Global Climate Change - Economics, Science, and Pol Homework Set 3

Homework #3 – Science of Climate Change (Part 2) This homework set must be completed on your own and you are not allowed to work in groups on it. Problem 1: 1-Box Carbon Model This problem will start with you being given a simple model of the Earth’s carbon system, as a single box (the Carbon in the atmosphere). The reason to consider such a model is to look at some of the impacts that anthropogenic emissions and natural sinks have on the amount of Carbon in the atmosphere. This is an important problem because as the amount of carbon in the atmosphere as CO 2 increase, so does its associated impact on the radiative balance of the Earth. To start this problem off, we assume that the amount of Carbon in the atmosphere can be modeled using the following simple equation, which only considers the anthropogenic emissions of Carbon (for example, as predicted by EPPA) and a natural sink for Carbon to the land and oceans. Equation 1a 𝑑𝑀 𝑀 =𝐸− 𝑑𝑡 𝜏(𝑀) In this equation, M represents the total stock of Carbon in the atmosphere (PgC), t is time (in years), E is the amount of anthropogenic Carbon emission (PgC/year), and τ is the lifetime of Carbon in the atmosphere due to the natural sinks of the land and ocean (in years). If you recall from PS#1 τ was defined as a constant. Here, we will expand upon that assumption. This equation has a few important aspects to it which make it more realistic and slightly more complex than just a simple additive expression. It is these aspects which make it realistic and will allow for deep insights to be made, while keeping the math relatively simple. The first to notice is that the loss of Carbon from the atmosphere is a function of the total amount of carbon in the atmosphere. This makes sense, since over time we expect the ability of the land and ocean to take up Carbon is not constant over time. Equation 1b 𝑀 − 𝑀0 𝜏 𝑀 = 𝜏0 + 𝜏1 𝑀0 2 In this equation, we are modeling this effect by stating that the residence time of the Carbon in the atmosphere will increase as the total amount of Carbon in the atmosphere increases. In specific, we are looking at two effects, a constant lifetime of Carbon in the atmosphere, based on today’s amount of Carbon in the atmosphere, and an increasing component, based on the amount of increase of the Carbon in the atmosphere from today. Note that this model only works to look at the lifetime of the Carbon uptake in a world in which the total Carbon in the atmosphere is increasing. Question 1: In your own words, first explain what equation 1a tells us, and then given two examples of how you could make equation 1a (a global atmospheric mass balance of Carbon) more realistic. Next, explain in your own words what equation 1b tells us, and at least one reason why we would expect both the land and ocean sinks to be less effective in the future. Now that we have looked at the ideas behind the equation, let us start to gather some numbers and use the equation to gain some insights. A recent report in Nature Magazine states that the present day annual emissions of Carbon are more than 9 gigatons of Carbon per year. Thus, we can take E ≈ 9 PgC/year as a rough estimate. Additionally, a current estimate of the stock of Carbon in the atmosphere can be found by starting with the current atmospheric concentration of CO2, which is about 382ppmv globally averaged over the surface of the Earth (figure 1) Figure 1 We can then multiple this number by the total number of moles of the atmosphere , to come up with the number of moles of CO2, and then multiplying this by the molecular weight of Carbon, come up with a stock of Carbon in the atmosphere. This procedure gives us a rough answer as 𝑀0 ≈ 8.4 ∗ 102 PgC. Question 2: Looking at equations 1a and 1b set up an equation for τ0. Next, using the data from figure 1 over the past few years, approximate the derivative (for example, using the slope of the line) term. Finally, solve this equation for τ0. Now, suppose that for a doubling of atmospheric Carbon, that the total lifetime of Carbon is expected to change by 20%, based on the current level of emissions and atmospheric Carbon stock. Using this information set up and solve an equation for τ1. The ultimate goal of climate policy will be to at least stabilize the atmospheric concentration (and hence stock) of carbon. Let us use this simple model to investigate this idea more closely. The first way to look at this is to say that the atmospheric stock of Carbon has stopped changing in time. In this model, we can do this by assuming that the model has come to steady state (Constant M). Question 3: Assume that the model has come to describe a system which has stabilized atmospheric Carbon. Explain why a steady state assumption is a valid way to represent this situation. Assuming that the atmospheric stock of Carbon is double today’s present value, what is the new lifetime of Carbon in the atmosphere, and what is the maximum level of emissions? Assuming that the atmospheric stock of Carbon is three times today’s present value, what is the new lifetime of Carbon in the atmosphere, and what is the maximum level of emissions in this case? Can you comment on what is happening to the lifetime of Carbon in the atmosphere as its total stock increases? What do you think will have to happen to the emissions of Carbon as the concentration in the atmosphere continue to increase, if we are to keep the stock in steady state (for this final example, you can use a mathematical abstract of the lifetime going to infinity, if it helps your thinking)? In reality, the lifetime of Carbon in the atmosphere will decrease faster, since the ability of the atmosphere and the ocean to uptake carbon will eventually stop, if the stock continues to remain at a very high level. How does the answer to what is required of emissions for stabilization based on this past statement compare with the answer of what is required of emissions for stabilization based on your previous question? Problem 2: Biofuels This problem will look at some of the scientific issues and impacts that using biofuels on a large scale can have on the Earth’s energy balance and climate. This is a very “hot topic” right now from a policy perspective, so the ability to perform back of the envelope calculations about its impacts can have vital importance for corporate and governmental strategy. The main idea of using biofuels is that the carbon associated with their combustion would be neutral, since it was previously taken from the atmosphere to grow the biofuel material previous to the material being harvested. However, this may not consider the changes to the albedo of the local region based on the different properties of the biofuel substance compared with what was there previously. This may not also consider the initial change in carbon associated with changing the landscape from its initial state to that in which the biofuel substance is being grown. There are many technological challenges and other logistical issues associated with changing the land on such a large scale as this problem will address, however we will not address them in this problem and will instead assume that these are not an issue. The first proposal would be to convert marginal desert regions of the world into grassland. The grassland would then be used to grow cellulosic material which would be converted into biofuels. For this problem, we will assume that 45% of the carbon in these cellulosic materials can be converted into biofuels, that all nations doing this will have access to sufficient water and technology as to get this done, and will not have any other constraints on achieving this change. Question 1: An approximate land surface area of the Earth’s marginal deserts (eg: Southwestern USA and Northern Mexico, Western China, and other regions which have some, but little rainfall) is about 11.5% of the total land surface area of the Earth. If the albedo of marginal deserts is given to be 0.37 (Ben Felzer, Private Communication) and the albedo of grasslands is given to be 0.20 (Ben Felzer), what is the expected local energy change to these regions in W/m2? What would be the global change in energy due to this local change in albedo? How do both these global and local changes compare with other processes that we have talked about in class? Can you give at least 2 effects that you would expect to happen locally, and connect these effects with the impact that may have on the newly transformed land? Now, let us look at this from the perspective of Carbon. If marginal deserts contain about 3 kgC/m2 (Ben Felzer) and grasslands contain about 4kgC/m2 (Ben Felzer), what is the net Carbon change in the atmosphere due to this change of the land? What type of impact would this make on the total change of atmospheric CO2 (hint: you may want to use one of the radiative models from the first problem set, or to look up what is called the “climate sensitivity” to help get some numbers for this one)? And finally, from a policy perspective, how should this change in Carbon be assigned between the different nations of the world? The second proposal would be to convert current forests of the world into grassland. The grassland again would be used for biofuels, with the same assumptions given above. This scenario would be easier to implement, and would also be able to be done in regions of the world where marginal “desserts” (Some food does sound better, does it not?!?!) do not exist. Naturally, both this and the above could be done in tandem. Question 2: Please answer the exact same questions as in Question 1, except for now we want to look at the tropical forests (eg: Brazil, Central Africa, South East Asia, etc.) and the temperate forests (eg: USA, China, Russia, etc.). For answering this problem, you are going to need the following information (all from Ben Felzer): albedo of tropical forest is 0.13, albedo of temperate forest is 0.14, the Carbon content of tropical forest is 30kgC/m2, and the Carbon content of temperate forests is 21 kgC/m2. Also, please comment on how much of the Earth’s total surface area is taken up by forests. Now, it is important to look at how these numbers and insights gained above may apply when we place a limit on the system, specifically how much land we would need to use for biofuels to power our economy. If we are to assume that in the future humans will consume roughly 20PgC/year for energy (a rough, if not low estimation based on current predictions), then we will need to produce biofuels to cover this entire amount, if it will be the only source of energy supply that we have to use and if we are interested in having a stabilized climate. For this assumption, we will further assume that dry biomass fuel would be usable as a substitution for liquid biomass fuel, and that the energy can be extracted without the need to add additional energy. Question 3: If we assume that biomass fuel extracted from grasslands contain 45% of the total Carbon content that the initial land contained for each year, how much land would we need to convert to grassland to run our world’s economy purely on biofuels? Assuming that we only converted the marginal deserts to grassland, how much total energy (as PgC) would this provide for the world to run on biofuels? What if we converted the marginal deserts and the forests? How large are these numbers compared with the present day Carbon emissions? Given some of the numbers in this example, how feasible do you think it is to have a policy which is focused exclusively on biofuel generation as the means of achieving atmospheric CO2 stabilization?