2.1. Model Development
[9] Our approach to modeling pond behavior is based on the simple idea that melting sea ice is porous and allows for a relatively rapid movement of water to overcome buoyancy. With this assumption, fresh water in ponds is at a level equal to the local sea level defined by the ice draft as shown schematically in Figure 1. Ponds appear on the ice whenever the local ice surface is below the ice freeboard, defined as the difference between the ice thickness and the ice draft. We model ice and ponds using a twodimensional grid with each grid cell having an ice thickness, h_{i} and water table height h_{f}. Ponds are defined as areas where the water table height is greater than the ice thickness, with pond depth defined as d = h_{f} − h_{i}. Gridded ice thickness can be initialized with topography derived either from direct observations (e.g., as in the work of Lüthje et al. [2006]), or by modeling the pond depth based on photographic images (described below). We also assume that the ice bottom is perfectly flat.
[10] Simulated pond growth results from bottom and sidewall melting, rather than through water flowing into the pond from surface melting. A number of assumptions are made to simplify the parameterization of pond melting. First, we assume that the pond water temperature is uniform within all ponds, which is partially justified by results from Skyllingstad and Paulson [2007] suggesting that even very light winds will thoroughly mix melt ponds. Pond water temperature is volume averaged every time step for all grid cells containing pond water. Averaging the pond water temperature across all ponds has the effect of increasing the growth of small ponds because of the depth dependence of solar radiation. Alternatively, we could calculate average pond temperatures for contiguous pond regions. However, we believe this approach would only have a minor effect on pond growth because the total absorbed heat is not directly affected by this simplification. Our next assumption is that the pond bottom for each grid cell is flat, allowing us to use the simplified bottom melt rate based on the area of the grid cell. Observed ponds, as shown in Figure 2, typically have a “U” shaped structure, with steep sidewalls and a uniform bottom, supporting this approach.
[11] Sidewall melting presents a more challenging problem in the grid box formulation. Because we only model a single pond depth and ice thickness for each grid cell, we need to have a method for partial melting of ice along the pond edge to account for pond side melting. Our strategy is to label each grid cell that is on the edge of pond as a “border” cell containing both pond water and ice. For each border cell, we define a variable for the fractional water content, f_{x}, and determine which pond cells are adjacent to the border cell. Melting of the border cell is calculated by assuming that water from adjacent pond cells transfers heat to the pond edge, increasing f_{x}. When f_{x} equals or exceeds the grid width, Δx, the cell is converted to a pond cell. Border cell depth is set to the average depth of the adjacent pond cells.
[12] Pond water temperature for grid cells not adjacent to the pond edge is simulated by solving a onedimensional heat budget:
where Q_{r} is the divergence of solar flux, F_{t} is the heat loss or gain through sensible, latent and long wave heat flux at the pond surface, w_{ice} is the ice melting velocity along the pond bottom defined below, and Q_{L} is the latent heat of fusion for ice [Skyllingstad and Paulson, 2007]. Ice melting rate, w_{ice}, is calculated using McPhee et al. [1987] as discussed below. A similar equation is used for border cells with the assumption that heat entering the fractional pond area in the border cell is distributed with neighbor pond cells along with heat loss from edge melting. For each border and neighbor cell, we compute a temperature change using
where u_{ice} is the ice edge melting velocity. This equation accounts for the heat input through the fractional area in the first term, and the heat loss through melting along the pond edge in the second term.
[13] Solar flux, F_{r}, is parameterized using a radiative transfer equation developed using observations from fresh water capped leads taken between 17 June and 4 August during the SHEBA experiment [Pegau, 2002]. Radiative fluxes are calculated using
where P_{m} is the proportion of shortwave energy in the band m, F_{rn} is the net shortwave radiation at the sea surface, K_{m} is the diffuse extinction coefficient, and z is the depth below the surface. Information on the band characteristics is provided in Table 1. Fluxes defined using (3) include the effects of both direct and diffuse solar radiation; however, we do not attempt to account for radiation reflected from the sidewalls of ponds, which may effect the pond melting rates.
Table 1. Band Characteristics Used to Determine the Shortwave Radiation Absorbed in a Freshwater Layer^{a}Wavelength Range  P_{m}  K_{m} 


350–700 nm (m = 1)  0.481  0.18 
700–900 nm (m = 2)  0.194  3.25 
900–1100 nm (m = 3)  0.123  27.5 
>1100 nm (m = 4)  0.202  300 
[14] Shortwave radiation reaching the pond bottom is either reflected upward by the ice under the pond or transmitted below the pond. Pond bottom albedo ranges from 0.7 to 0.2 depending on the thickness of the underlying ice and ice characteristics [Podgorny and Grenfell, 1996]. We modeled the bottom albedo as linearly decreasing from 0.5 to 0.3 in our basic 40 day simulations, and also present simulations that examine the sensitivity of the model to this parameter. Absorption of radiation reflected off the bottom is parameterized using (3) as if the depth were continuing to increase back to the surface:
where F_{r}(z_{b}) is the radiation intensity at the bottom of the pond with depth z_{b}, and α_{b} is the pond bottom albedo. In (4), the depth dependence of wavelength properties in the formula are retained. Heating of the pond water is calculated as a function of pond depth by integrating the downwelling and upwelling radiation:
Equations (4) and (5) can be used to estimate the effective pond albedo by calculating the total absorbed radiation and accounting for the radiation transmitted through the pond bottom:
[15] A plot of the predicted albedo integrated over the solar spectrum as a function of pond depth is presented in Figure 3 for three different pond bottom albedo values. For ponds with depths of ∼0.4 m, albedo ranges from 0.1 to 0.4, which is in agreement for first year pond albedo measured by Hanesiak et al. [2001] and for pond albedo measured by Barber and Yackel [1999]. Measured albedo values for shallow ponds (0.1 m) reported by Fetterer and Untersteiner [1998] ranged from 0.37 to 0.47 depending on the pond bottom albedo, also in general agreement with the modeled values shown in Figure 3.
[16] By using the onedimensional empirical formula presented in (3) and (4), we ignore the effects of sunlight that enters the pond at an angle and interacts with the pond sidewall. For small ponds, this could be significant, but will in general be much smaller than the integrated sunlight defined by the pond surface area. We also assume that the pond radiative properties, cloud conditions and overall weather are similar to those of the fresh water capped lead that was used to formulate (3) and (4).
[17] Surface fluxes of longwave radiation along with sensible and latent heating, F_{t}, are prescribed at the pond surface. For the simulations presented here, F_{t} is calculated using bulk formula for sensible and latent heat, along with measured downward longwave radiative fluxes and estimated upward infrared flux, using data collected during the SHEBA experiment. Methods for calculating fluxes are presented below in the results section.
[18] Melting along the pond bottom and sidewalls is parameterized using transfer coefficients developed by McPhee et al. [1987] that account for the differing diffusivities of salt and heat. Ice melting rate is calculated using
where S_{w} is the pond salinity at the ice edge, S_{i} is the ice salinity, w_{ice} = ρ_{i}/ρ_{o}m_{i} represents a velocity of the ice surface associated with ice growth or melting, ρ_{o} is the density of the pond water, 〈u_{i}T_{o}〉 and 〈u_{i}S_{o}〉 represent the subgrid heat and salinity boundary fluxes into the pond near the ice edge, respectively, m_{i} is the ice melting or growth rate, ρ_{i} = 920 kg m^{−3} is the ice density, and Q_{L} = L/c_{p}, where L = 3.34 × 10^{5} J Kg^{−1} is the latent heat of fusion and c_{p} = 4000 J (Kg °C)^{−1} is the specific heat of water. Scaling (7) and (8) with friction velocity u_{*} and integrating vertically results in two nondimensional functions
where T(x_{i}) and S(x_{i}) are the temperature and salinity at the nearest grid point to the pond bottom or edge.
[19] Equations (9) and (10) can be combined and simplified by replacing the wall temperature with the freezing temperature at S_{w}, or T_{w} = −mS_{w}, where m = −0.054 yielding
Ice melting rate, w_{ice}, is calculated by solving (10) with exchange functions defined as
where z_{o} = 0.002 m, ν = 1.4 × 10^{−6} m^{2} s^{−1} is the molecular viscosity, and κ_{T,} = 1.4 × 10^{−7} m^{2} s^{−1} and κ_{S,} = 7.4 × 10^{−10} m^{2} s^{−1} are the molecular diffusivities for heat and salinity, respectively. For most ponds, the salinity is low (<4 psu) [Eicken et al., 2002], consequently the salinity of the pond water does not have a large influence on the edge and bottom melting rates. Simulation of pond temperature is simplified by assuming that heat is instantly redistributed in ponds by wind forcing and buoyant convection, leading to a uniform pond temperature. As mentioned above, we accomplish this redistribution by computing an average pond water temperature for all grid locations containing pond water.
[20] A key parameter in the melting rate parameterization is the friction velocity, which is used to scale the exchange rates in (9)–(10). In the work of Skyllingstad and Paulson [2007], friction velocities were found to vary from pond sidewalls to pond bottom, with values ranging from 0.0005 m s^{−1} to 0.001 m s^{−1}. These were calculated for ponds having very weak wind forcing and basically smooth sides. Direct estimates of u_{*} ranging from 0.009 to 0.012 from beneath sea ice have been reported by McPhee [1992] for currents between 0.09 and 0.17 m s^{−1}. Here, to select representative pond sidewall and bottom friction velocities (v_{*} and u_{*}, respectively) we performed a sensitivity analysis over a range of v_{*} and u_{*}. Results from this analysis, presented in more detail in the results section, provide a range of pond depth and fractional area values similar to observed ponds during SHEBA. Based on the sensitivity results, we chose values of v_{*} = 0.014 and u_{*} = 0.005 m s^{−1} for the “best guess” control experiment presented below.
2.2. Pond Initialization
[21] Area observations of ponds are typically limited to aircraft surveys with limited information regarding pond depth or significant ice characteristics such as thickness and surface topography. Direct pond observations [e.g., Barber and Yackel, 1999; Fetterer and Untersteiner, 1998; Perovich et al., 2003] provide a snapshot of pond depth and ice thickness, but are lacking in a true field description of pond features. In the modeling study of Lüthje et al. [2006], 2.5 m resolution ice topography estimated from aerial laser altimetry data [Hvidegaard and Forsberg, 2002] were used as initial data for modeling ponds. However, they did not have a direct set of pond measurements to verify their results.
[22] Here we apply an indirect method for estimating an initial pond state given aerial photographs of observed ponds. Our method is similar to a technique developed by Sneed and Hamilton [2007] for estimating pond depths on the Greenland Ice Sheet. In our method, we rely on the relationship between pond albedo and pond depth, and on direct measurements of pond depths taken during the SHEBA experiment, whereas Sneed and Hamilton [2007] use the radiative transfer based on the spectral properties of pure water.
[23] Pond albedo can be calculated from (3) and (4) by comparing the input radiation with the total flux that is absorbed in the pond or transmitted through the pond bottom. Examples of albedo as a function of pond depth are shown in Figure 3 for range of α_{b} values between 0.7 and 0.2. Using aerial photographs converted to gray scale, we can estimate pond albedo, α_{p}, by computing the ratio of the gray scale of ponds with the gray scale of ice
where ice albedo, α_{i} = ∼0.6, for bare summer ice [Perovich et al., 2002a; Hanesiak et al., 2001] and B_{p} and B_{i} represent image brightness of ponds and ice, respectively.
[24] Solving for α_{p} requires an estimate of the ice brightness intensity, which can be determined by examining the histogram of brightness values for a gray scale image. We apply this method to an early melt season photograph taken during the SHEBA experiment (Figure 4) having resolution of 0.6 m. Brightness values for this image define a bimodal distribution, as shown in Figure 5, which has a nearnormal appearance centered around brightness value 150, with an extended tail over the low end of brightness intensity. The tail of brightness values represent ponds, suggesting a value of B_{i} = ∼130 as a cutoff for pond water versus ice.
[25] Calculation of the pond depth from the estimated pond albedo requires a value for the pond bottom albedo. As Figure 3 shows, changes in the pond bottom albedo have a significant impact on the relationship between pond depth and pond albedo. Pond albedo is a better indication of pond depth when the pond bottom albedo is high. Because our goal in this study is to build a realistic pond initial condition, we treat the pond bottom albedo as an adjustable parameter and select a value of α_{b} = 0.5 through trial and error. Are strategy basically calibrates the pond albedo to depth for the SHEBA conditions, and is not necessarily applicable to other aerial photograph surveys of melt ponds. Using (3)–(4), we generated a look up table of pond depth as a function of albedo and transform estimated albedo calculated with (13) to pond depths as presented in Figure 6. Pond depths in the plot are for the most part between 0.1 and 0.2 m, with a few ponds having deeper regions near 0.5 m. Unfortunately, we do not have an areal image that covers the SHEBA albedo line presented in Figure 2 near the beginning of the melt season. Nevertheless, since the 30 June image was taken very near the location of the albedo line, we believe the pond depth map shown in Figure 6 is a reasonable estimate of the actual conditions.
[27] One significant issue with the pond initialization method concerns the assumption of a flat ice surface. In reality, the ice surface in the region around ponds is likely to be somewhat lower than the average, which will lead to ponds expanding as the overall ice melts and the water table moves above the ice surface. By assuming a flat ice surface, pond expansion can only occur through sidewall melting. As a partial compensation for this simplification, we do not account for melting of the ice edge above the pond, but assume that the sidewall melting leads directly to pond expansion.