4. Zero-dimensional Model of Earth s Temperature

Transcription

1 4. Zero-dimensional Model of Earth s Temperature Introduction The term zero-dimensional implies that the earth is treated as a single point having a single temperature. Later we will consider a one-dimensional model in which temperature variation in one dimension is determined. For example, the single dimension could be latitude, in which case the model would attempt to explain the changes in temperature as one moves from equator to pole. A threedimensional model of the atmosphere would attempt to explain temperature variations as one moves north or south, east or west, and up or down. A time dependent three-dimensional model would attempt to explain temperature variations in time as well as three-dimensional space. For now we will start with the simplest model the zero-dimensional, single point, model. We begin with a box model, and consider the earth to be the box. It contains a certain amount of thermal energy, which can change because of the gain of solar energy or the loss of energy by radiation. The same process determines the input and output of energy: the process of black-body radiation. We may not be accustomed to thinking of it that way because the sun does not look black, and our eyes can not see radiation emitted by the earth. A black body means that an object is a good absorber of all radiation that strikes it, and therefore a good emitter of radiation as well. A black body is black in the sense that if you shine a flashlight on it, none of the light from the flashlight is reflected. In terms of reflected light, the object is black. In terms of radiated light, the object is bright and colorful with the brightness and spectrum of colors dependent on the temperature. Here we are combining the ideas from previous models. We have something akin to the bank account model, in which the storage of some quantity depends on the inputs and outputs. Here the inputs and outputs are not simply postulated, as they were in the bank account model, but are determined by definite natural physical processes, namely black-body radiation (see Figure 4.1). Solar Input Energy Content of the Earth Terrestrial Output Figure 4.1 Schematic of the box model.

2 Over a time interval τ, we have [Energy at t + τ] = [Energy at t] + [τ solar input per second] - [τ terrestrial output per second]. The temperature of earth is proportional to the energy content of earth, so we can write where: T τ α S c A + BT T t + τ = T t + τ 1 α S 4 τ A + BT t /c (4.1) Temperature Time increment Albedo (fraction of sunlight reflected) Solar constant Heat capacity Terrestrial radiation emitted to space Exercise 4.1 Justify Equation (4.1). The factor of 4 in the solar heating term is the ratio of the total surface area of earth to its cross sectional area, or shadow area. At any given instant sunlight only shines on part of the earth but terrestrial radiation is emitted from all over, therefore we must include the factor of 4. Next, make a spreadsheet model involving the constants A and B, the control parameters α and S, and the initial temperature T(0). It turns out that c and τ only affect the rate at which the temperature adjusts to equilibrium, but not the equilibrium temperature itself, so the actual values are not important for this model. Start with c = 1 and τ = 0.1, and then try some other values. Try other values for the initial temperature T(0), including some that are larger and some that are smaller. Is the equilibrium stable or unstable? Another argument for temperature of earth Another argument for estimating the earth s temperature comes from proportional reasoning. A square meter of the surface of the sun emits radiation at a rate F that is proportional to the fourth power of the surface temperature. This is an impressively high rate. For any black body, the radiation emitted per square meter of the surface, F, is F = σt (4.2) where T is the surface temperature and σ is a constant, known as the Stefan- Boltzmann constant. For now it is enough to know that an object at the freezing

3 temperature T = 273 K radiates F = 320 Wm -2. Then we can determine the flux F at any temperature T by proportional reasoning: F = F. (4.3) In particular, for the sun F = F. (4.4) The flux F through any sphere of radius R that contains the sun must be constant, and therefore F 4πR = F 4πR (4.5) where R is the radius of the sun. We can use this equation and relate F to the radiation emitted by an object at the freezing temperature, F, using Equation (4.4): F = F. (4.6) If we take R to be the radius of earth s orbit, then F is the flux of energy arriving at earth. A fraction α of this energy is reflected, so only 1 α F is used to heat the earth. In equilibrium, this energy input must be balanced by the energy radiated out to space by the earth. Thus (1 α)f πr = F 4πR. (4.7) We can cancel the common terms on both sides and substitute in for F and F using Equation (4.6) to get an expression for the earth s temperature T : T = T = T 1 α. (4.8) where D is the diameter of the sun, R is the distance from the earth to the sun, and therefore is the angular diameter of the sun as viewed from earth. This angle is approximately 1/2 degree, or 1/120 radians, so the square root is 1/11. The albedo is approximately 0.3, so we obtain T T /24, which is 250 K = -23 C. This number is a little bit cold, but still pretty impressive considering that the only information we supplied was the angular diameter and color/temperature of the sun, and the albedo of earth. This argument demonstrates the power of proportional reasoning.

4 Box 4.1 Observations of Global-Average Surface Temperature Figure 4.2 Estimate of annual-mean global-mean temperature of the surface of earth. Prior to the 1980s observations come from weather stations and occasional measurements from ships and buoys. Since the 1980s satellites have been measuring surface temperature. Data from NASA GISS Surface Temperature Analysis 1. Take home points The zero-dimensional model of the earth gives a temperature of about -20 C. This value depends on the temperature of the sun, the distance between the earth and the sun, and the albedo of earth. In the zero-dimensional model the equilibrium temperature is stable. If you perturb the system by increasing the temperature slightly then it will respond by cooling off, and vice versa. The sensitivity to albedo is about 1 C per 0.01 change in albedo. The sensitivity to solar constant is about 1 C per 20 Wm -2 change in solar constant.

5 Exercises 4.2 Check which equations require T in units of degrees Celsius and which require degrees Kelvin. The temperature of the surface of the sun is T 6000 K. Compare the results of the model of Equation (4.1) and Equation (4.8) using S = 1400 Wm Make a graph showing the equilibrium temperature versus solar constant. Show three lines on the same graph for albedo values of 0.2, 0.3 and 0.4. How sensitive is the temperature to changes in albedo and solar constant? Your answer should have the form: a change in albedo of 0.01 causes a change in temperature of, and a change in solar constant of 1 Wm -2 causes a change in temperature of. We will refer to these numbers later, so make sure to keep them. 4.4 Address the following questions in your concluding statement: Is the temperature predicted by the model close to the true temperature (see Box 4.1)? How sensitive is the calculation to the values of the input parameters? What physical principles are used to make this estimate of the earth s temperature? What does the model teach us? Can we use it to estimate how temperatures will change in response to greenhouse gases? Why or why not? References 1. GISTEMP Team, 2016: GISS Surface Temperature Analysis (GISTEMP). NASA Goddard Institute for Space Studies. Dataset accessed at (Note: to get an estimate of the global-mean surface temperature we added 14 C to the global-mean surface temperature anomaly provided by GISTEMP, as suggest by the authors of the dataset.)