Calculation of Antarctic sea ice melt rate enhancements for "The influence of ocean waves on Antarctic sea-ice albedo and seasonal melting, and potential coupled physical and biological feedbacks"

These simple Excel spreadsheets contain data to be able to reproduce the melt rate Figures 3a,b and c from the paper "The influence of ocean waves on Antarctic sea-ice albedo and seasonal melting, and potential coupled physical and biological feedbacks" by Massom et al., 2026. Also attached is the relevant sections (2.2 and 3.3) from within the paper. This text is also included below, but without the equations. 2.2 Computing radiative transfer and wave-driven melt-rate enhancement Here, we consider the effect of wave flooding and wave pulverisation on only the shortwave properties of an ice floe/area of wave slush, which is to reduce the sea-ice albedo. The longwave (infrared) emissivities of snow, ice and water are all close to 100% (Warren, 1982, 2019), so the only longwave effect of wave flooding is to increase the longwave emission slightly by increasing the surface temperature of the ice, bringing it up to ~0°C. Our computation starts with the daily-average incident solar flux at the top of the atmosphere, FTOA (in W m-2), and then multiplies it by the atmospheric transmittance over Antarctic sea ice, τa, to obtain the solar flux at the surface (Fsfc, in W m-2) as (1) Following Fitzpatrick and Warren (2005; FW05), and as τa is not directly available, we take two steps to obtain Fsfc. We first obtain the transmittance, τclr, for clear sky (shown in Fig. 7 of FW05 as a lin ear fit versus solar zenith angle θ), such that (2) with θ in degrees. Then we apply the “cloud radiative forcing” CRF as a correction, to obtain Fsfc as (3) where CRFd is the downward shortwave cloud radiative forcing in W m-2 (measured by FW05 and plotted in their Fig. 10), which is a negative quantity, indicating that clouds reduce the downward solar flux at the surface. We estimate the albedo change (decrease), Δα, caused by wave flooding of a snow-covered floe of wave pulverisation to be (4) where αs is the albedo of a snow-covered floe of first-year ice (FYI), and αw is the albedo of wave-flooded and/or wave-pulverised ice. Multiplying Fsfc by Δα gives the radiative forcing, RFw, of wave-flooded or wave-pulverised ice as (5) which is in units of W m-2. Equation (5) holds for a single idealised sea-ice slab or small area of (unconsolidated) wave slush at a given latitude and day of year. Due to current lack of important spatio-temporal information e.g., on the extent and fractional coverages of the different wave-flooded types and wave-pulverised ice, it is beyond the scope of this study to estimate regional averages of radiative forcing (RFw,avg) as a function of latitude, long itude and month. Our aim is simply to show that wave flooding and wave pulverisation could enhance seasonal sea-ice melting – arguing that it is worthy of further investigation. To estimate the change in melt rate of the idealised wave-flooded small floe and/or area of wave slush due to the radiative forcing, i.e., the wave-driven melt-rate enhancement, we divide the radiative forcing (Eq. 5) by the sea-ice density, 𝜌 = 905 kg m-3 (Fang et al., 2022), and the latent heat of fusion, 𝐿 = 334 J g-1 (Fang et al., 2022), to obtain (6) where h is the sea-ice thickness and t is time. 3.3 Wave enhancement of sea-ice vertical melt rate We first compute the wave-driven enhancement in vertical melt rate (dh/dt) based on the radiative effects of both wave flooding and wave pulverisation on the December solstice (the approximate midpoint of the austral-summer melt season) and for single parcels of ice at 65°S (a representative latitude of the MIZ e.g., for much of East Antarctica and the Ross Sea at that time of year [Massom et al., 2013]). In order to illustrate the technique and the magnitude of the effect, we initially focus on Type A (wave-overwashed ice, wave compression-flooded FYI and wave slush), where Δα = 0.38 (Table 2). On the December solstice at 65°S, the downward shortwave (SW) flux at the top of the atmosphere (FTOA) is 510 W m-2 (Fig. 2.7 of Hartmann, 2016). This corresponds to an effective daily-average solar zenith angle of 68° (solar constant is 1360 W m-2; cos‑1 (510/1360) = 68°). For solar zenith angle 68°, we get the downward SW at the surface, Fsfc, as described in Sect. 2.2. The corresponding atmospheric transmittance over sea ice is τa = 0.52, so Fsfc = 0.52 × 510 = 264 W m-2. Referring back to Eq. (5), the radiative forcing caused by wave flooding (RFw) is then: (7) = 0.38 × 264 = ~100 W m-2. For an idealised fully-flooded single floe (snow-free sea-ice slab) or a small area of wave slush, both the fraction of ice area affected (fw) and the ice concentration (C) are taken here to be 1.0, and we can compute the melt-rate enhancement by Eq. (6) (Sect. 2.2). In this case, dh/dt ≈ 2.9 cm day-1. We next carry out a broader sensitivity analysis of the relative enhancements in daily melt rates (dh/dt) for the four different classes of wave-modified (-darkened) ice surfaces A-D given in Table 2 (with albedo reductions Δα = 0.38, 0.48. 0.54, 0.64), and for three wave-flooding and/or wave-slush coverage fractions (fw = 0.33, 0.5, 1.0, based on visual inspection of photographs acquired from icebreakers (Fig. 1a–j)), over November through January and as a function of latitude. Results for 60°S, 65°S and 70°S, i.e., the zone typically covered by sea ice at some time during the annual retreat phase (e.g., Massom et al., 2013), are plotted in Fig. 3. These results and values for the wider latitudinal range of 55–75°S given in Appendix Tables A1–A3 show that wave-enhanced rates of vertical ice melting steadily build up through November, and that they generally increase with decreasing latitude and increasing fraction of coverage. In all cases, values of dh/dt converge towards a broad seasonal peak around the December solstice, before slowly decreasing through January. Estimated values of annual-maximum wave-driven dh/dt (in mid-December) for the different wave-darkening and coverage fraction scenarios for three latitudes are shown in Table 3. For ice types A and C which are not greened by algae, vertical melt-rate enhancement ranges from 0.9 cm day-1 (for type A at 70°S where fw = 0.33) to 4.3 cm day-1 (type C at 60°S where fw = 1.0), with algal greening increasing these values by 0.2–0.8 cm day-1. There is, in general, a slight decrease in dh/dt with increasing latitude south (Tables A1–A3 and Fig. 3), which is steepest in November. Sea-ice coverage south of 70oS is largely confined to the Weddell, Amundsen and Ross seas, whereas sea ice across East Antarctica mainly occurs equatorward of 67oS (Massom et al., 2013). The highest melt-rate enhancements are at 55oS (Tables A1–A3), but sea ice usually attains that latitude only in the eastern limb of the Weddell Gyre at ~5–25oE and parts of East Antarctica around ~80oE, and only in winter through October (cf., Massom et al., 2013; Comiso et al., 2017a). The values shown in Table 3 are for sea ice with fixed density 905 kg m-3 (see Eq. 6), which is taken here to be an approximation for a cold FYI floe. The densities of the four wave-affected ice-surface types are unknown, but they may be lower than 905 kg m-3 and are likely to decrease through late-spring and summer as melting progresses and ice permeability and porosity increase. This would be the case if the pore spaces are filled with air, but the density would increase if the pores filled with water. Given these uncertainties, we now investigate the effect of lowering ice density on dh/dt, using a value of 750 kg m-3 measured in Lützow-Holm Bay in East Antarctica by Urabe and Inoue (1988). Results in Table 4 show that decreasing the ice density increases the melt-rate enhancement by an extra 0.1–1.0 cm day-1 compared to ice with density 905 kg m-3, pushing dh/dt up to 1.0–6.1 cm day-1 (again depending on surface type, latitude, fw, and greening).