Numerical climate models, also called climate system models, are math models that show how important parts of Earth's climate interact. These parts include the atmosphere, oceans, land, and ice. Scientists use these models to study how the climate works and to predict future climate changes. Some models are not numerical and instead describe possible future climates in words.

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

Climate models differ in how detailed they are. A simple model treats Earth as a single point and averages energy leaving. More complex models divide Earth into layers (vertically) and regions (horizontally). The most detailed models include the atmosphere, oceans, and sea ice. These models use full equations for movement of mass, energy, and radiation. Other models, like Earth System Models, also include land use and changes in land use. This helps scientists study how climate and ecosystems affect each other.

Climate models use math equations based on physics, fluid motion, and chemistry. Scientists divide Earth into a 3D grid and apply equations to each section. Atmospheric models calculate wind, heat, radiation, humidity, and water movement in each grid square and how these interact with nearby squares. These models are linked with ocean models to show how climate changes over time because of ocean currents and the ocean's ability to store heat. Other factors, like ice sheets, are added to better predict long-term effects, such as rising sea levels.

Uses

Climate models help scientists study extreme weather events and determine how much human-caused climate change affects their frequency, strength, and effects. This process, called extreme event attribution, compares the role of human-caused global warming to natural weather patterns or random climate changes.

Climate models are developed and used in three main types of organizations:

• National meteorological services: These services often have sections that study climate patterns.
• Universities: Departments like atmospheric sciences, meteorology, climatology, and geography work on climate models.
• National and international research labs: Examples include the National Center for Atmospheric Research (NCAR) in Boulder, Colorado, the Geophysical Fluid Dynamics Laboratory (GFDL) in Princeton, New Jersey, Los Alamos National Laboratory, the Hadley Centre in Exeter, UK, the Max Planck Institute for Meteorology in Hamburg, 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. Real-world observations, such as data from satellites, must still be studied to understand current conditions. These models combine all available data to create detailed analyses and help make predictions. Simple models are sometimes used but can be misleading because they often leave out important parts, like the water cycle.

General circulation models (GCMs)

A general circulation model (GCM) is a type of climate model. It uses a mathematical system to study how air and water move in the atmosphere or ocean of a planet. These models apply complex math equations on a rotating sphere, including terms for energy sources like sunlight and heat from water changes. These equations are used in computer programs to simulate Earth's atmosphere or oceans. Models that study the atmosphere (AGCM) and models that study the ocean (OGCM) are important parts of climate systems, along with models for sea ice and land surfaces.

GCMs and global climate models help scientists predict weather, study climate patterns, and forecast future climate changes.

Atmospheric GCMs (AGCMs) focus on the atmosphere and use sea surface temperatures as starting points. Coupled models (AOGCMs), such as HadCM3, EdGCM, GFDL CM2.X, and ARPEGE-Climat, combine atmosphere and ocean models. 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 connect to other systems, like 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 and centuries were developed by Syukuro Manabe and Kirk Bryan at the Geophysical Fluid Dynamics Laboratory in Princeton, New Jersey. These models use a mix of equations from physics, chemistry, and sometimes biology to study Earth's systems.

Energy balance models (EBMs)

Before large computers were available in the 1960s, it was not possible to create detailed 3-D models of Earth's climate system. To study how Earth's climate changed in the past, scientists simplified the system by reducing its complexity. A basic model that balanced incoming and outgoing energy was created for Earth's atmosphere in the late 1800s. Other models, called Energy Balance Models (EBMs), also simplify Earth's climate by focusing on energy conservation in parts of the Earth-atmosphere system.

EBMs are useful because they are simple to understand and can sometimes provide clear mathematical answers. Some models include how oceans, land, or ice affect Earth's surface. Others include how water or carbon cycles work. These simplified models help scientists understand parts of Earth's climate that are hard to study with more complex models like General Circulation Models (GCMs).

Zero-dimensional models treat Earth as a single point, like a small dot in space. While this view is limited, it is helpful for applying physics rules to unknown objects if their main properties are known. For example, scientists know that most planets have a solid or liquid surface covered by a gas layer.

A simple model of Earth's radiative balance is:

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

Key values in this model include:

• S: The solar constant, which is the amount of sunlight energy reaching Earth per square meter (about 1367 Watts).
• r: Earth's radius (about 6.371 million meters).
• π: A mathematical constant (3.141…).
• σ: The Stefan-Boltzmann constant (about 5.67×10 J·K·m·s).

By simplifying the equation, the model becomes:

• The left side shows the energy from the Sun per square meter.
• The right side shows the energy Earth emits per square meter.

Other values specific to Earth include:

• a: Earth's average albedo (reflectivity), measured at 0.3.
• T: Earth's average surface temperature (about 288 Kelvin as of 2020).
• ε: Earth's effective emissivity (a value between 0 and 1, calculated as about 0.61).

This model helps scientists understand how Earth's temperature changes with sunlight, reflectivity, or emissivity. The emissivity also shows how much heat is trapped by Earth's atmosphere, which is the greenhouse effect.

Scientists compare calculated emissivity with real-world data. Earth's surface typically has high emissivity (0.96 to 0.99), but clouds (which cover half the planet) have lower emissivity (about 0.5). Considering these factors gives an effective emissivity of about 0.64.

Other models include layers of the atmosphere. The simplest is a one-layer model, which can be expanded to many layers. Each layer has its own temperature and emissivity. These models solve equations that describe how energy moves between layers.

These models are called multi-compartment models. They help estimate Earth's surface and atmospheric temperatures more accurately. They also explain how heat moves through the atmosphere, which is important for understanding the greenhouse effect. A version of this model was first used by Svante Arrhenius in 1896.

Water vapor affects Earth's emissivity. It influences how heat moves through the atmosphere and is affected by air movement. Scientists improved the zero-dimensional model to a one-dimensional model that includes both radiation and air movement:

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

These models use a detailed view of the atmosphere and help predict how Earth's temperature changes with elevation. They also explain how adding gases like carbon dioxide affects temperature.

Other factors, like ice-albedo feedback, are studied using models that consider how ice reflects sunlight. The zero-dimensional model can also be expanded to include horizontal energy movement, like the Budyko-Sellers model. This model, developed by scientists like Mikhail Budyko and William D. Sellers, showed how feedbacks in Earth's climate system work. It is a key example of how energy balance models help scientists study climate.

Earth systems models of intermediate complexity (EMICs)

There are different types of climate models, depending on the questions being studied and the time scales involved. At one end, there are simple models that use general ideas and patterns. At the other end, there are highly detailed models that can show the Earth's climate with the most precise spatial and time details possible today. Models with medium complexity fall between these two extremes. One example is the Climber-3 model. Its atmosphere uses a simplified version of a statistical-dynamical model with a grid covering 7.5° by 22.5° and a time step of half a day. Its ocean part uses a specific type of ocean model called MOM-3, which has a grid of 3.75° by 3.75° and 24 layers vertically.

Box models

Box models are simple representations of complex systems, showing them as boxes connected by flows. Each box represents a storage area for types of matter and energy that are evenly mixed. This means the concentration of any substance inside 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 bringing it in or taking it out, or because of processes inside the box that create, use, or change the substance.

Simple box models, which have few boxes and do not change in size or properties over time, are often used to create formulas that describe how substances change over time or reach a steady state. These formulas, called governing equations, are based on rules like the conservation of mass and energy. More complex models, which include many substances and equations, are studied using numerical methods to understand how systems behave.

Box models are widely used to study environmental systems and ecosystems. In 1961, Henry Stommel first used a simple two-box model to examine how large-scale ocean currents remain stable. A more detailed 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 area of science that helps scientists better understand complex systems. Using network theory in climate science is a new and growing field. To find and examine patterns in global climate, scientists create models that represent climate data as complex networks.

In most real-world networks, nodes and edges are clearly defined. However, in climate networks, nodes are points in a spatial grid that represents global climate data, which can be shown at different levels of detail. An edge connects two nodes based on how similar the data is between two sets of climate records over time. This method helps scientists gain new understanding of how the climate system changes across different areas and time periods.

History

In 1956, Norman Phillips created a mathematical model that accurately showed monthly and seasonal changes in the troposphere. This was the first successful climate model. After this, several groups started working to build general circulation models. The first general circulation climate model combined ocean and air processes and was developed in the late 1960s at the Geophysical Fluid Dynamics Laboratory, which is part of the U.S. National Oceanic and Atmospheric Administration.

By 1975, Manabe and Wetherald had built a three-dimensional global climate model that gave a reasonably accurate picture of Earth’s current climate. When the model’s atmosphere had double the amount of CO₂, it predicted a temperature rise of about 2 °C. Other computer models also showed similar results: any model that could resemble the real climate would also show higher temperatures if CO₂ levels increased.

In the early 1980s, the U.S. National Center for Atmospheric Research created the Community Atmosphere Model (CAM). This model could be used alone or as part of the Community Climate System Model. The latest version of the standalone CAM (version 3.1) was released on February 1, 2006. In 1986, scientists began modeling soil and plant types, leading to more realistic weather predictions. Coupled ocean-atmosphere climate models, such as the Hadley Centre’s HadCM3 model, are now used in climate change studies. Reviews of past climate models show they were mostly accurate, though they often underestimated how much warming would occur.

The Coupled Model Intercomparison Project (CMIP) has helped improve general circulation models (GCMs) and climate understanding since 1995. In 2010, the IPCC reported greater confidence in climate model predictions.

Coordination of research

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

A 2012 report from the U.S. National Research Council talked about how the many different U.S. climate modeling groups could work more effectively together. The report said that using a shared software system for all U.S. climate researchers and holding an annual meeting where climate modelers can share ideas could help improve efficiency.

Issues

Cloud-resolving climate models are now run on powerful supercomputers that use a lot of energy and produce carbon dioxide emissions. These models need exascale computing, which means performing a quintillion calculations every second. For example, the Frontier exascale supercomputer uses 29 megawatts of power. It can simulate one year of climate data at cloud-resolving scales in just one day.

Ways to save energy include using less precise math calculations, creating machine learning tools to skip unnecessary steps, and designing new algorithms that increase the speed of simulations.

Parametrization in weather or climate models is a way to simplify processes that are too small or complex to model directly. This is different from large-scale processes, like air movement, which are shown clearly in models. Parametrization uses specific values, such as how fast raindrops fall, how clouds form, and how light interacts with the atmosphere. These methods are important for both weather and ocean models. Emissions from sources like factories or cars in specific areas must also be simplified to study their effects on air quality.