Global Climate Change - Economics, Science, and Pol Homework set 1

Homework #1 – Science of Climate Change This homework set must be completed on your own and you are not allowed to work in groups on it. This homework set will not require any complex or difficult mathematics, and is more a means of testing your understanding of the basic science of the climate system. Please, do not be worried about your ability to perform the math required! I recommend that you start early, as the problems may not take too long to solve once the numbers are in hand, but often require your doing some simple research to obtain the numbers needed to solve the problems. Problem 1: Simple Climate Model This problem will start with your constructing a simple model to compute the energy balance of the Earth. In the most basic sense of conservation of energy, the energy from the sun which enters the Earth is balanced by the energy from the sun which is reflected from the Earth and the Infrared Radiation (IR) emitted by the Earth. If this were not the case, then the temperature of the Earth, on average, would either increase or decrease, until such time as the energy entering the Earth balanced the energy leaving the Earth. Luckily, the laws of physics determine that the warmer that a blackbody gets (what the Earth is assumed to be) the more radiation it emits, so that in the case of an imbalance, the Earth would warm, emit more IR, and thus bring itself back into equilibrium. To start this problem off, we assume that the flux of radiation from the Sun is a constant value of S [W/m2] (Look This Up!). Note that this flux is not the average value that the Earth receives from the sun. In fact, on the dark side of the planet, it is assumed that the average value is zero. Thus, the average value that the Earth receives must be smaller than this amount. To compute this average, look at Figure 1 and follow these words: the flux of intercepted radiation from the sun is bounded by a disk, the radius of the Earth, however this same flux of radiation must impinge upon the entire surface area of the Earth. Figure 1 Question 1: What is the average flux of radiation from the sun that impinges upon the Earth's surface, given in the units of [W/m2]? This constant will now be called Ω. Thus, to be in energy balance, the amount of energy leaving the Earth must equal Ω. First we will assume that the radiation impinging upon the Earth can be assumed to be in the visible portion of the electromagnetic spectrum, or can behave as if it were in the visible portion of the spectrum. This assumption will be relaxed later in this problem. Then, with the assumption applied, we want to look at the fraction of radiation that is immediately reflected back to space. To do this, we will ignore the impacts of radiation scattering, bending, and all other effects that a real atmosphere may impinge on the incoming radiation scheme. This value, called the Planetary albedo, is the fraction reflected back to space, and will be denoted as α (Look up a reasonable value for this!). The second source of radiation leaving the Earth is going to be assumed to be completely in the IR. To explain this, let us make the basic assumption that the Earth is a blackbody. This means that the Earth absorbs all electromagnetic energy which impinges its surface and radiates electromagnetic energy only as a function of its temperature. This is not an exact assumption to make about the solid Earth and it is a weaker assumption for the atmosphere, but for now, it will do. Also, to assume that all of the radiation emitted is in the IR is an incorrect assumption, but since most of the radiation emitted in the case of the Earth is in the IR, it is a pretty good assumption. This assumption dictates that the energy emitted is given by W = σT4 where σ = 5.67x10-8 is the Steffan-Boltzmann constant. Question 2: Given the incoming energy flux Ω, and the outgoing energy fluxes αΩ and W, set up an equation representing the energy balance of the Earth. Next, solve the equation for the average Temperature of the Earth. Finally, look up the average present-day temperature of the Earth and compare your value with this value. Do you have any guess why your value is so far off (hint: what are you comparing)? It is clear from the above results that our model does not do a good job at representing the actual surface temperature of the Earth. So, now we will try to improve our model. The first thing that we will do is to add in a very simple atmosphere. First, we will assume that this very simple atmosphere is only capable of transferring energy by means of radiation. Second, we will assume that the atmosphere behaves as a black body in the IR. Third, we will assume that the atmosphere is transparent to visible radiation, except for the effect of the Earth's albedo, as given above, which we will assume will be a function of clouds and the surface, and thus having no impact on the atmosphere. Fourth, we will assume that the atmosphere is composed of two layers. If the IR emitted from each layer is assumed to be emitted equally upwards and downwards, then we can set up a model following Figure 2 (the fact that for each layer of the model, the energy in must balance the energy out). Figure 2 Question 3: Given the incoming energy flux Ω, the outgoing energy fluxes αΩ and W, and that the atmosphere is two layers thick, set up the set of equations representing the energy balance of the Earth Te and the two atmospheric layers T1 and T2. Next, solve the set of equations . Finally, look up the average present-day temperature of the Earth and compare your value with this value. Do you have any guess why your value is so far off? Finally, we are going to consider building a more complex and realistic model. To do this, many assumptions made above could be weakened, or physical and chemical processes not yet considered could be taken into consideration. Some of these aspects to be considered include a more realistic treatment of the incoming electromagnetic radiation scheme, a more realistic treatment of the outgoing radiation scheme, a more realistic treatment of the atmosphere's behavior, and a more realistic treatment of the Earth's surface behavior. As a few examples to get you thinking in this new way: the incoming electromagnetic radiation stream is not purely visible radiation, the outgoing IR radiation is distributed over different portions of the IR spectrum, the atmosphere itself moves, and the fact that more than 70% of the Earth's surface is water. Question 4: Think about some of the ways in which this model could be improved, using the above paragraph as an initial guide. Can you come up with 4 additional fluxes of energy that should be incorporated into our simple climate model? For each of these fluxes of energy, please explain the scientific rationale for this flux of energy, and indicate in terms of words and an approximate value [W/m2], how this would change the energy balance of each box of our climate model. For example, if we found that there was a flux of energy from the surface to the upper atmosphere, X, which took energy from the surface and transported it to the upper atmosphere, then we could subtract X from the surface box and add X to the upper atmosphere box. What are your new temperatures now? How do they compare? Finally, to address future unsolved problems, we should think about how effects of greenhouse gases, aerosols, changes to the land, and other changes to the Earth can change the net equilibrium surface temperature. Question 5:By what distance would the Earth, if it were assumed to have a circular orbit around the sun, have to move closer to the sun and further from the sun respectively, to cause a change of 1C at the Earth's surface? You can use the simple model from Question 3 to solve this! How would emissions of greenhouse gases change the structure of the model from question 4 (give 2 different direct ways in which you would change the model, but you do not have to actually change the model)? How would changes of aerosol emissions change the structure of the model from question 4 (give 2 different direct ways in which you would change the model, but you do not have to actually change the model)? Can you think of any indirect changes to the model from question 4 that would occur due to small changes in the greenhouse gas and aerosol concentrations? How would these change the structure of the model from question 4? Problem 2: GWPs The GWP of a radiatively active gas is defined as the ratio of the time integrated radiative forcing of the release of 1kg of the given gas to the time integrated radiative forcing of the release of 1kg of CO2. In simple words, this looks at the impact on the radiation stream of the release of a kilogram of a given gas, summed over how long that gas will survive in the atmosphere, as compared with the impact on the radiation stream of the release of a kilogram of CO2, summed over how long the CO2 will survive in the atmosphere. In terms of an equation, it is represented as: (IPCC TAR) where ax is the radiative efficiency of the gas of choice, x(t) is an equation of the concentration of the pulse of the gas (usually of the form: exp(-t/τ)), τ is the lifetime of the gas of choice, and TH is the time horizon over which we are integrating. All variables have the same meaning for the case of CO2 (the reference gas). In this formulation ar = 0.01548 Wm-2/ppmv for a small perturbation from a CO2 concentration from 364 ppmv, and the lifetime τCO2 = 80 years. To solve for the remainder of the parts of this problem, you are going to need to know the current concentration of CO2 in the atmosphere (look it up!). Question 1: Using the information given, compute the GWP of CH4, N2O, and SF6 over the following Time Horizons: 20 years, 100 years, and 500 years. ACH4 = 3.7x10-4, AN2O = 3.1x10-3, ASF6 = 0.52, AHFC-23 = 0.16, τCH4 = 12 years, τN2O = 114 years, τSF6 = 3200 years, τHFC-23 = 260 years. All of these are from the IPCC TAR. Why would we want to look at this problem over such different time horizons (are there any policy implications or scientific rationales for choosing each of these values)? Now that we have computed what these GWPs are, we will look at some of the advantages and uncertainties associated with using this approach. The largest advantage is that it places a comparison of the different greenhouse gases on an equal footing, providing a quantitative way to assess the abatement of the emissions of two different gases. However, this method has many possible pitfalls associated with it. The first is that it is only applicable to species which have a long enough atmospheric lifetime to be uniformly distributed around the Earth. Secondly, it does not take into consideration the fact that linear changes to the concentrations of greenhouse gases may not correspond to a linear response to the radiative forcing. Thirdly, it does not take into consideration that the lifetimes of the greenhouse gases themselves may change over the time horizon of integration. Question 2: It has been recently shown that emissions of CO effect the radiative forcing of the atmosphere through a few different processes: CO ultimately chemically converts to CO2, CO removes OH from the atmosphere, thereby increasing the lifetime of CH4, and CO forms Ozone (a powerful greenhouse gas which itself does not have a GWP associated with it because of its highly heterogeneous spatial and temporal distributions in the troposphere). If increases in future CO emissions are assumed to increase the lifetime of CH4 to 14 years, what would be the new GWP for CH4 at each time horizon? What if future CO emissions increased the lifetime of CH4 to 16 years? A different way to look at this problem would be to assign a GWP to CO instead of to looking at the changes in the CH4 gwp. However, the lifetime of CO is on order of 1-2 months in the troposphere. Could we use the approach in this section to form a reasonable GWP for CO (give some reasons why or why not, you do not need to come up with an approximate GWP for CO – This is an active field of research!)? Can you give a minimum value for a GWP for CO, and a reason for why you have chosen this as a minimum value? Additionally, the lifetime of CO2 is highly variable. Changes to the land surface, types of vegetation, ocean mixing, or ocean capacity to store carbon, are a few changes which could change the lifetime of CO2 in the atmosphere. Question 3: If the lifetime of CO2 in the atmosphere were to increase by 50%, what would be the associated changes to the GWPs given in Question 1? Question 4: Aerosol effects on climate are also not treatable from the perspective of GWPs because of their heterogeneous distributions in space and time and their highly variable effects on the radiation balance of the Earth. However, explain in your own words, the consequences to the climate of reducing aerosol emissions on the global scale, based on at least the class readings and lectures (you may use additional sources of information, such as your personal experience or other readings). This is a current policy that governments are taking because of the harmful human health effects of aerosols. Problem 3: Carbon Cycle Assuming that the Earth's carbon cycle is in balance (a reasonable estimate before large-scale anthropogenic perturbation of the carbon cycle through fossil fuel burning and land use change), we will use stock and flow analysis to decipher different time scales of the Earth's carbon cycle. Let us define the residence time of carbon in a reservoir as τ = (total amount in the reservoir) / (rate of outflow of carbon from the reservoir). We can also state that the rate of flow of carbon into the reservoir can be substituted for the rate of outflow, if the system is at steady state. Given the assumption that the system is at steady state as given in Figure 3 Figure 3 Composite Model for the Global Carbon Cycle 615 G tons Atmosphere 62 731 G tons Biosphere 62 62 60 60 1238 G tons Soil 842 G tons Surface Ocean 52 9 9744 G tons Intermediate Ocean 162 43 205 26,280 G tons Deep Ocean 0.23 90,000,000 G tons Sediments Reservoir contents are in units of 109 tons C; transfer rates are in 109 tons C yr-1, Carbon is deposited in sediment both as CaCO3 and as organic matter. There is a small release of CO2 in steady state from the ocean; this source is employed in weathering of crustal rocks. Question 1: What is the residence time for carbon in: the atmosphere; the biosphere; the soil; the surface ocean; the combined system of the atmosphere, biosphere, soil, and surface ocean; the intermediate ocean; the deep ocean? After computing these values, explain why we may want to look at the residence time of the combined system of the atmosphere, biosphere, soil, and surface ocean? Is there a reasonable scientific rationale for assuming that a combined system exists? One of the consequences of a warming planet is that vegetation generally grows faster and dead vegetation decays faster. If this were to hold true, let us look at how this may effect the residence time of Carbon in the atmosphere, the biosphere, and the soils, assuming that there is no net change in the actual stocks of any of these systems. Question 2: What is the residence time for carbon in the atmosphere, biosphere, and the soil, if we were to make the following assumptions: there is no net change in the stock of any of these systems, and the assumed changes to the flows are to increase them by 50% (case 1) and by 100% (case 2)? The flow from the deep ocean box into the sediment box is the major means of removal of carbon into the solid Earth. However, with the burning of fossil fuels, we have been emitting carbon at a much faster rate that it can be removed from the system. There additionally is a missing natural source of carbon sinks and sources due to weathering, volcanoes, and other long-scale changes from the sediments back to the atmosphere. However, for the next question, we will look at the time scale associated with this removal mechanism. Question 3: What is the time scale associated with this removal mechanism and for comparison, how does it compare with the time scale of human existence on Earth? When looking at local measurements of surface CO2 for the past few decades, a few interesting things are noticed. Although we always assume CO2 to have a constant, average value, over the surface of the Earth, it is in fact not constant. Look up some of these monthly averaged concentrations (find them!) and notice that they are composed of an increasing trend and a seasonal variation. Using data from three locations on the Earth, one tropical, one Arctic, and one Antarctic (eg: Hawaii, Alaska, and the South Pole would suffice), we will set out to investigate some of these relationships. Question 4: What is the time scale of this oscillation at each of the sites? What is an approximate magnitude of this seasonal oscillation at each of the sites? What is the trend in the annual average of the CO2 at each of the sites? What is the reason for why the annual means are different between these three sites? What is the reason why the seasonal changes are different between these three sites? If a change of 1ppm of CO2 in the atmosphere is about 2.1x109 tons CO2, how does this compare with the stocks and flows given in Question 1? Finally, we will look at the question of CO2 entrainment through the growth of new biomass (for example, forests). The fact is that for a given emissions of CO2, some goes into the oceans, some remains in the atmosphere, and some goes into the biosphere is one that can not be changed. Thus, it is impossible to sequester all of the emitted CO2 in the form of biomass, once the CO2 is released into the atmosphere. However, if we assumed that humans emitted 6.3x109 tons of CO2 in a year, and that 1.4x109 tons ultimately ended up in the biosphere, 1.7x109 tons of CO2 ultimately ended up in the surface ocean, and the remainder in the atmosphere, it would be a reasonable assumption, based on past values. Whether or not the land can continue to uptake so much carbon, or can uptake even more carbon, is still a scientific question open to consideration. However, let us explore this a little more deeply: Question 5: If a forest can hold 3.5 tons of CO2 per hectare per year, and if forests were to take up 1.4x109 tons of CO2/year, into the indefinite future, how many hectares would be needed? Is even this minimal approach to carbon uptake by the land a feasible policy to implement? Note that even with this policy in place, that the CO2 in the atmosphere would continue to increase in concentration.