Simulation of the climate system in full 3-D space and time was impractical prior to the establishment of large computational facilities starting in the 1960s. In order to begin to understand which factors may have changed Earth's
paleoclimate states, the constituent and dimensional complexities of the system needed to be reduced. A simple quantitative model that balanced incoming/outgoing energy was first developed for the atmosphere in the late 19th century. Essential features of EBMs include their relative conceptual simplicity and their ability to sometimes produce
analytical solutions. Some models account for effects of ocean, land, or ice features on the surface budget. Others include interactions with parts of the
water cycle or
carbon cycle. A variety of these and other reduced system models can be useful for specialized tasks that supplement GCMs, particularly to bridge gaps between simulation and understanding.
Zero-dimensional models Zero-dimensional models consider Earth as a point in space, analogous to the
pale blue dot viewed by
Voyager 1 or an astronomer's view of very distant objects. This
dimensionless view while highly limited is still useful in that the laws of physics are applicable in a bulk fashion to unknown objects, or in an appropriate
lumped manner if some major properties of the object are known. For example, astronomers know that most planets in our own solar system feature some kind of solid/liquid surface surrounded by a gaseous atmosphere.
Model with combined surface and atmosphere A very simple model of the
radiative equilibrium of the Earth is :(1-a)S \pi r^2 = 4 \pi r^2 \epsilon \sigma T^4 where • the left hand side represents the total incoming shortwave power (in Watts) from the Sun • the right hand side represents the total outgoing longwave power (in Watts) from Earth, calculated from the
Stefan–Boltzmann law. The constant parameters include •
S is the
solar constant – the incoming solar radiation per unit area—about 1367 W·m−2 •
r is Earth's radius—approximately 6.371 million meters (m) •
π is the mathematical constant (3.141...) •
\sigma is the
Stefan–Boltzmann constant—approximately 5.67×10−8 J·K−4·m−2·s−1 The constant \pi\,r^2 can be factored out, giving a nildimensional equation for the equilibrium :(1-a)S = 4 \epsilon \sigma T^4 where • the left hand side represents the incoming shortwave energy flux from the Sun in W·m−2 • the right hand side represents the outgoing longwave energy flux from Earth in W·m−2. The remaining variable parameters which are specific to the planet include •
a is Earth's average
albedo, measured to be 0.3. •
T is Earth's
average surface temperature, measured as about 288
Kelvin (K) as of year 2020 •
\epsilon is the
effective emissivity of Earth's combined surface and atmosphere (including clouds). It is a quantity between 0 and 1 that is calculated from the equilibrium to be about 0.61. For the zero-dimensional treatment it is equivalent to an average value over all viewing angles. This very simple model is quite instructive. For example, it shows the temperature sensitivity to changes in the solar constant, Earth albedo, or effective Earth emissivity. The effective emissivity also gauges the strength of the atmospheric
greenhouse effect, since it is the ratio of the thermal emissions escaping to space versus those emanating from the surface. The calculated emissivity can be compared to available data. Terrestrial surface emissivities are all in the range of 0.96 to 0.99 (except for some small desert areas which may be as low as 0.7). Clouds, however, which cover about half of the planet's surface, have an average emissivity of about 0.5 (which must be reduced by the fourth power of the ratio of cloud absolute temperature to average surface absolute temperature) and an average cloud temperature of about . Taking all this properly into account results in an effective earth emissivity of about 0.64 (earth average temperature ).
Models with separated surface and atmospheric layers Dimensionless models have also been constructed with functionally distinct atmospheric layers from the surface. The simplest of these is the
zero-dimensional, one-layer model, which may be readily extended to an arbitrary number of atmospheric layers. The surface and atmospheric layer(s) are each characterized by a corresponding temperature and emissivity value, but no thickness. Applying radiative equilibrium (i.e conservation of energy) at the idealized interfaces between layers produces a set of coupled equations which are solvable. These multi-layered EBMs are examples of
multi-compartment models. They can estimate average temperatures closer to those observed for Earth's surface and troposphere. They likewise further illustrate the radiative
heat transfer processes which underlie the greenhouse effect. Quantification of this phenomenon using a version of the one-layer model was first published by
Svante Arrhenius in year 1896.
Radiative-convective models Water vapor is a main determinant of the emissivity of Earth's atmosphere. It both influences the flows of radiation and is influenced by convective flows of heat in a manner that is consistent with its equilibrium concentration and temperature as a function of elevation (i.e.
relative humidity distribution). This has been shown by refining the zero dimension model in the vertical to a one-dimensional radiative-convective model which considers two processes of energy transport: • upwelling and downwelling radiative transfer through atmospheric layers that both absorb and emit infrared radiation • upward transport of heat by air and vapor convection, which is especially important in the lower
troposphere. Radiative-convective models typically use a
distributed model of the atmosphere versus elevation. This has advantages over the lumped models and also lays a foundation for more complex models. They can estimate both surface temperature and the temperature variation with elevation in a more realistic manner. In particular, they properly simulate the observed decline in upper atmospheric temperature and the rise in surface temperature when
trace amounts of other non-condensible greenhouse gases such as
carbon dioxide are included.
Higher-dimension models The zero-dimensional model may be expanded to consider the energy transported horizontally in the atmosphere. This kind of model may well be
zonally averaged. This model has the advantage of allowing a rational dependence of
local albedo and emissivity on temperature – the poles can be allowed to be icy and the equator warm – but the lack of true dynamics means that horizontal transports have to be specified. Early examples include research of
Mikhail Budyko and
William D. Sellers who worked on the
Budyko-Sellers model. This work also showed the role of
positive feedback in the climate system and has been considered foundational for the energy balance models since its publication in 1969. == Earth systems models of intermediate complexity (EMICs) ==