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The daily envelope shrinks as e^(−z/d): below about 3d (35 cm) the day-night swing has practically vanished. Green dot: the sleeping chamber.
The science
Heat conduction in soil. Temperature obeys the diffusion equation ∂T/∂t = α ∂²T/∂z², α = k/ρc ≈ 5×10-7 m²/s for moist loam. It is solved with a Crank–Nicolson scheme on a 2 cm grid (second order, unconditionally stable) with the surface temperature as the top boundary and the undisturbed periodic wave at 4 m as the bottom boundary.
Damping depth. For a surface forced by A cos ωt the periodic solution is T = Tm + A e−z/d cos(ωt − z/d) with d = √(2α/ω): about 12 cm for the daily wave and 2.2 m for the annual wave. The wave shrinks by e at z = d and is delayed by z/(dω): the day-night swing is gone below ~40 cm, and a 1 m chamber lags the seasons by about 4 weeks. This is why burrows are a thermal refuge.
Animal energetics. Heat production M = max(BMR·a, C(Tb − Ta)) plus a panting term above 36 °C, with body temperature Tb = 34 °C and a narrow thermoneutral zone (lower critical 25 °C). BMR = 2.04 m0.75 W (about 0.6 of Kleiber's 3.4 m0.75; xenarthrans have low metabolic rates) and the minimum conductance C = BMR/(Tb − Tlc). The animal rests in the chamber (soil temperature + bedding) and forages in a chosen window if the air is warm enough; intake is a bell-shaped prey-activity factor times a peak rate. All physiological constants are illustrative values tuned for plausible behaviour, not measurements of a particular population; the fat reserve is capped at 18 MJ and ignores growth, reproduction and water balance.
Checks. (1) numerical amplitude vs 10 e−z/d at 10 and 20 cm; (2) phase lag vs z/d; (3) the erfc solution for a surface step T = T0 + ΔT erfc(z/2√(αt)); (4) three years of the annual wave through the production grid vs the analytic wave at 1 m; (5) continuity of the metabolic model at the lower critical temperature. Simplifications: 1-D vertical conduction, constant α, no evaporation or rain, surface temperature = air temperature × exposure factor, a single chamber.
Armadillo, grubs, plants, rocks, burrow mound and ground are models made in Blender; the temperature section is drawn from the simulated field.