The important advantage of this approach is that it allows the separation of the role of high differentiability and that of algebraic geometry in a smooth setting: all the geometrically relevant phenomena appear already for polynomial mappings. The geometric properties obtained are "stable with respect to approximation", and can be imposed on smooth functions via polynomial approximation.

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Post a Review. This could have been because wind shear forced eddies to stay in their respective thin horizontal layers instead of stretching vertically. Whatever the reason, though, the analysis showed that turbulent energy began flowing from smaller scales to larger scales, perhaps feeding Isabel from below.

Their work suggests that turbulence may offer hurricanes an extra source of fuel, perhaps explaining why some storms maintain strength even when conditions suggest they should weaken. Zhang now plans to use uncrewed flights and better sensors to help bolster that case. On Jupiter, a much larger world with an even flatter atmosphere, researchers have also pinpointed where turbulence switches between two-dimensional and three-dimensional behavior.

But in , Peter Read , a physicist at the University of Oxford, and Roland Young , his postdoc at the time, made a wind speed map using data from the space probe Cassini , which swung past Jupiter in on its way to Saturn.

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They saw energy flowing into larger and larger eddies, the hallmark of two-dimensional turbulence. But nothing about Jupiter is simple.

On smaller scales — across patches of surface about the distance between New York and Los Angeles or less — energy dissipated instead, indicating that other processes must also be afoot. The data suggest that not just fluid dynamics but magnetic fields sculpt the cloud bands.

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At the end of , Bouchet and Eric Woillez , also at ENS, sketched out their own theoretical account of how two-dimensional fluid flow can describe a rotating system such as the atmosphere of a planet. Their work shows how flows built from smaller turbulence can match the enormous pattern of alternating bands visible on Jupiter through a backyard telescope.

He tried to write out how the flow of energy would balance between small turbulent eddies and a bigger flow feeding on them in a simple case: a flat, square box. A single term, related to pressure, stood in the way of a solution. So Falkovich just dropped it.

## Tame Geometry with Application in Smooth Analysis

By discarding that troublesome term and assuming that the eddies in this system are too short-lived to interact with each other, Falkovich and his colleagues tamed the equations enough to solve the Navier-Stokes equations for this case. Then he tasked Jason Laurie, his postdoc at the time, with running numerical simulations that proved it. Hoping for payoff in the pure math of fluids and for insight into geophysical processes, physicists have also pushed the formula outside a simple square box, trying to figure out where it stops working.

Just switching from a square to a rectangle makes a dramatic difference, for example. In this case, turbulence feeds river-like flows called jets in which the formula starts to fail. If they vary enough, as they might in other situations, these fluctuations will overwhelm the stable flow. But their hope, ultimately, is to describe a far richer reality. His name is Corentin Herbert, not Herbert Corentin. Get highlights of the most important news delivered to your email inbox. Abusive, profane, self-promotional, misleading, incoherent or off-topic comments will be rejected.

Moderators are staffed during regular business hours New York time and can only accept comments written in English. Mathematicians Tame Turbulence in Flattened Fluids. Read Later. By squeezing fluids into flat sheets, researchers can get a handle on the strange ways that turbulence feeds energy into a system instead of eating it away.