grid.md

Spatial Grid

The finite difference grid uses geometrically increasing layer thicknesses, following Hayne et al. (2017), Appendix A2.2 (Eqs. A31--A33).

Thermal Skin Depth

The thermal skin depth $z_s$ is the characteristic depth of penetration of a periodic temperature wave:

$$ z_s = \sqrt{\frac{\kappa P}{\pi}} $$

where $P$ is the forcing period (e.g., the diurnal period) and $\kappa = K / (\rho c_p)$ is the thermal diffusivity.

For the Moon, z_s ≈ 4–7 cm depending on the H-parameter and temperature-dependent thermal properties (Hayne et al., 2017). The grid is constructed using surface-minimum properties (~3 cm), which ensures adequate resolution near the surface where gradients are steepest.

Grid Construction

The grid starts at $z = 0$ (the surface) with an initial layer thickness:

$$ \Delta z_0 = \frac{z_s}{m} $$

where $m$ is the number of layers within the first skin depth (default: 10).

Layer thickness grows geometrically with depth:

$$ \Delta z_{i+1} = \Delta z_i \left(1 + \frac{1}{n}\right) $$

where $n$ controls the growth rate (default: 5). Larger $n$ gives more uniform layers; smaller $n$ gives faster growth.

The grid extends to a total depth of $b$ skin depths (default: 20), ensuring that the bottom boundary is far enough below the surface that diurnal temperature variations are negligible.

Grid Parameters

Parameter	Default	Description
$m$	10	Layers per skin depth
$n$	5	Growth factor (dz[i+1] = dz[i]*(1+1/n))
$b$	20	Total depth in skin depths

For the Moon, these defaults produce approximately 45 layers extending to a depth of about 1 m.