Agronomy 405/505
Spring 2013
Problem Set 7
Due THURSDAY, March 7, 2013.
Assigned February 28, 2013.
1. (a) Using Equations 6.3 and 8.6 from Campbell and Norman, derive an expression for
the heat flux into the soil at the soil surface. Your result should match Equation
8.9. Use the identity sin(A + B) = sinA cosB + cosA sinB with A = ω(t − to )
and B = π4 .
(b) Plot Equation 8.6 and 8.9 at the soil surface for one diurnal cycle using A(0) =
10◦ C, Tave = 15◦ C, and to = 10 hours for a clay soil of water content θ = 0.1
and θ = 0.3. Start with the m–file “ps7 1b.m” on the course website if you would
like. Use Figures 8.2, 8.3, and 8.4 in Campbell and Norman to find appropriate
numerical values for soil constitutive properties instead of computing them with
the models given in the text.
(c) OPTIONAL: if you are interested. Derive an expression for the total heat input
into the soil for one diurnal cycle. Do this√by integrating Equation 8.9 over the
time that G ≥ 0 to obtain the expression 2 D ρs cs A(0) as given in the text.
(d) Use appropriate values to get numerical answers for the total heat input during
one diurnal cycle for the clay soil at both water contents.
2. Download the m–file “ps7 2.m” from the course website. This m–file plots a three–
dimensional graph of soil temperature as a function of time and depth into the soil. For
the clay soil in Problem 1, create temperature plots for both water contents, θ = 0.1
and θ = 0.3. Experiment with the “view([70,15])” command at the end of the m–file
if you would like. Then chose another type of soil, and create two more temperature
plots for θ = 0.1 and θ = 0.3. Be careful with the units of ω so that you get the correct
damping depth, D.
3. 8.1 in Campbell and Norman. Be sure to answer all three parts of the question.
4. 8.3 in Campbell and Norman. Assume the rocks are made of granite.
5. 8.4 in Campbell and Norman. Use Figure 8.5. Assume thermal admittance does not
depend on temperature. Assume a bed of leaves is similar to dry organic soil. Note
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that there is a mistake in (8.21) in the text. See the “list of book corrections” on the
course webpage.
6. 50 mm of irrigation is applied to a 10 hectare (25 acre) field.
(a) How many kilograms of water is this?
(b) How much water drains through the first 20 cm of the soil if the volumetric water
content of the first 20 cm changes from 0.15 m3 m−3 to 0.30 m3 m−3 ? Assume no
runoff.
7. 9.1 in Campbell and Norman. In 9.1b, there is an error. Compute the water potential
at the wetting front using Equation 9.9 instead of 9.10.
8. Consider two or more conductances in series.
(a) If one conductance is 2 mol m−2 s−1 and the other is 0.1 mol m−2 s−1 , what is
the overall conductance?
(b) If one conductance is 2 mol m−2 s−1 , another conductance is 1 mol m−2 s−1 , and
the other is 0.1 mol m−2 s−1 , what is the overall conductance?
(c) In terms of the overall (equivalent) conductance, does the order of the conductances (which one is first, second, or third) in series matter?
(d) If there are two or more conductances in series, and one conductance is much
smaller than the other conductances, what is a good approximation of the overall
(equivalent) conductance?
9. Read “Soil Microbes Produce Less Atmospheric Carbon Dioxide Than Expected With
Climate Warming” on the course website.
(a) How is carbon dioxide produced in the soil?
(b) What happens to the production of carbon dioxide as soil temperature increases?
(c) What is the next step in this research?
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