Does anyone know the amount of thermal energy that the Earth's mantle and core possess? I don't mean the maximum limit of electrical power we could generate with geothermal plants, but rather: if you took the Earth and magically cooled it down so that its temperature became homogeneously 0K, what would the change in the Earth's internal energy be? (ignoring the Sun)
If we don't have that data, does anyone have an idea how to perform an order of magnitude estimate?
The amount of thermal energy in the Earth's core and mantle is determined primarily by the temperature. The three-dimensional temperature field inside the earth is imprecisely known. One dimensional "Onion skin" models of the earth's interior are based upon empirical evidence from seismology, geodesy, and mineral physics, but there are almost certainly lateral variations in temperature which are important to understand.
The article mentioned in the comments can provide you with estimated temperatures at important discontinuities inside the earth. Connect these with constant gradients, or find an alternative model temperature profile.
You should be able to find some specific heat capacity measurements for periodite (mantle), perovskite, and liquid iron. To an order of magnitude these should be like $10^2$ - $10^3$ $\text{J} \text{ kg}^{-1} \text{K}$.
You'll also need the density profile, and can get that from a one dimensional model of the density of the earth. You might try Preliminary Reference Earth Model (PREM) or The Reference Earth Model Website. The latter reference models include three dimensional models.
A reasonable estimate for the total internal heat of the Earth is $Q_E\sim 2 \times 10^{31}\,\mathrm{J}$ (see below).
The value most often found online, however, is "$12.6 \times 10^{24}\,\mathrm{MJ}$" (=$1.26 \times 10^{31}\,\mathrm{J}$). This can be traced back to the estimate of $3 \times 10^{27}\,\mathrm{kgcal}$ given in Chapter 4 of Armstead's 1983 text on Geothermal Energy. (Given that the precision of the source is only 1 significant figure ("3"), the value would be better reported as $\sim 1.3 \times 10^{31}\,\mathrm{J}$ or $\sim 1 \times 10^{31}\,\mathrm{J}$.) Unfortunately, Armstead gives neither the calculation underlying this estimate nor cites a source.
This value is consistent with a quick Fermi estimate that the Earth's total internal heat is $$Q_E \sim T_E\,M_E\,c_E \sim 2 \times 10^{31}\,\mathrm{J}$$ where $T_E\sim 3000\,\mathrm{K}$ and $c_E\sim 1000\,\mathrm{J/(kg\,K)}$ are plausible guesses for the average internal temperature and average specific heat capacity of the Earth's interior, and $M_E$ is the Earth's mass.
A consistent estimate of $2 \times 10^{31}\,\mathrm{J}$ can be found on page 376 and Table 4.1 of Detlev Möller's "Chemistry of the Climate System - Vol. 2: History, Change and Sustainability, 3rd Ed (2020)." (A version from an earlier edition can be found in the answer to another question.) This breaks down the calculation layer by layer within the Earth, but the heat content of the individual layers seem inconsistent, so I have recalculated a similar table below:
| Region | Depth | T | Density | Mass | Matter | Specific Heat | Heat |
|---|---|---|---|---|---|---|---|
| (km) | (km) | (°C) | ($g/cm^3$) | (kg) | (J/kg/K) | (J) | |
| crust | 0-30 | ~350 | ~3 | 4.6$\times10^{22}$ | rocks | 1000 | 1.6$\times10^{28}$ |
| outer mantle | 30-300 | ~2000 | ~3.5 | 4.6$\times10^{23}$ | rocks | 1100 | 1.0$\times10^{30}$ |
| inner mantle | 300-2890 | ~3000 | ~4.7 | 3.6$\times10^{24}$ | rocks | 1100 | 1.2$\times10^{31}$ |
| outer core | 2890-5150 | ~5000 | ~10.5 | 1.8$\times10^{24}$ | Fe-Ni | 800 | 7.1$\times10^{30}$ |
| inner core | 5150-6371 | ~6000 | ~12.5 | $\times10^{22}$ | Fe | 800 | 4.6$\times10^{29}$ |
| Total | 6.0$\times10^{24}$ | 2.0$\times10^{31}$ |
I used eyeball averages of the geothermal gradient and Preliminary Earth Reference Model (PREM) density profile. One can find different values for these and the specific heats, but nothing that changes the total heat value significantly.
