Appendix 3. Derivation of Ice Melting Time

APPENDIX 3
Derivation of Ice Melting Time We shall begin by first rewriting Equation 6.2, which applies to the ablation season:

QM = FR + FE + FC(A3.1) To solve this equation, FR must be broken up into its solar and infrared radiation components. Therefore, Equation A3.1 becomes:

QM = F(1-a) - FI + FE + FC(A3.2) where F is the solar radiation that reaches the surface per unit of time, a is the albedo, or reflectivity of the surface, and FI is the net infrared radiation balance at the surface, in unit time. FI is the difference between outgoing infrared radiation from the surface, and infrared radiation, either reflected back by clouds and/or reemitted downward by the atmosphere. The first two terms on the right of Equation A3.2, the radiation balance, are the dominant terms in the equation (Paterson, 1981, p. 313). These two terms are illustrated in Figure A3.1. The last two terms on the right are impossible to calculate with any precision, because they vary considerably from glacier to glacier, and with the weather pattern (Kuhn, 1979, p. 264). They are smaller terms than the radiation balance, and will be treated by finding their average value compared to the radiation balance.

Figure A3.1 <C:\Program Files\e-Sword\Graphics\ICE\218.jpg>The Main Component in Melting Snow The main component in melting snow: Solar radiation (solid lines) minus the net infrared radiation (wavy lines). Dots are flowing water.

Data for an entire summer on the Karsa Glacier in Sweden, and for 24 glaciers from a wide variety of climatic regimes, indicate FC is about 30% of the radiation balance, and FE is about 10% (Paterson, 1969, pp. 56-61). Kuhn (1979, p. 270) reports that 57% of the melting during a 100-day period on a glacial in the Alps was due to the radiation balance; the other 43% came from FE and FC. During a 17-day period on a glacial in New Zealand, 52% of the melted ice was accounted for by net absorption of radiation (Hay and Fitzharris, 1988, p. 149), but the observational period was too short for an accurate average. The relationship reported by Paterson, for a wide variety of glaciers, seems like a good approximation, and will be used here. Equation A3.2 then simplifies to the following:

QM = 1.67[F(1-a)- F1]        (A3.3)

Central Michigan was chosen as the location to estimate the terms in Equation A3.3. The ice depth of the Laurentide ice sheet likely varied considerably, being thickest near the Atlantic Coast and thinnest in the western portion. Central Michigan would represent a good average ice depth for the Northern Hemisphere, as calculated in Chapter 5. This location is also about 450 kilometers inward from the edge of the ice sheet at maximum extension, but still close enough to represent the periphery. Central Michigan is fairly cloudy, and has about the lowest yearly average radiation in the United States. For this area, any postulated increase in cloudiness after glacial maximum, should not significantly alter the results. The melt season for the ice sheet was assumed to begin May 1 and to end September 30, which is probably too short, and, therefore, conservative. The first of May was chosen because the winter snow, which would be lighter than today in the drier late glacial climate, should have melted by then. I also assumed that a sufficient depth of the ice sheet near the periphery, had been raised to 0°C by May 1, so that additional heat would be expended in melting snow and ice. The melt-season temperature was assumed 10°C colder than a comparable present average, which seems like a good ice-age estimate for summer, on an ice sheet about 700 meters thick, and is relatively consistent with model results (Pollard, 1980, p. 385). After subtracting 10°C from the current average (as shown in Table A3.1), the post-Flood warm season temperature averages 8.4°C.

Some readers may think the winter snow would not melt before May 1, with the temperature still below 0°C in spring. Net melting actually can occur well below freezing in spring and summer, so the ablation season could begin around March 1. The reason snow can melt below freezing, is because solar radiation, during spring and summer, is intense at mid and high latitudes. This is well known to inhabitants of northerly latitudes, when, after a spring snow storm, the sun may start melting the snow at air temperatures well below freezing. Pickard (1984) has observed melting on the Antarctic ice sheet at 15°C below zero, and considers the ablation season to begin when the air temperature warms above -10°C. The average cloud-base temperature, which determines the infrared back-radiation to the ice sheet, was assumed to average 5°C during the ablation season. This value was chosen because there would be a strong inversion over the ice sheet, and upper air temperatures would be controlled more by the general circulation of the mid-latitude atmosphere (Manabe and Broccoli, 1985b, pp. 2179-2181). Consequently the lapse rate, or change in temperature with altitude, would be less than the assumed average of -6°C/1,000 meters in current-day models. Some researchers believe such a lower lapse rate is highly likely over an extensive snow cover (Williams, 1979, p. 448). To simplify the calculations, other weather variables, like relative humidity, cloudiness, and surface wind speed, were assumed similar to their present values. In Pollard’s (1980, p. 385) estimation, during an ice age there would be more cloudiness (which is questionable), higher winds, and more humidity. The latter two effects would add a significant amount of heat to the ice sheet. However, these differences will be neglected, and, consequently, the results will be even more conservative. The solar and infrared radiation balance depend especially on the amount of clouds and the surface albedo. Clouds have a high albedo. They not only reflect solar radiation back to space, they also absorb and retransmit infrared radiation back to the ice sheet. Although the effect of clouds is very complicated, the net result is less snowmelt than under sunny skies. Since the cloud and albedo conditions over the melting ice sheet in central Michigan are not known, extreme ranges for these variables will be used in the calculations. Maximum melting would occur under completely clear skies, and minimum melting under completely cloudy skies. The melting rate under variable-cloudiness similar to the current weather conditions will be interpolated using the present value of solar radiation. The interpolation is necessary because the net infrared radiation at the surface is impossible to estimate with any precision under variable cloud conditions for a five-month period. In completely dear skies, the infrared radiation from the atmosphere depends upon the vertical temperature and moisture profile of the atmosphere. However, the lower layers of the atmosphere can be used for the estimate, since they have the warmest temperatures and a greater moisture content than the middle and upper atmosphere (U. S. Army Corp of Engineers, 1956, p. 157). In general, the infrared radiation balance causes a loss of about 4.2 langleys/hour under clear skies, at an average temperature of 8.4°C (U. S. Army Corp of Engineers, 1956, pp. 141-191, Plate 5-3, Figure 4). A langley Isa 1:1-31 cal/cm². The infrared radiation balance, under cloudy skies, is simply the difference in the black-body radiation at the temperature of the ice-sheet surface (0°C) and the bottom of the clouds (5°C). The average solar radiation, under current conditions of cloudiness for East Lansing, Michigan, is presented in Table A3.1, along with average monthly temperatures (Crabb, 1950, p. 37).

The albedo of fresh snow Isaiah 0.7 to 0.9, for firm or melting snow, 0.4 to 0.6, and for glacial ice, 0.2 to 0.4 (Paterson, 1981, p. 305). The lower albedo for melting snow (0.4) is reached after about two weeks of melting (U. S. Army Corp of Engineers, 1956, Plate 5-2, Figure 4). The reason for the drop in albedo is because of the large increase in size of the ice crystals (Wiscombe and Warren, 1980). If ice is exposed, which is very likely as the ablation season progresses, the albedo will drop below 0.4. From these considerations, the maximum albedo for the five-month period will be assumed to be 0.4.

Due to several other mechanisms, the albedo can drop significantly below 0.4 along the periphery while the ice melts. One of these is dust (loess) that would collect at the surface from strong, dry winds during deglaciation. As previously discussed, a good indication of this is the loess sheets found just south of, and even within, the ice sheet periphery (Flint, 1971, pp. 251-266). The albedo over a permanent snow field in Japan was observed to drop to 0.15 during the summer, after 4,000 ppm of pollution dust had collected on the surface (Warren and Wiscombe, 1980, p. 2736). During winter, dust would be covered by fresh snow. But, after snow melt in the spring, the dust layer is reexposed. And while the ice melts during summer, the dust tends to remain concentrated at the surface, resulting in a low albedo (Warren and Wiscombe, 1985, p. 469). Anyone living in a snowy climate where the streets are sanded and plowed, can observe this. When the snow is piled up by the side of the road and melts, the sand becomes concentrated at the surface of the melting mound. A second mechanism that lowers the albedo is a positive feedback mechanism caused by crevassing. In crevasses, more ice is exposed, and causes multiple reflection of solar and atmospheric infrared radiation, so that more radiation is absorbed. Because of the tripling of the surface area of the Jakobshanvs Glacier in West Greenland due to crevassing, the observed average melting rate was 0.3 meters/day, for a total ablation-season thinning of 55 meters (Hughes, 1986). Crevassing would likely be extensive along the periphery of the Laurentide and Scandinavian ice sheets due to its temperate locality, rapid velocity with surges, and the many proglacial lakes and marine bays. Paterson (1981, p. 169) writes:

Entry of the sea into areas that are now Hudson Bay and the Baltic Sea played a large part in the rapid disintegration of the Laurentide and Fennoscandian Ice Sheets. Calving of icebergs into extensive ice-dammed lakes around the southern margins also contributed significantly.

All factors considered, I chose a minimum value of 0.15 for the albedo. The best estimate for albedo would lie between the extremes, but probably closer to the minimum value.

Table A3.2 presents the results from Equation A3.3 for East Lansing, Michigan, for the maximum and minimum values of cloudiness and surface albedo. A melting rate of about 10.4 meters/year, with cloudiness equivalent to present Michigan, and with an albedo interpolated between the extremes, is considered to be the best estimate. But no matter what the cloud conditions or the albedo, the melting rate is still high. In Chapter 5, the average ice depth for the Northern Hemisphere was calculated to range from 515 meters to 906 meters. At 10.4 meters/year, the ice would melt in 50 to 87 years. This is very short, and based on a conservative estimate of the melting rate. Even if the albedo was 0.4, melting would still occur in about 105 years. The interior of the ice sheets would, of course, disappear more slowly, due to colder temperatures and less solar radiation. However, the radiation would not be greatly reduced, because, in summer, the sunshine lasts for a longer period of the day than it does farther south. The albedo would be higher than at the periphery, because the blowing dust likely would not significantly penetrate that far north. A value of 0.4 seems like a good value for the average albedo. Table A3.2 shows that with an albedo of 0.4, and with the radiation equivalent to cloudy skies in central Michigan, the ice would melt at about seven meters/year. Observations from glaciers in the high Arctic support a melting rate near this value. Beget (1987, p. 85) states: A compilation of ablation gradients on active glaciers... suggests that glaciers at 69 to 71°N latitude typically have ablation rates of 5 to 7 m yr¹ at low elevations, and ablation rates of 1.5 to 3.0 m yr¹ at elevations of a kilometer....

Altogether, an estimate of 200 years to melt the interior ice is conservative, especially considering that in this model, the interior ice was much thinner than specified in uniformitarian models. The total time for a rapid post-Flood ice age, from the end of the Flood to when most of the ice disappeared, is on the order of 700 years.