Turbulence yields to topology

NSE PhD candidate Lucio Milanese expands a theory of turbulence to include both ionized and non-ionized fluids.

PSFC News

January 3, 2022

As a plasma physicist in the MIT Department of Nuclear Science and Engineering, PhD candidate Lucio Milanese is interested in understanding the behavior of plasmas, the gas-like flow of ions and electrons that comprise 99% of the visible universe, including the earth’s ionosphere, interstellar space and the solar wind. His study of turbulence in plasmas resulted last year in the discovery of a new mechanism called “dynamic phase alignment,” which shows how energy transfers (cascades) from large to small scales in a turbulent plasma when net helicity – the degree to which vortices twist and spiral – is present in the system. While seemingly chaotic, plasma turbulence still obeys certain laws: not only energy must be conserved during the turbulent cascade, but also helicity; and this imposes constraints on the dynamics of turbulence that underlie the phenomenon of dynamic phase alignment. Now, in an article published in Physical Review Letters, Milanese shows that the phenomenon of dynamic phase alignment is an essential ingredient in the turbulent dynamics of both plasma and nonionized fluids like water and air.

What questions are you addressing in this new paper?

My work focuses on understanding turbulence, its nature and structure, in both plasmas, which are ionized fluids where electrons have been stripped from the nuclei, and non-ionized fluids. In particular, I’m interested in understanding how energy is transferred from large scales to small scales in turbulent systems. Suppose you stir a cup of coffee. You will create a vortex or whirl at the size of the system. This large-scale structure will then break into smaller and smaller structures, until it reaches sufficiently small scales that the collisions between molecules dissipate the kinetic energy of the fluid movement into thermal energy. This transfer of energy from large scale to small scales is often referred to as the “Richardson-Kolmogorov cascade,” from the names of the two scientists who pioneered the concept. The systems I studied in this new paper are governed by the Navier-Stokes equations, which can describe a broad variety of physical systems, from atmospheric and oceanic flows to the flow of water in pipes and blood in our bodies, to air moving around a car or a sub-sonic aircraft.

The turbulent dynamics of fluids and plasmas is highly complex and chaotic, but not completely lawless. Similar to our previous work on plasmas, we studied particular systems in which two conservation “laws” must be obeyed. The first law is conservation of total energy, and the second is conservation of helicity, which describes the degree to which vortices twist and spiral. Helicity in Navier-Stokes turbulence is related to the product of velocity and vorticity, the latter quantity loosely describing the local level of “rotation” in the fluid. You can imagine a steady, unidirectional flow will not have significant vorticity. On the other hand, a rotating or helical flow will have significant rotational movement and thus vorticity. A tornado, for example, has a large amount of vorticity (i.e., rotation) aligned with its velocity, and plenty of helicity. Helicity is a topological constraint on the dynamics, as it relates to how turbulent structures are organized geometrically in space.

While it has been known for decades that both energy and helicity are transferred from large to small scales in the turbulent cascade, this empirical result did not rest on a robust theoretical foundation. This is because if you develop a model based purely on conservation of energy, as Richardson and Kolmogorov did, you end up violating conservation of helicity. They did not know that, like energy, helicity is an invariant that must also be conserved in the turbulent cascade. This was only discovered later, and how to factor that into their theory remained a mystery until now.

What were your findings?

In the paper we demonstrate that, as the turbulent cascade progresses from large to small scales, velocity and vorticity become progressively more correlated, thus allowing for a progressive reduction in the helicity invariant, which prevents the system from violating helicity conservation. We termed this phenomenon “dynamic phase alignment” as it manifests itself as a progressive alignment between the correlation of velocity and vorticity fluctuations: a form of self-organization that allows turbulence to obey the conservation laws the equations impose on it. In some sense, turbulence is yielding to topology as the topological constraint, namely conservation of helicity, is determining important aspects of the turbulent dynamics, such as the correlation between velocity and vorticity. Dynamic phase alignment thus provides the first fully self-consistent picture of the turbulent cascade of energy and helicity, offering a solution to a long-standing theoretical problem.

How does it build on previous work?

We had first discovered, via theory and numerical simulations, the existence of dynamic phase alignment in certain plasma systems such as the Earth’s ionosphere, the solar corona and some instances of the solar wind, as well as electron-positron (matter-antimatter) plasmas in regimes relevant to upcoming experiments. The remarkable result is that this exact same phenomenon is at play in non-ionized turbulent fluids described by the Navier-Stokes equation, such as tornadoes and hurricanes, turbulent flows of water in pipes or air moving around bodies, such as a car or aircraft.

Dynamic phase alignment is an important building block in a general theory of turbulence. In light of our recent result it acquires the distinction of being a unifying paradigm in the turbulence of fluids (both ionized and non-ionized), characterizing the turbulent dynamics in the presence of both energy and helicity.

Why is it important?

In addition to proposing a solution to a long-standing problem in the theory of turbulence, these sorts of “unifications” and cross-pollinations between adjacent but separate fields are particularly exciting as they shine light on fundamental truths about the nature of turbulence, irrespective of the details of the dynamics found in different physical systems.

What is the next step in understanding turbulence?

There are a couple of directions we could go. 

One is to try to better understand how dynamic phase alignment arises from the interactions of turbulent structures. We obtained the results on the scale-dependent correlation between the velocity and vorticity fields from a statistical analysis, but we don't yet understand how such scale-dependent correlation exactly comes to be from vortex structures interacting with each other and breaking into smaller ones (i.e., cascading).

The other direction is looking at plasma systems governed by magnetohydrodynamics (MHD), such as the plasmas in stars and interstellar and interplanetary media. Such systems exhibit three invariants, namely energy, kinetic helicity (measure of twist of velocity fields) and magnetic helicity (measure of twist of magnetic field lines). It would be very interesting to see if the dynamic phase alignment that we’ve uncovered survives then.

 

*The research is funded, in part, by the U.S. Department of Energy, the National Science Foundation, and Loureiro’s Professor Amar G. Bose Research Grant. Milanese was a recipient of the Manson Benedict Fellowship of the MIT Department of Nuclear Science and Engineering.

 

Topics: Magnetic fusion energy, Plasma theory & simulation, Plasma turbulence, Nuno Loureiro