1. An Introduction to Heat Flow
Lecture 10/15/2009
GE694 Earth Systems Seminar

2. Basic Concepts
Heat is a form of energy, and so the basic equations that describe heat and heat flow come from the “conservation of energy” law of physics.
Heat Transfer: Heat can be transferred by thermal conduction, thermal convection and radiation. In the solid Earth, the most important form of heat transfer is thermal conduction.

3. Fourier’s Law of Heat Conduction:
The above solution is for a linear change of temperature with distance. The change of T with y can be nonlinear, which means that q can vary with y.

4. The thermal conductivity of rocks is relative low, and is fairly similar for many different rock types:

5. North America Surface Heat Flow
Average continental heat flow = 65 ± 1.6 mW/m2
Average oceanic heat flow = 101 ± 2.2 mW/m2
Total continental heat flow = 1.3 x 1013 W
Total oceanic heat flow = 3.13 x 1013 W
(Heat flow values are measured by drilling into rock, measuring the temperature at different depths, and then calculating q from Fourier’s law. In order to apply Fourier’s law, the thermal conductivity constant k must be measured in the laboratory).

6. Heat Generation by Radioactive Decay:

7. Distribution of Radioactive Elements in the Earth:
In the crust and mantle, radioactive elements are not distributed uniformly but rather concentrate in some continental rocks. The radioactive elements in crustal rocks also show a marked difference when compared with the amounts of these elements in chondritic meteorites.

8. Steady-State Heat Flow when There is Heat Production
flow of heat out of slab = flow of heat into slab + heat production in slab
(conservation of energy)

9. The above differential equation can be solved by integration with respect to y.
Note: T increases with y2 due to the internal heat production.

10. Example: Continental Geotherms

11. 3-D Steady-State Heat Flow with Heat Production

12. 1-D Time Dependent Heat Flow (no Heat Production)
Equations (4-67) and (4-68) are 1-D forms of the “diffusion equation”, which shows up in many different kinds of problems in physics, chemistry, geophysics, geology, etc.

13. Instantaneous Heating or Cooling of a Half-Space
These are the initial conditions at t=0 and the boundary conditions at y=0 and y=infinity.
Equation (4-94) is the 1-D diffusion equation rewritten in the new coordinate system.

14. The solution to the differential equation (4-100) involves a special function called the “error function” or erf(x). The solution is
Here, eta is called the “similarity variable”.
In terms of eta, the diffusion equation becomes
The boundary conditions become

16. Cooling of the Oceanic Lithosphere: A Half-Space Model
Equation (4-125) can be used to estimate the temperature T at depth y as a function of x (or t since steady plate spreading is assumed). The thickeness of the thermal boundary layer yL is x y

17. Cooling of the Oceanic Lithosphere: A Plate Model
The boundary conditions for a plate heated from below, where the plate thickness at large time is yL0, is

18. Thus, at small times (i.e., near the spreading ridge), this solution can be manipulated into the half-space cooling model solution, while at large times the solution becomes a simple linear temperature gradient between the surface and the bottom of the plate. For this latter case, the heat flow is a simple conduction solution (4-134).

19. Cooling of Melts: The Stefan Problem
The phase change releases heat, and so it acts as a heat source.
= ym
= 

20. Solutions for this problem are found by matching the values of the left and right hand sides of equation by trial and error.

21. Cooling of Melts: The Solidification of a Dike or Sill
This problem starts with a temperature T=Tm in the dike and at its boundary, and the rest of the country rock is at T=T0. As time increases, the temperature of the dike cools and the heat diffuses into the country rock, as in our earlier problem of a sudden temperature increase at the edge of a half-space. Skipping the steps of the derivation to the solutions, we get a transcendental equation to be solved by trial and error.