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American researchers were the first to measure and demonstrate persistence of helicity in real fluid.

Energy conservation law, law of conservation of momentum, law of conservation of angular momentum are well known since school years. Conserved quantities (often referred to as “invariants”) are the basis of our understanding of the physical world. However, while energy, mass and momentum are familiar to almost everyone, few are aware that some systems, such as streams of fluid or plasma, have another invariant: helicity.

Essentially, helicity is a measure of flow topology showing whether the flow has any vortices and to what extent they are connected, entangled, located relative to each other1,2. Vortex rings may be convolved, interlocked, even tied in a bundle3. We are familiar with such vortex phenomena and these include not only hurricanes and tornados but also air circulation generated by airplane wings (the so called “wingtip vortices”), turbulence and even so seemingly trivial everyday occurrences such as smoke clouds, water vortices in the sink and in a cup of tea after stirring.

Helicity has long been an object of scientific interest. However, its detailed properties were only described in the second half of the 20th century1,2. Nevertheless, one hundred years before, Lord Kelvin was fascinated by elegance and stability of smoke rings4 and suggested the daring vortex theory of atoms. According to it, all atoms are interconnected and knotted vortex rings in aether, a mystic fluid which was thought to fill the whole Universe2,5. Though the idea was not confirmed (and neither did the aether hypothesis), Kelvin was one of the originators of numerous disciplines, e.g. mathematical knot theory and vortex dynamics5,6.

Until recently, helicity was thought to be conserved (invariant along with energy and momentum) only for ideal fluids, i.e. those with zero viscosity. If the flow of such fluid contains interconnected vortices (vortex rings), their nature does not change over time, they are sort of “frozen” in the flow2,3. In case of real fluid, such state is impossible due to the viscosity: friction affects whirls and modifies helicity3. For a long time researchers did not manage to investigate these issues empirically, as no accurate method measuring helicity was available. However, recently a group of scientists from Chicago University developed an elegant method that allowed them to demonstrate that water, the most real fluid, may retain helicity for a long period. The researchers published their studies in Science3, one of the most prestigious scientific journals.

However, before investigating stability of interconnected vortex rings in water, one had to first “knot” and “visualize” such rings. This obviously challenging task was addressed by Dustin Kleckner and William Irvine from Chicago University using a 3D printer, gas bubbles and good imagination7,8. According to the scientists themselves, the idea was suggested by dolphins playing with air rings in water5,9. The knots were “modeled” by various contoured structures, or “hydrofoils,” which were fabricated using a 3D printer and had the shape of an airplane wing in the cross section (such shape makes the flow circulate and form vortices)6,8,9. The scientists put these hydrofoils into water that was entrained with microscopic bubbles and then they quickly accelerated the hydrofoils. Thus, a knotted vortex was obtained in water; it had the shape of the countered structure put into water, and the bubbles allowed it to be visualized using laser scanning and high-speed camera (76,000 shots/second!)4-9.

Therefore, the first photos of knots tied in water without ends, unlike our shoelaces, were obtained. A trefoil (an ordinary knot with its free ends connected) and Hopf link (two linked circles) were generated.

hydrofoils and knot

Left: “hydrofoils” for generating knotted water vortex rings. The countered structures were 3D-printed. Right; a “knot” (water trefoil) generated using hydrofoils. Source Tying Water into Knots.

The scientists demonstrated that, once generated, these knots move in water, rotate, can untie on their own and finally disperse4-9. This study was published in Nature Physics and was hotly debated as it concerns in-depth questions from various physics disciplines including turbulence, plasma physics, flow of real and more exotic superfluids, transferring them from theory to experimental testing. In addition to scientific role of the study, knotted water vortices entrained with bubbles are just so beautiful and charming. The video of the study has gained more than 100,000 YouTube views.

After learning how to “knot” water and generate vortices interlocked in water experimentally, the scientists started to investigate peculiarities and lifecycles of such vortices in details3,10. For the first time, the physicists were able to measure total helicity of viscous fluid experimentally in laboratory and demonstrate that it may persist longer than theoretically believed3.

Once again, creative approach to experiments helped the researchers. As the bubbles were not suitable for the challenge anymore (they gave no adequate resolution), the physicists decided to visualize water vortices using a Sharpie marker 11. The experimental scheme was the same, but, instead of bubbles, the edges of hydrofoils were labeled with dots made with red marker containing rhodamine dye fluorescing at laser irradiation. The dye began to dissolve in water and stained the newly created water vortex after abrupt acceleration of the hydrofoil3,11,12.

The researchers generated such water ring vortices whose helicity was described by three topological forms: twisting, linking and writhing.3,11,12.

All these forms are easy to imagine if the vortex is visualized as a bundle of filaments like in a rope. If we twist the bundle, it will be twisting; if we do two rings and connect them as chain links, it will be linking; and if the bundle is coiled up like a telephone cord or a hosepipe, it will be writhing3.


Visual demonstration of three topological forms of vortex ring helicity: twisting, linking and writhing. Source Scheeler et al., 2017

These forms are interconnected and may pass from one to another3, 10. If we stretch the folded rope, we will obtain a twisted rope.

Due to a higher resolution, the scientists managed to simultaneously measure evolution of all the three forms of helicity, modification of vortex shape as a whole and its details from 1 mm to 30 cm3,11. Irvine group demonstrated that all helicity forms depend on each other, e.g. increased writhing reduces twisting (and vice versa), thus ensuring continuity of overall system helicity3,10,11. In the experiment of generating two vortex rings (one twisted and the other ordinary), the scientist observed such rings change in size and shape, and jump over each other as if playing leapfrog3.

The obtained results play an important role for both theoretical and applied physics. They will promote understanding generation of turbulence, mechanism and nature of tornados and other atmospheric phenomena3,10-12.


1.    Moffatt HK (1969) Degree of knottedness of tangled vortex lines. J Fluid Mech 35(Pt 1):117–129.

2.    Moffatt HK (2014) Helicity and singular structures in fluid dynamics. Proc Natl Acad Sci USA 111(10):3663–3670.

3.    Scheeler MW, van Rees WM, Kedia H, Kleckner D, Irvine WTM. (2017) Complete measurement of helicity and its dynamics in vortex tubes. Science. 4: 357(6350): 487-491.

4.    https://www.uchicago.edu/features/fragile_knots_untangle_deep_physics_questions/

5.    https://news.uchicago.edu/story/vortex-loops-could-untie-knotty-physics-problemsFirst fluid knots created in the lab

6.    https://www.newscientist.com/article/dn23227-first-fluid-knots-created-in-the-lab/

7.    https://www.nature.com/articles/495009b

8.    Kleckner, D., Irvine, W.T.M. Creation and Dynamics of Knotted Vortices. Nature Physics, 9, 253-258.

9.    https://www.americanscientist.org/article/tying-water-into-knots

10.    Scheeler MW, Kleckner D, Proment D, Kindlmann GL, Irvine WTM (2014). Helicity conservation in vortex reconnections. Proceedings of the National Academy of Sciences. 111(43): 15350-15355.

11.    https://news.uchicago.edu/story/clever-experiment-documents-multi-scale-fluid-dynamics

12.    https://www.newscientist.com/article/2142815-teacup-tornados-inner-twists-and-writhes-seen-for-first-time/