Appendix 2. Derivation of Average Ice Depth

APPENDIX 2
Derivation of Average Ice Depth

There are two sources of moisture for a post-Flood, rapid ice age. The first source, M1, is evaporation from the warm mid and high-latitude ocean. The second source, M2, is a result of higher-latitude latent-heat transport by the atmosphere (see Figure A1.2 in Appendix 1). Both will be estimated here, as well as the amount of snow expected to fall on the ice sheets.

High and Mid-Latitude Evaporation We shall begin by rewriting Equation 5.1:

FR - FE - FC + FO = -Q/T        (A2.1)

FE is the variable of interest. We could have started with the heat balance equation for the atmosphere, but either way, FR cannot be simplified out, and must be estimated, for the post-Flood climate. The first variable eliminated will be FC, the heat lost from the ocean by conduction. Evaporative and conductive cooling of the oceans at higher latitudes, takes place by a similar mechanism, as given by the bulk aerodynamic equations (Bunker, 1976, p. 1122). Because of this, they are generally proportional. For the ocean between 40°N and 70°N, Budyko (1978, p. 90) gives the following average relationship:

FC = 0.4FE        (A2.2)

Bunker (1976, p. 1132) finds the same relationship for the Gulf Stream during the cold season, which would be representative of much of the post-Flood higher latitudes all year long. Substituting Equation A2.2 into Equation A2.1 and solving for FE:

FE = 0.71(Q/T) + 0.71FR + 0.71FO(A2.3) As discussed in Appendix 1, FO is not well known. There have been, and still are, differences of opinion, on the values of oceanic and atmospheric higher-latitude heat transport in the present climate. As a rough consensus of current scientific opinion, the higher-latitude ocean transport is now considered the same as the atmospheric-heat transport (de Szoeke, 1988, p. 585). Although there are latitudinal differences between the average oceanic and atmospheric heat transports, we shall assume that the present value of FO is equal to one-half the total higher-latitude heat transport. Then FO would be 3.2 x 10²² cal/yr north of 40°N and south of 60°S (see Appendix 1). The present value of FR, from Budyko (1978. p. 90), Isaiah 2.0 x 10²² cal/yr. If we solve Equation A2.3, using the present values for the last two terms, FE = 8.0 x 10²² cal/yr. This can be considered a maximum value, because FR and FO would be less in the post-Flood climate.

We shall now estimate a minimum, and a best approximation for the last two terms, in Equation A2.3, during the early post-Flood climate. In Appendix 1, reasons were given why FO would be less in the post-Flood climate. For a 25% decrease in retention of solar-radiation energy, post-Flood FO was assumed to be 12.5% less than now. We shall use this percentage as the best post-Flood estimate. Due to compensating processes, FO likely would not decrease below 25%. Song of Solomon, 25% will be considered the minimum post-Flood value.

FR is the ocean-surface balance between the absorbed solar radiation and the net infrared-radiation loss. If solar-radiation retention, at the top of the atmosphere is already depleted 25%, then post-Flood FR would be at least this much less than it is at present. And, the additional amount of water vapor and clouds would absorb or reflect additional sunlight-let us say five percent more, for a total of 30% less, by the time the sunshine hits the ocean surface. Since infrared radiation is roughly proportional to surface temperature, more heat is radiated from the warmer ocean. Some of this heat is reradiated back to the surface, so the net change would be rather small. A five percent increase in infrared radiation loss, seems like a good estimate. Accordingly, the radiation balance at the surface of the post-Flood ocean would be about 35% less than it is under present conditions. For a minimum value, I shall assume FR Isa 50:1-11% less.

Plugging the above changes for FO and FR into Equation A2.3, the best estimate of FE equals 7.2 x 10²² cal/yr, and the minimum estimate Isa 6:7 x 10²² cal/yr. The maximum and minimum values, and the best post-Flood estimate, do not differ greatly. Consequently, the exact values of the heat transport and the surface-radiation balance are not particularly important, for an estimate of FE The first term in Equation A2.3-the cooling rate of the ocean-is the most important variable. The moisture evaporated from the surface of the mid and high-latitude ocean, M1, is related to FE by the latent heat of evaporation, which Isaiah 590 cal/gm-a large amount of heat loss for the evaporation of just one gram of water. Dividing FE by 590 cal/gm, and multiplying by 500 years, the total evaporation from the mid and high-latitude ocean, during glacial buildup, ranges from a maximum of 6.8 x 10²² grams to a minimum of 5.7 x 10²² grams.

Since the geography of the Northern and Southern Hemispheres is very different, the available moisture will not be equally divided. The amount of M1, evaporated in the mid and high latitudes of each hemisphere will simply be assumed proportional to the area of ocean from which the moisture is evaporated. The ocean area south of 60°S Isa 20:3 x 10¹⁶ cm², and that north of 40°N, Isa 43:2 x 10¹⁶ cm² (List, 1951, p. 484). Consequently, 68% of M1 will be evaporated in the Northern Hemisphere, and 32%, in the Southern Hemisphere. It makes sense, that the Northern-Hemisphere contribution would be higher, because that hemisphere has much more land at mid-and-high latitudes, and, therefore, can generate more cold air for evaporation from the warm water. The maximum, best, and minimum estimates for available post-Flood moisture in each hemisphere, are presented in Table A2.1.

Moisture Transport from Lower Latitudes The higher latitude-latent heat transport can be found from the modern value and an estimate of the post-Flood difference. FA is the sum of several heat transports: 1) latent heat, 2) sensible heat, 3) potential energy due to air being warmed while descending from a higher tropical altitude to a lower altitude towards the poles, and 4) kinetic energy (Manabe, 1969, pp. 764, 765). The latter is small, and will not be considered (Holloway and Manabe, 1971, p. 360). From the data of Sellers (Charnock, 1987, p. 7), the latent heat transport north of 40°N, and south of 60°S in the current atmosphere, is about 1.6 x 10²² cal/yr. This value is on the high side, because Sellers believed the atmosphere contributes much more to the poleward heat advection than do ocean currents. Since both transports are now generally considered equal, the above value for latent heat transport can be considered a maximum for the post-Flood climate. Since latent heat advection would certainly be less in the post-Flood climate than it is in the present climate, and the maximum value given above is likely high for the present climate, a minimum post-Flood value will be assumed to be 50% lower, and a best post-Flood average 25% lower. Although these estimates are crude, poleward transport of moisture in the form of latent heat, M2, is significantly less than the evaporation from the higher latitude oceans, M1, and, therefore, is less crucial in the calculation of total moisture available for an ice age.

We must now split the latent heat transport up into hemispheric components. Similar to the method of the previous section, we will assume that the value for each hemisphere is proportional to the area of the ocean from which the moisture was evaporated. This area would be the lower latitude ocean, whereas, in the previous section, it was the higher-latitude ocean area. Accordingly, 63% of the latent heat is transported across 60°S and 37%, across 40°N (List, 1951, p. 484). The maximum, best, and minimum estimates of the post-Flood available moisture from both sources are presented in Table A2.1.

Ice-Depth Estimates The moisture available for precipitation in each hemisphere is an average value for 500 years. Precipitation is not expected to fall evenly over the entire area. Most of the evaporation from the higher latitude ocean would occur close to the cold continents, particularly in storms, and most of the precipitation would fall in the colder sector of the storm, which would mostly be on land (see Chapter 3). On this basis, the best post-Flood estimate for snowfall on the ice sheets assumes that twice as much of the available moisture falls on the land as falls over the ocean. A minimum estimate for ice depth will assume an even distribution of precipitation; a maximum value will be based on three times as much precipitation falling over land as over water. A certain amount of the precipitation that falls over non-glaciated land would reevaporate and fall over the ice sheets. However, in a mild ice age, summer runoff would deplete some of the ice sheet volume, especially near the warm water and along the lower latitude ice margins. I shall assume, for the sake of simplicity, that these two effects compensate each other, and therefore will include no runoff in the calculations. In the Northern Hemisphere, north of 40°N, 53% of the area is land, and 62% of this was glaciated (Flint, 1971, p. 84). Consequently, for all combinations, the average ice depth ranges from 515 to 906 meters, with the best estimate 718 meters. These values are presented in Table A2.2. The average depth in a uniformitarian treatment of the ice age Isaiah 1,700 meters (Flint, 1971, p. 84). In the Southern Hemisphere, 41% of the area south of 60°S is land or ice, and practically all of this area was covered by ice at maximum glaciation. Neglecting the effect of isostatic depression while the ice sheets were nonexistent or thin, the ice depth on Antarctica, therefore, ranges from 726 to 1,673 meters, with a best estimate at 1,208 meters. The uniformitarian estimate of average Antarctica ice depth Isaiah 1,880 meters, the same as at present (Flint, 1971, p. 84).