Lesson Plan: The Cascade

Subject areas: Physics, Earth and environmental science, mathematics, history of science Recommended level: Ages 14 to 18 (upper secondary), adaptable for introductory undergraduate Duration: One to two 50 minute lessons

Learning objectives

By the end of this lesson, students will be able to:

1. Explain what turbulence is and describe the energy cascade using Richardson's "big whirls" image.

1. Distinguish the three dimensional (forward) energy cascade from the two dimensional (inverse) cascade, and connect each to real systems such as a stirred cup, a river, the ocean and the atmosphere.

1. Describe how Lewis Fry Richardson attempted the first numerical weather forecast by hand, why it failed, and why his method later succeeded with electronic computers.

1. Summarise the 2025 to 2026 finding that the direction of the turbulent energy flux can be controlled by the geometry of the forcing, and explain why that challenges an assumption held since 1941.

1. Discuss how the same scientific tool can be turned toward harm or toward good, using Richardson's own choices as a case study.

Background for the teacher

Turbulence is the chaotic, swirling motion of a fluid. In 1922 the British scientist Lewis Fry Richardson, a Quaker pacifist who served in an ambulance unit in the First World War, published the first method for forecasting weather by solving the governing equations numerically. His own trial forecast, computed by hand over about six weeks, was wrong by roughly a hundredfold because of imbalances in the starting data. His vision of a "forecast factory" of 64,000 human calculators is essentially what a modern weather supercomputer does today.

Richardson also captured turbulence in a rhyme: large eddies break into smaller eddies, which break into smaller ones still, until the energy is lost as heat. In 1941 Andrey Kolmogorov turned this picture into a precise law for three dimensional flows, in which energy always moves from large scales to small scales. In 1967 Robert Kraichnan showed that in two dimensional flows, like the thin shell of the ocean and atmosphere, the cascade runs in reverse, from small scales to large. For decades the direction seemed to be dictated by the dimensionality of the system. In 2025, a team led by Lei Fang (University of Pittsburgh) with collaborators in Turin, including Guido Boffetta, showed in Science Advances that by changing the geometry of the force driving a two dimensional flow they could make the energy run either way. The direction is a controllable setting, not a fixed law.

Warm up (10 minutes)

Pour cream or milk into a clear cup of coffee or tea and stir once. Ask students to sketch and describe what they see: large swirls breaking into smaller ones, then blurring away. Introduce Richardson's rhyme and ask what "and so on to viscosity" means.

Core activities

Activity 1: Modelling the cascade (15 minutes). Students use a shallow tray of water with a sprinkle of pepper, glitter or dye as tracers. Stir gently and observe how large swirls form and shed smaller ones. Discuss why the ocean and atmosphere behave more like a thin two dimensional sheet than a deep three dimensional volume.

Activity 2: The hand computer (15 minutes). Give small groups a simplified arithmetic task to perform together, each student responsible for one step and passing the result to the next. Time it. Then discuss Richardson's forecast factory and why computing speed, not the method, was the obstacle in 1922.

Activity 3: Direction as a knob (15 minutes). Using arrows on a diagram, students mark forward (large to small) and inverse (small to large) cascades. Pose the central question: if energy flux is really just force acting through a displacement, how could the angle between force and motion change the direction of the flow? Relate to the idea of work done being positive or negative.

Discussion questions

1. Why was Richardson's forecast wrong even though his method was correct? What does this teach us about the difference between a good idea and a working technology?

1. The ocean and atmosphere are described as two dimensional even though they exist in three dimensional space. Why?

1. If we could steer ocean currents or atmospheric flows, who should decide how that power is used?

1. Richardson destroyed unpublished work when he learned it could aid chemical weapons designers. Was that the right choice? When, if ever, should a scientist withhold a discovery?

Extension

• Research how modern weather forecasting works and estimate how many calculations per second a forecast requires, comparing this with Richardson's 64,000 human computers.

• Investigate other "dual use" discoveries in science and debate the responsibilities of researchers.

Glossary

• Turbulence: irregular, chaotic motion of a fluid.

• Energy cascade: the transfer of energy between different sizes of eddies in a turbulent flow.

• Forward cascade: energy moving from large scales to small scales (typical in three dimensions).

• Inverse cascade: energy moving from small scales to large scales (typical in two dimensions).

• Viscosity: the internal friction of a fluid, which turns the smallest eddies into heat.

• Numerical weather prediction: forecasting weather by solving the equations of the atmosphere with computation.