Initial Vortex Research Articles

Vortex dynamics impact on the wake flow of a marine rudder with leading-edge tubercles

The impact of two tubercle leading-edge (TLE) modifications on the turbulent wake of a reference marine rudder at Reynolds number 2.26 × 106 was analysed numerically using Detached Eddy Simulations (DES). This paper studies the counter-rotating vortex pair formation around the TLE and their impact on the wake structures behind the rudder to find out if the vortex interaction can accelerate the tip vortex dissipation. According to the results, the tubercles enhanced lift for angles of attack (AOA) 10º and above, but at the cost of a drag penalty which reduced the rudders’ lift-to-drag ratio. The formation of the distinctive stream-wise counter-rotating vortex pairs occurred behind the tubercles, which then interacted with the dominant tip vortex. Due to the inherent spanwise flow component of finite-span lifting surfaces the counter-rotating vortex pairs were generated at unequal strength and soon merged into singular vortices co-rotating with the tip vortex. The vortices facilitated flow compartmentalisation over the rudder suction side which broke up the trailing-edge vortex sheet and confined the spanwise flow separation over the rudder surface as AOA increased. The tubercles confined flow separation closer to the rudder tip which reduced the lift generation in the tip area and minimised the initial tip vortex strength. Large elements of stream-wise counter-rotating vorticity formed around the localised stall cells of the TLE rudders that interacted with the tip vortex downstream, introducing elliptical instabilities further weakening the tip vortex and changing its trajectory.

On the Existence of Weak Solutions for the 2D Incompressible Euler Equations with In–Out Flow and Source and Sink Points

Well-posedness for the two dimensional Euler system with given initial vorticity is known since the works of Judovi\v{c}. In this paper we show existence of solutions in the case where we allowed the fluid to enter in and exit from the boundaries and from some points of the fluid domain. In particular we derive the equations of the model as the limit when we replace the points by some small holes. To do that we extend the DiPerna-Lions theory with non-tangent velocity field on the boundary to the case of time-dependent domain, we extend the existence result for the two dimensional Euler system with in-out flow to time-dependent domain and finally we derive the system that models a fluid which is allowed to enter in and exit from the boundary and some points. The solutions are characterized by the presence of source, sink and vortex points.

New numerical methods for calculating statistical equilibria of two-dimensional turbulent flows, strictly based on the Miller–Robert–Sommeria theory

New numerical methods are proposed for the mixing entropy maximization problem in the context of Miller–Robert–Sommeria’s (MRS) statistical mechanics theory of two-dimensional turbulence, particularly in the case of spherical geometry. Two of the methods are for the canonical problem; the other is for the microcanonical problem. The methods are based on the original MRS theory and thus take into account all Casimir invariants. Compared to the methods proposed in previous studies, our new methods make it easier to detect multiple statistical equilibria and to search for solutions with broken zonal symmetry. The methods are applied to a zonally symmetric initial vorticity distribution which is barotropically unstable. Two statistical equilibria are obtained, one of which has a wave-like structure with zonal wavenumber 1, and the other has a wave-like structure with zonal wavenumber 2. While the former is the maximum point of the mixing entropy, the wavenumber 2 structure of the latter is nearly the same as the structure that appears in the end state of the time integration of the vorticity equation. The new methods allow for efficient computation of statistical equilibria for initial vorticity distributions consisting of many levels of vorticity patches without losing information about all the conserved quantities. This means that the statistical equilibria can be obtained from an arbitrary initial vorticity distribution, which allows for the application of statistical mechanics to interpret a wide variety of flow patterns appearing in geophysical fluids.

Asymmetric vortex dynamics in two-dimensional Bose–Einstein condensate with harmonic trap potential

Based on the two-dimensional Gross–Pitaevskii equation model, we investigated the asymmetric vortex evolution of two-dimensional Bose–Einstein condensates in a harmonic potential trap with polar direction perturbation. This corresponds to the generation of an asymmetric vortex evolution mode under initial vortex light manipulation together with polar direction perturbation. Unlike most of the prior work on this topic that uses a pure numerical method for the system under study, we use an exact analytical method rather than numerical simulation to investigate the key features of the system evolution dynamics in this study. Based on the variational method, and for different system parameter settings including the strength of the harmonic oscillator potential and the nonlinear interaction, we derived two evolution modes, namely, the periodic evolution oscillation mode and the monotonic decay mode, and pictorially demonstrated the evolution patterns of the system. In addition, we investigated the scenario when the system is in the rotating state, which corresponds to the action of the quadratic centrifugal potential. Notably, we also identified the damping effects for the two modes with increasing angular velocity, whose upper limit value corresponds to the constant vortex pattern of the system. Our theoretical results can be used to guide the experimental investigation of asymmetric vortex evolution in two-dimensional Bose–Einstein condensates.

Multi‐Scale Kelvin‐Helmholtz Instability Dynamics Observed by PMC Turbo on 12 July 2018: 2. DNS Modeling of KHI Dynamics and PMC Responses

AbstractKjellstrand et al. (2022), https://10.1029/2021JD036232 describes the evolution and dynamics of a strong, large‐scale Kelvin‐Helmholtz instability (KHI) event observed in polar mesospheric clouds (PMCs) on 12 July 2018 by high‐resolution imagers aboard the PMC Turbulence (PMC Turbo) stratospheric long‐duration balloon experiment. The imaging provides evidence of KH billow interactions and instabilities that are strongly influenced by gravity waves at larger scales. Specific features include initially separated regions of KHI, secondary convective and KH instabilities of individual billows, and “tubes” and “knots” that arise where billow cores are mis‐aligned or discontinuous along their axes. This study describes a direct numerical simulation of KH billow interactions in a periodic domain seeded with random initial noise that enables excitation of multiple KH billows exhibiting variable phase structures that capture multiple features of the observed KHI dynamics. Variable KH billow phases along their axes yield initial vortex tubes having diagonal alignments that link adjacent, but mis‐aligned, billow cores. Weak initial vortex tubes and billow cores having nearly orthogonal alignments amplify, interact strongly, and drive intense vortex knots at these sites. These vortex tube and knot (T&K) dynamics excite “twist waves” that unravel the initial vortex tubes, and drive increasingly strong vortex interactions and a cascade of energy and enstrophy to successively smaller scales in the turbulence inertial range. The implications of T&K dynamics are much more rapid and intense breakdown and decay of the KH billows, and significantly enhanced energy dissipation rates, where these interactions occur.

Global well-posedness to the two-dimensional incompressible vorticity equation in the half plane

This paper is concerned with the global well-posedness of the two-dimensional incompressible vorticity equation in the half plane. Under the assumption that the initial vorticity ω0∈Wk,p(R+2) with k≥3 an integer and 1<p<2, it is shown that the two-dimensional incompressible vorticity equation admits a unique solution ω∈C([0,T];Wk,p(R+2)) for any T>0. An elementary and self-contained proof is presented and delicate estimates of the velocity and its derivatives are obtained in this paper. It should be emphasized that the uniform estimate on ∫0t‖∇u(τ)‖L∞(R+2)dτ is required to complete the global regularity of the solution. To do that, the double exponential growth in time of the gradient of the vorticity in the half plane is established and applied. This is different from the proof of global well-posedness of the Euler velocity equations in the Sobolev spaces, in which a Kato-type or logarithmic-type estimate of the gradient of the velocity is enough to close the energy estimates.

Uniqueness of Yudovich’s solutions to the 2D incompressible Euler equation despite the presence of sources and sinks

In 1962, Yudovich proved the existence and uniqueness of classical solutions to the 2D incompressible Euler equations in the case where the fluid occupies a bounded domain with entering and exiting flows on some parts of the boundary. The normal velocity is prescribed on the whole boundary, as well as the entering vorticity. The uniqueness part of Yudovich’s result holds for Hölder vorticity, by contrast with his 1961 result on the case of an impermeable boundary, for which the normal velocity is prescribed as zero on the boundary, and for which the assumption that the initial vorticity is bounded was shown to be sufficient to guarantee uniqueness. Whether or not uniqueness holds as well for bounded vorticities in the case of entering and exiting flows has been left open until 2014, when Weigant and Papin succeeded to tackle the case where the domain is a rectangle. In this paper we adapt Weigant and Papin’s result to the case of a smooth domain with several internal sources and sinks.

Impacts of Radio Occultation Data on Typhoon Forecasts as Explored by the Global MPAS-GSI System

Global Navigation Satellite System (GNSS) radio occultation (RO) provides plentiful sounding profiles over regions lacking conventional observations. The Gridpoint Statistical Interpolation (GSI) hybrid system for assimilating RO data is integrated in this study with the Model for Prediction Across Scales–Atmosphere (MPAS) to improve tropical cyclone forecasts. After the MPAS-GSI assimilation cycles, dynamical vortex initialization (DVI) that may effectively spin up the initial inner typhoon vortex through cycled model integration is implemented to improve the initial analysis fit to the best track position as well as maximum wind or pressure intensity for Typhoon Nepartak (2016) that moved northwestward toward southern Taiwan. During the cycling assimilation, assimilation with RO data improves the temperature and moisture analysis, and largely reduces the forecast errors compared to those without RO data assimilation. The two RO operators that assimilate local bending angle or refractivity produce similar analyses, but the temperature and moisture increments from bending angle assimilation are slightly larger than those from refractivity assimilation. The MPAS forecasts at 60-15 km resolution show that the typhoon track prediction is improved with RO data, especially using bending angle data. The reduction in track deviations is explained by the wavenumber-one potential vorticity budget for several forecasts associated with the track deflection near southern Taiwan. Assimilation of RO data has fewer impacts on the typhoon intensity forecast compared to the DVI that largely improves the initial and thus forecasted intensity of the typhoon but at the cost of a slightly degraded track. Use of the enhanced 3 km resolution in the typhoon path also further improved the forecasts with and without the DVI. The feasible performance of the MPAS-GSI system with the RO data impact is also illustrated for Typhoon Mitag (2019), that passed around northern Taiwan.

Tropical Cyclone Size Is Strongly Limited by the Rhines Scale: Experiments with a Barotropic Model

Abstract Recent work found evidence using aquaplanet experiments that tropical cyclone (TC) size on Earth is limited by the Rhines scale, which depends on the planetary vorticity gradient β. This study aims to examine how the Rhines scale limits the size of an individual TC. The traditional Rhines scale is first reexpressed as a Rhines speed to characterize how the effect of β varies with radius in a vortex whose wind profile is known. The framework is used to define the vortex Rhines scale, which is the transition radius that divides the vortex into a vortex-dominant region at smaller radii, where the axisymmetric circulation is steady, and a wave-dominant region at larger radii, where the circulation stimulates planetary Rossby waves and dissipates. Experiments are performed using a simple barotropic model on a β plane initialized with a TC-like axisymmetric vortex defined using a recently developed theoretical TC wind profile model. The gradient β and initial vortex size are each systematically varied to investigate the detailed responses of the TC-like vortex to β. Results show that the vortex shrinks toward an equilibrium size that closely follows the vortex Rhines scale. A larger initial vortex relative to its vortex Rhines scale will shrink faster. The shrinking time scale is well described by the vortex Rhines time scale, which is defined as the overturning time scale of the circulation at the vortex Rhines scale and is shown to be directly related to the Rossby wave group velocity. The relationship between our idealized results and the real Earth is discussed. Results may generalize to other eddy circulations, such as the extratropical cyclone. Significance Statement Tropical cyclones vary in size significantly on Earth, but how large a tropical cyclone could potentially be is still not understood. The variation of the Coriolis parameter with latitude is known to limit the size of turbulent circulations, but its effect on tropical cyclones has not been studied. This study derives a new parameter related to this concept called the “vortex Rhines scale” and shows in a simple model how and why storms will tend to shrink toward this size. These results help explain why tropical cyclone size tends to increase slowly with latitude on Earth and can help us understand what sets the size of tropical cyclones on Earth in general.

Time-Asymptotic Study of a Viscous Axisymmetric Fluid without Swirl

We study the long-time behaviour of axisymmetric solutions without swirl for the threedimensional Navier-Stokes equations in the whole space. Assuming that the initial vorticity is sufficiently localised, we compute explicitly the leading terms in the asymptotic expansion of the solution, both for the vorticity and the velocity field. In particular, we identify optimal temporal decay rates depending on the spatial localisation of the initial data. Our approach relies on accurate $L^p$-$L^q$ estimates for the linearised evolution equation and its Taylor expansion in self-similar variables.

Dynamics of Rotating Spin-Orbit-Coupled Spin-1 Bose-Einstein Condensates With In-Plane Gradient Magnetic Field in an Anharmonic Trap

We investigate the dynamics of rotating spin-orbit-coupled spin-1 Bose–Einstein condensates (BECs) in an in-plane gradient magnetic field, which is confined in an anharmonic trap. In the case of rotating spin-orbit-coupled spin-1 BECs with given parameters, the system evolves from initial disk-shaped condensates into drastic turbulent oscillations and ghost vortices on the surface of the component densities due to surface wave excitations and then into two final vortex necklaces along the azimuthal direction with an irregular density hole, in which the vortices differ by one quantum number in turn. For the case of rotating spin-orbit-coupled spin-1 BECs with in-plane gradient magnetic field, with the dynamic evolution, the system undergoes a transition from an initial central polar-core vortex to violent turbulent oscillations and then to a final vortex chain along the diagonal of BECs, with the remaining vortices symmetrically distributed on both sides in the individual component. In addition, the corresponding spin texture undergoes a transition from plane-wave phase to double half-antiskyrmion necklaces for the former case and a transition from a structure similar to a quadrupole magnetic field to a half-antiskyrmion chain with the rest of the half-antiskyrmions on both sides. During the dynamic evolution process, the angular momentum increases gradually and then approaches a convergent value.

Convective storms in closed cyclones in Jupiter: (II) numerical modeling

On May 31, 2020 a convective storm appeared in one small cyclone in the South Temperate Belt (STB) of Jupiter. The storm, nicknamed as Clyde's Spot, had an explosive start and quickly diminished in activity in a few days. However, it left a highly turbulent cyclone as a remnant that evolved to become a turbulent segment of the STB in a time-scale of one year. A very similar storm erupted on August 7, 2021 in another cyclone of the STB with a similar initial phenomenology. In both cases, the outbreaks started in cyclones that were the result of the merger of pre-existing vortices. In a previous paper we presented an observational study of these storms compared with a similar cyclonic convective system observed during the Voyager 2 flyby [Hueso et al., Convective storms in closed cyclones in Jupiter's South Temperate Belt: (I) Observations, Icarus, 380, 2022]. Here we present numerical simulations of these vortices and storms with the Explicit Planetary Isentropic-Coordinate (EPIC) numerical model. We simulate mergers of cyclones in Jupiter's STB and investigate the deep structure of the resulting cyclone and its capability to uplift material from the water condensation level. Convection is introduced in the model imposing heating sources whose vertical extent, horizontal size and duration are free parameters that we explore. Our simulations reproduce the cloud field of both storms after short episodes of a few hours of intense convection. The evolution of the morphology of the convective cyclone after the convective pulse stopped shows a strong relation between the convective energy released and the initial vorticity in the cyclone. Similar results are obtained for the cyclonic storm observed during the Voyager 2 flyby. We also compare our simulations of these storms with numerical simulations of a storm that developed in the STB in 2018 inside an elongated cyclone known as the South Temperate Belt Ghost [Iñurrigarro et al., Observations and numerical modelling of a convective disturbance in a large-scale cyclone in Jupiter's South Temperate Belt, Icarus, 336, 2020]. In addition, we also simulate one of the large-scale storms that develop in the South Equatorial Belt comparing our simulations with Voyager 1 observations of one of those events. From these simulations, we establish a relative scale of energies associated to these convective storms. As coherent cyclones isolate the local atmosphere from their surroundings, we propose that the availability of condensables inside closed cyclones limits the duration of active convection, allowing larger convective outbursts in larger cyclones. Our simulations of the short and intense convective pulse associated to the 2020 and 2021 STB suggest a minimum local water abundance of 1.0–1.2 times solar at the location of the storms. The lower number considers a significant contribution of ammonia condensation, and the larger number considers only water moist convection with a negligible role of ammonia.

Phase memory of optical vortex beams

Optical vortex beams are under considerable scrutiny due to their demonstrated potential for applications ranging from quantum optics to optical communications and from material processing to particle trapping. However, upon interaction with inhomogeneous material systems, their deterministic properties are altered. The way these structured beams are affected by different levels of disturbances is critical for their uses. Here, for the first time, we quantify the degradation of perfect optical vortex beams after their interaction with localized random media. We developed an analytical model that (1) describes how the spatial correlation and the phase variance of disturbance affect the phase distribution across the vortex beams and (2) establishes the regimes of randomness for which the beams maintain the memory of their initial vorticity. Systematic numerical simulations and controlled experiments demonstrate the extent of this memory effect for beams with different vorticity indices.

Probabilistic representation of helicity in viscous fluids

It is shown that the helicity of three dimensional viscous incompressible flow can be identified with the overall linking of the fluid's initial vorticity to the expectation of a stochastic mean field limit. The relevant mean field limit is obtained by following the Lagrangian paths in the stochastic Hamiltonian interacting particle system of [S. Hochgerner, Proc. R. Soc. A 474:20180178].

Local well-posedness for the motion of a compressible gravity water wave with vorticity

In this paper we prove the local well-posedness (LWP) for the 3D compressible Euler equations describing the motion of a liquid in an unbounded initial domain with moving boundary. The liquid is under the influence of gravity but without surface tension, and it is not assumed to be irrotational. We apply the tangential smoothing method introduced in Coutand-Shkoller [10,11] to construct the approximation system with energy estimates uniform in the smooth parameter. It should be emphasized that, when doing the nonlinear a priori estimates, we need neither the higher order wave equation of the pressure and delicate elliptic estimates, nor the higher regularity on the flow-map or initial vorticity. Instead, we adapt the Alinhac's good unknowns to the estimates of full spatial derivatives.

Long Term Vortex Flow Evolution around a Magnetic Island in Tokamaks.

We present a gyrokinetic analysis of the vortex flow evolution in a magnetic island in collisionless tokamak plasmas. In a short term ω[over ¯]_{D}t<1, where ω[over ¯]_{D} is the secular magnetic drift of the orbit center, initial monopolar vortex flow approaches to its residual level determined by the neoclassical enhancement of polarization shielding after collisionless relaxation. The residual level depends on the location inside an island and is higher than the Rosenbluth-Hinton level [M.N. Rosenbluth and F.L. Hinton, Phys. Rev. Lett. 80, 724 (1998)PRLTAO0031-900710.1103/PhysRevLett.80.724] due to finite island width. In a long term ω[over ¯]_{D}t>1, the residual vortex flow evolves to a dipolar zonal-vortex flow mixture due to toroidicity-induced breaking of a helical symmetry. The mixture forms localized flow shear layers near the island separatrix away from X points. The deviation of the streamlines of the mixture flows from magnetic surfaces allows turbulence advection across the island. We expect a small island w≲qρ_{Ti}/s[over ^] provides a favorable condition for this mixture flow formation, while the monopolar vortex flow persists for a larger island.

Long time fate of two-dimensional incompressible high Reynolds number Navier–Stokes turbulence: A quantitative comparison between theory and simulation

Predicting the long time or late time states of two-dimensional incompressible, high Reynolds number, slowly decaying turbulence has been one of the long-standing problems. Using “point vortices” as “inviscid” building blocks, which do not respect incompressibility, statistical mechanical models conserving only total energy and zero total circulation result in the well-known sinh-Poisson relation between vorticity and stream function. On the other hand, statistical mechanics of “inviscid patch” vortices, which respects incompressibility by conserving regions of zero and nonzero vorticity, predicts a generalized relaxed state, which has never been systematically compared with direct numerical simulations (DNS). In this study, starting from highly packed regions of nonzero initial vorticity, we demonstrate using high resolution, high Reynolds number DNS that the late time states agree with predictions from patch vortex models. As total circulation is reduced or diluted, we show that late time states of our DNS systematically and unambiguously lead to the sinh-Poisson relationship between vorticity and stream function. We believe that our quantitative findings solve one of the long-standing problems in two-dimensional turbulence.

Evolution of double-eye wall hurricanes and emergence of complex tripolar end states in moist-convective rotating shallow water model

A detailed investigation of linear instabilities of double-eyewall hurricane-like vortices with double maxima (“walls”) of azimuthal velocity and vorticity around the central minimum (“eye”) and of their nonlinear saturation is carried out in the framework of the moist-convective rotating shallow water model. It is shown that developing barotropic instability leads to inward displacement and gradual disappearance of the inner wall and to the formation of a nontrivial tripolar quasi-stationary structure, provided the second, outer, wall of the initial vortex is high enough. The effects of moist convection lead to the enhancement of the outer wall. In comparison, under the influence of the same instability, vortices with an inner wall stronger than the outer one evolve toward monopolar single-eyewall structures.

Fast switching of magnetic vortex state under an alternating magnetic field

Vortex is considered to be a promising candidate for nonvolatile magnetic random storage devices due to its unique binary state and high thermal stability. The control of vortex polarity and chirality is key to devices based on vortex. In this work, the switching of the magnetic vortex state in Permalloy nanodisks has been investigated by micromagnetic simulation employing in-plane alternating magnetic fields. During the switching of magnetic vortex state, an antivortex is formed, and the antivortex and vortex will not annihilate together. The formation of antivortex and the annihilation of initial vortex lead to the switching of magnetic vortex state. The switching of vortex state depends on magnetic field and disk size. The switching time of magnetic vortex state declines with the enhancement of magnetic field amplitude or magnetic field frequency. All the switching can be completed within 1.4 ns, indicating that the alternating magnetic field driving can induce the rapid switching of vortex state. This work provides an effective method for the fast switching of magnetic vortex state, which benefits the potential storage applications in vortex-based spintronic devices.

Formation of small-scale vortices in the core of a large merged vortex

The merger of two surface quasi-geostrophic vortices is examined in detail. As the two vortices collapse towards each other in the merging process, they trap their external fronts between them; these fronts are inserted into the final merged vortex, where they form a central, nearly parallel, sheared velocity strip, sensitive to barotropic instability. As a result, this strip breaks up into an alley of small vortices. Subsequently, these small vortices may undergo merger and grow in size in the core of the large merged vortex. The number of small trapped vortices decreases correspondingly. Finally, a single or two small vortices remain. These processes are analysed using a numerical model of the surface quasi-geostrophic equations. The sensitivity of this process to the initial vortex characteristics is explored. A parallel is drawn between this problem and the instability of a rectilinear strip of temperature with a central gap. The application of this problem to the Ocean is discussed.