This Crustal Heat Gain stuff was way off-topic... sorry,
http://www.scienceagogo.com/forum/ubbthreads.php?ubb=showflat&Number=25270#Post25270so I'll just respond here with the Abstract and some quotes.
....
They don't have all the raw data in this "Letter," but they talk a bit about where their equation came from.
"We use an analytical representation derived from equations presented by Ingersoll et al. [1948] and Carslaw and Jaeger [1959] and employed by Lachenbruch et al. [1982]."
Beltrami, H., J. E. Smerdon, H. N. Pollack, and S. Huang (2002), Continental heat gain in the global climate system, Geophysical Research Letters, 29(8), 1167.
Table 1. Mean Heat Flux into the Continental Lithosphere and the Corresponding Heat Gained in Fifty Year Intervals Over the Last Five Hundred Years
Time Interval Mean Heat Flux (mW m-2) Heat Gain (10^21 J)
1950–2000 39.1 9.1
1900–1950 29.1 6.8
1850–1900 18.0 4.2
1800–1850 14.2 3.3
1750–1800 10.0 2.3
1700–1750 7.6 1.8
1650–1700 4.9 1.1
1600–1650 3.5 0.8
1550–1600 1.9 0.4
1500–1550 1.0 0.2
Total 30.0 x10^21 J.
Abstract
Recent estimates have shown the heat gained by the ocean, atmosphere, and cryosphere as 18.2 · 10^22 J, 6.6 · 10^21 J, and 8.1 · 10^21 J, respectively over the past half-century. However, the heat gain of the lithosphere via a heat flux across the solid surface of the continents (29% of the Earth's surface) has not been addressed. Here we calculate that component of Earth's changing energy budget, using ground-surface temperature reconstructions for the continents. In the last half-century there was an average flux of 39.1 mW m-2 across the land surface into the subsurface, leading to 9.1 · 10^21 J absorbed by the ground. The heat inputs during the last half-century into all the major components of the climate system — atmosphere, ocean, cryosphere, lithosphere-reinforce the conclusion that the warming during the interval has been global.
...
We also recognize the possibility of additional uncertainty associated with the chosen value of average thermal conductivity that is required for this formulation. We have used the value 3.0 W m-1 K-1 which is very typical of a wide variety of common crustal rocks at near-surface temperatures [Cermak and Rybach, 1982]. A formal estimate of the uncertainty in the average thermal conductivity is difficult, but a ±10% range would encompass most educated estimates. .... With specific attention to the 1950–2000 interval we calculate the mean value of the flux for the last half of the 20th century as 39.1 ± 3.5 mW m-2. Integrated over the area of the continents for the fifty-year interval, this flux deposited 9.1 ± 0.8 · 10^21 J into the continental crust. ....
.... The mean fluxes for each fifty year interval are also given in Table 1 along with the corresponding heat gained by the continental lithosphere during each period. The total energy deposited into the continental lithosphere over the last five centuries sums to 3.0 · 10^22 J. These fluxes indicate that 30% of the heat gained by the ground in the last five centuries was deposited during the last fifty years, and over half of the five-century heat gain occurred during the 20th century. However, the total energy gained by the continental lithosphere over the five-hundred year interval is less than the oceanic gain in the last fifty years.
.... Our conclusions about the heat gain of the lithosphere, the last major component of Earth's climate system, are consistent with those presented by Levitus et al. [2001]. While the estimates may eventually be refined, their fundamental implication remains clear: all major components of the Earth's climate system have warmed over the last half century. This further supports the conclusion that the observed warming of Earth during the last fifty years has been truly global and extends upward into the atmosphere as well as downward into the Earth's oceans, cryosphere and continental crust.
fyi...
They have data from 616 boreholes, on six continents. Probably less than 100 are evenly (globally) distributed, and the rest are heavily concentrated in NW N. America, Europe, S. Africa, Mongolia, and Indonesia.
fyi, my interpretation of their equation #1...
Heat Flux, (in milliWatts/square meter), equals twice the thermal conductivity, divided by the square-root of [Pi times the thermal diffusivity (times the change in temperature/depth)]; and summed over the range of measurements multiplied by the "slope of a surface temperature ramp over a given uniform time interval, dt" (times the change in temperature/depth) (adjusted for the range of measurements).
Wow, who figured that out!Anyway....
As you can see from the above table, it really has a dramatic rise, shaped a bit as a "hockey stick."
Right?
Now I can go ponder the significance of the LIA & MWP in relation to Beltrami's 500 year snapshot.