Appendix 1. Derivation of Time for Maximum Glaciation

APPENDIX 1
Derivation of Time for Maximum Glaciation To solve Equation 5.1, we need to first divide the ocean surface into areas of net heating and net cooling in the post-Flood climate. In the present and post-Flood climate, the tropical and subtropical oceans gain heat, while the mid and higher-latitude oceans lose heat. Net cooling of the ocean surface is accomplished by evaporation and conduction. The heat lost by conduction occurs by a similar process, and over a similar area as that for evaporative cooling. The areas of net cooling in the post-Flood climate correspond closely to the areas of moderate-to-strong evaporation, which are estimated for the Northern Hemisphere in Figure 3.9. These areas are for the first half of glaciation, and an average for the entire period of glaciation is needed. As the ice age progressed, areas of moderate and high evaporation would spread out slightly. Another feature to note in Figure 3.9, is that areas of net cooling and heating are not latitudinally symmetrical. High evaporation takes place off the east coast of the continents, while generally light evaporation occurs off the west coasts. For the purpose of estimating the variables in Equation 5.1, the dividing line, between areas of net surface cooling and warming, was assumed to average 40°N in the Northern Hemisphere.

Since the Southern Hemisphere has a much larger ocean, and the continent of Antarctica is the predominant source of cold air, moderate-to-strong evaporative and conductive cooling of the ocean surface was assumed to have taken place in a 10°-latitude strip around Antarctica. This strip of ocean has an average latitude of about 60°S. Therefore, 60°S was assumed to be the boundary latitude between net oceanic warming to the north and net cooling to the south, during the post-Flood ice age. Figure A1.1 shows the heat balance for the partitioned ocean. The only difference for the ocean heat-balance equation-the terms of which are illustrated in Figure A1.1 and Figure 5.2-is that heat flow by ocean currents must be included. Figure A1.1 illustrates this heat flow, and the spreading of the colder water from the higher latitude surface to the deeper ocean.

Figure A1.1 <C:\Program Files\e-Sword\Graphics\ICE\200.jpg>Heat balance in a post-Flood ocean. The heat balance of the ocean partitioned into areas of net surface heating and cooling in the post-Flood climate. Symbols the same as in Figure 5.1. In addition, double arrows represent the flow of colder water, F0 is the heat flux by surface ocean currents, FD is the heat flux from deeper water and q/T is the rate of heat loss in the higher latitude surface layer. For the two surface areas of net cooling (shown by the solid boxes in Figure A1.1), Equation 5.1 becomes:

FR - FE - FC + FO + FD = -q/T(A1.1) where FO is the net rate of higher-latitude heat transport by ocean currents, FD is the rate of heat added from the deep warm ocean at mid and high latitudes, and q is the heat lost in the higher latitude surface layer in time T. FR, FE, and FC were previously defined in Chapter 5.

Equation A1.1 can be further simplified. FD is equal to Q/T- q/T added from the deep ocean. This heat from below is equal to the total cooling of the ocean, Q/T, minus the small contribution from the surface layers, q/T. The reason the surface layer value is small, is because it has a much smaller volume, compared to the total volume of the ocean. As previously stated in Chapter 5, the signs of the terms in the balance equations must be watched carefully. If a variable adds heat to the volume under consideration, the sign is positive, and vice versa for a variable that subtracts heat. This is why FV is positive in Equation A1.1, while Q/T is negative in Equation 5.1. Substituting the above equality and solving for T, Equation A1.1 becomes:

T = -Q/(FR - FE - FC + FO)(A1.2) In order to solve this equation, the heat balance of the atmosphere must be included, because FE and FC transfer heat from the ocean to the atmosphere. The heat added to the atmosphere will, in turn, influence the rate of heat loss from the ocean. In other words, the atmosphere acts like a regulator, or negative feedback, to the heat flux. If the continental air becomes too cold, evaporative and conductive heating, FE and FC, are stronger, as colder, drier, continental air encounters the warm ocean. More atmospheric heating over the ocean, spread around mid and higher latitudes by the atmospheric circulation, will cause the continental air to not be so cold, which then reduces FE and FC. Because of the warmth of the mid and high-latitude oceans and the bulk aerodynamic equations for FE and FC applied to areas adjacent to cold continents, the transfer of heat to the atmosphere is high. This heat, injected into the higher latitudes, would result in milder, wet winters, especially for areas along the coast. Continental interiors at higher latitude, would, of course, be kept cool by the mechanisms described in Chapter 3. The equation for the heat balance of the atmosphere north of 40°N, and south of 60°S, in the post-Flood climate, is (Budyko, 1978, p. 86):

-FRA + FE + FC + FA = -QA/T(A1.3) where FRA is the average radiative heat balance for the atmosphere per unit of time, which is negative at higher latitudes; FA is the higher latitude rate of heat transport by the atmosphere; and QA is the total change in heat content of the higher latitude atmosphere, in time T. The atmospheric heat balance for the mid and high latitudes of the Northern Hemisphere is depicted in Figure A1.2. Analogous to FO, in the ocean heat balance, FA represents the resultant of the lower-latitude atmospheric-heat balance. QA is negative, because the higher latitude atmosphere will gradually cool in the post-Flood climate. Considering the low specific heat and the light weight of air, QA will be small compared to the other terms in the equation, and can be neglected.

Figure A1.2 <C:\Program Files\e-Sword\Graphics\ICE\203.jpg>Heat Balance for Northern Hemisphere Atmosphere The atmospheric heat balance for the Northern Hemisphere north of 40°N.

FRA is difficult to estimate, so can be simplified by the following equation (Budyko, 1978, p. 86):

FRA = FRE + FR(A1.4) where FRE is the average radiative heat balance per unit of time at the top of the atmosphere, equal to the difference between the absorbed solar radiation by the earth and atmosphere and the outgoing infrared radiation to space. Substituting into Equation A1.3, and setting QA equal to zero:

-FRE- FR + FE + FC + FA = 0(A1.5) FRE and FR are negative, because they cause the higher-latitude atmosphere to lose heat. The second, third, and fourth terms in Equation A1.5, are the same terms as in the oceanic-heat-balance Equation A1.2. However, the terms apply to different volumes. Equation A1.5 includes the total volume of atmosphere above the higher latitude oceans, and also the total volume above the higher latitude continents. In order to substitute an expression for the atmospheric heat balance into the oceanic heat balance equation, only those terms common to both the atmosphere and ocean can be exchanged. In other words, we can only substitute FR, FE, and FC for the atmospheric volume above the ocean into Equation A1.2, simplifying the latter. Separating Equation A1.5 into over-land and over-ocean volumes, and rearranging terms, Equation A1.5 becomes:

(-FR + FE + FC)LAND + (-FR + FE + FC)OCEAN = FRE - FA(A1.6) The first term in parentheses cancels, because over-land FR equals FE + FC (Budyko, 1978, pp. 86, 90). Equation A1.6 then becomes:

(-FR + FE + FC)OCEAN = FRE - FA(A1.7) Substituting Equation A1.7 into Equation A1.2:

T = -Q/(-FRE +FA + FO)        (A1.8)

Equation A1.8 is as far as we can simplify the ocean-heat-balance equation. In today’s climate, which is in a steady state, Q is zero, and the terms in the parentheses balance (Carissimo et al., 1985, p. 83). This will later help us estimate post-Flood values of FA and FO, together. Now we must estimate the four terms on the right for the post-Flood climate. The values for the terms in parentheses will be in calories/year.

Oceanic Cooling

Q is the total heat lost to the ocean between the Flood and ice-age maximum. It depends upon the initial average temperature and the threshold temperature at glacial maximum. The initial ocean temperature depends upon the average temperature of the pre-Flood ocean, and on the average temperature and volume of subterranean water added to this ocean from the “fountain of the great deep.” The temperature of the pre-Flood ocean was probably warm, but depended upon many factors. The temperature of subterranean water, today, varies from the warm temperature of some hot springs, to the very hot temperature of 350°C for many oceanic hydrothermal vents that have been recently discovered along ocean ridges (Kerr, 1987). In general, the average temperature of the subterranean water will depend upon the average depth of the released water.

Since the pre-Flood atmosphere was likely warmer than at present (see Chapter 2), the pre-Flood deep ocean must have been warmer than the current average of 4°C. Therefore, I shall assume the average pre-Flood ocean temperature was 10°C. Since, in the present crust of the earth, the temperature warms with depth at 30°C per kilometer, I will assume the average temperature of the subterranean water was 200°C. This temperature corresponds to a depth of about six kilometers, assuming the same vertical-temperature gradient in the pre-Flood crust. The volume of the ocean now Isa 1:5 x 10⁹ km³ with an average depth of four kilometers (Considine, 1983, p. 2045). If the “fountains of the deep” added 10% more water to the pre-Flood ocean, the height of the ocean would have risen 363 meters, if contained only in the ocean basins, with no change in the elevation of the ocean bottom. The average temperature of the ocean, at the end of the Flood, would then be 27°C.

These figures are rough estimates. Possibly, an average temperature of 200°C for subterranean water is too hot. Maybe 100°C is more reasonable. On the other hand, the highest pre-Flood mountain could have been much higher than 363 meters, requiring a greater volume of ocean water to cover the tops of all the highest mountains (Gen 7:19). Disregarding the assumption that the extra water was contained only in the ocean basin, 363 meters would reduce to around 200 meters, depending upon the proportion of land versus ocean, and the average height of the pre-Flood continents. Although there is no way of knowing the variables that determined the average ocean temperature immediately after the Flood, I can envisage a very warm ocean. The fountains of the deep are a large heat source. Therefore, 30°C, likely an upper boundary, will be assumed as the initial ocean temperature. Initial ocean temperatures of 25°C and 20°C will also be used in calculating the time to reach glacial maximum. A temperature warmer than 30°C would have been a serious threat to marine life as we know it. Although the ocean would be well mixed, small regional differences in ocean temperature would be likely, and surface cooling at higher latitudes would begin soon after the Flood, relieving stress on marine life. In general, a uniformly warm temperature would occur from top to bottom, and from pole to pole, following the Flood. A source of moisture adequate for an ice age depends upon the ocean surface temperature near the area of ice buildup, which, in turn, depends upon the average temperature of the entire ocean. In today’s climate, the ocean temperatures are too cool to supply sufficient moisture for a continental snow cover that would last through a summer. This is because the solar radiation, in summer, as shown in Chapter 6, is very evident in melting snow. Thus, ice-age maximum would certainly occur at an average ocean temperature higher than at present. Therefore, I will assume the ice-age maximum occurred at a threshold ocean temperature of 10°C. Since the specific heat of ocean water Isa 1:1-31 cal/gm, a 20°C change in temperature, from the end of the Flood to glacial maximum, represents a heat loss of 3.0 x 10²⁵ calories.

Although this loss would be from higher-latitude surface cooling, one significant effect that would slow the oceanic cooling is the upwelling of colder water along the west coast of the continents in the Northern Hemisphere, especially along North America (Budyko, 1978, p. 90). Figure 4.2 shows the estimated areas of upwelling in the North Atlantic. In the present climate, upwelling transports colder, deep water, to near the surface, where it is warmed by solar radiation. At the beginning of the post-Flood ice age, upwelling would transport warm water to the surface. Only after a sufficient pool of cold water developed below the surface, would the upwelling become important. This would occur towards the later stages of the ice buildup, and would, consequently, not be too significant. Therefore, upwelling will be ignored. The effect of including upwelling, would be to make the ice age a little longer. Thus, Q will be set equal to 3.0 x 10²⁵ calories in Equation A1.8.

Radiation Balance at Top of Atmosphere

FRE is the difference between the incoming solar radiation that is absorbed by the earth-atmosphere system and the outgoing infrared radiation at the top of the atmosphere. Both will be strongly affected by the unique post-Flood climate. The amount of solar-radiation energy directed back into space will depend upon the amount of volcanic dust and aerosols trapped in the upper atmosphere, the snow cover, and cloudiness during time T. Since the average reflectivity is unknown, I shall assume a range of values, depending mainly on the volcanic dust and aerosol loading. The eruption of Krakatoa reduced sunlight four percent (Oliver, 1976, p. 936), and the April 1982 eruption of El Chichón, in Mexico, reduced it five percent at Fairbanks, Alaska, during a three-month period the following winter (Wendler and Haar, 1986). Mass and Portman (1989, p. 567) estimate modern-day volcanoes reduce the total sunlight five percent to seven percent for a limited time at higher latitudes. The extensive post-Flood volcanism must have reduced sunlight much more, since eruptions observed during the past 200 years are quite small, compared to ice-age eruptions (Kerr, 1989, p. 128). For a minimum average, I will assume 10% of the present solar radiation energy is redirected back to space. For a maximum possible average, which is unrealistic, I will assume 75% is lost from Earth. Values for a 25% and a 50% loss will also be calculated. Table A1.1 presents the values for the yearly average solar radiation at the top of the atmosphere, for each 10°-latitude band. The data for the present atmosphere was taken from Budyko (1978, p. 91). The infrared-radiation loss, at the top of the atmosphere, is generally proportional to the surface temperature (Budyko, 1978, pp. 93, 94). Infrared radiation will also depend, to a lesser extent, upon the atmospheric distribution of water vapor, clouds, and Col 2:1-23. Since latitudes poleward of 60° were significantly warmer in the post-Flood climate, infrared radiation loss must have been higher Accordingly, I adjusted current values for the high latitudes upward 20%. Values for the latitude band, 40°N to 60°N, were assumed the same as present values, because a warmer average temperature over the ocean would likely be balanced by colder average annual temperatures over the continents. Table A1.2 presents the yearly average infrared-radiation loss in the present and the post-Flood climates, for each 10° latitude band.

Table A1.3 presents the difference between the absorbed solar radiation and the outgoing infrared radiation for the four volcanic-dust and aerosol-loading scenarios. For instance, in Table A1.2, the value for post-Flood outgoing radiation in latitude band Matthew-50°N, which Isaiah 160.2 kcal/cm2-yr, is subtracted from the values for the four dust loading scenarios for that latitude band, from Table A1.1. The resulting four values are then multiplied by the area of the earth within that latitude band, which Isa 31:5 x 10¹⁶cm². The values in the latitude bands are totaled for each dust-and-aerosol-loading scenario, and these four values represent the range in FRE that will be used to solve Equation A1.8.

Poleward Heat Transport

FA and FO are the higher-latitude transports of heat by the atmosphere and ocean, respectively. These variables for the post-Flood climate need estimating. Both are residuals of the heat balance of the atmosphere and ocean at lower latitudes. Unfortunately, FA and FO are not precisely known, in the present climate, although the general features are understood (Charnock, 1987). The sum of the transports is reasonably known in the present climate, since the sum must balance the higher-latitude-heat deficit, which is FRE (de Szoeke, 1988, p. 585). The sum is what we need for solving Equation A1.8. Minimum and maximum values will again be estimated for the post-Flood climate. The minimum estimate will be zero. The maximum values of FA and FO for each hemisphere in the present climate, will be used as the maximum post-Flood estimates, because the post-Flood values would be less than today, as will be shown. An attempt will be made to find a best estimate for the post-Flood heat transports.

Satellite measurements, potentially, are the best method of observing the higher-latitude-heat balance (Vonder Haar and Oort, 1973). However, these estimates do have errors (Hall and Bryden, 1982; Luther and Herman, 1987, p. 134). Recently, improved satellite measurements have been employed to estimate FA and FO (Carissimo et al., 1985). The new estimates are not much different from the previous estimates of Sellers (Charnock, 1987), but are a little higher than those of Budyko (1978). The most recent satellite measurements are probably a little high, but will do for our estimate of maximum-heat transport.

Higher latitude heat transport varies with latitude, and is maximum at 40°N and 40°S. The values for the present atmosphere will be taken from Carissimo et al. (1985, p. 91), based on recently improved satellite estimates. The maximum higher latitude transport for both hemispheres Isa 8:4 x 10²² cal/yr, which is the assumed maximum for the post-Flood climate. The current transport poleward, across 40°N and 60°S, Isa 6:4 x 10²² cal/yr. With an estimate of FA and FO for the post-Flood atmosphere and ocean, we will be ready to solve Equation A1.8. The solar and infrared radiation balance in the tropics and subtropics during early post-Flood time, would be smaller than at present, due to volcanic dust. Less radiation available for heating, results in smaller values of FE and FC, and, also, probably smaller values for the higher-latitude-heat transports. (See Figure A1.1 for an illustration of the lower-latitude oceanic-heat balance.) The values for FA and FO would also vary, according to which dust and aerosol-loading scenario is used. And, the mean latitudinal atmospheric temperature difference would be less than at present, because of the much warmer temperature at higher latitudes. This would cause FA to be smaller, since it is proportional to the north-south temperature difference (Budyko, 1978, p. 94). A similar argument applies to FO since the north-south ocean-temperature difference would be small, especially at the beginning of the ice age. Consequently, I assume the best post-Flood estimate of FA and FO to be one-half the percent change in the post-Flood average solar radiation. For instance, if post-Flood solar radiation was reduced 25%, on the average, the higher-latitude-heat transport is assumed reduced 12.5%.

Table A1.4 presents the results for the various combinations of the terms in Equation A1.8. The estimated time for glacial maximum ranges from as little as 174 years, to as much as 1,765 years-both unrealistic-and both very short compared to geological time. In view of the solar-radiation decrease caused by modern volcanic eruptions, the best estimate for an average solar-radiation reduction until glacial maximum, is probably 25%. Therefore, from Table A1.4, the time to reach glacial maximum would be around 500 years. In the above calculations, Q, and the infrared radiation loss, were not given maximum and minimum ranges. Further calculations, with different values for these variables, have been tried. For instance, if I had used present values for post-Flood infrared radiation loss (Table A1.2), which would be a minimum, in the post-Flood climate, the best estimate for the time to reach post-Flood glacial maximum, would be about 640 years. By reducing the initial ocean temperature immediately after the Flood to 25°C and to 20°C, instead of 30°C, and using the best post-Flood estimates for the other variables, the time to reach glacial maximum would be 370 and 245 years, respectively. A corresponding decrease in ice volume would also occur with these temperatures (see Appendix 2). No matter which values are used for the variables in Equation A1.8, the main conclusion is still the same-glacial maximum is reached in a very short time.