Researchers have discovered that magnetic field fluctuations at the Sun’s poles follow a fundamental law of turbulence, first identified in 1941 by Soviet mathematician Andrey Kolmogorov. They analyzed fifty years of direct measurements accumulated by the Wilcox Solar Observatory in the United States and applied a modern wavelet analysis method to this data. It turned out that the energy of these fluctuations is distributed between large and small waves according to a strict mathematical rule. The exponent of this distribution was found to be close to the famous value of –5/3, which precisely describes the universal law of turbulence for liquids and gases.
This work was carried out by a team of researchers from Perm Polytechnic University, the Institute of Continuous Media Mechanics of the Ural Branch of the Russian Academy of Sciences, and the Institute of Terrestrial Magnetism, Ionosphere, and Radio Wave Propagation of the Russian Academy of Sciences. They believe this discovery provides physicists with a new tool for testing models of the Sun’s convective zone—the very region within the star where its magnetic field originates. The scientists published their findings in the journal Monthly Notices of the Royal Astronomical Society. The research was supported by the Russian Science Foundation.
Solar activity begins deep within the Sun’s interior. There, in a layer roughly two hundred thousand kilometers thick, hot plasma continuously moves: rising from the depths to the surface as heated currents, while cooler ones sink downward. This chaotic boiling generates the magnetic field, which then manifests on the surface as dark spots, brilliant flares, and coronal mass ejections. It is impossible to directly observe this interior—no telescope or probe can penetrate its opaque bulk. Therefore, solar turbulence was previously studied mainly through indirect evidence, such as counting and analyzing statistics of sunspot emergence, which is linked to the horizontal (toroidal) component of the magnetic field encircling the star.
However, Russian scientists chose a different approach and focused on an area that had long been overlooked: the Sun’s poles. To do this, they utilized unique data collected by the magnetograph telescope at the Wilcox Solar Observatory since May 1976. This vast dataset spans four and a half solar cycles. At the poles, the magnetic field behaves differently than at the equator: here, it emerges as vertical lines of force. This is the so-called poloidal component, which shapes the star’s magnetic poles themselves, and they swap places every eleven years, as if on command.
To make sense of these fifty years of regular measurements, mathematicians applied wavelet transformation. This method is akin to dissecting a complex musical composition: it allows the overall signal to be broken down into individual oscillatory notes, isolating vibrations of different periods and examining how they change over time.
Conducting the research. Author and source: press service of PNIPU
As a result, for the first time in the history of direct observations of the Sun’s magnetic field, scientists observed a power-law spectrum. This pattern covered a range of scales spanning nearly a hundredfold—from one month to six and a half years. Within this range, the slope of the spectrum, which indicates how rapidly high-frequency oscillations decay, was extremely close to the famous “–5/3” law derived by Kolmogorov for developed turbulence.
Project leader Rodion Stepanov, a Doctor of Physical and Mathematical Sciences and professor at the Department of Mathematical Modeling of Systems and Processes at Perm Polytechnic University, explained:
The spectral characteristics obtained are not merely a confirmation of the universality of Kolmogorov turbulence. Now, any realistic computer model of the solar dynamo must reproduce not only the 11-year cycle but also the full spectrum of oscillations with an index close to –5/3.
Understanding these mechanisms opens the door to more accurate predictions of space weather. The current 25th solar cycle, which began in 2019, has turned out to be far more intense than forecasts predicted. Older models failed: scientists used to gauge the strength of an upcoming cycle by the intensity of the polar field during the quiet period, when sunspots are scarce, but this method missed the mark for this cycle.
Researchers believe that models need urgent improvement, including through the use of artificial intelligence. As Rodion Stepanov noted, the variability of the polar field is especially valuable for machine learning, as it contains a broad spectrum of oscillations generated by nonlinear processes in the convective zone.
In the future, new space probes will help verify these findings. The European Solar Orbiter is already operational, and polar observatories are on the horizon, including the Chinese SPO project and Russia’s “Interheliosonde.” These will provide even more detailed images and measurements of the fine structure of the Sun’s magnetic poles.
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