Atmospheric Vertical Structure and the First Law of Thermodynamics

Tony Hansen, St. Cloud State University
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Diagram from Answer key - TH Atmosphere Vert. Structure This sequential set of in-class and homework problems concerns applications of the First Law of Thermodynamics. In the homework, students are first asked to compute and plot potential temperatures of specified adiabats. In a second assignment, the potential temperature from an observed sounding is computed and plotted to develop a framework for beginning to understand the stratification of the atmosphere. These activities are intended to help students discover the importance and utility of conservation principles derived from the First Law of Thermodynamics. In addition, they provide a first step in evolving from the p-V diagrams the students have seen in their physics coursework toward the thermodynamic diagrams used in meteorology.

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Learning Goals

  • Introduce the concept of potential temperature used to characterize dry adiabatic processes in meteorology
  • Apply elementary calculus skills in interpeting atmospheric vertical structure
  • Apply basic graphing skills

Context for Use

These assignments are intended to be used in the first part of a sophomore/junior level Atmospheric Thermodynamics course. Students will have learned the equation of state and been introduced to the First Law before these problems are assigned. They must have completed a year of calculus and the first semester (preferably both semesters) of calculus-based physics, as well as a 200 level introductory meteorology course.

Description and Teaching Materials

In-class Preliminary Activities (Microsoft Word PRIVATE FILE 31kB Jul17 04)
Homework Assignments (Microsoft Word PRIVATE FILE 66kB Jul17 04)
Homework Solutions (Microsoft Word PRIVATE FILE 186kB Jul16 04)
Instructors Notes (Microsoft Word PRIVATE FILE 26kB Jul16 04)

Teaching Notes and Tips

  • Students entering this course typically do not make the connections between the mathematics and physics that they've had and the atmospheric processes they can observe. Making these connections requires a concerted effort at linking the First Law as a statement of energy conservation to its implications for atmospheric structure, stability, and the construction of thermodynamic diagrams. Relating the basic principles to the current weather is an important and useful way to pique the students' interest.
  • This activity is designed as a sequence beginning with in-class exercises (In-class Preliminary Activities (Microsoft Word PRIVATE FILE 31kB Jul17 04)) followed by homework exercises (Homework Assignments (Microsoft Word PRIVATE FILE 66kB Jul17 04)).
  • The homework assignments are relatively short and should be due the next class period after they are assigned (2 days later typically). I assign questions 1-5 of the first set of problems initially, followed by questions 6-8 which are assigned on the next class day. The second set of questions can be assigned concurrently with questions 6-8, but I make them due a class period later.
  • Parts of this exercise can be done in a spreadsheet program, but they can also be done by hand. If a spreadsheet is required of the students, they may need to be given more time to complete the assignment.


  • Once the assignment is completed, students can be asked in any subsequent class what they think the vertical profile of Θ is on that day based upon the weather they can observe out the window. For example, for an afternoon class on a day with fair weather cumulus, you'd expect a dry mixed layer with constant potential temperature below the cloud base. After the introduction of the equivalent potential temperature, Θe, later in the course, this oral exercise can be extended to that variable as well.
  • In addition, actual soundings taken on a given class day can be investigated for evidence of dry ascent/mixing, moist ascent, subsidence, nocturnal inversions, etc..
  • On a subsequent exam, I ask questions like the following using qualitative weather information from a recent event:
    1. Yesterday in central MN (or wherever), the sky was clear all day with bright sunshine. Based on this information, sketch below what you think the vertical profiles of w (the water vapor mixing ratio), Θ and Θe looked like (below 800 mb) in the Chanhassen sounding from 00Z last night. Be sure to THINK! Briefly explain your reasoning.
    2. Yesterday in Portland, ME (or wherever), the sky was overcast and light to moderate to heavy rain fell from the afternoon through the evening. Based on this information, sketch below what you think the vertical profiles of w, Θ and Θe looked like in the Portland sounding from 00Z last night from the cloud base upward. Be sure to THINK! Briefly explain your reasoning.
    I have found that students spend a considerable amount of time answering questions like these. The situation in Question 1 exhibits a mixed layer in which Θ, Θe and w are all constant with height. The temperature lapse rate in this layer would be the dry adiabatic lapse rate. In the second case, since it is raining, the saturated ascent and associated diabatic heating due to condensation will lead to Θ increasing with height, w decreasing with height and Θe constant with height. The sounding in this case would exhibit a saturated, moist adiabatic temperature lapse rate above the cloud base.