivity depends the ice salinity in order to include effects of brine pockets.
where kice is the conductivity for pure ice, β is an empirical constant, Sice is the bulk salinity of the ice, and Tf is the freezing temperature (see Björk  for details). In the present model formulation there are several ice categories with different thicknesses and internal temperatures, each one of them occupying a certain area fraction. For every ice category surface heat balance (B1) must be satisfied and the individual surface temperature, Tsi, is adjusted to
where the upper index, i, denotes the ice category. The area averaged surface temperature is given by summation over the ice categories, including the open-water fraction, a0.
where Tm is the ocean mixed layer temperature, ai is the area fraction for category i, and n is the total number of ice categories. We assume that the different heat fluxes at the surface are effectively smoothed out in the atmosphere by advection so that the atmosphere only “feels” the area averaged surface temperature. The relation between the longwave radiation, air temperature, and the mean surface temperature is derived by inserting (B4) into (B5).
Inserting (A16), and (A17) into (B6), gives the mean horizontal surface temperature for the atmosphere in thermal equilibrium
The mean surface temperature is analytically computed from the ice thickness distribution, the incoming solar radiation, D, and N. Equation (B9) shows the direct relation between the processes within in the atmospheric column and the surface. The air temperature and the longwave radiation are given by equations (A17) and (A16), respectively, by inserting the computed average surface temperature. The surface temperature for the different ice categories are computed from the radiation balance (B4). An ice category is melting if the surface temperature for that particular ice category is greater than the melting point. These surface temperatures are set to 0°C in (B5), which means that 〈T0〉 and 〈C0〉 changes. It should be mentioned that ε1 and ε2 can be neglected without any serious error since the turbulent heat flux only affects the lower part of the atmosphere (ζ/N ≪ 1). ε1 and ε2 are kept in the model computations in order to check the heat conservation.
 The authors thank P. Winsor whose comments improved the manuscript and also for processing ice draft data from the submarine cruises. We also want to thank G. Walin for valuable discussions and guidance throughout the work. Financial support was given by the Swedish Natural Science Research Council (NFR) under grant G 650-19981511/2000.