Albedo Modification

Albedo — Albedo is the reflective power of a surface. In the case discussed in this article, it is the reflective power of the atmosphere, and it describes how much of the incoming solar radiation is reflected by the atmosphere.

Abstract: This article describes my calculations related to the albedo enhancement of the Earth's atmosphere. I have found that if we wanted to compensate for the increase in global warming caused by cleaning the atmosphere of human-made pollution, we would have to deposit 1-2 Tg of reflective material in the stratosphere per year at the annual cost of US$23-45 (using 1992 prices). The range of values depend on the residence time of the reflective material. If in addition we would like to protect ourselves from the increased global warming between now and 2050, caused by our continuing use of fossil fuels, we would have to deposit a total of 2-6 Tg of reflective material per year at the estimated annual cost of US$45-135 billion. I have derived a simple mathematical formula, which describes the mechanism of construction of the albedo enhancing shield. I also give formulae, which can be used to calculate the cost of construction using rockets. The cost of construction using hydrogen-filled balloons is approximately the same as for rockets, but cost of the construction using helium filled balloons is about 4 times higher.

Introduction

As if we did not have enough trouble with global warming, we have now created a problem of global dimming. Through our interference with nature we have placed ourselves in a trap without an easy way to escape.

The use of fossil fuels creates two forms of atmospheric pollution: pollution with greenhouse gasses, mainly carbon dioxide, and pollution with aerosols (small particles). Greenhouse gases reside in the atmosphere for a long time. Their residence time is unknown but is estimated at hundreds of years. Aerosols, on the other hand, stay in the atmosphere for a short time. Their residence time depends on their size, chemistry, and location in the atmosphere, but typically they are removed from the lower part of the atmosphere within days. For instance, much of human-made atmospheric pollution contains sulphur. The residence time of sulphur dioxide is 0.6-2 days and of sulphates 2.7-7.2 days. Various natural processes contribute to the production of aerosols but human activities have now a significant contribution.

Aerosols have a beneficial cooling effect on the Earth's surface. The effect is either direct or indirect. The direct effect occurs when aerosols scatter solar radiation, which otherwise would have reached the Earth's surface. The indirect effect is through the formation of clouds. Droplets of water are formed around aerosols and the created clouds reflect the solar radiation. The cooling effect of human-made aerosols is called global dimming.

The problem with aerosols is not only because they are quickly removed from the atmosphere and thus have to be replaced by new aerosols, but also because they cause a variety of ecological and health problems. When aerosols containing sulphur or nitrogen are incorporated in the clouds they return to the Earth's surface in the form of acid rain. Aerosols can be carried by the air currents and deposited in places far removed from places of their production. Harmful effects of aerosols are discussed in my book, The Little Green Handbook.

So here we have Catch 22: we need aerosols to keep us cool but we don't need them because they cause ecological and health problems. To reduce the effects of global warming we have to keep on polluting the atmosphere because aerosols do not stay there long, but by polluting the atmosphere we are also increasing the concentration of greenhouse gases, which stay in the atmosphere much longer and cause global warming. Thus to protect ourselves from global warming we have to increase global warming. This is what I call the trap of global dimming.

Maybe we could escape the trap by abandoning the use of fossil fuels. Without replacing the aerosols, the surface temperature would increase but we might hope that it would not increase too high. However, we cannot try this solution because we depend almost entirely on fossil fuels.

It has been suggested that we might solve the problem, at least temporarily, by creating a protective shield made of aerosols deposited in the stratosphere. The advantage of having aerosols in the stratosphere is that they remain there over much longer time than in the troposphere, for about 1-2 years. Also, we would not be in an immediate contact with them, so hopefully we would not suffer health problems. It would also appear, that depending on the type of aerosols they would probably be environmentally harmless. However, all these issues would have to be carefully discussed and assessed. Aerosols would provide a reflective shield for the sunrays and would compensate for the increased global warming caused by the cleaning the lower part of the atmosphere of the human-made pollution.

The apparent need for geoengineering illustrates the immensity of the problem we have created. There is no simple solution. We are already involved in uncontrolled geoengineering in the form of changing the composition of the Earth's atmosphere through the use of fossil fuels. Maybe there is no other way out but to fight fire with fire and use a well-designed and well-controlled geoengineering project.

I have carried out the presented here calculations in connection with the proposed albedo enhancement by injecting sulphur into the stratosphere. I was fascinated by this idea and I wanted to understand how such a massive geoengineering project would work. Part of my calculations involved an estimation of the costs, which interested Professor Dr Paul Crutzen and which he used in his publication, "Albedo Enhancement by Stratospheric Sulfur Injections: A Contribution to Resolve a Policy Dilemma?", Climate Change 77(2006)211-220.

THE DYNAMICS OF ALBEDO MODIFICATION

Preliminary considerations

To understand the process of construction of an albedo enhancing shield in the stratosphere we have to include in our calculations not only the amount of the reflective material we can deploy in a given span of time but also the gradual deterioration of the shield because of the limited residence time of the deposited material. The amount of material removed by natural processes from the atmosphere is proportional to the amount of material present at any given time:

where N is the amount of material, t is the time and λ is a constant characterising the considered material.

Solution of this simple equation gives:

which gives:

The residence time is defined by N/N₀=1/e, i.e. it is the time when the original amount of material is reduced to 1/e, which is approximately 37%. Thus, the residence time T is given by:

The residence time of aerosols in the stratosphere is estimated at between 1 and 2 years, which corresponds to λ = 1 and 0.5, respectively.

Gradual deployment

The amount of albedo enhancing material is so large that it is impossible to deposit it in one shot. We have to do it gradually. But this creates the problem because while we are depositing new material, the material we have already deposited decays. It is like filling in a leaking container with water.

So now the questions are, how long and how fast do we have to keep on deploying the albedo enhancing material into the stratosphere to create a protective shield of required concentration? Once we have created it, how much shall we have to keep on adding to compensate for its continuing deterioration? The remaining calculations are aimed at answering these questions.

Let us assume that we measure time in years. Let the total injection period be t years. To calculate how much material we have to deploy we can divide the total injection period t into small sections and calculate how much of the injected material during this small section of time will remain in the atmosphere at the end of the total injection period t.

Let m be the number of assumed small injection periods per year and M the total number of injection periods during the total injection time t. The length of time τ₀ of each assumed small injection period is of course:

and the total length of injection period is:

Let W be the total amount of material deployed during the time t. This time could be one year or more. The amount of material deployed during each small period τ₀ is:

and the amount of material deployed per year is:

Let us label each consecutive small section of time τ₀ using index i. Let L_i be the amount of material that will remain in the atmosphere at the end of time t from the material injected in the section i. Using the basic equation for the decay, we can see that this amount is given by:

The total amount of the material that will remain in the atmosphere at the end of time t is given by:

Therefore

which can be reduced to:

After integration we get the following simple formula:

which can be rewritten more conveniently as:

where

— the thickness of the protective albedo enhancement layer after the total injection time t.
— The amount of albedo-modifying material injected per year.
λ  — The decay constant (e.g.1 if residence time is 1 year or 0.5 if residence time is 2 years)

We have derived a simple but interesting formula. It shows that if we continue injecting a fixed amount of albedo-modifying material, the thickness of the protective layer will be at first increasing rapidly but after a certain time it will gradually level off and continue approaching an asymptotic value of . This asymptotic value is the maximum level of protective layer we can construct by annual injections of a fixed amount of the albedo-modifying material.

The figure below illustrates this time-dependent behaviour. (Tg =10¹² grams, that is, trillion grams.)

The dynamics of albedo modification of the Earth's atmosphere

So for instance, by regularly injecting 5 Tg per year, we can construct a 10 Tg protective layer if the residence time is 2 years (λ = 0.5) or 5 Tg if the residence time is 1 year (λ = 1). To construct a 10 Tg layer for the residence time of one year we would have to increase the regular annual injections to 10 Tg.

We can also calculate how much time we need to have to construct a shield of a required concentration. Here we have to consider the ratio

The number of years t’ required to reach a satisfactory level of construction is given by:

Thus for instance, the construction will reach 99% of the saturation level in 4.6 years if the residence time is 1 year or 9.2 years if the residence time is 2 years. This might sound counterintuitive but we should remember, that with fixed injections and longer residence time we shall be contracting a thicker protective layer (see the enclosed figure).

We could deposit larger amounts of material to build the protective shield faster but this approach would be unadvisable. Depending on the delivery system, fast deployment could create environmental, technical, and financial problems.

We can also see that in order to maintain a protective layer at a required concentration we would have to continue injecting large quantities of the albedo enhancing material forever. Clearly, this is not a solution to global warming but only a temporary solution while we try to clean up the atmosphere and to reduce emissions of greenhouse gases.

ESTIMATION OF THE COSTS

Approximate calculations

Carbon concentration in the atmosphere increases. If we want to protect ourselves from the expected increase in global warming, we have to consider not only the increase caused by removing human-made aerosols from the atmosphere but also the additional increase caused by our continuing use of fossil fuels.

Let us first calculate the thickness of the protective shield we would have to construct to compensate for the increase in global warming caused by cleaning the atmosphere of the human-made pollution. The protective power of aerosols is labeled in scientific literature as negative radiative forcing or negative climate forcing, and is measured in Watts per square metre (W/m²). We can change "negative radiative forcing" to a more convenient label radiative cooling, which is then represented by positive numbers.

Estimations of the radiative cooling of atmospheric aerosols vary typically between 1.4 and 2.2 W/m² , which give the mean value of 1.8 W/m². Crutzen (private communication) used 1.4 W/m² and estimated that to compensate for the removal of human-made aerosols we would need to construct a 1.9 Tg sulphur shield.

I used a different approach to estimate the thickness of the protective shield. My estimation, which does not depend on the type of the reflective material, were made in the following steps.

First, I have analysed carbon dioxide concentrations dating back to 1765 and the corresponding increase in radiative forcing of carbon dioxide. I have found that at the beginning of the past century, for each 46 ppmv increase of carbon dioxide concentration, radiative forcing increased by 1W/m². At the end of the last century, this value increased to 60 ppmv. Using a conservative linear approximation gives me a value of 55 ppmv. Thus at present, for each 55 ppmv increase of carbon dioxide concentration in the atmosphere, radiative forcing increases by 1 W/m².

If the radiative cooling by human-made aerosols is indeed 1.4 W/m², as assumed by Crutzen, then by removing them from the atmosphere we would increase the effective radiative forcing of greenhouse gases by 1.4 W/m², which is like increasing carbon dioxide concentration by 77 ppmv.

Next, to calculate the thickness of the protective shield we also need to know the relationship between carbon dioxide concentration expressed in ppmv and the concentration and the weight of carbon. This relationship can be calculated using the mass of the troposphere. The calculations give the following prescription:

1ppmv =2.13 GtC

The equivalent increase in carbon dioxide concentration by 77 ppmv is like adding an extra 164 GtC to the atmosphere .Finally before we take the final step, we need to know what thickness of the albedo enhancing shield can mitigate this amount of carbon.

In 1992, National Academy of Science published a voluminous book entitled Policy Implications of Greenhouse Warming: Mitigation, Adaptation, and the Science Base. Let's abbreviate it as NAS-1992. The book contains many small errors (as one could expect in such a big volume) but with proper care it's possible to navigate around them. The authors claim that a 10 Tg (Tg stands for teragram or trillion grams) albedo enhancing shield can mitigate 1 TtC (TtC stands for trillion tonnes of carbon) in the atmosphere, which means that 1 Tg albedo enhancing shield mitigates 100 GtC (GtC stands for gigatonne of carbon or billion tonnes of carbon).Thus to compensate for the increase in surface warming caused by the removal of human-made aerosols we would need to construct a 1.6 Tg protective shield. This value is close to the value of 1.9 Tg of sulphur estimated by Crutzen.

If we assume that radiative cooling of aerosols is 1.8 W/m², then to compensate for the increased radiative forcing caused by cleaning the atmosphere of aerosols we would have to construct a 2.1 Tg protective shield. For the sake of cost estimates we can assume that we need a 2 Tg shield.

To construct and maintain such a shield we would have to deposit 1 Tg of reflecting material per year (indefinitely) if its residence time is 2 years or 2 Tg per year if its residence time is 1 year.

Now we can also calculate the additional thickness of the protective shield if we also wanted to protect ourselves from an increase in global warming between now and say 2050 caused by our continuing use of fossil fuels. We have to understand that while we continue burning fossil fuels, our problems increase. The longer we wait the more difficult it will be to solve them.

At present, the estimated carbon dioxide concentration is 375 ppmv (parts per million per volume). The projected concentrations vary over a large range of values (see my book, The Little Green Handbook). My calculations indicate that by 2050 we shall have 450-500 ppmv. IPCC estimations give 480-580 ppmv and UNEP gives 450-500 ppmv. Thus, by 2050, we can expect an extra 75-205 ppmv of carbon in the atmosphere.

The Earth is already too warm and the increasing concentration of carbon dioxide will make it even warmer. Indeed, IPCC estimates that by 2050, the average surface temperature of our planet will increase by 1.1-3.7⁰C. This should be compared with the average increase of only 0.6⁰C in the last century, which is already not only uncomfortable but also dangerous. Thus, by 2050, the temperature increase will at least double as compared with the increase in the last century.

Thus, the additional 75-205 ppmv of carbon concentration in the atmosphere by 2050 would mean an additional 160-437 GtC. To mitigate it we would need 1.6-4.4 Tg albedo enhancing shield, which gives the combined thickness of 3.6-6.4 Tg (approximately 4-6 Tg).

Using the1992 prices given in NAS 1992 we can calculate that the approximate cost of deploying albedo enhancing material in the stratosphere is $25 billion per Tg. To construct and maintain a 4-6 Tg shield we would have to deploy 2-6 Tg of albedo enhancing material (depending on the residence time) per year (indefinitely) at the annual cost of $50-$150 billion.

If our emissions of greenhouse gases are going to continue to increase after 2050, we will have to increase the thickness of the protective shield. However, maybe by then we shall manage to reduce the use of fossil fuels.

Formulae for the exact calculations of the costs.

Two ways have been suggested to deploy albedo modifying material in the stratosphere: rockets and or balloons. According to NAS-1992, the cost of delivery for both is the same as long as balloons are filled in with hydrogen. If they are filled in with helium, the cost is about four times higher. As an example of exact cost calculations I shall consider here the rockets. I base my formulae and calculations on the data supplied in NAS -1992.

The number of rocket launches (naval rifles or guns) N needed to deliver a total weight of the required material per year can be calculated using the following simple formula:

where
w – weight of deployed material per shot (per rocket)
f – number of shots per hour for each rocket launcher
t – number of hours of shooting per day
d – number of shooting days per year

Calculations of the costs have to include a number of components as listed below:

The cost P₁ of rockets:

where p₁ the cost of a rocket and the albedo enhancing material it will carry to the stratosphere.

The cost P₂ of barrel replacement:

where

p₂ the cost of a gun barrel
n₁ – the number of shots per gun before the barrel needs to be replaced.

The rockets are arranged is stations so we have to calculate the cost P₃ of running the stations.

where

q – a factor accounting for extra guns that will be out of action for maintenance (1<q<2).
N – as before, the number of working guns at any given time.
n₂ – the number of guns per station
p₃ – the cost running a single station. This cost includes overheads, electricity, etc. We can also include here the capital cost for constructing a station and spread it over the number of years of expected operation.

In the calculations, the total number of guns, qN, is rounded up to an the nearest higher integer number. In my calculations, I round it up to the higher 10 (see below).

The salary P₄ for the personnel working at the stations:

where
n₃ – the number of people per rocket launcher per shift
s – the number of shifts
S – the annual salary per person
D – number of working days per year excluding weekends and holidays.

As an example I use input data as suggested in NAS-1992. They are summarised in the table below.

Results of the calculations are presented in the table below:

Environmental implications of atmospheric geoengineering

To be continued