heat flow-2
TRANSCRIPT
Department of Earth SciencesKFUPM
Heat Flow
Solid Earth Geophysics Parks and Plates
©2005 Robert J. LillieNATIONAL PARKLANDSNATIONAL PARKLANDS
Over Hotspot -Highly
Elevated
Off of Hotspot -
Lower
Farther from Hotspot and Eroded
- Even Lower
What could you say about the qualitative behavior of the Heat
Conduction equation after so many things?
1D Heat Conduction Equation
pp c
AT
c
k
t
T
2
3D Heat Conduction Equation
p
c
pp c
A
z
T
c
k
t
T
2
2
Specific heat
Specific heat is defined as the amount of heat required to raise 1 kg of material by 1C. Thus, Wkg-1 oC-1 is the unit for specific heat.
Calculation of Simple Geotherms –Equilibrium Geotherms
pp. 279 of Fowler, 2005
Equilibrium Geotherms
• The temperature vs. depth profile in the Earth is called the geotherm.
• An equilibrium geotherm is a steady state geotherm.• Therefore:
2
20,
T T Aand
t z k
pp. 275 of Fowler, 2005
time
rate A per unit volume per unit time
Thermal Conductivity
Boundary conditions
• Since equation above is a second order differential equation, we should expect to need 2 boundary conditions (bc) to obtain a solution.
• A possible pair of bc’s is:– Temparature T=0 on z=0
– Surface heat flow Q= =-Q0 on z=0
– Then
pp. 277 of Fowler, 2005
2
20,
T T Aand
t z k
zT
kzQ
)(
kQzT
/0
Solution
• Integrate the differential equation once:
• Use the second bc to constrain c1
• Substitute for c1:
1
T Azc
z k
01
Qc
k
0QT Az
z k k
pp. 277 of Fowler, 2005
Solution
• Integrate the differential equation again:
• Use the first bc to constrain c2 which is the constant of integration.
• Substitute for c2:
• Note: Q0 will usually be a negative number in z-positive-downward frame.
20
22
Q zAzT c
k k
2 0c
20
2
Q zAzT
k k
pp. 278 of Fowler, 2005
Oceanic Heat FlowHeat flow is higher over and more scattered over young oceanic crust, which is formed by intrusion of basaltic magma from below.
The heat drives water convection due to very permeable of the fresh basalt despite the fact that ocean crust is gradually covered by impermeable sediment and water convection ceases.
Ocean crust ages as it moves away from the spreading center. It cools and it contracts.
pp. 289 Fowler, 2005
d = 5.65 – 2.47e-t/36
d (km) = bathymetric depth
t (Ma) = Lithosphere age
Q=Heat Flow
pp. 289-290 Fowler, 2005
These data have been empirically modeled in two ways:
d = 2.6 + 0.365t1/2For ages <20 my:
For ages >20 my:
For ages >55 my:For ages <55my: Q = 510 t-
1/2
Q = 48+ 96e-t/36
Oceanic Heat Flow, Mean Depth,t1/2, t-1/2 Age Law
For ages <70 my:
For ages <120 my:
See for detail on Table 7.5, pp. 296, Fowler, 2005
Half Space Model: Specified temperature at top boundary. No bottom boundary condition. Cooling and subsidence are predicted to follow square root of time as discussed by:
Plate Model: Specified temperature at top and bottom boundaries. Cooling and subsidence are predicted to follow an exponential function of time. Roughly matches Half Space Model for first 70 my.
pp. 294 Fowler, 2005
Lith
osph
ere
pp. 298 Fowler, 2005
Plate Motion
The base of the mechanical boundary layer is the isotherm chosen to represent the transition between rigid and viscous behavior. The base of the thermal boundary layer is another isotherm, chosen to represent correctly the temperature gradient immediately beneath the base of the rigid plate. In the upper mantle beneath these boundary layers, the temperature gradient is approximately adiabatic. At about 60-70 Ma, the thermal boundary layer becomes unstable, and small-scale convection starts to occur. With a mantle heat flow of about 38x10-3 Wm-2 the equilibrium thickness of the mechanical boundary layer is approximately 90 km.
Thermal Structure of oceanic lithospheri
c plate.
Radioactive Heat Generation
Radioactive elements:
Uranium (238U, 235U), Thorium (232Th)
and Potassium (40K)
Continental Heat Flow
Heat flow versus crustal age for the continents. The heights of the boxes indicate the standard deviation about the mean heat flow, and the widths indicate the age ranges (After Sclater et al., 1980)
pp. 299 Fowler, 2005
Heat Flow Provinces from Eastern USA
pp. 299 Fowler, 2005
Internal heat generation
Measured Heat Flow
The model of plate cooling with age generally works for continental lithosphere, but is not very useful.
Variations in heat flow in continents is controlled largely by changes in the distribution of heat generating elements and recent tectonic activity.
Continental
Heat Flow
pp. 302 Fowler, 2005
Range of Continental and Oceanic Geotherms in the crust and upper
mantle
pp. 303 Fowler, 2005
Oceanic Lithosphere
Thermal models of the lithospheric plates beneath oceans and continents. The dashed line is the plate thickness predicted by the PSM plate model; k (values of 2.5 and 3.3) is the conductivity in Wm-1 C-1.