3.1. Control Simulation
 A series of experiments are presented beginning with a control simulation representing forcing conditions from the SHEBA melt season. As stated in the introduction, we assume that the ice is porous throughout the experiments, with pond levels equaling the local sea level. In the SHEBA experiment, ponds began forming around mid-June as surface snow melted. Pond fractional coverage for the albedo line increased to about 28% and then decreased in the latter half of June as the ice became permeable. Pond drainage continued until late June when the pond fraction reached ∼12–15% on 26 June. Thereafter, pond coverage on the albedo line increased gradually to a maximum of ∼38% in early August. Pond coverage in the region around the SHEBA camp estimated from aerial photographs ranged from ∼14% at the end of June to ∼23% in the beginning of August [Perovich et al., 2002a]. We assume that the late June minimum in pond fractional area represents the time when pond levels reached equilibrium with the local sea level, hence our choice of the 30 June image for initializing the model.
 Model forcing for the SHEBA control case is based on observations of the temperature and specific humidity taken between 27 June 1998 and 6 August 1998 representing the time period between the beginning of pond drainage and the maximum pond coverage in August. Direct observations of heat and moisture flux from ponds, for example by eddy covariance methods, were not made during the SHEBA field program. As a substitute, we employ flux estimates based on simple bulk formula for sensible and latent heat [see Stull, 1988, p. 262]
where ρa = 1.29 kg m−3 is the air density, Cp = 4000 J K−1 kg−1 is the air heat capacity, CH = 1.5 × 10−3 is the transfer coefficient for heat and moisture, U is the wind speed, Tr is the measured atmospheric reference temperature from 2.5 m height, Tp is the pond temperature, Lv = 2.5 × 106 J kg−1 is the latent heat of vaporization for water, qs is the saturation specific humidity and qr is the measured specific humidity from 2.5 m height. Although more sophisticated methods for estimating CH are available, tests of the pond model suggest that results are relatively insensitive to this parameter. Downwelling longwave fluxes were measured during SHEBA. We estimate upwelling longwave flux using
where the emissivity, ɛ = 0.97, and σ = 5.67 × 10−8 W m−2 K−4 is the Stefan-Boltzmann constant.
 Plots of the radiative and surface heat flux terms using (16) and (17) are shown in Figure 7. Short wave fluxes during the 40 day period ranged from daily high values of about 500 W m−2 to evening minimum values of ∼50 W m−2. Net long wave flux was typically a function of cloud cover, with clear conditions generating surface cooling of 50–60 W m−2. Cloud cover often produced a downward flux with surface warming between 10 and 20 W m−2. Both sensible and latent heat varied over the time of record with magnitudes frequently less than 10 W m−2, except in late July/early August when conditions began cooling with the onset of autumn.
Figure 7. Flux of (a) shortwave radiation, (b) longwave radiation, (c) sensible heat flux, and (d) latent heat flux for the SHEBA experiment for the 40 day period starting on 27 June 1998 (day 178). Integrated solar flux is 210 W m−2; combined longwave, sensible, and latent fluxes yield an integrated value of −8.2 W m−2.
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 Background ice melting rates for the control experiment were set according to SHEBA ice mass measurements, which indicated surface melting rates of ∼1.5 cm d−1 and ice bottom melting rates of ∼0.5 cm d−1 between late June and early August [Perovich et al., 2003]. Ice surface melting is only performed in regions not covered by a pond, whereas the bottom melting is uniformly applied.
 Two key parameters needed to simulate the pond heat flux and melting rates accurately are the pond bottom and sidewall friction velocities, u* and v*, respectively. Measurement of these parameters in actual ponds has not been conducted and only limited, idealized large eddy simulation results exist for estimating pond friction velocities [see Skyllingstad and Paulson, 2007]. In the absence of measurements, we treat u* and v* as adjustable parameters and perform a series of pond sensitivity experiments to determine values for friction velocity that yield pond fractional area and pond depths similar to observations from the SHEBA albedo line presented in Figure 2. Figure 8 shows contour plots of pond fractional area and pond depth for a range of u* and v* between 0.001 and 0.016, calculated for increments of 0.001, and using the 40 day forcing described above. In general, results show that increasing friction velocity leads to increased pond bottom or wall flux. Variations between u* and v* lead to counteracting changes in the pond depth and fractional area; large fractional area coincides with shallow ponds, whereas small fractional area corresponds with deep ponds. For our control case, we selected values of u* and v* (0.005 m s−1 and 0.014 m s−1, respectively) that yield a pond fractional area similar to observations from SHEBA aircraft surveys and pond depths consistent with the SHEBA albedo line shown in Figure 2. As shown by Figure 8, our selection is somewhat arbitrary given that a range of u* and v* will yield similar fractional area and pond depth.
Figure 8. Contours of (a) pond fraction and (b) pond depth as a function of pond bottom friction velocity, u*, and pond sidewall friction velocity, v*, after 40 days. Forcing is the standard SHEBA control case.
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 Pond depth for the control simulation at 40 days is shown in Figure 9 for comparison to the initial state shown in Figure 6. Overall, pond depths and sizes increase in the simulation with typical initial pond depths of ∼0.15–0.2 m increasing to ∼0.4–0.6 m. Many of the ponds merge to form large, complex features by the end of the simulation, agreeing with aerial observations [Perovich et al., 2002b]. We do not produce new ponds in the simulation because of the imposed, flat ice surface, which is in contrast to observations that show small pond formation from bare ice. Consequently, simulated ice fractional area is likely to underestimate actual conditions.
 A sample cross section from the model at x = 200 m (Figure 10) shows the pond characteristics at three different times in the simulation. Comparison of this plot with Figure 2 provides a measure of the ability of the model to simulate pond attributes assuming pond water clarity equivalent to fresh water capped leads. Many of the modeled ponds show deepening from about 0.1 m to 0.4 m over the ∼40 day time period, in comparison to observed depths of about 0.45 m over the same time period. Lateral growth of the simulated ponds also appears to underestimate observed pond behavior with the model tending to maintain pond widths similar to the initial state. As mentioned above, this aspect of the simulation is produced by the flat ice surface assumption. Actual ponds grow from both sidewall melting and flooding of surrounding low ice surfaces, due to decreasing ice thickness and ice draft. Deeper ponds are evident in the modeled case, but these are for ponds that are initialized with a greater depth than the observed cases. Both observed and modeled ponds that begin in close proximity tend to merge and form larger ponds, for example as shown by the pond between 60 and 70 m in the model, and near 175 m in the observations.
 Average simulated pond temperature (not shown) ranges from ∼0.6°C at the beginning of the simulation to ∼0.35°C after 40 days, decreasing as the solar forcing is reduced. These temperatures are in the same range as observed values (T. Grenfell, personal communication, 2006) and results from large eddy simulation of ponds [Skyllingstad and Paulson, 2007].
 Plots of the horizontally averaged pond fractional area, depth, albedo, and total ice surface albedo (assuming a bare ice albedo of 0.6) are shown in Figure 11 for the control simulation with constant pond bottom albedo of 0.5 and with variable pond bottom albedo decreasing linearly from 0.5 to 0.3 over the 40 day simulation. The variable pond bottom albedo is based on the observation that ice under ponds generally darkens because air bubbles escape from the ice, and the ice often thins more rapidly under ponds because of more rapid bottom melting. Overall, pond fraction increases at a nearly constant rate from an initial minimum of 0.1 to a final value of 0.28. Pond fraction is nearly the same in both the constant and variable bottom albedo cases, suggesting that pond bottom albedo is secondary in determining the melting rate of sea ice by pond water. Average pond depth is initially about 0.12 m and increases to a stable value of about 0.28 m at 30 days. As with the fractional area, pond bottom albedo has only a small influence on the pond depth. Additional heat from reflected solar radiation off the pond bottom in the constant albedo case produces a slight increase in pond fraction and depth.
Figure 11. Control simulation average (a) pond fraction, (b) pond albedo, (c) total albedo (assuming ice albedo of 0.6), and (d) pond depth (m) for variable pond bottom albedo (solid) and constant pond bottom albedo (dashed).
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 In the constant bottom albedo case, pond albedo gradually decreases from 0.29 to a near constant value of 0.24 on about day 30, representing a total change of about 0.05. Pond albedo in the variable case basically follows the linear decrease forced by the assigned variable bottom albedo combined with the depth dependent albedo change, yielding a final pond albedo of ∼0.15, and a total pond albedo change of ∼0.14. Although pond bottom albedo is not critical for pond growth, it is important to note that solar radiation that penetrates pond bottoms (low albedo) is absorbed in the underlying ice and ocean boundary layer. Ultimately this heat can go toward increasing the ice bottom melting rates.
 Albedo of the total ice surface is a function of both the individual pond albedo and the pond areal coverage. As pond fraction increases, the total albedo decreases. Consequently, plots of the average total albedo (Figure 11c) show a more pronounced influence of the pond lateral growth in comparison to the depth-dependent pond albedo. For example, in the constant bottom albedo case, the total albedo decreases from 0.57 to almost 0.5 for a total change of about 0.07. This compares with an average pond albedo change of 0.05, indicating the important role that pond coverage has in setting the ice surface albedo. Results from the variable bottom albedo case indicate a total albedo decrease of ∼0.1, which is less than the change in the average pond albedo. The effects of decreasing pond albedo in this case are reduced because of the averaging with ice albedo of 0.6.
 These results indicate that the effects of pond depth on pond albedo are not always the main aspect of ponds that affect the ice surface albedo. While the pond albedo reaches a steady value around day 30 in the constant bottom albedo case, total albedo continues to decrease because of the lateral melting of ponds and increased pond area. Consequently, the expansion of ponds is a key factor in the decrease of total ice surface albedo.
 Comparison of our results with observations from Perovich et al. [2002a] suggest that the pond model underestimates pond and total ice albedo for the variable pond bottom albedo case. For example, observed pond albedo for “dark” ponds in the SHEBA albedo line decreased from ∼0.4 to ∼0.1 during the July time period. Total albedo observed from aircraft indicated a decrease from ∼0.55 to ∼0.44 [Perovich et al., 2002b], however these estimates include the substantial increase in lead coverage in the beginning of August. Pond fraction estimates varied significantly in the observations with the albedo line showing an increase from about 15% to 38%, whereas aerial estimates suggest an increase from about 15% in late June to 24% in early August, which is more in line with the pond model results. In our simulations, the initial ice fraction is slightly lower because of the image selected for the initial condition. However, we predict an increase in coverage of about 19%, which falls between the observed albedo line increase and aircraft estimates. Overall, the linear behavior of pond coverage and total albedo predicted by the model is consistent with observed trends over the main summer melting season. Likewise, the more rapid decrease in pond albedo at the beginning of the simulation is similar to observed albedo from the albedo line showing a steep decline in pond albedo in late June followed by an almost constant minimum value later in July [Perovich et al., 2002a, Figure 5].
 We note that our results do not agree with photographic-based observations of ponds on deformed multiyear ice presented by Fetterer and Untersteiner . Pond levels in deformed, thick ice often have a level above sea level (which is assumed to be the pond level in our model), and therefore can lose water through drainage via percolation or channels. Consequently, the pond schematic presented by Fetterer and Untersteiner  shows decreasing pond size as the summer progresses, in reverse of the behavior modeled here.
 In addition to the ice surface assumptions used in the model, it is important to note that the radiative transfer coefficients used in (3)–(4) may not be representative for melt ponds. These coefficients are derived from measurements taken in a lead capped by nearly fresh water (salinity of about 2 psu), which did not have significant impurities. Melt ponds, however, can contain sediment and atmospheric aerosols, which accumulate on the ice below the pond. Both of these contaminants will change the effective transmittance of the pond water and could affect absorbed solar radiation both within the pond water and along the pond bottom and perimeter. At this time, we are not aware of measurements to support parameterizing this effect.