  Thermal phase curves of exoplanet atmospheres have revealed temperature maps
as a function of planetary longitude, often by sinusoidal decomposition of the
phase curve. We construct a framework for describing two-dimensional
temperature maps of exoplanets with mathematical basis functions derived for a
fluid layer on a rotating, heated sphere with drag/friction, which are
generalizations of spherical harmonics. These basis functions naturally produce
physically-motivated temperature maps for exoplanets with few free parameters.
We investigate best practices for applying this framework to temperature maps
of hot Jupiters by splitting the problem into two parts: (1) we constrain the
temperature map as a function of latitude by tuning the basis functions to
reproduce general circulation model (GCM) outputs, since disk-integrated phase
curve observations do not constrain this dimension; and (2) we infer the
temperature maps of real hot Jupiters using original reductions of several
Spitzer phase curves, which directly constrain the temperature variations with
longitude. The resulting phase curves can be described with only three free
parameters per bandpass -- an efficiency improvement over the usual five or so
used to describe sinusoidal decompositions of phase curves. Upon obtaining the
hemispherically averaged dayside and nightside temperatures, the standard
approach would be to use zero-dimensional box models to infer the Bond albedo
and redistribution efficiency. We elucidate the limitation of these box models
by demonstrating that negative Bond albedos may be obtained due to a choice of
boundary condition on the nightside temperature. We propose generalized
definitions for the Bond albedo and heat redistribution efficiency for use with
two-dimensional (2D) temperature maps. Open-source software called kelp is
provided to efficiently compute these phase curves.
