Numerical climate models, also called climate system models, are mathematical tools that help scientists understand how different parts of Earth's climate work together. These parts include the atmosphere, oceans, land, and ice. Scientists use these models to study how the climate system changes over time and to predict future climate conditions. Some climate models are not numerical and instead use descriptive stories to explain possible future changes.

These models consider energy from the Sun and energy leaving Earth. When more energy comes in than goes out, Earth's temperature changes. Energy from the Sun is mostly visible light and short-wave infrared radiation. Energy leaving Earth is long-wave infrared radiation. These processes are part of the greenhouse effect.

Climate models range in complexity. A simple model might treat Earth as a single point and average outgoing energy. More complex models divide Earth into sections vertically and horizontally. The most detailed models, called coupled atmosphere–ocean–sea ice global climate models, solve complex equations for how mass, energy, and radiation move. Other models, like Earth System Models, include land use and how it changes. This helps scientists study how climate and ecosystems interact.

Climate models use math equations based on physics, fluid motion, and chemistry. Scientists divide Earth into a 3D grid and apply these equations to each section. Atmospheric models calculate wind, heat, radiation, humidity, and water movement in each grid and how these factors interact with nearby areas. These models are linked with ocean models to study climate changes caused by ocean currents and the ocean's ability to store heat. Scientists can also add ice-sheet models to better predict long-term effects like rising sea levels.

Uses

Advanced climate models help scientists study extreme weather events by showing how much human-caused climate change affects how often these events happen, how strong they are, and their effects. This science tries to find out if extreme events are caused by human-made global warming or if they are just part of natural weather patterns and random changes in the climate.

There are three main types of organizations where climate models are created, used, and studied:

• National weather services: These often have sections that study climate patterns.
• Universities: Departments like atmospheric science, meteorology, climatology, and geography work on climate models.
• Research labs: Examples include the National Center for Atmospheric Research (NCAR) in Colorado, the Geophysical Fluid Dynamics Laboratory (GFDL) in New Jersey, Los Alamos National Laboratory, the Hadley Centre in the UK, the Max Planck Institute in Germany, and the Laboratoire des Sciences du Climat et de l'Environnement (LSCE) in France.

Large climate models are important tools, but they are not perfect. Scientists must still pay attention to real-world observations to understand what is happening and why. These models help combine data from satellites and other sources to create detailed summaries of current conditions, which can then be used to make predictions. Smaller, simpler models are sometimes used, but they often leave out important parts, like the water cycle, which can lead to incomplete results.

General circulation models (GCMs)

A general circulation model (GCM) is a type of climate model. It uses a mathematical model to study how air or water moves around the Earth. These models use equations that describe how air or water flows on a spinning Earth, along with terms that explain how energy is added (such as heat from sunlight or heat from water changes). These equations help create computer programs that simulate Earth's atmosphere or oceans. Models that study the atmosphere (AGCM) and models that study the ocean (OGCM) are important parts of climate models, along with parts that study sea ice and land surfaces.

GCMs and global climate models are used to predict weather, study climate patterns, and predict future climate changes.

Atmospheric GCMs (AGCMs) focus on the atmosphere and use sea surface temperatures as starting points for their calculations. Coupled models (AOGCMs), which combine atmosphere and ocean models, include examples like HadCM3, EdGCM, GFDL CM2.X, and ARPEGE-Climat. The first model that combined ocean and atmosphere processes was created in the late 1960s at the NOAA Geophysical Fluid Dynamics Laboratory. AOGCMs are the most complex climate models and include as many natural processes as possible. However, they are still being improved, and some uncertainty remains. These models can also be connected to other models, such as those that study the carbon cycle, to better understand how different parts of Earth interact. These combined models are sometimes called "earth system models" or "global climate models."

Models designed to study climate changes over decades to centuries were created by Syukuro Manabe and Kirk Bryan at the Geophysical Fluid Dynamics Laboratory in Princeton, New Jersey. These models use combinations of equations from physics, chemistry, and sometimes biology to simulate Earth's systems.

Energy balance models (EBMs)

Before large computers were available in the 1960s, scientists could not simulate Earth's climate in full 3-D space and time. To study how Earth's climate changed in the past, scientists had to simplify the system. A basic model that balanced incoming and outgoing energy was created for the atmosphere in the late 1800s. Other models, called Energy Balance Models (EBMs), use the rule that energy must be conserved to estimate surface temperatures by focusing on individual parts of the Earth-atmosphere system.

EBMs are simple to understand and sometimes allow scientists to solve problems mathematically. Some models include how oceans, land, or ice affect surface temperatures. Others consider parts of the water cycle or carbon cycle. These models help scientists study specific climate questions and support more complex models called General Circulation Models (GCMs).

Zero-dimensional models treat Earth as a single point, like the "pale blue dot" seen from space. While this view is very basic, it is still useful because it applies physical laws to unknown objects or groups major properties together if they are known. For example, scientists know most planets have a solid or liquid surface covered by a gaseous atmosphere.

A simple model of Earth's radiative balance is:

• The left side shows the total energy from the Sun (in Watts).
• The right side shows the total energy Earth emits into space, calculated using the Stefan–Boltzmann law.

Key values include:

• S: The solar constant, about 1367 W·m² (energy from the Sun per square meter).
• r: Earth's radius, about 6.371 million meters.
• π: A math constant (3.141…).
• σ: The Stefan–Boltzmann constant, about 5.67×10⁻⁸ J·K⁻⁴·m⁻²·s⁻¹.

By simplifying the equation, scientists can compare incoming and outgoing energy per square meter.

Other values depend on Earth's characteristics:

• a: Earth's average albedo (reflectivity), about 0.3.
• T: Earth's average surface temperature, about 288 K (15°C) as of 2020.
• ε: Earth's effective emissivity (how well Earth emits energy), about 0.61. This value depends on clouds and the atmosphere.

This model helps scientists understand how Earth's temperature changes with factors like solar energy, albedo, or emissivity. Emissivity also shows how the atmosphere traps heat, known as the greenhouse effect.

Scientists compare the model's emissivity to real-world data. Earth's surface emits about 0.96–0.99 of energy, but clouds (covering half Earth) emit about 0.5 energy. Using these values, scientists estimate Earth's effective emissivity as about 0.64.

Other models divide the atmosphere into layers. The simplest is a one-layer model, which can be expanded to include more layers. Each layer has a temperature and emissivity, but no thickness. These models use energy balance rules to solve equations.

These layered models are called multi-compartment models. They estimate temperatures closer to real-world values and explain how heat moves through the atmosphere. A version of this model, called the one-layer model, was first used by Svante Arrhenius in 1896 to study the greenhouse effect.

Water vapor affects Earth's emissivity. It influences how radiation moves and is affected by heat movement in the atmosphere. This is studied using a one-dimensional model that considers:

• Radiation moving up and down through the atmosphere.
• Heat movement through air and vapor, especially near Earth's surface.

These models divide the atmosphere into sections based on height, allowing better estimates of surface and atmospheric temperatures. They show how adding gases like carbon dioxide changes temperatures in the atmosphere and at Earth's surface.

Other factors, like ice-albedo feedback, are studied using models that focus on specific processes. For example, ice reflects more sunlight, which can change global temperatures.

The zero-dimensional model can be expanded to study energy movement across Earth's surface. These models often average temperatures by latitude, allowing scientists to link local albedo and emissivity to temperature. However, they do not simulate how energy moves dynamically across Earth's surface.

Early scientists like Mikhail Budyko and William D. Sellers developed models that showed how feedbacks in the climate system affect temperature. Their work, published in 1969, helped create modern energy balance models.

Earth systems models of intermediate complexity (EMICs)

There are two main types of models used in science, depending on the questions being studied and the time scales involved. On one end, there are conceptual models that use observations and patterns to make predictions. On the other end, there are general circulation models that use the most detailed data possible in space and time. Models with intermediate complexity fall between these extremes. One example is the Climber-3 model. Its atmosphere uses a 2.5-dimensional statistical-dynamical model with a grid size of 7.5° × 22.5° and a time step of half a day. Its ocean component uses the MOM-3 (Modular Ocean Model) with a grid size of 3.75° × 3.75° and 24 vertical layers.

Box models

Box models are simple representations of complex systems, showing parts as boxes connected by flows. Each box holds a reservoir, or collection, of matter and energy that is mixed evenly. The concentration of any substance in a box is the same throughout the box at any time. However, the amount of a substance in a box can change over time because of flows into or out of the box, or because the substance is created, used up, or changed within the box.

Simple box models, which have a small number of boxes with fixed sizes, are often used to create formulas that describe how the amount of a substance changes over time or remains stable. These formulas, called governing equations, are based on rules like the law of conservation of mass or energy. More complex models with many substances and equations are studied using computer calculations to understand how the system behaves.

Box models are widely used to study environmental systems and ecosystems. In 1961, Henry Stommel created a simple two-box model to examine the stability of large ocean currents. Another model has been used to study how ocean currents interact with the carbon cycle.

Networked data models

The study of complex networks has become an important part of science, helping researchers understand how complex systems work. Using network theory in climate science is a new and growing area of research. To find and study patterns in global climate, scientists create models that represent climate data as complex networks.

In most real-world networks, the points (called nodes) and connections (called edges) are clearly defined. However, in climate networks, nodes are based on locations within a grid that represents global climate data. This grid can be shown at different levels of detail. An edge connects two nodes if the climate data from those locations show a strong statistical similarity, which might indicate a relationship. This method helps scientists gain new understanding about how the climate system changes over different areas and time periods.

History

In 1956, Norman Phillips created a mathematical model that showed accurately how the atmosphere changes monthly and seasonally. This was the first successful climate model. Soon after, other groups started working on general circulation models, which simulate how air and water move globally. The first general circulation climate model combined ocean and air processes and was created in the late 1960s at the Geophysical Fluid Dynamics Laboratory, part of the U.S. National Oceanic and Atmospheric Administration.

By 1975, Manabe and Wetherald developed a three-dimensional model that showed a realistic picture of Earth’s current climate. When the model’s atmosphere had double the amount of carbon dioxide, global temperatures rose by about 2 degrees Celsius. Other models also showed similar results: any model that accurately represented the real climate would predict higher temperatures if carbon dioxide levels increased.

In the early 1980s, the U.S. National Center for Atmospheric Research created the Community Atmosphere Model (CAM), which can be used alone or as part of a larger climate system model. The most recent version of the standalone CAM was released on February 1, 2006. In 1986, scientists began including soil and plant types in models, leading to more accurate predictions. Coupled ocean-atmosphere models, like the Hadley Centre’s HadCM3 model, are now used to study climate change. Reviews of past models show they were generally correct, though they often predicted less warming than actually occurred.

The Coupled Model Intercomparison Project (CMIP) has helped improve global climate models and understanding of climate change since 1995.

In 2010, the IPCC said it had greater confidence in predictions made by climate models.

Coordination of research

The World Climate Research Programme (WCRP), managed by the World Meteorological Organization (WMO), organizes climate modeling research around the world.

A 2012 report by the U.S. National Research Council talked about how the many different U.S. climate modeling efforts could become more unified. The report suggested that creating a shared software system for all U.S. climate researchers and holding an annual meeting about climate modeling could help save time and resources.

Issues

Cloud-resolving climate models are now run on very powerful supercomputers that use a lot of electricity and produce carbon dioxide emissions. These models need exascale computing, which means they perform a very large number of calculations—up to a quintillion—each second. For example, the Frontier exascale supercomputer uses 29 megawatts of power. It can simulate a full year of climate conditions at cloud-resolving scales in just one day.

Methods that may help reduce energy use include: using less precise calculations to save power; creating machine learning tools to skip unnecessary steps in computing; and designing new, efficient numerical methods that increase the number of simulated years completed each day.

Parametrization in an atmospheric model (such as a weather or climate model) is a way to replace processes that are too small or complex to be shown directly in the model with simpler representations. This is different from processes like the large-scale movement of air, which are shown directly in the models. These simplified methods use specific values, or parameters, to represent the processes. Examples include how fast raindrops fall, how convective clouds form, simplified ways to calculate how radiation moves through the atmosphere based on existing codes, and how clouds behave at a microscopic level. Radiation-related parametrization is important for both weather and ocean models. Emissions from different sources in specific areas of the model also need to be represented using parametrization to understand their effects on air quality.