Rock Cover and Thermal Damping

What is modelled

Rock conducts heat slowly. When the surface warms and cools in a regular cycle, a heat wave runs into the rock, getting weaker and later with depth. Below a few damping depths the temperature hardly changes, which is why rooms cut deep into rock stay cool in the day and warm at night. A longer cycle, such as the year, reaches much deeper than the day.

Equations

• Heat equation ∂T/∂t = α ∂²T/∂x² with surface T = Tm + A cos(ω t).
• Damping depth δ = √(2 α / ω), solution T = Tm + A e−x/δ cos(ω t − x/δ).
• Time lag x / (δ ω) and 10 % depth δ ln 10.
• Surface heat flux amplitude √2 k A / δ.

Checks run in the page

• A Crank-Nicolson finite-difference run against the analytic amplitude and phase at the room.
• The 10 % penetration depth found by bisection.
• The surface heat flux amplitude and its zero cycle mean.
• The scaling of damping depth with period and diffusivity.

Honest simplifications

The rock is uniform and semi-infinite, the surface temperature is a pure sine, the room is only a marker at one depth and does not disturb the heat flow, and moisture, air circulation, solar radiation and rain are left out. Real churches also depend on ventilation, openings and the shape of the carved space, so this shows only the conduction part.

• Press Thin 20 cm cover, daily to see the room follow the surface.
• Press Annual cycle, 2.5 m cover to watch the heat wave reach many metres deeper and arrive months later.
• Press Porous scoria, monthly to see a low-diffusivity rock damp the swing faster.