Understanding turbulence, which is characterized by growing chaos at multiple scales

Mathematics proves turbulence's chaos creates super diffusion

Turbulent flow is responsible for the rapid scattering of dandelion seeds, volcanic ash, and untied balloons. Its swirling behavior accelerates particle scattering faster than other mixing methods, as particles acquire energy from the interactions of eddies.

Turbulence has been called the most important unsolved problem in classical physics

Despite being the most common type of fluid flow in our world, scientists have struggled to model turbulent flow accurately. Though behaving unpredictably, understanding inertia and viscosity can help determine whether turbulence will occur.

View the transition from laminar to turbulent flow

Using a dye released in the water, streamlines, which are parallel to one another and consistent at low flow velocities, become visible. As the water moves more quickly, viscosity cannot suppress the velocity change, and streamlines become chaotic, creating turbulence.

When experiencing turbulence, fluid elements undergo stretching and folding

Simulations show patches of fluid particles, such as splashes of milk, bend under turbulent flow, creating curvature that grows linearly then exponentially. This accelerated deformation enhances folding and mixing faster than other methods, such as stirring or beating.

Laminar flow moves in smooth, parallel layers, while turbulent flow is chaotic

In laminar flow, particles that make up a fluid move in consistent paths that do not cross each other. Contrastingly, differences across the speed of these particles cause them to mix, creating turbulence when viscosity cannot suppress the differences.

Learn how natural systems exhibit characteristics of chaos theory

In biology, some mathematical models of ecosystem populations and heart irregularities show strong sensitivity to initial conditions, which scientists can simulate to determine quantitative tipping points at which these systems become unstable. Within the social sciences, crowd behavior and the virality of social media posts have shown how human behavior fits within the framework of chaos theory.

Chaos theory—informally known as the butterfly effect—is the reason why weather forecasts are presented with probabilities rather than certainties.

The origin of chaos theory, colloquially known as the 'butterfly effect'

The theory explains that systems following well-understood physical laws can still produce unpredictable outcomes due to small variations in their initial conditions. Originally identified in weather data, elements of chaos have been observed in stock market fluctuations, cardiac arrhythmias, and the evolution of online social movements.

View the Lorenz attractor, a graphical solution to chaotic equations that resembles a butterfly

The discovery of chaos theory came from Edward Lorenz's computer model of the atmosphere, which used three equations to quantify atmospheric circulation. However, the chaotic evolution of this system creates wandering figure-8-like shapes—some resembling butterfly wings—as it repeatedly attempts to settle on one of two values along unique trajectories.

The 'butterfly effect' is the informal name for chaos theory

In mathematics, chaos theory involves complex systems that are highly sensitive to their initial conditions. Originally discovered when analyzing weather data, slight variations in the data input into such systems create drastically different results, despite following known physical laws.

Turbulence is crucial to the formation of stars in molecular clouds

Within simulations of these clouds, gravity, jets, supernovae, and galactic motion-induced stirring have been shown to develop turbulence that creates shocks. These turbulent shocks move quickly in low-density regions and slow in high-density areas, forming lumps of matter with sufficient gravitational force to collapse and initiate fusion, birthing a star.

A mathematical and visual analysis distinguishing chaos from randomness

Random systems lack any underlying pattern or causal relationships, while chaos is mathematically deterministic because the behavior follows established laws of nature. In 3D plots, chaotic systems produce outputs that approximate a mathematical function, while random points appear as statistical noise.

Horsehead-shaped features seen on the sun and the clouds of Earth and Jupiter are examples of Kelvin-Helmholtz instabilities, where a faster-moving fluid rolls atop a slower-moving fluid.

The life of Edward Lorenz, who originally used a seagull in his 'butterfly effect' metaphor

By accidentally finding that rounding one variable to its third decimal place and re-running a weather simulation produced a drastically different two-month forecast, the American mathematician established the field of chaos theory. His paper—"Predictability: Does the Flap of a Butterfly's Wings in Brazil Set off a Tornado in Texas?"—introduced the well-known metaphor for the behavior of chaotic systems. (Some readers may experience a paywall.)

Turbulence is needed to make airplanes more stable and golf balls fly farther

Vortex generators—small fins placed along airplane wings—create small eddies that prevent airflow separation, maintaining lift and preventing stalls. Golf balls have dimples that create a turbulent boundary layer on their surface, minimizing drag forces.

How chaos helps explain why small market corrections can produce major economic instability

Oxford University complex systems professor Doyne Farmer discusses how predictions from many economic models are overly weighted toward simple relationships, such as supply and demand, and external shocks, such as the COVID-19 pandemic. However, real-world macroeconomic systems involve public and private actors interacting, producing feedback loops and cascading effects that resemble the oscillatory behavior seen in chaotic systems.

Pilots are familiar with multiple types of turbulence, each with varying severity

Each type is defined based on what disrupts airflow in the atmosphere, including thunderstorms, mountains, and wakes produced by aircraft that previously took off. Pilots classify turbulence as light, moderate, severe, or extreme based on how it affects aircraft movement.

The arrow of time—the concept that time only moves in the forward direction—is thought to result from the universe progressing toward greater disorder, or entropy.

The turbulence experienced in airplanes results from unexpected changes in airflow

Pilots rely on forecasts, onboard tools, and air traffic control to avoid regions of severe turbulence during flight. Turbulence is generally weaker for those sitting in the front of the main cabin and for morning flights that follow cooler nighttime temperatures.

Atmospheric processes amplify the strength of storms by coalescing clouds

Research has shown that turbulent mixing of air at the edges of clouds and other small-scale thermal activity beneath them help clouds group together, intensifying short bursts of rainfall, especially in tropical regions. Simulations reveal this produces more powerful storms than predicted by older models, though they may occur less frequently.

An explanation distinguishing the butterfly effect from causality, necessity, and sufficiency

In pop culture, the analogy has conveyed the impression that the flapping of a butterfly's wings has a cascading effect that directly produces a tornado somewhere else on Earth. However, butterfly-wing flapping alone is insufficient to cause tornadoes, as the complexity of atmospheric systems requires many more conditions to produce them, any of which can be the tipping point to initiate tornadogenesis.

Explore interactive simulations of strange attractors, the graphs that help visualize chaos

Although simple, linear equations reach an explicit value for a given input, such as *f(x) \= x* + 1 equaling 4 when *x* equals 3, some chaotic, nonlinear systems tend to evolve towards multiple values simultaneously. These simulations allow users to observe the unusual mathematical artifacts created by systems as their outputs evolve toward these distinct values.

An interview with James Yorke, the man who coined the mathematical term 'chaos'

Along with co-author Tien-Yien Li, the former University of Maryland mathematics department chair published the paper, "Period Three Implies Chaos" in 1975. "Chaos" then became the terminological placeholder for systems with sensitive dependence on initial conditions, which had been well known to mathematicians for over a decade.

Climate change is increasing the frequency and intensity of severe turbulence

Severe turbulence over the North Atlantic has increased by 55% over the past 40 years, driven by strengthened jet streams and storm systems. Ongoing increases in global temperatures, which fuel these changes, may triple severe turbulence incidents in the coming decades.

Swimming in schools helps some fish conserve energy in turbulent flows

Scientists studying groups of giant danios swimming in turbulent waters found that the fish used up to 79% less energy than they did when swimming individually in the same conditions. The same fish were found to use the same amount of energy in turbulent and calm water when in schools.

Researchers identify a hidden instability pattern in simple ocean wave models

Stokes waves—a solution to the mathematics of water waves—were once assumed to be stable. Italian mathematicians have proven disturbances cause these waves to collapse and such collapses follow an infinite repeating pattern. To complete the proof, the team offered a $100 donation to solicit online help.

The Navier-Stokes equations are fundamentally accurate but unsolved in 3D

The equations model fluids and have been shown to have solutions for all systems in two dimensions. Averaging and approximations let the equations be used in climate models, aerodynamics, and medicine, though their use in some systems produces nonsensical answers.

Because some models of the universe do not incorporate time, it may be an emergent property born from our limited perception of the universe, rather than a fundamental property of reality.

The Navier-Stokes equations include the fluid version of Newton's second law

The first equation is the fluid version of the conservation of mass, which prevents incompressible fluids from gaining or losing matter. The second equation replaces acceleration and inertia, which is resistance to acceleration, with fluid pressure and viscosity.

Learn why modern scientific patterns have been described as theories rather than laws

After confirming that the accuracy of some laws of physics is limited to certain domains, science became more hesitant to categorize identified mathematical relationships, such as quantum mechanics, as unchanging laws. Instead, these recent, complex frameworks are designated as theories to acknowledge that they are conditional and may be part of a larger, more complete future framework.

Visualize chaos with the orbit of three or more objects bound by gravity

In this simulation, users can simulate randomized and preset orbital configurations for up to 10 celestial bodies to observe the system's evolution in an interactive 3D space. The same configurations can produce different orbital behavior—a feature of chaos—by making small changes to an object's mass under advanced settings.

View a collection of sound animations, which depict how a source of sound waves pushes against particles to carry energy away without matter being displaced.

In a tornado outbreak, a single severe weather system can spawn multiple twisters

By maintaining the ingredients for tornadoes—differences in wind speed and direction across altitudes, moisture, and rising air—for long periods of time as it moves across a region, a severe thunderstorm can continuously produce tornadoes by lifting multiple columns of horizontally rotating air.

Tornado formation relies on wind shear to create a tube of rotating air

Once a horizontal vortex is created by air circulating between winds of different directions and speeds, an updraft, such as those seen in powerful thunderstorms, tilts the vortex vertically. The updraft then strengthens the vortex, which becomes a tornado upon touching the ground.

Quantum field theory reveals that particles are ripples in universal fields

Instead of being made of indivisible particles, all visible matter in the universe is built from underlying quantum fields—smooth, invisible entities that behave like waves. The theory suggests these fields are fundamental, and particles are disturbances that move through them.

Explore a simulation of ideal gas behavior, which can be explained by kinetic molecular theory

The theory describes particles as moving in random directions, with a distribution of velocities determined by the gas's temperature. The particles collide elastically like billiard balls, are negligibly small, and do not exert forces on one another. The gas laws can be derived from these principles using statistics or observed via this interactive.

Various types of tornadoes are distinguished by their shape, size, and location

Waterspouts form over water as long, thin columns of mist and air, while elephant trunks are tornadoes that elongate vertically as they become tilted during their formation, resembling the mammal's appendage. Wedge tornadoes are among the most powerful, often spanning a mile or more in width.

Wave tanks can be used to create and study the impacts of tsunamis by building tension in an underwater object before allowing it to slip and rapidly displace water.

The weakening of dark energy suggests a dynamic universe with evolving forces

The discovery, which came from analyzing extensive data from a major cosmological survey, challenges the assumed constancy of dark energy in modern physics. Understanding dark energy's behavior is crucial for predicting the universe's long-term evolution.

The Reynolds number is the ratio of viscous and inertial forces in fluids

It can approximate fluid behavior in simple systems by non-dimensionalizing—removing the units within—the Navier-Stokes equations. A value close to zero indicates that the fluid will exhibit laminar flow, while a large number suggests that turbulence will occur.

The ultraviolet catastrophe is the infinite energy prediction of classical physics

When scientists first tried calculating the radiation emitted by a heated object, their assumption was that energy was equally distributed across wavelengths and produced infinite energy at low wavelengths. This contradicted observations, which showed near-zero energy intensity at those values.

An analysis of the logistic map, which shows how chaos can emerge from simple equations

When modeling systems such as the predator-prey cycle of an organism's population in an ecosystem, the logistic map function shows the population normally settles into an equilibrium value. However, at high reproduction and starvation rates, multiple such values arise, making it difficult to accurately model long-term population growth in some environments.

Explore an interactive map of the unified theories of everything

This interactive concisely describes many highly complex theories ranging from quantum gravity and black holes to dark matter and energy. Supplementary resources, key questions, and possible solutions accompany each description. As of 2026, no developed theory provides an accepted explanation of all observed phenomena in the universe at all scales.

Vincent Van Gogh's 'The Starry Night' accurately depicts turbulent flow

Analysis of the swirling patterns in the painting found an alignment with the mathematics used to describe turbulence, despite Van Gogh's unfamiliarity with the science. The depiction may indicate he possessed an uncanny ability to capture nature's details.

Lightning is the sudden flow of electricity in the atmosphere as two oppositely charged regions seek to become balanced via the movement of electrons.

Probability waves govern particles like electrons in the quantum space

Just as waves are extended objects whose heights change across space, quantum waves dictate that the probability of an object existing in different locations across space varies. Like water waves, the probability waves can combine, changing the heights—and therefore probabilities—in different locations.

Chaos gardening involves scattering seeds without any planning

The trend is intended to produce colorful, dense gardens by mixing various seeds. While encouraging experimentation and lower stakes, the success of these gardens depends on choosing plants with similar needs and incorporating native, nectar-rich flowers.

Fast-changing conditions during their formation make tornado forecasting difficult

The final steps of tornadogenesis—the process of forming a tornado—depend on near-surface details, such as local geography, which are difficult for radar to observe and affect how funnel clouds make contact with the ground. Inconsistencies in storm behavior have limited the amount of advance warning people receive before tornadoes touch down.

Emergent quantum phenomena can produce nonfundamental quasiparticles

While physicists have discovered new particles by colliding known ones at high speeds, placing known particles in extreme environments or configurations can allow quantum mechanics to produce bizarre behaviors, making them seem like new types of matter.

Explore a simplified breakdown of the Einstein field equations

Often depicted as one formula, Einstein's equation represents a suite of 16 nonlinear partial differential equations, which can be simplified to six unique ones, that relate the geometry of spacetime to the presence of mass and energy.