Numerical **climate models** use quantitative methods to simulate the interactions of the important drivers of climate, including atmosphere, oceans, land surface and ice. They are used for a variety of purposes from study of the dynamics of the climate system to projections of future climate. Climate models may also be qualitative (i.e. not numerical) models and also narratives, largely descriptive, of possible futures.^{[1]}

Quantitative climate models take account of incoming energy from the sun as short wave electromagnetic radiation, chiefly visible and short-wave (near) infrared, as well as outgoing long wave (far) infrared electromagnetic. An imbalance results in a change in temperature.

Quantitative models vary in complexity. For example, a simple radiant heat transfer model treats the earth as a single point and averages outgoing energy. This can be expanded vertically (radiative-convective models) and/or horizontally. Coupled atmosphere–ocean–sea ice global climate models solve the full equations for mass and energy transfer and radiant exchange. In addition, other types of modelling can be interlinked, such as land use, in Earth System Models, allowing researchers to predict the interaction between climate and ecosystems.

Box models are simplified versions of complex systems, reducing them to boxes (or reservoirs) linked by fluxes. The boxes are assumed to be mixed homogeneously. Within a given box, the concentration of any chemical species is therefore uniform. However, the abundance of a species within a given box may vary as a function of time due to the input to (or loss from) the box or due to the production, consumption or decay of this species within the box.^{[citation needed]}

Simple box models, i.e. box model with a small number of boxes whose properties (e.g. their volume) do not change with time, are often useful to derive analytical formulas describing the dynamics and steady-state abundance of a species. More complex box models are usually solved using numerical techniques.^{[citation needed]}

Box models are used extensively to model environmental systems or ecosystems and in studies of ocean circulation and the carbon cycle.^{[2]}
They are instances of a multi-compartment model.

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. A useful application of the law of conservation of energy to such point objects is the construction of simple Energy Balance Models (or EBM's).

A very simple model of the radiative equilibrium of the Earth is

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 6.371×10^{6}m*π*is the mathematical constant (3.141...)*is the Stefan–Boltzmann constant—approximately 5.67×10*^{−8}J·K^{−4}·m^{−2}·s^{−1}

The constant can be factored out, giving a nildimensional equation for the equilibrium

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

*is Earth's average albedo, measured to be 0.3.*^{[3]}^{[4]}*is Earth's average surface temperature, measured as about 288 K as of year 2020*^{[5]}*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.^{[6]}

The calculated emissivity can be compared to available data. Terrestrial surface emissivities are all in the range of 0.96 to 0.99^{[7]}^{[8]} (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^{[9]} (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 258 K (−15 °C; 5 °F).^{[10]} Taking all this properly into account results in an effective earth emissivity of about 0.64 (earth average temperature 285 K (12 °C; 53 °F)).^{[citation needed]}

Dimensionless models have also been constructed with functionally separated atmospheric layers from the surface. The simplest of these is the zero-dimensional, one-layer model,^{[11]} 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 interfaces between layers produces a set of coupled equations which are solvable.^{[12]}

Layered models produce temperatures that better estimate those observed for Earth's surface and atmospheric levels.^{[13]} 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.^{[14]}

The zero-dimensional model above, using the solar constant and given average earth temperature, determines the effective earth emissivity of long wave radiation emitted to space. This can be refined in the vertical to a one-dimensional radiative-convective model, which considers two processes of energy transport:^{[15]}

- upwelling and downwelling radiative transfer through atmospheric layers that both absorb and emit infrared radiation
- upward transport of heat by convection (especially important in the lower troposphere).

The radiative-convective models have advantages over the simple model: they can determine the effects of varying greenhouse gas concentrations on effective emissivity and therefore the surface temperature. But added parameters are needed to determine local emissivity and albedo and address the factors that move energy about the Earth.^{[citation needed]}

Effect of ice-albedo feedback on global sensitivity in a one-dimensional radiative-convective climate model.^{[16]}^{[17]}^{[18]}

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.^{[19]}

Main article: Earth systems model of intermediate complexity |

Depending on the nature of questions asked and the pertinent time scales, there are, on the one extreme, conceptual, more inductive models, and, on the other extreme, general circulation models operating at the highest spatial and temporal resolution currently feasible. Models of intermediate complexity bridge the gap. One example is the Climber-3 model. Its atmosphere is a 2.5-dimensional statistical-dynamical model with 7.5° × 22.5° resolution and time step of half a day; the ocean is MOM-3 (Modular Ocean Model) with a 3.75° × 3.75° grid and 24 vertical levels.^{[20]}

Main article: General circulation model |

General Circulation Models (GCMs) discretise the equations for fluid motion and energy transfer and integrate these over time. Unlike simpler models, GCMs divide the atmosphere and/or oceans into grids of discrete "cells", which represent computational units. Unlike simpler models which make mixing assumptions, processes internal to a cell—such as convection—that occur on scales too small to be resolved directly are parameterised at the cell level, while other functions govern the interface between cells.

Atmospheric GCMs (AGCMs) model the atmosphere and impose sea surface temperatures as boundary conditions. Coupled atmosphere-ocean GCMs (AOGCMs, e.g. HadCM3, EdGCM, GFDL CM2.X, ARPEGE-Climat)^{[21]} combine the two models. The first general circulation climate model that combined both oceanic and atmospheric processes was developed in the late 1960s at the NOAA Geophysical Fluid Dynamics Laboratory^{[22]} AOGCMs represent the pinnacle of complexity in climate models and internalise as many processes as possible. However, they are still under development and uncertainties remain. They may be coupled to models of other processes, such as the carbon cycle, so as to better model feedback effects. Such integrated multi-system models are sometimes referred to as either "earth system models" or "global climate models."

There are three major types of institution where climate models are developed, implemented and used:

- National meteorological services. Most national weather services have a climatology section.
- Universities. Relevant departments include atmospheric sciences, meteorology, climatology, and geography.
- National and international research laboratories. Examples include the National Center for Atmospheric Research (NCAR, in Boulder, Colorado, US), the Geophysical Fluid Dynamics Laboratory (GFDL, in Princeton, New Jersey, US), Los Alamos National Laboratory, the Hadley Centre for Climate Prediction and Research (in Exeter, UK), the Max Planck Institute for Meteorology in Hamburg, Germany, or the Laboratoire des Sciences du Climat et de l'Environnement (LSCE), France, to name but a few.

The World Climate Research Programme (WCRP), hosted by the World Meteorological Organization (WMO), coordinates research activities on climate modelling worldwide.

A 2012 U.S. National Research Council report discussed how the large and diverse U.S. climate modeling enterprise could evolve to become more unified.^{[23]} Efficiencies could be gained by developing a common software infrastructure shared by all U.S. climate researchers, and holding an annual climate modeling forum, the report found.^{[24]}