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Mikhail I. Budyko's Ice-Albedo Feedback Model

Mikhail I. Budyko's article, "The effect of solar radiation variation on the climate of the Earth," published in 1969 was one of the first theoretical investigation of the ice-albedo feed back mechanism; i.e., the more snow and ice the more solar radiation is reflected back into space and hence the colder Earth grows and the more it snows. Budyko notes that over the last 200 million years the temperature difference between the poles and the equator was comparatively small compared what it is today. He says that during that time there were no "zones of cold climate."

It was only in Quaternary times that Earth's polar regions became covered with glaciers and a large temperature differential developed between the poles and the equator. His article reports on his investigation of the mechanism for the initiation of glaciation.

He and his group at the Main Geophysical Observatory at Leningrad found that over the period from the end of the 19th century until 1940 there was a rise in the average temperature of the Earth of 0.6° C. From 1940 until the mid1950's there a fall in temperature of 0.2° C.

One major purpose of the article was to provide quantitative analysis of how to explain the variation of glaciation in the Quaternary Period. He considered two sources of variation in the amount of heat absorbed from the sun:

Budyko concludes that variations in volcanic dust could account for the Quaternary variations in glaciation, but the Milankovitch Theory can not. An important aspect of Budyko's analysis is that the latitude of the lower limit of glaciation is a key variable. At the present time the polar glaciations extend to an average of about 72°. The Miklankovitch theory, according to Budyko, would only account for a change in that lower limit of 1°; whereas the variation in volcanic dust would lead to a change in that lower limit of glaciation of 10° to 18° which corresponds to the actual variation in glaciation.

A key result of Budyko's analysis is that there is a crucial latitude such that if the ice coverage reaches that level it will, due to the linkage of ice/albedo/radiation absorption, go on to cover the Earth. Budyko estimates that crucial latitude to be 50°.

Budyko's Model

Based upon his extensive previous work on the heat budget of the Earth Budyko estimates the outgoing rate of radiation energy per unit area, I (kcal/cm²/month), at a location by the following equation:

I = 14.0+0.14T - n(3.0+0.1T)

where T is temperature of Earth's surface at that location in °C and n is the proportion of cloud coverage.

Budyko is taking top-of-the-atmosphere outgoing radiation to be linear functions of the surface temperature for the non-cloud covered (say a0+B0T) and the cloud covered (say a1+B1T). Then outgoing radiation energy per unit area I would be given by:

I = (1-n)(a0+B0T) + n(a1+B1T)
which then reduces to
I = a0+B0T - n[(a0-a1)+(B0-B1)T]

Therefore Budyko is taking the outgoing radiation from cloud coverage surface to be 11.0+0.4T.

The first equation can be used to compute outgoing radiation energy for a latitude band or for the whole surface of the Earth. For later work let the first equation be expressed as

I = a0+B0T - n(Δa+ΔBT).

The second equation of Budyko's model is the heat balance equation for the Earth:

A = Q(1−α) - I

where A is the net gain in heat energy per unit area (often presumed to be zero), Q is the solar radiation per unit coming in at the top of the atmosphere, α is the albedo and I is again the outgoing radiation enery per unit area.

In effect, this second equation for the whole Earth (A=0) is really

Ip = Qp(1−αp)

where the subscript p for a variable indicates that it is the average for the surface of the Earth.

The first equation, expressed for the Earth's surface, is

Ip = a0+B0Tp - n(SΔa+SΔBTp)

where Tp is the average temperature of the Earth.

Combining the two equations and solving for Tp gives:

Tp = [Qp(1−αp) - (a0 - nΔa)]/(B0−nΔB)

Budyko wants to compute the average temperatures for latitude bands but for a full analysis that would require consideration of the horizontal heat transfers by the atmosphere and the hydrosphere. This was not practical given the state of the art at the time he was doing his investigation so he opts for a simplified scheme in which the net heat outflow is given by:

A = β(T-Tp)
A = Q(1-α) − a0+B0T - n(Δa+ΔBT)

Combining these two equations and solving for T gives

T = [Q(1-α) − (a0 - nΔa) + βTp]/(β + (B0−nΔB))

The value Budyko uses for β is 0.235 kcal/cm²/month-degree. He uses 1.92 kcal/cm²/minute for the calculation of both Qp and Q. However Q for a particular latitude band depends upon the cosine of latitude. The average cloudiness for the Earth is taken to be 0.5 and this same value is used for all latitude bands. Cloudiness is presumed to be independent of temperature.

Albedo does vary with latitude in Budyko's model. The variation with latitude is

0° - 60°0.32

In the analysis using the model if the glaciation changes the albedo of the glaciated area is set equal to 0.62 and the band immediately closer to the equator is given an albedo of 0.5.

The model is symmetric with respect to hemispheres.

Variation in Parameters

Budyko derived from his fundamental relations an equation for determining the average surface temperature of the Earth; i.e.,

Tp = [Qp(1−αp) - (a0 - nΔa)]/(B0−nΔB)

In his analysis Budyko considers some exogenous variations in Qp due to factors such as changes in the characteristics of Earth's orbit or variation in the transparency of the atmosphere due to volcanic dust. There are endogenous changes in the average albedo for the Earth. Suppose the changes are ΔQp and Δαp and that they are relatively small. Then the change in Tp is given by

ΔTp = [ΔQp(1−αp−Δαp) − QpΔαp]/(B0−nΔB)
which Budyko chooses to express as
ΔTp = [Qp/(B0−nΔB)][(ΔQp/Qp)(1−αp−Δαp) − Δαp]

The change in albedo Δαp can be computed as

Δαp = 0.3lq

where l is the ratio of the area that changed from non-glacier to glacier to the total area of the Earth, q is the ratio of the average radiation in the zone of change to the average radiation of the Earth. The 0.3 comes from the difference in the albedo of a glacier, 0.62, and the albedo of non-glaciated Earth, 0.32.

When this expression for Δαp is substituted into the equation above for ΔTp the result is Budyko's Equation (6).

Budyko's equation for the average temperature of a latitudinal band, when some of the variables have changed from their current values, is

T = [(1-α)(Q+ΔQ) - (a0-nΔa) + β(Tp + ΔTp)]/(β+B0-nΔB)

The term (1-α)(Q+ΔQ) Budyko chooses to express as Q(1-α)(1+ΔQ/Q) and takes ΔQ/Q as being equal to ΔQp/Qp. Note that in this term the albedo at a particular latitude is a function of temperature because it depends upon glaciation. The latitudinal temperature profile must be solved for iteratively. The preliminary computatins of temperature indicate the latitudes where the temperature falls below the critical level at which glaciation occurs. To predict the post-glaciation temperature at a latitude Budyko's model must continue the iterations until the albedo profile stabilizes.

The graph below shows Budyko's computed temperatures based upon current parameter values and the observed latitudinal temperatures.

The Results of Budydo's Analysis

Budydo considers decreases in radiation of 1 percent and 1.5 percent. His model predicts decreases in the average surface temperature of the Earth of 5°C and 9°C, respectively. The lower limits of ice covering are displaced 10° and 18° of latitude toward the equator.

When the radiation input is decreased by 1.6 percent the lower limit of ice covering preliminarily reaches 50° and then continues until it reaches the equator.

Budyko found that if the latitude of ice covering reaches 50° of latitude it will continue on until it reaches the equator even if the level of radiation is returned to the present level.

Budyko used his model to predict what the latitudinal temperature profile would be if the polar ice cover were removed.

The results indicated that there would be relatively small increase in temperature for latitudes below 60°, on the order of 2°C, but in the polar regions the increase would be on the order of 10°C. This would mean temperatures in the polar regions of only a few degrees below zero and possibly not cold enough to generate an ice covering for the region. Budyko emphasizes the instability of such an ice-free regime in the Arctic.

Budyko compares the variation in radiation input necessary to produce the variation in ice covering of the past geologic era and concludes that the variation involved in the Milankovitch theory is insufficient to account for past glaciations. However, he concludes that the variation in heat absorbed by the Earth due to variation in the volcanic dust content of the atmosphere is sufficient to explain the periods of glaciation.

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