GGR 203F - Introduction to Climatology
Problem Set One
Assigned: 6 Jan 2025
Due: 11:59 PM, Jan 27th, 2025, using the Assignments tab on the course Quercus site.
Taken Up: (via recording that will be released): around 12:00 PM, Jan 30th
If you have not done so already, and have no previous experience using Excel for calculations, you should review the Excel guide that has been posted and spend some time exploring the various features and options with your own trial computations, before starting this problem set. Doing so systematically now will save you a lot of time over the course of the 4 problem sets in this course. An Excel guide has been posted along with this problem set (suggestions on how to improve the guide are always welcome).
Meteorology and Climate
1. This first problem is a simple mathematical parable illustrating the difference between meteorology
and climate. We shall consider a series of numbers generated by the formula,
Xn+1 = (Xn - 2)2
(a) In the first column of an Excel worksheet, set up numbers from 0 to 60. In the next column, next to the row with the zero, enter the value 0.7000. This is X0. Next, compute the next 60 terms of the series in the remaining rows of the second column of your Excel worksheet. Next, plot the time series (where the numbers in the first column are the X-values, the numbers in the second column the Y-values). How would you describe the resulting plot? Do you see any obvious cycles? Marks: Calculations, 2; Word answers, 2; Graphs, 4
(b) Starting with X0 = 0.7002 in the third column, compute the next 60 terms of the series, as in (a), and plot on the same graph. Compute the difference between the first two series in the 4th column.
In predicting the weather, we start from some observed initial conditions and project into the future using equations based on the deterministic laws of physics. The simple equation we've used here illustrates some of the properties of the real atmosphere. Suppose that X0 = 0.7000 was the exact initial value, but that the value X0 = 0.7002 was the observed initial value, containing an error of +0.0002.
What happens to this error as n increases (that is, for successive terms in the series)? 2 marks
(c) Let us suppose that we reduced our initial error by a factor of two. Recompute the above series starting with X0 = 0.7001 in the 5th column and plot on the same graph as the first two series, and compute the difference between the 1st and 3rd series in the 6th column. Letting the series starting with X0 = 0.7000 be the "true" series, and letting the series starting with X0 = 0.7001 and X0 = 0.7002 be the "predicted" series, has our ability to predict into the future improved by cutting the initial error in half? If so, by how long?
2 marks
(d) Climate involves the average and variability of day to day weather events, but it is also only a sample so, continuing our parable, prepare a single table that gives
- the means based on the 1st 21 terms, the 1st 41 terms, and all 61 terms.
- number of values less than 1.0 for all 61 terms, and
- number of values greater than 3.0 for all 61 terms
for each of the above three series, and comment on what this might imply about our ability to predict future climates compared to future weather. 3 marks
Hand in a printout of your graph (only one, with 3 curves) and the first page of your Excel worksheet, the summary table from (d) and your comments to (a)-(d). [15 marks total Q1]
Planck Function
2. The Planck function in terms of wavelength λ is given by
which can be written as
where C1 = 2πhc2 = 3.7419 x 108 W m-2 (μm)4 and C2 = ch/k = 1.439 x 104 μm K (values and names of c, h, and k are given at the end of the problem set for the sake of completeness).
(a) In the above expression, the term C2/λT is said to be the “argument” of the exponential function. What do you notice about the dimensions (units) of the argument of an exponential function (at least in this case)? 2 marks
(b) Using the given values of C1 and C2, compute B(λ,T) in an Excel worksheet for T = 260 K. Put the constants that you will use and the temperature in separate rows at the top of the spreadsheet, along with the name of the constants or variable and the units. Below that, do the following (4 marks):
- in the first column, set up the boundaries of the various subintervals that will be used. These boundaries should go from 3.0 μm to 20.0 μm in 0.5-μm intervals, from 20.0 μm to 40.0 μm in 1-μm intervals, from 40.0 μm to 60.0 μm in 2-μm intervals, and from 60.0 μm to 80.0 μm in 5-μm intervals
- in the 2nd column, compute the midpoints for each of the intervals (you will have one fewer midpoint value than boundary value)
- in the 3rd column, compute the of B(λ,T) at the λs given in the 2nd column. Note that you only need to type the formula once, in the first row, then you can drag it down, but don’t forget to put in $ signs where they are needed (see the handout on using Excel).
Do not type the values of the constants into your equation; rather, set the values of the constants in various cells near the top of the worksheet (and type the symbol for each constant, the name of the constant if it has a name, and the units of the constant next to each constant). Then, have the equation that you are typing use the cells at the top of the spreadsheet where the values are assigned, by clicking on that cell as you type the equation, and putting a $ before the letter for the column and before the raw number.
- Plot your results.
- In the 4th column, compute the widths of the intervals centred at the λs given in the 2nd column
- Use the 5th column to estimate the total emission graphically (that is, estimate the area under the curve shown in your plot), with the answer at the bottom of this column, then compute σT4 (the total emission according to the Stefan Boltzman Law) below that, where σ = 5.673 x 10-8 W m-2 K-4. How close are the two values? What could be done
(without cheating) to make the two values closer to one another? 2 marks
(c) Confirm that the wavelength of peak emission agrees with Wien's law (compute the expected value from Wien’s Law somewhere in your worksheet). 2 marks
(d) Copy the worksheet two times, then change the temperature to 280 K for one sheet and 300 K for another sheet. Plot the results for all three cases on the same graph (5 marks). Be sure to label both axis and to include units in the labels, to add a title to the graph, and to adjust the size of all fonts to about 14 or 16 so that everything is easily legible. Comment on how the emission of radiation and its distribution change with temperature. 2 marks
You should hand in the following: The graph of your 3 Planck function curves (all on one chart), the first worksheet, and a single sheet with a table giving the three graphical estimates of total radiation, the total radiation according to the Stefan-Boltzman law for each case, the wavelength of peak emission, and your comments or answers to the above questions. [17 marks total Q 2]
Emission and absorption of radiation
3. Based on the fundamental units,
(i) Show that 1 joule = 107 ergs, and
(ii) Show that 1 newton = 105 dynes.
(iii) Given the density of water as 1 g cm-3, what is the density as kg m-3? 3 marks
4. What is the wavelength of electromagnetic radiation emitted by electric oscillators of frequencies 1.2 x
1015 s-1, 1.2 x 1014 s-1, and 3.6 x 1013 s-1? In what part of the electromagnetic spectrum do each of these oscillators lie? What kind of oscillator is each likely to involve (i.e.: oscillating electron, vibrating dipole molecule, or rotating dipole molecule)? 6 marks
5. The earth is 1.7% closer to the sun than average on 3 January, and 1.6% further than average on 5 July. Using a solar constant of 1370 W m-2, by how much would the solar flux density (W m-2) on a plane perpendicular to the sun's rays vary between these two dates? Given a global mean albedo of 0.3, what would be the change in absorbed energy from January to July, averaged over the entire globe? 5 marks
6. If the incident solar flux at a surface is 520 W m-2, what would be the change in amount of absorbed solar energy (in W m-2) if the surface changed from snow-free grassland with surface albedo αs = 0.25, to a deep snow cover with αs = 0.80? 3 marks
Investigating the atmospheric ("greenhouse") effect
7. Consider the simple one-layer atmosphere + surface model developed in class.
(a) Compute the infrared emission of radiation to space for a surface temperature of 300 K and an atmospheric temperature of 250 K for atmospheric emissivities of 0.0, 0.1, 0.5, and 1.0. Explain your results briefly.
(b) Compute the infrared emission of radiation to space for a surface temperature of 300 K, an atmospheric emissivity of 0.3, and atmospheric temperatures of 290 K, 300 K, and 310 K. Explain your results for each case in comparison to the surface emission. Which of these cases corresponds to a "greenhouse" effect and which to an "anti-greenhouse" effect.
(c) We can mimic the direct effect of a CO2 increase by increasing the atmospheric emissivity slightly, from 0.3 to 0.305. Compute the effect that this increase has on the infrared emission to space for a surface temperature of 300 K and atmospheric temperatures of 290 K and 310 K, and comment on your results.
Do all your calculations in Excel, and summarize your results in a single table.
13 marks total for Q 7
62 marks total
Constants :
Stefan-Boltzman constant: 5.673 x 10-8 W m-2 K-4
Speed of light, c: 2.998 x 108 m s-1
Stylistic Issues Concerning Graphs
1. There should be a non-vague title at the top of the graph
2. Make sure that each axis has a clear label with units where appropriate.
3. Omit any unnecessary zeros after the decimal point on the axis tick labels (i.e., labels should be 5, 10, etc, and not 5.000, 10.000).
4. Use a constant precision in the axis tick labels (i.e., 0.0, 0.5, 1.0, 1.5 … not 0, 0.5, 1, 1.5)
5. Use a good sized font (such as 14 or 16 pt) for all tick and axis labels and for legend entries.
6. For graphs with a y-axis spanning positive and negative values, make sure that the x-axis and axis labels align with the lowest point on the y-axis, not with the zero point on the y-axis.
7. Legends should go in a box within the graph area, not at the bottom or side of the graph. Click on the legend once you’ve moved it and select “Fill”, “White” to block out the lines behind the legend, and put a boundary around it.
Marks will be deducted if these rules are not followed.
Marks will also be deduced for spelling mistakes and improper grammar.
The bottom line is that your work should look professional.
Eliminating Excess Digits in Worksheets
Sometimes Excel gives the result of calculations with 8-9 or more decimal digits, far more than can be justified or are necessary. These make it harder on the eyes and require wider columns, and so should be eliminated.
To do this, highlight the cells to be changed, then right-click and select Format Cells, then Number, and in the window “Decimal Places”, select some reasonable number.
After, make columns narrower if necessary in order for a table to fit within one screen width (for example).
Format for submission of problem set:
Please make sure that you submit your problem set as a single pdf file, with all the answers to your questions in the correct order. The easiest way to do that is to type the answers to the word questions in the excel sheet right below your calculations. You can use the "save as adobe pdf" command under "File", and Excel will create a single pdf file with separate pages for each worksheet - so make sure that your worksheets are in the correct order. Please verify that your resulting pdf file looks good (i.e., without the image taking up a small corner of the page instead of the entire page). You might need to highlight the part of the page that should go into the pdf (under the Page Layout tab and then "Print Area"), so that only that part is shown (that is, without wasted space at the end).
Bottom line: would you want to mark 30 assignments that look like yours?