Problem Set #3

Activity

The documents associated with this problem set, including a formatted answer sheet, can be found on CANVAS.

1. Worksheet successfully downloaded! Be sure also to download the answer sheet from CANVAS (Files > Problem Sets > PS#3).
    

   Each problem (#2 to #4) will be graded on a quality scale from 1 to 10 using the general rubric as a guideline. Problems #3 and #4 will each be worth twice as much as #2, meaning each of those two problems will be given a quality score as high as 10, which then will be multiplied by 2. Thus, a score (up to 10 points, 20 points, and 20 points from #2, #3, and #4, respectively) as high as 50 is possible, and that score will be recorded in the grade book.

   The objective of this problem set is for you to work with some of the concepts and mathematics around zero-dimensional energy-balance models (0-D EBMs) covered in Lesson 4. You may find Excel useful in this problem set, but you may use any software you wish, keeping in mind that the instructor only can provide help with Excel.

2. The following is the equation for planetary surface temperature under zero-dimensional energy balance, which is equation (5) on this page from Lesson 4:

   $$TS=\frac{\frac{S\left(1-\alpha \right)}{4}-A}{B}$$

   Re-read the derivation of that equation in Lesson 4. After re-reading the derivation, explain what each variable in the equation is. Summarize also the assumptions behind the derivation; in other words, explain what a 0-D EBM is. This is as straightforward as it seems, but this problem is here to ensure that you understand the underlying concepts because they are referenced often in climate science. Report your discussion on the answer sheet.

3. Report your responses to all parts of this problem on the answer sheet.
   1. Determine the planetary surface temperature T_S for a blackbody Earth in both Kelvin and degrees Celsius, rounding to two decimal places. For the input values, let “A” be 315 W m^-2 and “B” be 4.6 W m^-2 °C^-1 and the solar constant be 1,370 W m^-2 and the planetary albedo value be 0.32, the “standard” values mentioned in Lesson 4. Although the answer to this question is given in Lesson 4, you MUST show explicitly all values substituted into the equation for T_S. Be mindful of the order of operations as you perform the calculation.
   2. Recall that the graybody assumption is that Earth’s atmosphere has a greenhouse effect, i.e., the atmosphere itself emits longwave radiation downward to the surface. We account for it by varying the values of terms “A” and “B”. Let “A” be 214.4 W m^-2 and “B” be 1.25 W m^-2 °C^-1 and the solar constant be 1,370 W m^-2 and the planetary albedo value be 0.32. Calculate the planetary surface temperature for this conception of a graybody Earth, rounding the result to two decimal places and expressing it in both degrees Celsius and Kelvin. Once again, you MUST show explicitly all values substituted into the equation for T_S.
   3. Using values of solar constant ranging from 0 to 2,000 W m^-2, incremented by 100 W m^-2, and assuming that the planetary albedo value is 0.32 and that term “A” is 214.4 W m^-2 and “B” is 1.25 W m^-2 °C^-1, calculate planetary surface temperature, displaying the results in a table, expressing them in Kelvin, and rounding them to two decimal places. Also, plot the results, with the horizontal axis giving values of solar constant and the vertical axis giving values of planetary surface temperature.
   4. Using values of planetary albedo ranging from 0 to 1, incremented by 0.05 W m^-2, and assuming that the solar constant is 1,370 W m^-2 and that term “A” is 214.4 W m^-2 and “B” is 1.25 W m^-2 °C^-1, calculate planetary surface temperature, displaying the results in a table, expressing them in Kelvin, and rounding them to two decimal places. Also, plot the results, with the horizontal axis giving values of planetary albedo and the vertical axis giving values of planetary surface temperature.
   5. Using values of term “A” ranging from 0 to 350 W m^-2, incremented by 50 W m^-2, and assuming that the solar constant is 1,370 W m^-2, planetary albedo value is 0.32, and term “B” is 1.25 W m^-2 °C^-1, calculate planetary surface temperature, displaying the results in a table, expressing them in Kelvin, and rounding them to two decimal places. Also, plot the results, with the horizontal axis giving values of term “A” and the vertical axis giving values of planetary surface temperature.
   6. Using values of term “B” ranging from 0 to 5.0 W m^-2 °C^-1, incremented by 0.5 W m^-2 °C^-1, and assuming that the solar constant is 1,370 W m^-2, planetary albedo value is 0.32, and term “A” is 214.4 W m^-2, calculate planetary surface temperature, displaying the results in a table, expressing them in Kelvin, and rounding them to two decimal places. Also, plot the results, with the horizontal axis giving values of term “B” and the vertical axis giving values of planetary surface temperature.
4. This problem will deal with the notion of climate sensitivity, which is described in Lesson 4 Report all responses to the parts of this problem on the answer sheet.
   1. Write the equation that approximates the change in downward longwave radiation forcing associated with a change in CO₂ concentration from a reference concentration to some new concentration value, defining each of the variables that are in the equation.
   2. Define equilibrium climate sensitivity in words, and write the equation that approximates it, defining each of the variables that are in the equation.  How is this equation related to the equation that you wrote down in part (a)?
   3. The European Union has defined 2°C warming relative to pre-industrial temperatures as the threshold for Dangerous Anthropogenic Interference (DAI) with the climate system. Estimate the concentration of carbon dioxide (in ppm) at which we would expect to breach the DAI amount of warming. Assume a “mid-range” graybody parameter setting, i.e., term “B” is 1.25 W m^-2 °C^-1. Also assume that the pre-industrial carbon-dioxide concentration is 280 ppm and that the pre-industrial average global temperature is 288 K. Show your work.
   4. The atmospheric carbon-dioxide concentration is currently at about 414 ppm and is increasing by about 2 ppm per year. If we continue to increase carbon-dioxide concentration at this rate, how many years will it take until we commit ourselves to DAI, based on the climate sensitivity (i.e., the graybody parameter setting) considered above? If you were advising policy makers, how many years would you tell them we have to stabilize carbon-dioxide emissions and why? Show your work.
   5. Some scientists have argued that the threshold of 2°C is actually too high for DAI and that it should be a lower value. Re-work (c) and (d) assuming an alternative DAI of 1.5°C, and report the results. Is it too late to avoid DAI at this threshold? Show your work.