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Since their theoretical discovery more than a half-century ago, vortices observed in bulk superconductors have carried a quantized value of magnetic flux determined only by fundamental constants. A recent experiment reported 'unquantized' quantum vortices carrying the same fraction of flux quantum in Ba$_{0.23}$K$_{0.77}$Fe$_2$As$_2$ in a small temperature range below its superconducting critical temperature ($T_C$). Here, we use scanning superconducting quantum interference device (sSQUID) microscopy with improved sensitivity to investigate the genesis of fractional vortices in Ba$_{0.23}$K$_{0.77}$Fe$_2$As$_2$. We report the direct observation of a single-flux quantum vortex splitting into two different fractions with increasing temperature. The flux of the two fractions has opposite dependence on temperature, while the total flux sums up to one flux quantum despite their spatial separation. Overall, our study shows the existence of different fractional vortices and their stability in temperature ranging from 0.1 to 0.99 $T_C$. Besides the implications of this observation for the fundamental question of quantum vorticity, the discovery of these objects paves the way for the new platform for anyon quasiparticles and applications for fractional fluxonics.

Ring seismology has recently revealed the presence of internal gravity waves inside Saturn that extend up to 60% of Saturn's radius starting from the center, in what is recognized today as Saturn's stably-stratified dilute core. Similarly, gravity measurements on Jupiter suggest the existence of a dilute core of still poorly constrained radial extent. These cores are likely in a double-diffusive regime, which prompt the question of their long-term stability. Indeed, previous DNS (Direct Numerical Simulations) studies in triply-periodic domains have shown that, in some regimes, double-diffusive convection tends to spontaneously form shallow convective layers, which coarsen until the region becomes fully convective. In this letter, we study the conditions for layering in double-diffusive convection using different boundary conditions, in which temperature and composition fluxes are fixed at the domain boundaries. We run a suite of DNS varying microscopic diffusivities of the fluid and the strength of the initial stratification. We find that convective layers still form as a result of the previously discovered gamma-instability which takes place whenever the local stratification drops below a critical threshold that only depends on the fluid diffusivities. We also find that the layers grow once formed, eventually occupying the entire domain. Our work thus recovers the results of previous studies, despite the new boundary conditions, suggesting that this behavior is universal. The existence of Saturn's stably-stratified core, today, therefore suggests that this threshold has never been reached, which places a new constraint on scenarios for the planet's formation and evolution., Comment: submitted to The Journal of Planetary Science

Recent theoretical progress using multiscale asymptotic analysis has revealed various possible regimes of stratified turbulence. Notably, buoyancy transport can either be dominated by advection or diffusion, depending on the effective P\'eclet number of the flow. Two types of asymptotic models have been proposed, which yield measurably different predictions for the characteristic vertical velocity and length scale of the turbulent eddies in both diffusive and non-diffusive regimes. The first, termed a `single-scale model', is designed to describe flow structures having large horizontal and small vertical scales, while the second, termed a `multiscale model', additionally incorporates flow features with small horizontal scales, and reduces to the single-scale model in their absence. By comparing predicted vertical velocity scaling laws with direct numerical simulation data, we show that the multiscale model correctly captures the properties of strongly stratified turbulence within regions dominated by small-scale isotropic motions, whose volume fraction decreases as the stratification increases. Meanwhile its single-scale reduction accurately describes the more orderly, layer-like, quiescent flow outside those regions., Comment: Resubmitted to JFM Rapids after referee review

Quantifying transport by strongly stratified turbulence in low Prandtl number ($Pr$) fluids is critically important for the development of better models for the structure and evolution of stellar and planetary interiors. Motivated by recent numerical simulations showing strongly anisotropic flows suggestive of scale-separated dynamics, we perform a multiscale asymptotic analysis of the governing equations. We find that, in all cases, the resulting slow-fast systems take a quasilinear form. Our analysis also reveals the existence of several distinct dynamical regimes depending on the emergent buoyancy Reynolds and P\'eclet numbers, $Re_b = \alpha^2 Re$ and $Pe_b = Pr Re_b$, respectively, where $\alpha$ is the aspect ratio of the large-scale turbulent flow structures, and $Re$ is the outer scale Reynolds number. Scaling relationships relating the aspect ratio, the characteristic vertical velocity, and the strength of the stratification (measured by the Froude number $Fr$) naturally emerge from the analysis. When $Pe_b \ll \alpha$, the dynamics at all scales is dominated by buoyancy diffusion, and our results recover the scaling laws empirically obtained from direct numerical simulations by Cope et al. (2020). For $Pe_b \ge O(1)$, diffusion is negligible (or at least subdominant) at all scales and our results are consistent with those of Chini et al. (2022) for strongly stratified geophysical turbulence at $Pr = O(1)$. Finally, we have identified a new regime for $\alpha \ll Pe_b \ll 1$, in which slow, large scales are diffusive while fast, small scales are not. We conclude by presenting a map of parameter space that clearly indicates the transitions between isotropic turbulence, non-diffusive stratified turbulence, diffusive stratified turbulence and viscously-dominated flows, and by proposing parameterisations of the buoyancy flux, mixing efficiency and turbulent diffusion coefficient., Comment: 28 pages, 1 figure

Shear instabilities can be the source of significant amounts of turbulent mixing in stellar radiative zones. Past attempts at modeling their effects (either theoretically or using numerical simulations) have focused on idealized geometries where the shear is either purely vertical or purely horizontal. In stars, however, the shear can have arbitrary directions with respect to gravity. In this work, we use direct numerical simulations to investigate the nonlinear saturation of shear instabilities in a stably stratified fluid, where the shear is sinusoidal in the horizontal direction, and either constant or sinusoidal in the vertical direction. We find that, in the parameter regime studied here (non-diffusive, fully turbulent flow), the mean vertical shear does not play any role in controlling the dynamics of the resulting turbulence unless its Richardson number is smaller than one (approximately). As most stellar radiative regions have a Richardson number much greater than one, our result implies that the vertical shear can essentially be ignored in the computation of the vertical mixing coefficient associated with shear instabilities for the purpose of stellar evolution calculations, even when it is much larger than the horizontal shear (as in the solar tachocline, for instance)., Comment: 26 pages, 8 figures, resubmitted to ApJ

Fingering convection (also known as thermohaline convection) is a process that drives the vertical transport of chemical elements in regions of stellar radiative zones where the mean molecular weight increases with radius. Recently, Harrington & Garaud (2019) used three-dimensional direct numerical simulations to show that a vertical magnetic field can dramatically enhance the rate of chemical mixing by fingering convection. Furthermore, they proposed a so-called "parasitic saturation" theory to model this process. Here, we test their model over a broad range of parameter space, using a suite of direct numerical simulations of magnetized fingering convection varying the magnetic Prandtl number, magnetic field strength, and composition gradient. We find that the rate of chemical mixing measured in the simulations is not always predicted accurately by their existing model, in particular when the magnetic diffusivity is large. We then present an extension of the Harrington & Garaud (2019) model which resolves this issue. When applied to stellar parameters, it recovers the results of Harrington & Garaud (2019) except in the limit where fingering convection becomes marginally stable, where the new model is preferred. We discuss the implications of our findings for stellar structure and evolution., Comment: 18 pages, 7 figures, accepted for publication in ApJ; v2 revisions: updates sec 5 & 6 to describe new, successful model; v3: primarily condenses/streamlines text, fixed minor issue in one figure

We demonstrate that superconductors with broken inversion symmetry support a family of stable, spatially localized configurations of the self-knotted magnetic field. These solutions, that we term ``toroflux,'' are the superconducting counterparts of the Chandrasekhar-Kendall states (spheromaks) that appear in highly conducting, force-free astrophysical and nuclear-fusion plasmas. The superconducting torofluxes are solutions of superconducting models, in the presence of a parity-breaking Lifshitz invariant associated with the $O$ point-group symmetry. These solutions are characterized by a non-vanishing helicity of the magnetic field, and also by a toroidal dipole moment of the magnetic field. We demonstrate that a magnetic dipole or a ferromagnetic inclusion in the bulk of a noncentrosymmetric superconductor sources finite-energy toroflux solutions., Comment: Added discussion about additional observables; Replaced with a version in print in Phys. Rev. B; 18 pages, 5 figures

We study the properties of oscillatory double-diffusive convection (ODDC) in the presence of a uniform vertical background magnetic field. ODDC takes place in stellar regions that are unstable according to the Schwarzschild criterion and stable according to the Ledoux criterion (sometimes called semiconvective regions), which are often predicted to reside just outside the core of intermediate-mass main sequence stars. Previous hydrodynamic studies of ODDC have shown that the basic instability saturates into a state of weak wave-like convection, but that a secondary instability can sometimes transform it into a state of layered convection, where layers then rapidly merge and grow until the entire region is fully convective. We find that magnetized ODDC has very similar properties overall, with some important quantitative differences. A linear stability analysis reveals that the fastest-growing modes are unaffected by the field, but that other modes are. Numerically, the magnetic field is seen to influence the saturation of the basic instability, overall reducing the turbulent fluxes of temperature and composition. This in turn affects layer formation, usually delaying it, and occasionally suppressing it entirely for sufficiently strong fields. Further work will be needed, however, to determine the field strength above which layer formation is actually suppressed in stars. Potential observational implications are briefly discussed., Comment: 24 pages, single column, 6 figures, accepted for publication in ApJ

We investigate the linear stability of a sinusoidal shear flow with an initially uniform streamwise magnetic field in the framework of incompressible magnetohydrodynamics (MHD) with finite resistivity and viscosity. This flow is known to be unstable to the Kelvin-Helmholtz instability in the hydrodynamic case. The same is true in ideal MHD, where dissipation is neglected, provided the magnetic field strength does not exceed a critical threshold beyond which magnetic tension stabilizes the flow. Here, we demonstrate that including viscosity and resistivity introduces two new modes of instability. One of these modes, which we call a resistively-unstable Alfv\'en wave due to its connection to shear Alfv\'en waves, exists for any nonzero magnetic field strength as long as the magnetic Prandtl number $Pm < 1$. We present a reduced model for this instability that reveals its excitation mechanism to be the negative eddy viscosity of periodic shear flows described by Dubrulle & Frisch (1991). Finally, we demonstrate numerically that this mode saturates in a quasi-stationary state dominated by counter-propagating solitons., Comment: 23 pages, 10 figures, submitted to JFM

In recent years, there were a number of experimental developments and discoveries of novel superconducting materials which exhibit multicomponent, many-body degrees of freedom. These superconductors, that are described by several superconducting condensates, feature many new interesting phenomena that are absent in their single-component counterparts. Several of these new aspects of multicomponent superconductivity are addressed in this report., Comment: 159 pages; Dissertation (M\'emoire d'Habilitation \`a Diriger des Recherches); In English, including a summary in French

Recent experiments [V.~Grinenko {\it et al.} {Nat. Phys. {\bf 17}, 1254 (2021)}; \url{http://doi.org/10.1038/s41567-021-01350-9}] reported the observation of a condensate of four-fermion composites. This is a resistive state that spontaneously breaks the time-reversal symmetry, leading to unconventional magnetic properties, detected in muon spin rotation experiments and by the appearance of a spontaneous Nernst effect. In this work, we derive an effective model for the four-fermion order parameter that describes the observed spontaneous magnetic fields in this state. We show that this model, which is alike to the Faddeev-Skyrme model can host skyrmions: magnetic-flux-carrying topological excitations., Comment: 17 pages, 6 figures; Additional results

A dilute gas of Bose-Einstein condensed atoms in a non-rotated and axially symmetric harmonic trap is modelled by the time dependent Gross-Pitaevskii equation. When the angular momentum carried by the condensate does not vanish, the minimum energy state describes vortices (or antivortices) that propagate around the trap center. The number of (anti)vortices increases with the angular momentum, and they repel each other to form Abrikosov lattices. Besides vortices and antivortices there are also stagnation points where the superflow vanishes; to our knowledge the stagnation points have not been analyzed previously, in the context of the Gross-Pitaevskii equation. The Poincar\'e index formula states that the difference in the number of vortices and stagnation points can never change. When the number of stagnation points is small, they tend to aggregate into degenerate propagating structures. But when the number becomes sufficiently large, the stagnation points tend to pair up with the vortex cores, to propagate around the trap center in regular lattice arrangements. There is an analogy with the geometry of the Kosterlitz-Thouless transition, with the angular momentum of the condensate as the external control parameter instead of the temperature., Comment: 18 pages, 4 figures, 2 figure tables; Discussion of analogy with the Kosterlitz-Thouless transition; Additionnal discussion of the fixed point structures (animations are available in ancillary files)

Zahn's widely-used model for turbulent mixing induced by rotational shear has recently been validated (with some caveats) in non-rotating shear flows. It is not clear, however, whether his model remains valid in the presence of rotation, even though this was its original purpose. Furthermore, new instabilities arise in rotating fluids, such as the Goldreich-Schubert-Fricke (GSF) instability. Which instability dominates when more than one can be excited, and how they influence each other, were open questions that this paper answers. To do so, we use direct numerical simulations of diffusive stratified shear flows in a rotating triply-periodic Cartesian domain located at the equator of a star. We find that either the GSF instability or the shear instability tends to take over the other in controlling the system, suggesting that stellar evolution models only need to have a mixing prescription for each individual instability, together with a criterion to determine which one dominates. However, we also find that it is not always easy to predict which instability "wins" for given input parameters, because the diffusive shear instability is subcritical, and only takes place if there is a finite-amplitude turbulence ``primer'' to seed it. Interestingly, we find that the GSF instability can in some cases play the role of this primer, thereby providing a pathway to excite the subcritical shear instability. This can also drive relaxation oscillations, that may be observable. We conclude by proposing a new model for mixing in the equatorial regions of stellar radiative zones due to differential rotation., Comment: 19 pages, 11 figures

Discoveries of ordered quantum states of matter are of great fundamental interest, and often lead to unique applications. The most well known example -- superconductivity -- is caused by the formation and condensation of pairs of electrons. A key property of superconductors is diamagnetism: magnetic fields are screened by dissipationless currents. Fundamentally, what distinguishes superconducting states from normal states is a spontaneously broken symmetry corresponding to long-range coherence of fermion pairs. Here we report a set of experimental observations in hole doped Ba$_{1-x}$K$_x$Fe$_2$As$_2$ which are not consistent with conventional superconducting behavior. Our specific-heat measurements indicate the formation of fermionic bound states when the temperature is lowered from the normal state. However, for $x \sim 0.8$, instead of the standard for superconductors, zero resistance and diamagnetic screening, for a range of temperatures, we observe the opposite effect: the generation of self-induced magnetic fields measured by spontaneous Nernst effect and muon spin rotation experiments. The finite resistance and the lack of any detectable diamagnetic screening in this state exclude the spontaneously broken symmetry associated with superconducting two-fermion correlations. Instead, combined evidence from transport and thermodynamic measurements indicates that the formation of fermionic bound states leads to spontaneous breaking of time-reversal symmetry above the superconducting transition temperature. These results demonstrate the existence of a broken-time-reversal-symmetry bosonic metal state. In the framework of a multiband theory, such a state is characterized by quartic correlations: the long-range order exists only for {\it pairs} of fermion pairs., Comment: 31 pages, 11 figures, supplemental information

The dynamics of fluids deep in stellar interiors is a subject that bears many similarities with geophysical fluid dynamics, with one crucial difference: the Prandtl number. The ratio of the kinematic viscosity to the thermal diffusivity is usually of order unity or more on Earth, but is always much smaller than one in stars. As a result, viscosity remains negligible on scales that are thermally diffusive, which opens the door to a whole new region of parameter space, namely the turbulent low P\'eclet number regime (where the P\'eclet number is the product of the Prandtl number and the Reynolds number). In this review, I focus on three instabilities that are well known in geophysical fluid dynamics, and have an important role to play in stellar evolution, namely convection, stratified shear instabilities, and double-diffusive convection. I present what is known of their behavior at low Prandtl number, highlighting the differences with their moderate and high Prandtl number counterparts., Comment: Invited review published in Physical Review Fluids

Preferential concentration is thought to play a key role in promoting particle growth, which is crucial to processes such as warm rain formation in clouds, planet formation, and industrial sprays. In this work, we investigate preferential concentration using 3D Direct Numerical Simulations adopting the Eulerian-Eulerian two-fluid approach, where the particles are treated as a continuum field with its own momentum and mass conservation laws. We consider particles with Stokes number $St \lesssim O(0.01)$ in moderately turbulent flows with fluid Reynolds number $Re \leq 600$. In our previous work (Nasab & Garaud, Physical Review Fluids. doi: 10.1103/PhysRevFluids.5.114308, 2020), we established scaling laws to predict maximum and typical particle concentration enhancements in the context of the particle-driven convective instability. Here we verify that the same results apply when turbulence is externally driven, extending the relevance of our model to a wider class of particle-laden flows. We find in particular that (i) the maximum particle concentration enhancement above the mean scales as $u_{rms}^2 \tau_p / \kappa_p$, where $u_{rms}$ is the rms fluid velocity, $\tau_p$ is the particle stopping time, and $\kappa_p$ is the assumed particle diffusivity from the two-fluid equations; (ii) the typical particle concentration enhancement over the mean scales as $(u_{rms}^2 \tau_p / \kappa_p)^{1/2}$; and (iii) the probability distribution function of the particle concentration enhancement over the mean has an exponential tail whose slope scales like $(u_{rms}^2 \tau_p /\kappa_p)^{-1/2}$. We conclude by discussing the caveats of our model and its implications in a relevant cloud application., Comment: 28 pages, 12 figures

The past 20 years have witnessed a renewal of interest in the subject of double-diffusive processes in astrophysics, and their impact on stellar evolution. This lecture aims to summarize the state of the field as of early 2019, although the reader should bear in mind that it is rapidly evolving. An Annual Review of Fluid Mechanics article entitled "Double-diffusive convection at low Prandtl number" (Garaud, 2018) contains a reasonably comprehensive review of the topic, up to the summer of 2017. I focus here on presenting what I hope are clear derivations of some of the most important results with an astrophysical audience in mind, and discuss their implications for stellar evolution, both in an observational context, and in relation to previous work on the subject., Comment: Proceedings of the summer school "Multi-Dimensional Processes In Stellar Physics". Edited by Michel Rieutord, Isabelle Baraffe and Yveline Lebreton

Vortices in a Bose-Einstein condensate are modelled as spontaneously symmetry breaking minimum energy solutions of the time dependent Gross-Pitaevskii equation, using the method of constrained optimization. In a non-rotating axially symmetric trap, the core of a single vortex precesses around the trap center and, at the same time, the phase of its wave function shifts at a constant rate. The precession velocity, the speed of phase shift, and the distance between the vortex core and the trap center, depend continuously on the value of the conserved angular momentum that is carried by the entire condensate. In the case of a symmetric pair of identical vortices, the precession engages an emergent gauge field in their relative coordinate, with a flux that is equal to the ratio between the precession and shift velocities., Comment: improved version; Replaced with a version in print in JHEP; Supplemental Material as ancillary file; 21 pages, 5 figures

Motivated by the dynamics in the deep interiors of many stars, we study the interaction between overshooting convection and the large-scale poloidal fields residing in radiative zones. We have run a suite of 3D Boussinesq numerical calculations in a spherical shell that consists of a convection zone with an underlying stable region that initially compactly contains a dipole field. By varying the strength of the convective driving, we find that, in the less turbulent regime, convection acts as turbulent diffusion that removes the field faster than solely molecular diffusion would do. However, in the more turbulent regime, turbulent pumping becomes more efficient and partially counteracts turbulent diffusion, leading to a local accumulation of the field below the overshoot region. These simulations suggest that dipole fields might be confined in underlying stable regions by highly turbulent convective motions at stellar parameters. The confinement is of large-scale field in an average sense and we show that it is reasonably modeled by mean-field ideas. Our findings are particularly interesting for certain models of the Sun, which require a large-scale, poloidal magnetic field to be confined in the solar radiative zone in order to explain simultaneously the uniform rotation of the latter and the thinness of the solar tachocline., Comment: Accepted to MNRAS, 14 figures

We use large-scale Monte-Carlo simulations to study thermal fluctuations in chiral $p$-wave superconductors in an applied magnetic field in three dimensions. We consider the thermal stability of previously predicted unusual double-quanta flux-line lattice ground states in such superconductors. In previous works it was shown that, neglecting thermal fluctuations, a chiral $p$-wave superconductor forms an hexagonal lattice of doubly-quantized vortices, except extremely close to the vicinity of $H_{c2}$ where double-quanta vortices split apart. We find dissociation of double-quanta vortices driven by thermal fluctuations. However, our calculations also show that the previous predictions of hexagonal doubly-quantized vortices, where thermal fluctuations were ignored, are very robust in the considered model., Comment: 21 pages, 12 figures, submitted to Physical Review B

Turbulent mixing in the radiative regions of stars is usually either ignored or crudely accounted for in most stellar evolution models. However, there is growing theoretical and observational evidence that such mixing is present and can affect various aspects of a star's life. In this work, we present a first attempt at quantifying mixing by horizontal shear instabilities in stars using Direct Numerical Simulations. The shear is driven by a body force, and rapidly becomes unstable. At saturation, we find that several distinct dynamical regimes exist, depending on the relative importance of stratification and thermal diffusion (viscosity can in principle also matter, but is usually negligible in most stellar applications). In each of the regimes identified, we put forward a certain number of theoretically motivated scaling laws for the turbulent vertical eddy scale, the typical turbulent diffusion coefficient, and the typical amplitude of temperature fluctuations (among other quantities). Based on our findings, we predict that the majority of stars should fall into one of two categories: high P\'eclet number stratified turbulence, and low P\'eclet number stratified turbulence. The latter is presented in detail in a related paper by Cope et al. (2020), while the former is discussed here. Applying our results to the best-known stellar shear layer, namely the solar tachocline, we find that it should lie in the high P\'eclet number stratified turbulence regime, and predict a substantial amount of vertical mixing for temperature, momentum and composition. Taken as is, the new turbulence model predictions are incompatible with the Spiegel & Zahn (1992) model of the solar tachocline. However, we also show that rotation and magnetic fields are likely to affect the turbulence, and need to be taken into account in future studies., Comment: Accepted for publication in the ApJ

The solar tachocline is a shear layer located at the base of the solar convection zone. The horizontal shear in the tachocline is likely turbulent, and it is often assumed that this turbulence would be strongly anisotropic as a result of the local stratification. What role this turbulence plays in the tachocline dynamics, however, remains to be determined. In particular, it is not clear whether it would result in a turbulent eddy diffusivity, or anti-diffusivity, or something else entirely. In this paper, we present the first direct numerical simulations of turbulence in horizontal shear flows at low Prandtl number, in an idealized model that ignores rotation and magnetic fields. We find that several regimes exist, depending on the relative importance of the stratification, viscosity and thermal diffusivity. Our results suggest that the tachocline is in the stratified turbulence regime, which has very specific properties controlled by a balance between buoyancy, inertia, and thermal diffusion., Comment: Invited review for the meeting Dynamics of the Sun and Stars: Honoring the Life and Work of Michael J. Thompson (Boulder, Colorado, 24-26 September 2019)

Superconducting materials with noncentrosymmetric lattices lacking space inversion symmetry exhibit a variety of interesting parity-breaking phenomena, including the magneto-electric effect, spin-polarized currents, helical states, and the unusual Josephson effect. We demonstrate, within a Ginzburg-Landau framework describing noncentrosymmetric superconductors with $O$ point group symmetry, that vortices can exhibit an inversion of the magnetic field at a certain distance from the vortex core. In stark contrast to conventional superconducting vortices, the magnetic-field reversal in the parity-broken superconductor leads to non-monotonic intervortex forces, and, as a consequence, to the exotic properties of the vortex matter such as the formation of vortex bound states, vortex clusters, and the appearance of metastable vortex/anti-vortex bound states., Comment: Added analytic expressions of the physical quantities in the London limit; Replaced with a version in print in Phys. Rev. B; 12 pages, 4 figures

Heavy particles in turbulent flows have been shown to accumulate in regions of high strain rate or low vorticity, a process otherwise known as preferential concentration. This can be observed in geophysical flows, and is inferred to occur in astrophysical environments, often resulting in rapid particle growth which is critical to physical processes such as rain or planet formation. Here we study the effects of preferential concentration in a two-way coupled system in the context of the particle-driven convective instability. To do so, we use Direct Numerical Simulations and adopt the two-fluid approximation. We focus on a particle size range for which the latter is valid, namely when the Stokes number is $\lesssim O(0.1)$. For Stokes number above $\sim 0.01$, we find that the maximum particle concentration enhancement over the mean scales with the rms fluid velocity $u_{\rm{rms}}$, the particle stopping time $\tau_p$, and the particle diffusivity $\kappa_p$, as $u_{\rm{rms}}^2 \tau_p / \kappa_p$. We show that this scaling can be understood from simple arguments of dominant balance. We also show that the typical particle concentration enhancement over the mean scales as $(u_{\rm{rms}}^2 \tau_p/ \kappa_p)^{1/2}$. We finally find that the probability distribution function of the particle concentration enhancement over the mean has an exponential tail whose slope scales as $(u_{\rm{rms}}^2 \tau_p / \kappa_p)^{-1/2}$. We apply our model to geophysical and astrophysical examples, and discuss its limitations., Comment: Paper submitted to Physical Review Fluids

We consider the dynamics of a vertically stratified, horizontally-forced Kolmogorov flow. Motivated by astrophysical systems where the Prandtl number is often asymptotically small, our focus is the little-studied limit of high Reynolds number but low P\'eclet number (which is defined to be the product of the Reynolds number and the Prandtl number). Through a linear stability analysis, we demonstrate that the stability of two-dimensional modes to infinitesimal perturbations is independent of the stratification, whilst three-dimensional modes are always unstable in the limit of strong stratification and strong thermal diffusion. The subsequent nonlinear evolution and transition to turbulence is studied numerically using direct numerical simulations. For sufficiently large Reynolds numbers, four distinct dynamical regimes naturally emerge, depending upon the strength of the background stratification. By considering dominant balances in the governing equations, we derive scaling laws for each regime which explain the numerical data., Comment: 37 pages, 9 figures

Superconducting materials with non-centrosymmetric lattices lacking the space inversion symmetry are known to exhibit a variety of interesting parity-breaking phenomena, including the anomalous Josephson effect. Here we consider a Josephson junction consisting of two non-centrosymmetric superconductors (NCSs) connected by a uniaxial ferromagnet, and demonstrate that it exhibits a direct analog of the Chiral Magnetic Effect observed in Dirac and Weyl semimetals. We propose to use this "Chiral Magnetic Josephson junction" (CMJ junction) as an element of a qubit with a Hamiltonian tunable by the ferromagnet's magnetization. The CMJ junction allows to avoid the use of an offset magnetic flux in inductively shunted qubits, thus enabling a simpler and more robust architecture. The resulting"`chiral magnetic qubit" is protected from the noise caused by fluctuations in magnetization when the easy axis of the uniaxial ferromagnet is directed across the junction., Comment: 7 pages, 3 figures

Fingering convection is a turbulent mixing process that can occur in stellar radiative regions whenever the mean molecular weight increases with radius. In some cases, it can have a significant observable impact on stellar structure and evolution. The efficiency of mixing by fingering convection as a standalone process has been studied by Brown et al. (2013), but other processes such as rotation, magnetic fields and shear can affect it. In this paper, we present a first study of the effect of shear on fingering (thermohaline) convection in astrophysics. Using Direct Numerical Simulations we find that a moderate amount of shear (that is not intrinsically shear-unstable) always decreases the mixing efficiency of fingering convection, as a result of the tilt it imparts to the fingering structures. We propose a simple analytical extension of the Brown et al. (2013) model in the presence of shear that satisfactorily explains the numerically-derived turbulent compositional mixing coefficient for moderate shearing rates, and can trivially be implemented in stellar evolution codes. We also measure from the numerical simulations a turbulent viscosity, and find that the latter is strongly tied to the turbulent compositional mixing coefficient. Observational implications and caveats of the model are discussed., Comment: Submitted to ApJ

Double-diffusive convection has been well studied in geophysical contexts, but detailed investigations of the regimes characteristic of stellar or planetary interiors have only recently become feasible. Since most astrophysical fluids are electrically conducting, it is possible that magnetic fields play a role in either enhancing or suppressing double-diffusive convection, but to date there have been no numerical investigations of such possibilities. Here we study the effects of a vertical background magnetic field (aligned with the gravitational axis) on the linear stability and nonlinear saturation of fingering (thermohaline) convection, through a combination of theoretical work and direct numerical simulations (DNSs). We find that a vertical magnetic field rigidifies the fingers along the vertical direction which has the remarkable effect of enhancing vertical mixing. We propose a simple analytical model for mixing by magnetized fingering convection, and argue that magnetic effects may help explain discrepancies between theoretical and observed mixing rates in low-mass red giant branch (RGB) stars. Other implications of our findings are also discussed., Comment: 8 pages, 4 figures

Convective overshooting in super asymptotic giant branch stars has been suggested to lead to the formation of hybrid white dwarfs with carbon-oxygen cores and oxygen-neon mantles. As the white dwarf cools, this core-mantle configuration becomes convectively unstable and should mix. This mixing has been previously studied using stellar evolution calculations, but these made the approximation that convection did not affect the temperature profile of the mixed region. In this work, we perform direct numerical simulations of an idealized problem representing the core-mantle interface of the hybrid white dwarf. We demonstrate that, while the resulting structure within the convection zone is somewhat different than what is assumed in the stellar evolution calculations, the two approaches yield similar results for the size and growth of the mixed region. These hybrid white dwarfs have been invoked as progenitors of various peculiar thermonuclear supernovae. This lends further support to the idea that if these hybrid white dwarfs form then they should be fully mixed by the time of explosion. These effects should be included in the progenitor evolution in order to more accurately characterize the signatures of these events., Comment: 12 pages, 7 figures; Accepted to ApJ

We study the dynamics associated with the extension of turbulent convective motions from a convection zone (CZ) into a stable region (RZ) that lies below the latter. For that purpose, we have run a series of three-dimensional direct numerical simulations solving the Navier-Stokes equations under the Boussinesq approximation in a spherical shell geometry. We observe that the overshooting of the turbulent motions into the stably stratified region depends on three different parameters: the relative stability of the RZ, the transition width between the two, and the intensity of the turbulence. In the cases studied, these motions manage to partially alter the thermal stratification and induce thermal mixing, but not so efficiently as to extend the nominal CZ further down into the stable region. We find that the kinetic energy below the convection zone can be modeled by a half-Gaussian profile whose amplitude and width can be predicted a priori for all of our simulations. We examine different dynamical lengthscales related to the depth of the extension of the motions into the RZ, and we find that they all scale remarkably well with a lengthscale that stems from a simple energetic argument. We discuss the implications of our findings for 1D stellar evolution calculations.

The failed "vortex-atoms" theory of matter by Kelvin and Tait had a profound impact on mathematics and physics. Building on the understanding of vorticity by Helmholtz, and observing stability of smoke rings, they hypothesised that elementary particles (at that time atoms) are indestructible knotted vortices in luminiferous aether: the hypothetical ideal fluid filling the universe. The vortex-atoms theory identified chemical elements as topologically different vortex knots, and matter was interpreted as bound states of these knotted vortices. This work initiated the field of knot theory in mathematics. It also influenced modern physics, where a close although incomplete analogy exists with the theory of superfluidity, which started with Onsager's and Feynman's introduction of quantum vortices. Indeed many macroscopic properties of superconductors and superfluids are indeed determined by vortex lines forming different "aggregate states", such as vortex crystals and liquids. While crucial importance of knots was understood for many physical systems in the recent years, there is no known physical realization of the central element of Kelvin theory: the stable particle-like vortex knot. Indeed, vortex loops and knots in superfluids and ordinary superconductors form as dynamical excitations and are unstable by Derrick theorem. This instability in fact dictates many of the universal macroscopic properties of superfluids. Here we show that there are superconducting states with principally different properties of the vorticity: where vortex knots are intrinsically stable. We demonstrate that such features should be realised near certain critical points, where the hydro-magneto-statics of superconducting states yields stables vortex knots which behave similar to those envisaged in Kelvin and Tait's theory of vortex-atoms in luminiferous aether.

Spin-orbit interaction (SOI) plays a key role in creating Majorana zero modes in semiconductor nanowires proximity coupled to a superconductor. We track the evolution of the induced superconducting gap in InSb nanowires coupled to a NbTiN superconductor in a large range of magnetic field strengths and orientations. Based on realistic simulations of our devices, we reveal SOI with a strength of 0.15-0.35 eV$\require{mediawiki-texvc}\AA$. Our approach identifies the direction of the spin-orbit field, which is strongly affected by the superconductor geometry and electrostatic gates., Comment: 7 pages, 4 figures. Supplemental Material as ancillary file

We study the effects of rotation on the growth and saturation of the double-diffusive fingering (thermohaline) instability at low Prandtl number. Using direct numerical simulations, we estimate the compositional transport rates as a function of the relevant non-dimensional parameters - the Rossby number, inversely proportional to the rotation rate, and the density ratio which measures the relative thermal and compositional stratifications. Within our explored range of parameters, we generally find rotation to have little effect on vertical transport. However, we also present one exceptional case where a cyclonic large scale vortex (LSV) is observed at low density ratio and fairly low Rossby number. The LSV leads to significant enhancement in the fingering transport rates by concentrating compositionally dense downflows at its core. We argue that the formation of such LSVs could be relevant to solving the missing mixing problem in RGB stars., Comment: 16pages, 20 figures, submitted to ApJ

Being able to account for the missing mixing in stellar radiative zones is a key step toward a better understanding of stellar evolution. Zahn (1974) argued that thermally diffusive shear-induced turbulence might be responsible for some of this mixing. In Part I and Part II of this series of papers we showed that Zahn's (1974, 1992) mixing model applies when the properties of the turbulence are local. But we also discovered limitations of the model when this locality condition fails, in particular near the edge of a turbulent region. In this paper, we propose a second-order closure model for the transport of momentum and chemical species by shear-induced turbulence in strongly stratified, thermally diffusive environments (the so-called low P\'eclet number limit), which builds upon the work of Garaud \& Ogilvie (2005). Comparison against direct numerical simulations (DNSs) shows that the model is able to predict the vertical profiles of the mean flow and of the stress tensor (including the momentum transport) in diffusive shear flows, often with a reasonably good precision, and at least within a factor of order unity in the worst case scenario. The model is sufficiently simple to be implemented in stellar evolution codes, and all the model constants have been calibrated against DNSs. While significant limitations to its use remain (e.g. it can only be used in the low P\'eclet number, slowly rotating limit), we argue that it is more reliable than most of the astrophysical prescriptions that are used in stellar evolution models today.

This paper continues the systematic investigation of diffusive shear instabilities initiated in Part I of this series. In this work, we primarily focus on quantifying the impact of non-local mixing, which is not taken into account in Zahn's mixing model \citep{Zahn92}. We present the results of direct numerical simulations in a new model setup designed to contain coexisting laminar and turbulent shear layers. As in Part I, we use the Low P\'eclet Number approximation of \citet{Lign1999} to model the evolution of the perturbations. Our main findings are twofold. First, turbulence is not necessarily generated whenever Zahn's nonlinear criterion \citep{Zahn1974} $J{\rm Pr} < (J{\rm Pr})_c$ is satisfied, where $J=N^2/S^2$ is the local gradient Richardson number, ${\rm Pr} = \nu/ \kappa_T$ is the Prandtl number, and $(J{\rm Pr})_c \simeq 0.007$. We have demonstrated that the presence or absence of turbulent mixing in this limit hysteretically depends on the history of the shear layer. Second, Zahn's nonlinear instability criterion only approximately locates the edge of the turbulent layer, and mixing beyond the region where $J{\rm Pr} < (J{\rm Pr})_c$ can also take place in a manner analogous to convective overshoot. We found that the turbulent kinetic energy decays roughly exponentially beyond the edge of the shear-unstable region, on a lengthscale $\delta$ that is directly proportional to the scale of the turbulent eddies, which are themselves of the order of the Zahn scale (see Part I). Our results suggest that mixing by diffusive shear instabilities should be modeled with more care than is currently standard in stellar evolution codes., Comment: Submitted to ApJ

Disorder in two-band superconductors with repulsive interband interaction induces a frustrated competition between the phase-locking preferences of the various potential and kinetic terms. This frustrated interaction can result in the formation of an $s+is$ superconducting state, that breaks the time-reversal symmetry. In this paper we study the normal modes and their associated coherence lengths in such materials. We especially focus on the consequences of the soft modes stemming from the frustration and time-reversal-symmetry breakdown. We find that two-bands superconductors with such impurity-induced frustrated interactions display a rich spectrum of physical properties that are absent in their clean counterparts. It features a mixing of Leggett's and Anderson-Higgs modes, and a soft mode with diverging coherence length at the impurity-induced second order phase transition from $s_{\pm}/s_{++}$ states to the $s+is$ state. Such a soft mode generically results in long-range attractive intervortex forces that can trigger the formation of vortex clusters. We find that, if such clusters are formed, their size and internal flux density have a characteristic temperature dependence that could be probed in muon-spin-rotation experiments. We also comment on the appearance of spontaneous magnetic fields due to spatially varying impurities., Comment: Added discussion of spontaneous magnetic fields due to spatially varying impurities; Replaced with a version in print in Phys. Rev. B; 17 pages, 8 figures

In multiband systems, such as iron-based superconductors, the superconducting states with locking and anti-locking of the interband phase differences, are usually considered as mutually exclusive. For example, a dirty two-band system with interband impurity scattering undergoes a sharp crossover between the $s_{\pm}$ state (which favors phase anti locking) and the $s_{++}$ state (which favors phase locking). We discuss here that the situation can be much more complex in the presence of an external field or superconducting currents. In an external applied magnetic field, dirty two-band superconductors do not feature a sharp $s_{\pm}\to s_{++}$ crossover but rather a washed-out crossover to a finite region in the parameter space where both $s_{\pm}$ and $s_{++}$ states can coexist for example as a lattice or a microemulsion of inclusions of different states. The current-carrying regions such as the regions near vortex cores can exhibit an $s_\pm$ state while it is the $s_{++}$ state that is favored in the bulk. This coexistence of both states can even be realized in the Meissner state at the domain's boundaries featuring Meissner currents. We demonstrate that there is a magnetic-field-driven crossover between the pure $s_{\pm}$ and the $s_{++}$ states., Comment: This is a follow-up, extedending arXiv:1707.06412; Replaced with a version in print in Phys. Rev. B; Minor changes; 8 pages, 5 figures

We present Direct Numerical Simulations of the transport of heat and heavy elements across a double-diffusive interface or a double-diffusive staircase, in conditions that are close to those one may expect to find near the boundary between the heavy-element rich core and the hydrogen-helium envelope of giant planets such as Jupiter. We find that the non-dimensional ratio of the buoyancy flux associated with heavy element transport to the buoyancy flux associated with heat transport lies roughly between 0.5 and 1, which is much larger than previous estimates derived by analogy with geophysical double-diffusive convection. Using these results in combination with a core-erosion model proposed by Guillot et al. (2004), we find that the entire core of Jupiter would be eroded within less than 1Myr assuming that the core-envelope boundary is composed of a single interface. We also propose an alternative model that is more appropriate in the presence of a well-established double-diffusive staircase, and find that in this limit a large fraction of the core could be preserved. These findings are interesting in the context of Juno's recent results, but call for further modeling efforts to better understand the process of core erosion from first principles., Comment: Accepted for publication in ApJ