Mathematics proves turbulence's chaos creates super diffusion

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

As energy is lost in fluids undergoing turbulent, non-linear flow, particles will collectively form eddies, or whirlpools, of various sizes that mix together. The energy cascade of larger eddies becoming smaller ones before dissipating completely varies based on the initial conditions of the fluid, a characteristic feature of chaotic systems.

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.

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.

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.

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.

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

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.)

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.

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.

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.

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.

Traffic flow can be improved through research into card shuffling

Card-shuffling mathematics has been applied to solving problems in biology to estimate the evolutionary distance between organisms and in computer science to determine the optimal distribution of files and folders in databases.

The universe only evolves toward increasing disorder, providing a direction to time

Within physics, the second law of thermodynamics states that entropy always increases, as seen by a concentrated drop of dye slowly diffusing throughout a cup of water, increasing disorder. Because such processes do not unfold in reverse, entropy may be what provides time with a single direction, unlike space.

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.

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.

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.

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.

Tiny water droplets can blur a rainbow's colors into a foggy white

When sunlight enters water droplets, it can reflect from the inside and spread out into different colors, producing a series of colorful arcs—supernumerary bows—which we refer to as rainbows. For droplets a few hundredths of a millimeter in size, these arcs overlap onto each other, producing fogbows of white light.

Explore chaos in this simulation of two double pendulums, where small differences in the release angles produce wildly different outcomes.

Stars appear to twinkle because turbulence in the motion of molecules in Earth's atmosphere causes light to bend and wiggle rapidly on its way to the surface—a phenomenon called scintillation.

Climate models use complex equations and supercomputers to predict future climate

The equations apply the laws of thermodynamics, fluid motion, and radiative transfer to a 3D grid along Earth's surface. After solving them for each cell in the grid over a time interval, the results carry over to neighboring cells, and the process repeats into the future.

Quantum mechanical principles are responsible for chemical properties

Besides quantization, the wave-particle duality, and probabilistic features, quantum theory's mathematics introduces the exclusion principle, quantum tunneling, quantum entanglement, and other phenomena foreign to our everyday experiences. Some suggest that all possibilities for a quantum system may also occur across the multiverse.

Collisions between cosmic rays and atmospheric particles dwarf particle accelerators

According to Einstein's model of gravity, confining immense energy into a sufficiently small region would warp space-time enough to create a black hole. If accelerators actually did create black holes, then trillions more would be produced daily in the atmosphere, where collisions are millions of times more energetic.

Particle detection has evolved from cloud and bubble chambers to digital 3D displays

As invisible particles passed through cloud and bubble chambers, they disrupted supersaturated vapor and superheated liquid, respectively, creating trails that served as proxies for cosmic rays. The transition from photographed events to digital sensors enhanced data collection and accuracy.

Dirty lightning is a thunderstorm created from friction between rising and falling flows of ash and smoke during volcanic eruptions.

Simulation of the coronavirus inside an aerosol droplet

A new simulation shows how the virus survives inside tiny airborne particles of water — and gives clues about how the Delta variant became dominant.

Diffusion models generate nontext media by adding and removing noise from a dataset

Unlike text-based AI tools, which rely on large language models to recognize patterns in text, diffusion models use a reward system to generate images, video, audio, and 3D models. They can also be customized to match a user’s style.

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.

Learn why magnets, like statically charged balloons, can deflect water

While the oxygen atom in water molecules strongly attracts hydrogen's electrons—producing an uneven charge distribution that pulls water toward balloons that have been rubbed against surfaces—the presence of a magnetic field alters the path of these electrons, which produce their own tiny magnetic fields. The interaction of these fields repels water away from magnets, making it diamagnetic.

Volcanic activity may have triggered the Black Death by spewing gases that led to crop failures in Europe, thus forcing the import of grain ships with plague-infested fleas.

The boiling temperature of water decreases with increasing elevation because the molecules need less energy to overcome the lower atmospheric pressure associated with thinner air.

How fusion—which releases energy when two lighter atoms combine to form a heavier atom—differs from fission

Most of the observable universe's volume consists of voids and supervoids—enormous regions of space with little to no galaxies, stars, or light.

Pure water does not conduct electricity because this flow of energy requires the movement of charged particles, and water is a neutral molecule.

Scientists testing how climate change affects plants found, in a years-long simulation study, that poison ivy grew 70% faster with the resulting extra carbon dioxide.

To reduce wrinkles in clothing, research recommends gentle wash cycles with more water and tumble drying with smaller loads and longer times.

As of 2025, the most cited papers in science include a review of graphene, methods to improve image recognition, streamlining electron density calculations, and optimizing protein studies.

How Amazon operates a 'flywheel' business model, a self-reinforcing cycle that drives continuous growth

Waves off the coast of Nazaré, Portugal, can reach over 100 feet and are caused by the area's deep underwater canyon geography

Some renewable energy sources may weaken the resiliency of electric grids by removing outage buffers provided by coal- and natural gas-fired plants, which provide the system with more time to find a new power source if a plant goes down.

Sunflowers have long been a symbol of nuclear disarmament because they can extract large amounts of radioactive cesium from soil.

In 1945, Surrealist artist Salvador Dalí and legendary animator Walt Disney began collaborating on a short film that was not completed until 2003, years after both men had died.

Exploring the Larry Bird punch that almost changed March Madness

The US dollar is the most widely circulated currency in the world, and is involved in nearly 90% of global transactions

The term 'algorithm' originated from the Latinized name of al-Khwārizmī, a 9th-century mathematician whose works introduced Arabic numerals and the decimal-based numeral system to Europe and whose treatise 'Al-Jabr' gave algebra its name.

Gerrymandering works through two techniques: cracking opposing voters across districts to dilute their power, and packing them into a few to waste their votes in lopsided majorities.

Art theft is the third-largest source of criminal revenues worldwide, thanks in part to the subjective nature of valuing a work of art.

Explore an interactive map built from more than 100,000 survey responses on where people find joy in their lives

Double world record holder Hunter Ewen helped conquer his fear of balloons, called globophobia, by training to win Guinness World Record titles for balloon inflation.

A small portion of The Winstons' 'Amen, Brother,' an obscure '60s gospel record, is the basis for thousands of hip-hop tracks, making it one of the most sampled records ever that's informally known as 'the Amen break.'