Climate model

Numerical climate models are mathematical models that can simulate the interactions of important drivers of climate. These drivers are the atmosphere, oceans, land surface and ice. Scientists use climate models to study the dynamics of the climate system and to make projections of future climate and of climate change. Climate models can also be qualitative models and contain narratives, largely descriptive, of possible futures.
Climate models take account of incoming energy from the Sun as well as outgoing energy from Earth. An imbalance results in a change in temperature. The incoming energy from the Sun is in the form of short wave electromagnetic radiation, chiefly visible and short-wave infrared. The outgoing energy is in the form of long wave infrared electromagnetic energy. These processes are part of the greenhouse effect.
Climate 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 and horizontally. More complex models are the coupled atmosphere–ocean–sea ice global climate models. These types of models solve the full equations for mass transfer, energy transfer and radiant exchange. In addition, other types of models can be interlinked. For example Earth System Models include also land use as well as land use changes. This allows researchers to predict the interactions between climate and ecosystems.
Climate models are systems of differential equations based on the basic laws of physics, fluid motion, and chemistry. Scientists divide the planet into a 3-dimensional grid and apply the basic equations to those grids. Atmospheric models calculate winds, heat transfer, radiation, relative humidity, and surface hydrology within each grid and evaluate interactions with neighboring points. These are coupled with oceanic models to simulate climate variability and change that occurs on different timescales due to shifting ocean currents and the much larger heat storage capacity of the global ocean. External drivers of change may also be applied. Including an ice-sheet model better accounts for long term effects such as sea level rise.

Uses

Complex climate models enable extreme event attribution, which is the science of identifying and quantifying the role that human-caused climate change plays in the frequency, intensity and impacts of extreme weather events. Attribution science aims to determine the degree to which such events can be explained by or linked to human-caused global warming, and are not simply due to random climate variability or natural weather patterns.
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, the Geophysical Fluid Dynamics Laboratory, Los Alamos National Laboratory, the Hadley Centre for Climate Prediction and Research, the Max Planck Institute for Meteorology in Hamburg, Germany, or the Laboratoire des Sciences du Climat et de l'Environnement, France.

Big climate models are essential but they are not perfect. Attention still needs to be given to the real world. The global models are essential to assimilate all the observations, especially from space and produce comprehensive analyses of what is happening, and then they can be used to make predictions/projections. Simple models have a role to play that is widely abused and fails to recognize the simplifications such as not including a water cycle.

General circulation models (GCMs)

Energy balance models (EBMs)

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. Other EBMs similarly seek an economical description of surface temperatures by applying the conservation of energy constraint to individual columns of the Earth-atmosphere system.
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
where

the left hand side represents the total incoming shortwave power from the Sun
the right hand side represents the total outgoing longwave power 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
π is the mathematical constant
' 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.
' is Earth's average surface temperature, measured as about 288 Kelvin as of year 2020
' is the effective emissivity of Earth's combined surface and atmosphere. 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. Clouds, however, which cover about half of the planet's surface, have an average emissivity of about 0.5 and an average cloud temperature of about. Taking all this properly into account results in an effective earth emissivity of about 0.64.

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 are each characterized by a corresponding temperature and emissivity value, but no thickness. Applying radiative equilibrium 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

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. 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.
Other parameters are sometimes included to simulate localized effects in other dimensions and to address the factors that move energy about Earth. For example, the effect of ice-albedo feedback on global climate sensitivity has been investigated using a one-dimensional radiative-convective climate model.