1. Heat flow in a Moving Lithosphere Due to a Mantle Plume
Colleen Hartsig
Mandi Markee

2. Introduction
Theoretical Constructs
Brief overview of the derivation

3. Basic Considerations
2 Dimensional Gaussian heat source at some depth zo.
The temperature of the Lithosphere is constant at the surface. Justifies use of ‘image’ heat source.

4. …Fun with Fourier Transforms
First start by deriving the relevant diffusion equation for a static source…

5. Fourier transform the equation into the wave-number/frequency domain.
Looking at the denominator, we can see that it will be appropriate to inverse Fourier transform using the calculus of residues.

6. We need to take the integral of:
(Looks scarier than it really is!)
Notice that there is a pole at:
Using Cauchy’s Residue theorem we can solve the integration over f.
The denominator is gone… now the rest of the integration is a piece of cake!

7. The remaining integration is Fourier transforms of Gaussians
The remaining integration is Fourier transforms of Gaussians. We obtain:

8. We want to solve the heat equation for a moving lithosphere with a zero temperature boundary condition at the surface.
Place an image heat source at zo above the surface.
Solve for temperature (total heat accumulated) at a point at time t.

9. After considering the additional imposed conditions we obtain.
The heat flow at the surface is:
(This whole derivation could have also been accomplished by taking the convolution of the heat source with the appropriate Green’s function.)

10. Surface Heat Flow for point source of 1 cal/s
y (km)
U = .01cm/year
x (km)
y (km)
U = .1cm/year
x (km)
y (km)
U = 1cm/year
x (km)

11. Isotherms in a vertical plane along x-axis for point source of 1 cal/s
U = .01cm/year
z (km)
x (km)
U = .1cm/year
z (km)
x (km)
U = 1cm/year
z (km)
x (km)
U = 10cm/year
z (km)
x (km)

12. Isotherms in a vertical plane added to a static temperature profile for oceanic lithosphere for point source of 109 cal/s
U = .01cm/year
z (km)
x (km)
U = .1cm/year
z (km)
x (km)
U = 1cm/year
z (km)
x (km)
U = 10cm/year
z (km)
x (km)
Shaded regions are hotter than 1300°C

13. References
Birch, F.S., Conductive heat flow anomalies over a hot spot in a moving medium, Journal of Geophysical Research, Vol 80, No 35, 1975.
Carslaw, H.S., and J.C. Jaeger, Conduction of Heat in Solids, 2nd ed., Oxford University Press, New York, 1959.