Snowball Earth

A Simulation by Jacob West
June 7, 2000

Subtopics:

Abstract
Introduction
A Simple Model
Algorithm
Results
Conclusions
References
Further Illustrations
Source Code

Abstract:

Recently, geological evidence has been found to suggest that early in Earth's history there was a period in which the Earth was completely covered in ice. This period was first termed "Snowball Earth" in 1992 by Joseph L. Kirschvink, a geobiologist at the California Institute of Technology. Kirschvink pointed out that during a period of global glaciation, the liquid water required for the erosion of rocks and consumption of atmospheric carbon dioxide would be trapped in ice, however the production of carbon dioxide would continue as would tectonic plate motion and the formation of volcanoes. This production without consumption would cause a surplus of carbon dioxide, a greenhouse gas, and Kirschvink hypothesized, grow thick enough to substantially heat the planet and thereby bring an end to the global glaciation. To date, there has been no mathematical model to support this qualitative theory of our early Earth. In this paper, I outline an initial attempt to simulate the "Snowball Earth Effect" based on a very simple climate model. Although this is a crude model, it does demonstrate the basic phenomenon behind the theory of Snowball Earth, and I believe will serve as a stepping stone for more accurate (and hopefully more complex) simulations.

Introduction:

In August of 1998, the theory of a "Snowball Earth" was first introduced in the journal Science by a group of researchers including Paul Hoffman, Alan Kaufman, and Daniel Schrag. The idea has had cautious support in the scientific community, and has recently been a topic of increasing interest. As the theory goes, between 750 million and 580 million years ago, Earth froze over like a cosmic snowball for some 10 million years. The ice covered planet lacked the water necessary for erosion and was thereby unable to remove atmospheric carbon dioxide. The levels of carbon dioxide in the atmosphere continued to build during this time as tectonic plate motion formed CO₂ producing volcanoes. Over time, this heat trapping greenhouse gas caused the surface temperature on Earth to rise until the ice began to melt. Once underway, the Earth thaws very rapidly on a geological time scale - only a few hundred years. The Earth's climate however has now shifted from one extreme to another, from completely covered in ice to an inferno of a greenhouse effect! This severe climate reversal is believed to have happened as many as four times before stabilizing into less dramatic climate fluctuations. This theory has been extensively developed over the past eight years as geological evidence was uncovered in thick layers of ancient rock. Among other clues, this rock suggested that the Neoproterozoic oceans and atmosphere contained little or no oxygen, and that there was a prolonged drop in biological activity, both indicating a prolonged period of global glaciation.
To date, this theory has only been developed qualitatively with researchers focusing on piecing together the geological evidence into a reasonable story for our Earth's history. However, a mathematical or numerical model is needed to solidify these ideas into a more persuasive form, and essential before the theory will have wide spread acceptance.

A Simple Model:

The goal of this project has been to devise the simplest climate model possible, while complex enough to still produce the "Snowball Earth Effect." As a result, there are many obvious factors that would undoubtedly contribute to the onslaught of a snowball Earth that have been intentionally left out, including oceanic currents, regional and seasonal surface temperature variations, and currently solar intensity (however I intend to add that later on). The model I have adopted includes only surface temperature, albedo, and optical depth. It is summarized in the following dependency diagram.

Here the arrows imply dependencies. It is from this basic structure that I designed my Snowball Earth simulation, motivated by the belief that these factors will have the most profound effect on the Earth's climate over large time scales. Naturally, the surface temperature is dependent on the Earth's albedo, a measurement of the surface reflectivity indicating how much of the Earth is covered in ice, and optical depth, an indirect measurement of atmospheric gas concentration. Consequently, optical depth has dependencies on CO₂ and H₂O, our most prominent greenhouse gases.

Algorithm:

The following is the preliminary algorithm I have devised based on the above to simulate the snowball Earth effect. This is a schematic representation for explanation purposes. If you are interested in viewing the actual source code, it is available upon request only. Please click here for more information.

begin main code // main loop for all dependent functions

set initial conditions for CO₂ , H₂O, To, and T (represented as Tg / To)

for past to present // for loop beginning 800 million years ago and iterating forward in increments of 10,000 years

           Albedo ( function of T )
        CO₂ ( function of T )
        H₂O ( function of T )
        Optical depth ( function of CO₂and H₂O )
        Surface Temperature ( function of Albedo and Optical Depth )

loop

end main code

Quite simply, the main code is a loop representing the forward progression of time. It contains functions for calculating the individual contributions of albedo and optical depth (and their dependencies) to the overall surface temperature. All of the above functions are assuming averages over the entire planet. In other words, albedo should be understood as the average albedo over the Earth, surface temperature being the average temperature of the planet, and so on. A more accurate model would perhaps calculate the above values for localized patches of the Earth and then integrate over the entire surface. Below the functions in the main code are outlined in greater detail.

begin function code // code for the functions outlined above

    begin Albedo
        sigmoid function to determine y             // used to represent critical points after which the Earth quickly freezes or thaws
           linear approximations to determine y   // used outside critical points
                                           ;                                          // y is a temperature (T) dependent function determined from the above
         A = 0.55 - 0.35 * tanh ( y )          // Albedo equation
           return A
     end

    begin CO₂
            dPCO₂ / dt = Sources - Sincs = C - C * exp ( 35 * ( T - 1.0214 ) ) // where C is a constant representing the current CO₂ concentrations
           CO₂= Initial Concentration + Rate (above equation)
           return log ( 10 * CO₂) // this is because optical depth has a logarithmic dependency on carbon dioxide concentration
      end

    begin H₂O
          ln ( P_now / P ) = (dH / R) * [ ( 1 / T ) - ( 1 / T_now ) ] // Clausian-Clapyeron
           return P // vapor pressure for water at the variable temperature T
     end

    begin Optical depth
          tau = 0.5 * CO₂ + 1.5 * H₂O + NO₂// here NO₂is representative of background gases and contributes a constant 0.0371
           return tau
    end

    begin Surface Temperature
          T = [ ( 1 - A ) * ( 1 + tau ) ] ^ ( 1 / 4 )
           return T
     end

end function code

These functions together form what I believe to be the simplest possible mathematical model for the "Snowball Earth Effect." A more complex model would undoubtedly include solar intensity (varying with time as it has increased approximately six percent from 800 million years ago), oceanic currents (redistributing heat over the plant), and seasonal variations. Unforntunately, the higher the order of complexity in climate modeling the more chaotic it becomes, to the point of making the project a study of Chaos Theory rather than one of an evolving Earth.

Results:

The following graphs show the results of simulations run under four sets of conditions; 1) a control run with conditions set at those of the present day Earth to demonstrate that our environment is a stable solution, 2) a cold situation beginning at 252 K with initial carbon dioxide concentrations equal to today's levels, 3) a hot situation beginning at 308 K again with initial carbon dioxide concentrations the same as today, and finally 4) a frozen situation beginning at 240 K with initial carbon dioxide concentrations equal to twice that of today's levels.

1) Control

This graph shows that by this model, the current day Earth is a stable solution, as we should hope. Here T = 1.0214 corresponds to a global surface temperature of 13 deg C, assuming a To = 280 K. As you can see, there is an initial fluctuation for approximately 1 million years before stabilizing at conditions analogous to those of today. On a geological time scale, this is a very small fluctuation.

2) Cold Situation

This graph shows a cold situation with an initial temperature of 252 K. It is difficult to see, however the temperature begins at T=0.9 before quickly droping within the first 10,000 years. Consistent with theory, a "Snowball Earth" forms here and maintains for approximately 16 million years (slightly longer than the theory predicts) while carbon dioxide levels build. The snowball slowly melts until it reaches a critical balance, after which the great majority of the ice quickly melts and the now thick carbon dioxide gas causes a temperature spike in a violent greenhouse backlash. This greenhouse effect dissipates within 20,000 years as heightened levels of erosion consume the carbon dioxide leaving the Earth slightly colder than it is today. As normal carbon dioxide and water levels return, residual ice is melted and the temperature slowly rises. There is another miniature fluctuation as ice, carbon dioxide, and water levels come into a balance, from which point a stable environment comparable to today's is reached. (Note: This stabilization maintains for all time afterwards, so only the interesting portions are shown here.)

3) Hot Situation

Here I demonstrate the model's response to a very high temperature corresponding to an intense greenhouse effect. As dictated by the rate equation for carbon dioxide described above, such high temperatures causes the production of atmospheric CO₂ to be overwhelmed by the Earth's erosive consumption of the gas. Consequently, the temperature quickly drops within a few million years and eventually returns to a stable solution.

4) Frozen Situation

In this situation, the Earth begins already in the snowball effect, however carbon dioxide levels are twice that of today's. Here we see a behavior very similiar to the cold solution with exception of reaching a stable solution faster, as we would expect given the increased carbon dioxide levels.

Conclusions:

In conclusion, although this is a relatively simplistic numerical model, it indeed supports the present day theory of "Snowball Earth," and in so doing serves to enhance the possibility that at one time the Earth was in fact a cosmic snowball for millions of years. I would hope that from this initial simulation, others might build a more thorough model to help better explain this very interesting phenomenon in our Earth's history.

References:

    1. Hoffman, P., Schrag D., "Snowball Earth," Scientific American, January (2000).

    2. Stevenson, D., "Geology 1," Caltech course, June (2000).

Last Modified: 06/10/00