Hi! You are in a museum...

… where you can learn, in 10 minutes, how mathematical models help us understand global warming, and predict changes in the climate.

Jump straight to whatever catches your interest and start exploring!

Or look at everything in order: start here and go around the museum in a counterclockwise direction following the numbers.

Do you want to know more? You can find more information about the content and background of this museum in this website.


This 10 Minute Museum was created and developed by IMAGINARY, a nonprofit organization dedicated to helping you explore the world through the eyes of mathematics.

Visit to learn more.

This project is enabled by the Klaus Tschira Stiftung.

Scientific advisors:

France Caron (University of Montreal, Canada)
Dietmar Kröner (University of Freiburg, Germany)
Chirag Dhara (Indian Institute of Tropical Meteorology, India)

Content contributors:

Hilary Weller
Oliver Labs
Judy Twedt
Cameron Beccario
Pavel Dobryakov

Design and Production:

Werkstatt für alles

Full credits

GCM - the mega model

General Circulation Models (GCM) are the most precise tool we have to assert that climate change is a reality. GCMs are detailed models of the whole Earth, divided in many small cells of just a few kilometers width. The models are initialized using current values of pressure, humidity and temperature in each cell and future values are calculated using numerical simulations of physical laws (such as the Navier-Stokes equations).

Mathematicians are continuously improving these models. This makes them more comprehensive and accurate. Complex processes are included in the calculations, such as carbon cycles, interactions between cloud droplets and aerosols, etc.

What is a climate model?

Climate models are tools to explore weather trends and dynamics across large spans of time. They show how human behavior affects climate change and help us understand, predict, and better respond to it.

A mathematical model is an approximation of reality. It includes all the relevant factors in a process and how they relate to each other (leaving out superfluous things).

Global warming, for example, becomes obvious through climate models. For this, we explore how much energy is contained in the Earth, which elements accumulate that energy, and how they are distributed over the planet.

Weather ≠ Climate

Weather: atmospheric conditions such as temperatures, wind and rain, which can change from minute to minute

Climate: the average weather of an entire region over a long time (30 years or more).

The Navier-Stokes Equations

The movement of balls, cars, and rockets can be explained mathematically:

A ball's bounce explained with x and y axes. Diagrams of a car and a rocketship with the force vectors they're subject to.

Similarly, the Navier-Stokes equations can explain and predict the movement of fluids, like water in the oceans, gases in the atmosphere, an oil spill, smoke, or Currywurstsoße:

A grid with vectors (arrows) on each square. The directions of the arrows suggest two small cyclones side by side.

We can input the current state of the atmosphere and oceans as numbers in the equations, and get the future weather or climate as the result.

The Navier-Stokes equations have many other applications. They are used to understand the flow of air around an airplane wing, blood in the circulatory system, pollution, and to simulate fire and smoke in video games.

The navier stokes equations with labels explaining what the different letters and symbols mean.

The Gulf Stream

The Gulf Stream is a current in the Atlantic Ocean that brings warmth from tropical waters to the coasts of northern Europe. As a result, temperatures there are higher than in other regions of the world that are similarly close to the Arctic.

Models show that as the ice caps melt due to global warming, they release unsalted water to the ocean. This slows down and could even stop the Gulf stream. North Europe could be much colder in the future!

So a trend toward global warming does not mean that temperatures will rise everywhere on Earth. Warming triggers complex reactions in the climate system that can lead to unexpected consequences.

Box Models

With a box model, we can predict the future by simulating the complex interactions between different processes. The results are often surprising, even though the connections between the processes are well known and predictable.

We put solar energy, CO2 emissions, population numbers, economic activity data (or whatever numbers we want to model) into individual "boxes." Then we draw connections between those boxes that are related to each other via physical laws or other formulas. The model is executed by calculating the step by step transfer of values between the boxes, simulating the passage of time.

See a Forecast Factory from 1922

Decades before the first supercomputers, mathematician Lewis Fry Richardson dreamed of a sort of factory for global climate modeling.

Today, computer systems with hundreds of thousands of computer cores have made that fantasy a reality. They process billions of weather observations in parallel to run a model.

Bigger and faster computers enable more accurate modeling because they can divide the world into smaller parts and simulate the passage of time using shorter increments.

A philosophy of models

We cannot observe the complex consequences of our actions at the scale of the entire planet or an entire society in the laboratory. But we can use models to handle that complexity.

A model...


to better understand the processes and their consequences in a system.


by removing all non-essential factors. What we lose in accuracy, we gain in insight.


It solves the tedious calculations so we can focus on interpreting the results.


It helps us find the causes of changes we observe, and deduce which factors are most relevant to the result.


Its predictions can help us make better decisions.

Do you hear the climate crying?

Data sonification makes changes happening on our planet audible. The numbers are not represented in tables or diagrams, but in notes. Songs are created from them!

In all tracks we assign a higher pitch to a higher value in our data, and a lower pitch to a lower value.


The Deafening Rise of Carbon Pollution (1:32).

The monthly concentration of CO2 from 1958 to 2017 is represented by the pitch of each note.

Sonification of Arctic Sea Ice (1:29).

An index of total ice cover in the Arctic Ocean from 1979 to 2016, each note represents one month of data.

The Sound of Earth’s Fever (1:03).

A sonic interpretation of the global mean surface temperature index. In 1977 there is a pause to draw attention to the warming that has occurred since then.

Modern global climate data starts in 1880. We can discover the climate of yesterday in many different ways.

Data crucial for our models. We need precise measurements of the magnitudes of the model (temperatures, speeds, masses) to connect the model to the real world.

Data is also used to verify the quality of a model. If the model predicts a result at odds with the observational data, then it must be inaccurate, incomplete, or simply wrong.

Modern mathematical models rely on massive amounts of data, also called Big Data. In climate models, this means satellite data that allows accurate weather and climate measurements from every point on Earth.

Explore the temperature trend

The values shown how much the temperature deviates from the average value in that country for the 1951-1980 period.


... appears in various forms in modeling:

Statistics and probability theory

help deal with data and uncertainty.

Dynamical systems

help running processes which change over time, often using differential equations.

Discrete geometry

is needed to describe specific shapes, such as the spherical Earth, the ocean floor or the structure of Earth's surface in a region.

Graph theory

helps to deal with complex networks, for example studying causal relationships between events.

Game theory

helps in proposing environmental policies.

Grid models

Even with the fastest supercomputers, we cannot model all climate processes in all regions of the world in full detail.

Therefore, we divide the world into small grid cells and calculate the climate processes for these cells in parallel.

The size of the cells is called the resolution of the grid. A common resolution is 100 square kilometers of Earth's surface per cell.

There are many criteria to pick a grids for our models:

Latitude - Longitude grid

The classic! It has the problem that the cells become smaller towards the poles.

Cubed sphere grid

It is created by "inflating" a cube. Its cells get distorted where the cube's edges meet.

Octahedral Gaussian Grid

Cells are triangles of similar sizes. Their angles, however, are not always the same.

Hexagonal Icosahedron Grid

Can you find the cells with five sides? These make it difficult to list adjacent cells.

A chat with scientists

Climate scientists from around the world share data and results, and integrate their models. Organizations help them collaborate, turn research results into policies, and establish international agreements.

The World Meteorological Organization (WMO) and the World Climate Research Programme (WCRP) set set research agendas and help coordination and information exchange.

The Intergovernmental Panel on Climate Change (IPCC) produces reports that communicate the latest climate science to policymakers.

The United Nations Framework Convention on Climate Change (UNFCCC) made over 190 countries agree on limiting global warming to 1.5ºC above pre-industrial levels.

Consequences of climate change

Models can anticipate consequences of climate change to our planet and our society before it is too late to avoid them.


Agriculture is very sensitive to climate. Climate change threatens access to food and the diversity of our diet.


Droughts and floods make access to clean water difficult for industry and the population.


Climate change causes migration of insects that carry contagious diseases to new regions.


Extreme weather and rising sea levels endanger lives and disrupt all human activities.

Many futures

Models don’t predict one future, but many alternate timelines. To calculate the climate 100 years from now, we need to know future human gas emissions, which depend on the economy, population growth, and environmental policies.

Scientists simulate future technologies and economic conditions using models to create different scenarios.

Some scenarios show us a future in which emissions increase at a steady rate; in others, global warming is slowed by strict legislation. By using the same scenarios, scientists can share climate models and compare conclusions.

Climate models run using worst-case scenarios show very hot futures with irreversible consequences. They give us the chance to act now and avoid living in a dark timeline.

Ocean acidification

The oceans store about one-third of the world's CO2, which would otherwise be in the atmosphere increasing the greenhouse effect.

Increases in CO2 emissions lead to more CO2 in the oceans. There it reacts with the sea water creating carbonic acid (H2CO3). This raises the acidity of the oceans, affecting sea life. For instance, coral reefs and the shells of marine animals get damaged by it.

One side of the seashells has been damaged by acid. Try this experiment at home by leaving some shells in vinegar for a few hours.

Tipping points

In complex systems, a tipping point is where a small change in the system makes big differences in their evolution. Often, these changes are irreversible, or it takes much more effort to reverse than was necessary for the triggering change.

Take permafrost, for example: this is the name given to the frozen earth in regions located very far north or south. It contains vast amounts of methane and CO2 that have been trapped in plants and animals for thousands of years. When the permafrost melts, these chemicals are released into the atmosphere. This increases the greenhouse effect and global warming. Once a certain temperature is reached, it becomes impossible to reverse the melting process.

Text on the wheel

You can push a rock up a hill, but after the Tipping point it rolls down on its own very quickly...

Before reaching a tipping point it’s possible to go back...
But once past it, it’s not possible to return, or it’s much harder.