Planetary surface energy balance

A planetary surface energy balance equates absorbed short-wave radiation and heat supplied from below with reflected short-wave radiation and emitted long-wave radiation. With incident flux $F_r$, Bond albedo $\alpha$, conductive upward flux $F_c$, and emissivity $ϵ$,

The solar constant is solar irradiance on a plane normal to incoming sunlight at a specified reference distance from the Sun, conventionally the mean Earth-Sun distance. Local horizontal irradiance includes solar geometry and atmospheric transmission.

At latitude $\varphi$ and solar declination $\delta$, integrate $S_0\max (0,\sin \varphi \sin \delta +\cos \varphi \cos \delta \cos H)$ over hour angle $H$. Sunset occurs at $H_0=\arccos (-\tan \varphi \tan \delta )$ when this lies in $[0,\pi ]$, with clipping for polar day or night. The displayed daily mean is at the top of the atmosphere; local absorbed heat flux also includes transmission and surface albedo.

Ocean heat content relative to a reference temperature $T_f$ is an integral through the water column. For a uniform column of depth $H$, $Q={\rho }_w{c}_{w}H(T-T_f)$ per horizontal area. Melting ice already at its fusion temperature requires its latent heat; warming cold ice first requires additional sensible heat. A surface temperature alone does not determine the column integral.

Surface albedo is the fraction of incident solar energy flux reflected by a surface, averaged over the specified spectrum and illumination geometry. The locally absorbed fraction is $1-\alpha$. It is distinct from geometric albedo, which measures brightness in a particular viewing configuration.