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Here we provide a theoretical framework revealing that the radius$R_{d}$of the top droplet ejected from a bursting bubble of radius$R_{b}$and$Bo\\leqslant 0.05$can be expressed as$R_{d}/R_{b}=K_{b}(1-(Oh/Oh_{c}^{\\prime })^{1/2})$for$Oh\\lesssim Oh_{c}^{\\prime }$or as$R_{d}\\approx 18\\,\\unicode[STIX]{x1D707}_{l}^{2}/(\\unicode[STIX]{x1D70C}_{l}\\unicode[STIX]{x1D70E})$for$Oh\\gtrsim Oh_{c}^{\\prime }$, with the numerically fitted constants$K_{b}\\approx 0.2$,$Oh_{c}^{\\prime }\\approx 0.03$,$Oh=\\unicode[STIX]{x1D707}_{l}/\\sqrt{\\unicode[STIX]{x1D70C}_{l}\\,R_{b}\\,\\unicode[STIX]{x1D70E}}\\ll 1$the Ohnesorge number,$Bo=\\unicode[STIX]{x1D70C}_{l}\\,g\\,R_{b}^{2}/\\unicode[STIX]{x1D70E}$the Bond number, and$\\unicode[STIX]{x1D70C}_{l}$,$\\unicode[STIX]{x1D707}_{l}$and$\\unicode[STIX]{x1D70E}$indicating the liquid density, dynamic viscosity and interfacial tension coefficient, respectively. These predictions, which do not only have solid theoretical roots but are also much more accurate than the usual 10 % rule used in the context of marine spray generation via whitecaps for$R_{b}\\lesssim 1$mm, agree very well with both experimental data and numerical simulations for the values of$Oh$and$Bo$investigated. Moreover, making use of a criterion which reveals the mechanism that controls the growth rate of capillary instabilities, we also explain here why no droplets are ejected from the tip of the fast Worthington jet for$Oh\\gtrsim 0.04$. In addition, our results predict the generation of submicron-sized aerosol particles with diameters below 100 nm and velocities${\\sim}\\unicode[STIX]{x1D70E}/\\unicode[STIX]{x1D707}_{l}$for bubble radii$10~\\unicode[STIX]{x03BC}\\text{m}\\lesssim R_{b}\\lesssim 20~\\unicode[STIX]{x03BC}\\text{m}$, within the range found in natural conditions and in good agreement with experiments – a fact suggesting that our study could be applied in the modelling of sea spray aerosol production.

Here we provide a unified theoretical description of two different physical situations in which liquid jets are expelled out of the bulk of a liquid as a consequence of the capillary collapse of a void. We demonstrate that the velocity field giving rise to the emergence of these jets can be calculated as the flow generated by a line of sinks with a length and an intensity that can be expressed in terms of the initial cavity radius and the wavelength and velocity of the capillary waves propagating along the cavity walls. The predicted jet speeds, which are expressed through algebraic equations, are in good quantitative agreement with those obtained from experiments and from the simulations of bubbles bursting on a free surface or after the implosion of the crater formed when a drop impacts a liquid pool.

Magnetoconvection at the solar surface governs the dynamics in the upper solar atmosphere and sustains the heliosphere. Properties of this fundamental process are poorly described near the solar poles. Here we report the first out-of-ecliptic remote-sensing observations of the south pole of the Sun from a high-latitude campaign of the Solar Orbiter spacecraft, which reveal spatial and temporal evolution of supergranular convective cells. The supergranular cells have spatial scales of 20–40 Mm. From 8 days of observations starting on 2025 March 16, our analysis shows that the magnetic network migrates poleward, on average, at high latitudes (above 60°), with speeds in the range of 10–20 m s−1, depending on the structures being tracked. These results shed light on the buildup of the polar magnetic field that is central to our understanding of the solar cycle and the heliospheric magnetic field.

Direct remote-sensing observations of the solar poles have been hindered by the restricted view obtained from the ecliptic plane. For the first time ever, Solar Orbiter with its remote-sensing instruments observed the poles of the Sun from out of the ecliptic in the spring of 2025. Here, we report the first measurements of the magnetic field of the solar poles taken when Solar Orbiter was at heliographic latitudes ranging between 14 .° 9 and 16 .° 7. The data sets were collected by the High Resolution Telescope of the Polarimetric and Helioseismic Imager on board Solar Orbiter (SO/PHI-HRT). Two sets of observations, approximately one month apart, for the south and north pole are considered in this work. The magnetic flux and flux density measured during these campaigns are reported as a function of the heliographic latitude observed by SO/PHI-HRT. The net fluxes show a different latitudinal distribution for the two polar caps. We also discuss the observed dependence of the measured fluxes on the viewing angle. These first results highlight the importance of high-resolution direct measurements of the polar field, paving the way for the high-latitude observations planned for SO/PHI-HRT in the coming years.

The occurrence of the elliptic instability in rings and helical vortices is analysed theoretically. The framework developed by Moore & Saffman (Proc. R. Soc. Lond. A, vol. 346, 1975, pp. 413–425), where the elliptic instability is interpreted as a resonance of two Kelvin modes with a strained induced correction, is used to obtain the general stability properties of a curved and strained Batchelor vortex. Explicit expressions for the characteristics of the three main unstable modes are obtained as a function of the axial flow parameter of the Batchelor vortex. We show that vortex curvature adds a contribution to the elliptic instability growth rate. The results are applied to a single vortex ring, an array of alternate vortex rings and a double helical vortex.

Calcium sulfate minerals such as gypsum play important roles in natural and industrial processes, but their precipitation mechanisms remain largely unexplored. We used time-resolved sample quenching and high-resolution microscopy to demonstrate that gypsum forms via a three-stage process: (i) homogeneous precipitation of nanocrystalline hemihydrate bassanite below its predicted solubility, (ii) self-assembly of bassanite into elongated aggregates co-oriented along their c axis, and (iii) transformation into dihydrate gypsum. These findings indicate that a stable nanocrystalline precursor phase can form below its bulk solubility and that in the CaSO₄ system, the self-assembly of nanoparticles plays a crucial role. Understanding why bassanite forms prior to gypsum can lead to more efficient anti-scaling strategies for water desalination and may help to explain the persistence of CaS0 4 phases in regions of low water activity on Mars.

In this paper, we analyse the curvature instability of a curved Batchelor vortex. We consider this short-wavelength instability when the radius of curvature of the vortex centreline is large compared with the vortex core size. In this limit, the curvature instability can be interpreted as a resonant phenomenon. It results from the resonant coupling of two Kelvin modes of the underlying Batchelor vortex with the dipolar correction induced by curvature. The condition of resonance of the two modes is analysed in detail as a function of the axial jet strength of the Batchelor vortex. In contrast to the Rankine vortex, only a few configurations involving $m=0$ and $m=1$ modes are found to become the most unstable. The growth rate of the resonant configurations is systematically computed and used to determine the characteristics of the most unstable mode as a function of the curvature ratio, the Reynolds number and the axial flow parameter. The competition of the curvature instability with another short-wavelength instability, which was considered in a companion paper (Blanco-Rodríguez & Le Dizès, J. Fluid Mech., vol. 804, 2016, pp. 224–247), is analysed for a vortex ring. A numerical error found in this paper, which affects the relative strength of the elliptic instability, is also corrected. We show that the curvature instability becomes the dominant instability in large rings as soon as axial flow is present (vortex ring with swirl).

Phase diversity techniques are widely employed in solar astronomy to evaluate and correct the aberrations stemming from atmospheric turbulences, the telescope, and its instruments. The method uses information provided by a pair of images. One of them is usually focused while the other one is defocused. The amount of defocus to be induced is somehow arbitrary, though. In this work we carry out a series of numerical experiments with artificial solar images to investigate the performance of phase diversity for different choices of the relative defocus among the two images. The experiments allow us to determine the amount of defocus that produces the best wave front restoration when changing: (1) the number of Zernike polynomials of the retrieved and/or incident wave fronts; (2) the signal-to-noise ratio of the images; (3) the amplitude of the incident aberrations; and (4) the observed scene. We find a correlation between the amount of defocus needed for optimal restorations and the number of Zernike polynomials employed in the optimization. Values larger than the typically accepted choice of 1λ (peak to peak) are obtained in most cases.

We review two instability mechanisms that may be active in wind turbine wakes and contribute to their downstream evolution, by considering simplified configurations of one or several spatially uniform helical vortices. One category of instabilities involves displacement perturbations of the vortices, with wavelengths that are large compared to the size of their cores; they can be analysed using a filament approach. Previous theoretical results, confirmed by our recent experiments, show that the predicted instability modes are related to the pairing phenomenon found in periodic arrays of vortices. A second group of instabilities involves internal perturbations of the vortex cores, with wavelengths scaling on the core size. They result from deformations of the cores due to curvature, torsion or the strain induced by neighbouring helix loops. Our experiments show that the non-linear evolution of the shortwave instabilities, combined with the pairing mechanism, leads to a rapid destruction of the helical wake vortices.

Phase diversity techniques are usually based on the comparison of synchronously acquired pairs of focused–defocused images. This way, differences between both images are avoided except from random pixel variations due to noise on the detector and the phase diversity itself. In some astronomical instruments, though, the two images are not taken simultaneously. This work studies the impact of carrying out phase diversity with pairs of asynchronously acquired images while observing an evolving solar scene. We evaluate the performance of this technique as a function of the time gap between the images through the use of a magnetohydrodynamical simulation of the solar scene as observed by an instrument. We describe the incident wave front with two numbers of Zernike polynomials (20 or 32) to explore their effect on the wave front sensing accuracy and we employ two levels of noise to study their impact in the object restoration. We find that a time gap among our simulation images smaller than ∼10 s has a negligible impact on the performance of the method. The rms error of the Zernike coefficients fitting worsens exponentially from there on, but the evolution is similar no matter the number of polynomials used in the fitting. Meanwhile, the quality of the object restoration benefits from lower noise levels, but it decreases linearly with the time gap independently of the amount of noise.