Human societies have dramatically changed the composition of the atmosphere, raising carbon dioxide levels from a preindustrial value of 280 parts per million to over 400 parts per million, primarily from fossil fuel burning and deforestation. Since carbon dioxide is a greenhouse gas, one expects an increase in global temperature due to the modified atmospheric composition, and temperatures have warmed about 0.8 ^∘C since the early twentieth century. Independent evidence from satellite data, ground stations, ship measurements, and mountain glaciers all show that global warming has occurred [ 1]. Other aspects of climate that have changed include atmospheric water vapor concentration, Arctic sea ice coverage, precipitation intensity, extent of the Hadley circulation, and height of the tropopause in accordance with predictions.

Because there is considerable interest in predicting future climate changes over the coming decades, there is a large international effort focused on climate modeling at high spatial resolution, with detailed treatment of complex physics. These models, known as general circulation models or global climate models (GCMs), incorporate physical effects such as the fluid dynamics of the atmosphere and ocean, radiative transfer, shallow and deep moist convection, cloud formation, boundary layer turbulence, and gravity wave drag [ 2, 3]. More recently, Earth system models have additionally incorporated effects such as atmospheric chemistry and aerosol formation, the carbon cycle, and dynamic vegetation.

Due to the complexity of GCMs, output from these models can often be difficult to interpret. To make progress in understanding, there has been a concerted effort among climate scientists and applied mathematicians to develop a hierarchy of dynamical models, designed to better understand climate phenomena. Held [ 4] has provided an argument for the usefulness of hierarchies within climate science. Four different classes of simplified dynamical models of climate change are discussed in this entry.

The essence of global warming can be elegantly expressed as a one-dimensional system, with temperature and atmospheric composition a function of height alone. The first necessary physical process is radiative transfer, which includes both solar heating and radiation emitted from the Earth, which is partially absorbed and reemitted by greenhouse gases. The second necessary ingredient is convection, which mixes energy vertically within the troposphere, or weather layer, on Earth. The first radiative-convective calculation of carbon dioxide-induced global warming was performed by Manabe and Strickler in 1964 [ 5]. Many example radiative-convective codes are publicly available.

GCMs typically use the primitive equations as their dynamical equations, which assume hydrostatic balance and the corresponding small-aspect ratio assumptions that are consistent with this. The primitive equations are where λ = longitude, θ = latitude, p = pressure, u = zonal wind, v = meridional wind, \(f = 2\varOmega \sin (\theta ) =\) Coriolis parameter, a = Earth radius, \(\varPhi = g\,z =\) geopotential with g = gravitational acceleration and z = height, \(\omega ={ Dp \over Dt} =\) pressure velocity, T = temperature, \(T_{v} = T/(1 - (1 - R_{d}/R_{v})q) =\) virtual temperature (which takes into account the density difference of water vapor), R _d = ideal gas constant for dry air, R _v = ideal gas constant for water vapor, \(\kappa ={ R \over c_{p}}\), and c _p = specific heat of dry air.

Much of the complexity of comprehensive GCMs comes from parameterizations of Q = heating and F = momentum sources. A realistic circulation can be produced from remarkably simple parameterizations of Q and F. Held and Suarez [ 6] used “Newtonian cooling” and “Rayleigh friction” in their dry dynamical core model: The equilibrium temperature T _eq is chosen essentially to approximate the temperature structure of Earth if atmospheric motions were not present. Rayleigh friction exists within the near-surface planetary boundary layer. The relaxation times τ _Q and τ _D are chosen to approximate the typical timescales of radiation and planetary boundary layer processes. Hyperdiffusion is typically added as a final ingredient.

The dry dynamical core model can be used to calculate dynamical responses to climate change by prescribing heating patterns similar to those experienced with global warming, for instance, warming in the upper tropical troposphere, tropopause height increases, stratospheric cooling, and polar amplification. An example of such a study, which examines responses of the midlatitude jet stream and storm tracks, is Butler et al. [ 7].

In order to study these impacts of climate change in an idealized model with an active moisture budget, Frierson et al. [ 8] developed the gray-radiation moist (GRaM) GCM. This model has radiation that is only a function of temperature, and simplified surface flux boundary layer, and moist convection schemes. In addition to the influences of water vapor listed above, this model has been used to study the effect on overturning circulations, precipitation extremes, and movement of rain bands.

A final class of simplified dynamical models of climate change are those with simplified vertical structure. Most famous of these are energy balance models, reviewed in North et al. [ 9], which represent the vertically integrated energy transport divergence in the atmosphere as a diffusion. The simplest steady-state energy balance model can be written as where S is the net (downward minus upward) solar radiation; \(L = A + BT\) is the outgoing longwave radiation, a linear function of temperature T; and D is diffusivity. Typically written as a function of latitude alone, the energy balance model can incorporate climate feedbacks such as the ice-albedo feedback and can calculate the temperature response due to changes in outgoing radiation (e.g., by modifying A). More recently, energy balance models have been used to interpret the results from both comprehensive GCMs and idealized GCMs such as those described above. This exemplifies the usefulness of a hierarchy of models for developing understanding about the climate and climate change.