The skeleton of turbulence

Brazilian researchers identify the coherent structures that make up the “skeleton” of turbulent flow. Articles were published in Astrophysical Journal Letters

7/04/2011

By Fábio de Castro

Agência FAPESP – Two studies led by Brazilian researchers and published in Astrophysical Journal Letters magazine identified the coherent structures that compose the “skeleton” of turbulence.

Even though turbulence may be a phenomenon characterized by the chaotic movement of fluid particles, there are techniques for identifying coherent structures and therefore predicting these movements.

According to the authors, studies on the dispersion of volcanic ash, cyclones, tornadoes, tsunamis, solar cycles, planetary and stellar formation, the primordial Universe and other areas as diverse as the circulation of blood in the cardiovascular system and controlled thermonuclear fusion will be able to benefit from the two studies.

Research was led by spatial physicist Abraham Chian of the National Institute for Space Research (Inpe), and computational mathematician Erico Rempel from the Institute of Aeronautical Technology (ITA), in São José dos Campos (SP), in cooperation with a colleague at Stockholm University (Sweden) and a doctoral student from Inpe.

One of the studies was concluded during Chian’s visit to the California Institute of Technology in the United States with funding from the Guggenheim Foundation. Inpe doctoral candidate Pablo Muñoz participated in the study, receiving best student project award at the 9th Latin American Spatial Geophysics Congress held in Costa Rica in April 2011.

The scientists studied the magnetic field related to coherent turbulence structures observed in solar plasma. According to Chian, utilizing data provided by instruments on board four space probes in the Cluster mission , the group detected two so-called coherent “current sheets” in front of an interplanetary magnetic cloud.

“Data analysis on magnetic fluctuations near these coherent structures demonstrated that solar wind exhibits well-developed Kolmogorov turbulence behavior similar to the turbulence found near a nuclear fusion machine, in the solar atmosphere, in the interstellar medium, in a wind tunnel and in the canopy of the Amazon rain forest, to give a few examples,” Chian told Agência FAPESP.

Characterizing the dynamics at the front boundary of an interplanetary magnetic cloud is fundamental for effective monitoring and forecasting of spatial climate, as there is evidence that a magnetic storm on Earth can be started by a magnetic cloud from a solar eruption.

“Extreme natural events such as cyclones, tsunamis, excessive rainfall in localized areas, sun spots and interplanetary coronal mass ejections are related to coherent structures that dominate the dynamics of turbulence and can cause great impact on Earth’s climate, the spatial climate, and the solar-terrestrial environment,” he explained.

The second project began during Rempel’s post-doctoral work at Cambridge University as a FAPESP fellow and involved the work of Chian and Axel Brandenburg, professor at the Nordic Institute for Theoretical Physics and Stockholm University (Sweden).

According to Chian, Brandenburg is one of the pioneers of the cosmic dynamo model. “The dynamo theory could explain the origin and evolution of solar cycles like, for example, the appearance of prolonged periods of calm in solar activity known as the Grand Minimas,” he said.

In the study, the group investigated the coherent Lagrangian structures of astrophysical turbulence based on numerical simulation of a non-linear dynamo model.

Coherent Lagrangian structures are material lines or surfaces that act as barriers to transportation in turbulence. Inspired by the chaos theory, the concept was introduced nearly ten years ago by George Haller, currently a mechanical engineering professor at Canada’s McGill University.

“The new non-linear technique allows for a more accurate visualization of the dynamics and the complex structure of fluids that wouldn’t be possible using traditional techniques based on Eulerian formalism,” said Chian.

These structures are determined by computing the Maximal Lyapunov Exponent of finite time, which gives the average value of the maximum rate of divergence or lengthening between the trajectories of particles in a certain time interval.

“This allows for identification of attractor and repeller trajectories in images from numeric simulations or in real images of the liquid velocity field, revealing the skeleton of turbulence that forms the barriers for transportation of particles. The crossings of these barriers are responsible for the chaotic mixture of particles,” he said.

Coherence in chaos

According to the authors, the study of coherent Lagrangian structures has applications in a number of areas such as predicting the movement of pollutants in the atmosphere or sea, phytoplankton migration in the ocean, aperiodic flow in tornadoes, the interaction between fluid and structure through cardiac valves and the thermonuclear plasma in magnetic confinement machines.

According to Rempel, the Brazilian group was the first to introduce this new technique to astrophysics. Using simulated images of turbulence to model the magnetic field generation in the convective layers of the Sun and other stars, the study proved that coherent Lagrangian structures are capable of clearly distinguishing the details of the complex spatial distribution of transport barriers between two different dynamo regimes.

“Since the concept was developed by Haller, the technique has been applied to fluid problems; both simulations as well as in observational data of the dispersion of pollutants in the oceans, for example. But they hadn’t yet been used in astrophysics for fluids with a magnetic field,” said Rempel.

These coherent structures mark certain preferential directions of the particles of fluids in motion. When a pollutant is carried by oceanic vortices and currents, the identification of coherent structures can allow individuals to detect lines of attraction and make predictions as to where the fluid will go. The same phenomenon can happen, for example, with volcanic ash in the atmosphere.

“In astrophysics, our goal was to understand the impact of the magnetic field on the turbulent movements of a star’s plasma,” said Rempel, who coordinated the project entitled “Numerical simulation and analysis of the transition to turbulence in spatial plasmas: a dynamic systems approach”, funded through FAPESP’s Research -Regular program.

According to him, in the Sun’s convective layer, an intensely turbulent region, particles move as if they were caught in vortices. Coherent Lagrangian structures mark the boundaries of these vortices, separating the fluid regions between which particles don’t mix.

“When we did the coherent structure study, we saw that some particles can cross to other regions of the fluid. In the case of a star, we observed that when the magnetic field was stronger, there were fewer crossings, in other words, less turbulence,” he said.

According to Rempel, these results were obtained from a very simplified simulation “Based on that academic model, we will now try to extend the application to more realistic models of the Sun’s convective layer,” he said.

The articles “Detection of current sheets and magnetic reconnections at the turbulent leading edge of an interplanetary coronal mass ejection”, (doi: 10.1088/2041-8205/733/2/L34), by Abraham Chian and Pablo Muñoz, and “Lagrangian coherent structures in nonlinear dynamos”, by Erico Rempel, Abraham Chian and Axel Brandenburg (doi: 10.1088/2041-8205/735/1/L9), can be read by subscribers to Astrophysical Journal Letters.