Fourier's Law of Heat Conduction
q = -kdT/dy
where q is heat flow (mW/m2), k is thermal conductivity (W/m-°C), T is temperature (°C) and y is depth (m).
Typical values of temperature gradient is 20-30 °C/km. Thermal conductivity of rocks range from 1 to 5 W/m-°C.
Near surface temperatures are influenced by climatic changes. Steady state thermal structure below about 300 m.
Thermal conductivity is measured in a divided bar apparatus. An unknown sample is placed between two brass disks. The top and bottom are held at fixed temperatures. Temperatures at the interface between the sample and the brass disk are measured until they remain constant. Thermal conductivity of the unknown sample is determined from the temperature gradient and the heat flow, which is known for the two brass disks.
Earth's surface heat flow
Heat heat flow is low (56.5 mW/m2 for the continents and 78.2 mW/m2 for the oceans).
Continental heat flow correlates with near surface radiogenic heat production.
Ocean heat flow correlates with age of the seafloor.
Heat Generation by Decay of Radioactive Elements
Some of the heat lost to space is replaced by radiogenic heat production cause by decay of U, Th, and K. Some of the heat loss results in cooling of the earth. (20% according to Turcotte and Schubert).
238U and 232Th are the primary heat producers today. 235U and 40K were more important in the Earth's early history. They decay faster and in much smaller abundance now.
Most radioactive isotopes are concentrated in the continental crust. There is a small amount in the oceanic crust and very little in the mantle. Radioactive isotopes decrease exponentially in abundance with depth.
One-dimensional Steady Heat Conduction with Volumetric Heat Production
-kd2T/dy2 = rH
where r is density and H is heat production per mass. This is a mathematical statement of conservation of heat energy. The left side of the equation is the net heat gain or loss from heat conduction, which must be precisely balanced by the heat generated.
Temperature is given by
T = Ts + (qs/k)y - (rH/2k)y2
The temperature gradient is not linear. It is a quadratic function of depth. Heat flow is higher at the surface than at depth.
Conduction Temperature Profile for the Mantle
If the near surface temperature gradient is projected downward into the mantle to 200 km depth. Temperatures are above 3000 °C and the entire mantle would be molten. Heat conduction is a good fit to temperature gradients in the lithosphere but not the asthenosphere.
Conductive temperature profiles are a good fit to observed temperatures in the continental crust if radioactive elements are concentrated in the upper part of the crust. The best fit is for radioactive elements to decrease exponential with depth. About one half of the surface heat flow is from radiogenic heat production within the upper crust and the other half comes from the mantle.
Subsurface Temperature due to Periodic Surface Temperature and Topography
Changes in topography or the presence of lakes produce changes in subsurface temperatures. Surface temperature variations decay exponentially with depth in a distance proportional to the horizontal wavelength of the surface temperature variation.
One-Dimensional, Time-Dependent Heat Conduction
rc¶T/¶t = k¶2T/¶t2 or ¶T/¶t = k¶2T/¶t2
where c is the heat capacity (how much energy does it take to raise 1 kg of something 1 °C) and k is the thermal diffusivity (k/rc) This is a mathematical statement of heat energy conservation. The net heat gain/loss due to conduction (right hand side) is equal to the net change in heat energy with time (left hand side). Thermal diffusivity relates how long it takes a temperature anomaly to propagate a given distance.
Periodic Heating of a semi-infinite half-space
Thermal diffusivity of earth materials is low so even long term changes (ice ages) only effect the upper few hundred meters.
Surface variations in temperature with time decrease in amplitude with depth. In addition, they are out of phase. It takes time for heat to diffuse downward, so the surface temperature may already be declining when a region in the subsurface starts to heat up due to an earlier increase in surface temperature.
T = To + DT exp(-y*sqrt(w/2k) cos(wt - y*sqrt(w/2k)
where To is the reference or mean surface temperature and w is the circular frequency (two pi times frequency).
Instantaneous Heating or Cooling of a Semi-Infinite Half-Space
Boundary conditions are constant temperature at t =0, fixed surface temperature, and initial temperature does not change at some very large depth.
Use a dimensionless similarity solution to get Temperature
(T-To)/(Ts-To) = erfc(y/2sqrt(kt))
where erfc is the complementary error function.
Thickness of the boundary layer
yT =2.32 sqrt(kt)
q = k (Ts-To)/(sqrt3.14159kt)
Cooling of Oceanic Lithosphere - substitute distance from ridge crest divided by spreading rate for time and mantle temperature for To in cooling half-space model.
(T-Tm)/(Ts-Tm) = erfc(y/2sqrt(kx/u))
where erfc is the complementary error function.
Thickness of the boundary layer - in good agreement with seismic data
YL =2.32 sqrt(kx/u)
Heat flow - in good agreement overall (heat flow correlates with age) but too high for young crust because of hydrothermal circulation and too low for very old crust implying that there is some constant heat flow from the asthenosphere.
qs = k (Tm-Ts) sqrt(u/(3.14159kx))
Cooling half space does not fit continental heat flow data.