Past exam of the physics of atmospheres and oceans course of the University of Oxford / 2018 / 4 / c

In the quasisteady diffusion approximation, the vapour-density disturbance outside a small stationary particle satisfies Laplace's equation, with a specified equilibrium value on its surface and a uniform far-field value. This has the same mathematical boundary-value problem as the electrostatic potential outside a conductor held at fixed potential. The integrated normal gradient is proportional to the conductor's geometrical capacitance.

Here $C$ is a length: a sphere of radius $r$ has $C=r$. The corresponding electrical capacitance in SI units would be $4\pi {ϵ}_{0}C$, not $C$ farads. Replacing the spherical diffusive conductance $4\pi r$ by $4\pi C$ accounts for the crystal's non-spherical geometry. Coupling the analogous heat-conduction problem to vapour diffusion supplies the ice-specific thermal and diffusional resistances.

The approximation assumes continuum diffusion, equilibration of the surface vapour concentration, an approximately isothermal particle, a slowly changing shape, weak perturbations and a far field sufficiently distant relative to the crystal size. It neglects ventilation by falling motion, interactions with neighbouring particles and facet-dependent attachment kinetics. Shape affects diffusive growth through capacitance, not merely through projected collecting area as in a collision calculation.