Vortex Patches Research Articles

Quantitative Hölder estimates for even singular integral operators on patches

In this paper we show a constructive method to obtain C˙σ estimates of even singular integral operators on characteristic functions of domains with C1+σ regularity, 0<σ<1. This kind of functions were shown in first place to be bounded (classically only in the BMO space) to obtain global regularity for the vortex patch problem [5,2]. This property has then been applied to solve different type of problems in harmonic analysis and PDEs. Going beyond in regularity, the functions are discontinuous on the boundary of the domains, but C˙σ in each side. This C˙σ regularity has been bounded by the C1+σ norm of the domain [8,14,16]. Here we provide a quantitative bound linear in terms of the C1+σ regularity of the domain. This estimate shows explicitly the dependence of the lower order norm and the non-self-intersecting property of the boundary of the domain. As an application, this quantitative estimate is used in a crucial manner to the free boundary incompressible Navier-Stokes equations providing new global-in-time regularity results in the case of viscosity contrast [12].

Vortex patch problem for steady lake equation

We study the vortex patch problem for the steady lake equation in a bounded domain and construct three different kinds of solutions where the vorticity concentrates in the domain or near the boundary. Our approach is based on the Lyapunov–Schmidt reduction, which transforms the construction into a problem of seeking critical points for a function related to the kinetic energy. The method in this paper has a wide applicability and can be used to deal with general elliptic equations in divergence form with Heaviside nonlinearity.

Stability of radially symmetric, monotone vorticities of 2D Euler equations

We consider the incompressible Euler equations in \( {\mathbb {R}}^{2} \) when the initial vorticity is bounded, radially symmetric and non-increasing in the radial direction. Such a radial distribution is stationary, and we show that the monotonicity produces stability in some weighted norm related to the angular impulse. For instance, it covers the cases of circular vortex patches and Gaussian distributions. Our stability does not depend on \(L^\infty \)-bound or support size of perturbations. The proof is based on the fact that such a radial monotone distribution minimizes the impulse of functions having the same level set measure.

Multiscale Steady Vortex Patches for 2D Incompressible Euler Equations

In this paper, we study the existence and qualitative properties of multiscale steady vortex patches for Euler equations in a 2D bounded domain. By considering certain maximization problems for the vorticity, we obtain the existence of double vortex patches which are trapped in a neighborhood of two points. Limiting localizations of these two points are determined by the Robin function and the boundary of the domain, rather than critical points of the Kirchhoff--Routh function $H_2$, which is quite different from all the known results. Moreover, the strengths of two components of vorticity are of different order. Multiscale vortex patches concentrating near $k$ points are also constructed for any integer $k\geq 2.$

Transition of Near-Ground Vorticity Dynamics during Tornadogenesis

Abstract Although much is known about the environmental conditions necessary for supercell tornadogenesis, the near-ground vorticity dynamics during the tornadogenesis process itself are still somewhat poorly understood. For instance, seemingly contradicting mechanisms responsible for large near-ground vertical vorticity can be found in the literature. Broadly, these mechanisms can be sorted into two classes, one being based on upward tilting of mainly baroclinically produced horizontal vorticity in descending air (here called the downdraft mechanism), while in the other the horizontal vorticity vector is abruptly tilted upward practically at the surface by a strong updraft gradient (referred to as the in-and-up mechanism). In this study, full-physics supercell simulations and highly idealized simulations show that both mechanisms play important roles during tornadogenesis. Pretornadic vertical vorticity maxima are generated via the downdraft mechanism, while the dynamics of a fully developed vortex are dominated by the in-and-up mechanism. Consequently, a transition between the two mechanisms occurs during tornadogenesis. This transition is a result of axisymmetrization of the pretornadic vortex patch and intensification via vertical stretching. These processes facilitate the development of the corner flow, which enables production of vertical vorticity by upward tilting of horizontal vorticity practically at the surface, i.e., the in-and-up mechanism. The transition of mechanisms found here suggests that early stages of tornado formation rely on the downdraft mechanism, which is often limited to a small vertical component of baroclinically generated vorticity. Subsequently, a larger supply of horizontal vorticity (produced baroclinically or via surface drag, or even imported from the environment) may be utilized, which marks a considerable change in the vortex dynamics.

Existence and regularity for vortex patch solutions of the 2D Euler equations

Existence and regularity for vortex patch solutions of the 2D Euler equations

The Centroid-Deformation Decomposition for Buoyant Vortex Patch Motion

The motion of a two-dimensional buoyant vortex patch, i.e. a vortex patch with a uniform density different from the uniform density of the surrounding fluid, is analyzed in terms of evolution equations for the motion of its centroid, deformation of its boundary and the strength of a vortex sheet which is essential to enforce pressure continuity across the boundary. The equations for the centroid are derived by a linear momentum analysis and that for the sheet strength by applying Euler's equations on the boundary, while the boundary deformation is studied in the centroid-fixed frame. A complicated coupled set of equations is obtained which, to the best of our knowledge, has not been derived before. The evolution of the sheet strength is obtained as an integral equation. The equations are also examined in the limit of a patch of vanishing size or a buoyant point vortex.

Helical Contour Dynamics

The equations of motion for an incompressible flow with helical symmetry (invariance under combined axial translation and rotation) can be expressed as nonlinear evolution laws for two scalars: vorticity and along-helix velocity. A metric term related to the pitch of the helix enters these equations, which reduce to two-dimensional and axisymmetric dynamics in appropriate limits. We take the vorticity and along-helix velocity component to be piecewise constant. In addition to this vortex patch, a vortex sheet develops when the along-helix velocity is nonzero. We obtain a contour dynamics formulation of the full nonlinear equations of motion, in which the motion of the boundary is computed in a Lagrangian fashion and the velocity field can be expressed as contour integrals, reducing the dimensionality of the computation. We investigate the stability properties of a circular vortex patch along the axis of the helix in the presence of a vortex sheet and along-helix velocity. A linear stability calculation shows that the system is stable when the initial vortex sheet is zero, but can be stable or unstable in the presence of a vortex sheet. Using contour dynamics, we examine the nonlinear evolution of the system, and show that nonlinear effects become important in unstable cases.

From Point Vortices to Vortex Patches in Self-Similar Expanding Configurations

The main result is that given a self-similarly expanding configuration of 3 point vortices that start sufficiently far out, we can instead take compactly supported vorticity functions, and the resulting solution to 2D incompressible Euler will evolve like a nearby point vortex configuration for all time, with the size of the patches growing at most as \(t^{1/4+\epsilon }\) and the distance between them growing as \(\sqrt{t}\).

Evolution of the Singularities of the Schwarz Function Corresponding to the Motion of a Vortex Patch in the Two-dimensional Euler Equations

Evolution of the Singularities of the Schwarz Function Corresponding to the Motion of a Vortex Patch in the Two-dimensional Euler Equations

Riemann–Hilbert problem on a torus and a vortex patch in a wedge

An exact formula for the conformal map from the exterior of two slits onto the doubly connected flow domain is obtained when a fluid flows in a wedge about a vortex. The map is employed to determine the potential flow outside the vortex and the vortex domain boundary provided the circulation around the vortex and constant speed on the vortex boundary are prescribed, and there are no stagnation points on the walls. The map is expressed in terms of a rational function on an elliptic surface topologically equivalent to a torus, and the solution to a symmetric Riemann–Hilbert problem on a finite and a semi-infinite segments on the same genus-1 Riemann surface. Owing to its special features, the Riemann–Hilbert problem requires a novel analogue of the Cauchy kernel on an elliptic surface. Such a kernel is proposed and employed to derive a closed-form solution to the Riemann–Hilbert problem and the associated Jacobi inversion problem. The final formula for the conformal map possesses a free geometric parameter and two model parameters. It is shown that the solution exists and the vortex has two cusps, while the solution does not exist when the wedge angle exceeds π.

Vortex Patches Choreography for Active Scalar Equations

This paper deals with the existence of $N$ vortex patches located at the vertex of a regular polygon with $N$ sides that rotate around the center of the polygon at a constant angular velocity. That is done for Euler and (SQG)$_\beta$ equations, with $\beta\in(0,1)$, but may be also extended to more general models. The idea is the desingularization of the Thomsom polygon for the $N$ point vortex system, that is, $N$ point vortices located at the vertex of a regular polygon with $N$ sides. The proof is based on the study of the contour dynamics equation combined with the application of the infinite dimensional Implicit Function theorem and the well--chosen of the function spaces.

Adiabatic behavior of an elliptical vortex in a time-dependent external strain flow

When vortex patches are subjected to externally imposed strain flows where the strain rate grows slowly in time, the system approximately preserves an adiabatic invariant related to oscillatory elliptical deformations of the vortices. Formulae are derived for the breaking of this invariant with respect to the rate and functional form of the strain ramp. The analytical results are compared to experimental data acquired using magnetized pure electron plasmas, which provide a close laboratory realization of the motion of two-dimensional ideal vortices.

Nonlinear Stability of Planar Vortex Patches in an Ideal Fluid

In this paper, we prove nonlinear stability of planar vortex patches concentrated near a non-degenerate minimum point of the Robin function in a general bounded domain. These vortex patches are stationary solutions of the two-dimensional incompressible Euler equations. The result is obtained by showing that these concentrated vortex patches are in fact isolated maximizers of the kinetic energy among isovortical patches.

Existence of Steady Symmetric Vortex Patch in a Disk

In this paper, we construct a family of symmetric vortex patches for the 2D steady incompressible Euler equations in a disk. The result is obtained by studying a variational problem in which the kinetic energy of the fluid is maximized subject to some appropriate constraints for the vorticity. Moreover, we show that these vortex patches “shrink” to a given minimum point of the corresponding Kirchhoff–Routh function as the vorticity strength parameter goes to infinity.

Existence of steady multiple vortex patches to the vortex-wave system

In this paper we prove the existence of steady multiple vortex patch solutions to the vortex-wave system in a planar bounded domain. The construction is performed by solving a certain variational problem for the vorticity and studying its asymptotic behavior as the vorticity strength goes to infinity.

Rotating vortex patches for the planar Euler equations in a disk

We construct a family of rotating vortex patches with fixed angular velocity for the two-dimensional Euler equations in a disk. As the vorticity strength goes to infinity, the limit of these rotating vortex patches is a rotating point vortex whose motion is described by the Kirchhoff-Routh equation. The construction is performed by solving a variational problem for the vorticity which is based on an adaption of Arnold's variational principle. We also prove nonlinear orbital stability of the set of maximizers in the variational problem under Lp perturbation when p∈[3/2,+∞).

Growth of perimeter for vortex patches in a bulk

We consider the two-dimensional incompressible Euler equations. We construct vortex patches with smooth boundary on T2 and R2 whose perimeter grows with time. More precisely, for any constant M>0, we construct a vortex patch in T2 whose smooth boundary has length of order 1 at the initial time such that the perimeter grows up to the given constant M within finite time. The construction is done by cutting a thin slit out of an almost square patch. A similar result holds for an almost round patch with a thin handle in R2.

On radial symmetry of rotating vortex patches in the disk

In this note, we consider the radial symmetry property of rotating vortex patches for the 2D incompressible Euler equations in the unit disk. By choosing a suitable vector field to deform the patch, we show that each simply-connected rotating vortex patch D with angular velocity Ω, Ω≥max⁡{1/2,(2l2)/(1−l2)2} or Ω≤−(2l2)/(1−l2)2, where l=supx∈D⁡|x|, must be a disk. The main idea of the proof, which has a variational flavor, comes from a recent paper of Gómez-Serrano–Park–Shi–Yao, arXiv:1908.01722, where radial symmetry of rotating vortex patches in the whole plane was studied.

Алгоритм идентификации вихревых пятен на основе моделей точечных вихрей

An algorithm for identifying a plane vortex fluid flow from known velocity vectors at a finite set of points in the flow region is proposed. To describe the approximation of the vortex structure, a mathematical model of the system of point vortices is used. The parameters to be determined and characterizing the flow are the number of point vortices, their coordinates, and intensity. Unknown parameters are determined as a result of minimizing the functional that relates the known velocity vectors and the velocity model flows of the vortex system at the same points in the region. An implementation of the algorithm for the simplest case, one vortex patch, is presented. Computational experiments were carried out, which showed the effectiveness of the developed method.