Theoretical estimation of cooling times

Steven R Dickman
Binghamton University
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Development of equations governing conduction of heat, culminating in a scaling analysis to easily estimate order-of-magnitude time for magma body to cool. Simple ultimate formulas and wide-ranging geoscience applications serve to promote student confidence in a quantitative approach to Earth science.

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upper-level survey course in geophysics for geology seniors and grad students

(an expanded version of this activity is presented in a second course, an advanced geophysics class for geophysics, physics, and engineering students)
Designed for a geophysics course

Skills and concepts that students must have mastered

Students should have previously seen and understood the meaning of ordinary derivatives, and be able to take simple derivatives (such as x^^2)

How the activity is situated in the course

This activity is pretty much the culmination of my course, both in substance and scheduling (usually we run out of time around this point). Ideally, including the peripheral topics and applications of heat flow, it requires about 2 weeks of lecture time (= 6 hours of lecture); but, if necessary, can be done in streamlined form in about 4 hours of lecture time.


Content/concepts goals for this activity

  • solving differential equations approximately (this is necessary for dealing quantitatively with the complex Earth System)
  • revealing the slowness and inefficiency of thermal conduction

Higher order thinking skills goals for this activity

  • being able to cast geoscience problems quantitatively
  • seeing the world in an order-of-magnitude hierarchy
  • some synthesis of ideas, concerning conduction versus convection
  • some data analysis

Other skills goals for this activity


Description of the activity/assignment

This combination lecture sequence / problem set activity takes a theoretical approach to the subject of conduction of heat. The lectures present Fourier's Law, its doubles (similar equations in other disciplines), and its use for measuring heat flow world-wide; derive, explain, and interpret the diffusion equation; discuss simple (1-D) solutions whose relevance to the Earth is questioned; and perform a simple scaling of the equation to obtain an approximate formula for cooling time. The problem set provides 'hands-on' experience with calculating the magnitude of heat flow, determining heat flow from temperature observations, and estimating cooling time for magma bodies. This activity gives the students essential knowledge about the transmission of heat; a perspective on conduction versus convection within the Earth; and an appreciation for geologic time. By its end, the students should have greater confidence dealing with equations; an exposure to partial derivatives; and an appreciation of the value of a quantitative approach to Earth science problems.
Addresses student fear of quantitative aspect and/or inadequate quantitative skills
Uses geophysics to solve problems in other fields

Determining whether students have met the goals

Grading the problem set and the exam corresponding to those lectures should allow for an accurate assessment.

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Supporting references/URLs

For data summarizing heat flow observations, any recent textbook in geophysics could be consulted. For the Verhoogen view of diffusion, which I learned as his student, it is possible that his 1974 textbook (The Earth) will suffice.

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