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We consider a Markov chain Monte Carlo method, known as Hamiltonian Monte Carlo (HMC), on compact manifolds in Euclidean space. It utilizes Hamiltonian dynamics to generate samples approximating a target distribution in high dimensions efficiently. The efficiency of HMC is characterized by its convergence property, called geometric ergodicity. This property is important to generate low-correlated samples. It also plays a crucial role in establishing the error estimate for the quadrature of bounded functions by HMC sampling, referred to as the Hoeffding-type inequality. While the geometric ergodicity has been proved for HMC on Euclidean space, it has not been established on manifolds. In this paper, we prove the geometric ergodicity for HMC on compact manifolds. As an example to confirm the efficiency of the proposed HMC method, we consider a sampling problem associated with the N-vortex problem on the unit sphere, which is a statistical model of two-dimensional turbulence. We apply HMC to approximate the statistical quantities with respect to the invariant measure of the N-vortex problem, called the Gibbs measure. We observe the organization of large vortex structures as seen in two-dimensional turbulence. [ABSTRACT FROM AUTHOR]

This paper shows that the topological structures of particle orbits generated by a generic class of vector fields on spherical surfaces, called the flow of finite type, are in one-to-one correspondence with discrete structures such as trees/graphs and sequences of letters. The flow of finite type is an extension of structurally stable Hamiltonian vector fields, which appear in many theoretical and numerical investigations of two-dimensional (2D) incompressible fluid flows. Moreover, it contains compressible 2D vector fields such as the Morse–Smale vector fields and the projection of 3D vector fields onto 2D sections. The discrete representation is not only a simple symbolic identifier for the topological structure of complex flows, but it also gives rise to a new methodology of topological data analysis for flows when applied to data brought by measurements, experiments, and numerical simulations of complex flows. As a proof of concept, we provide some applications of the representation theory to 2D compressible vector fields and a 3D vector field arising in an industrial problem. [ABSTRACT FROM AUTHOR]

We construct point vortex equilibria with strengths quantized by multiples of 2π in a fixed background vorticity field on the surface of a curved torus. The background vorticity consists of two terms: first, a term exponentially related to the stream function and a second term arising from the curvature of the torus, which leads to a Liouville-type equation for the stream function. By using a stereographic projection of the torus onto an annulus in a complex plane, the Liouville-type equation admits a class of exact solutions, given in terms of a loxodromic function on the annulus. We show that appropriate choices of the loxodromic function in the solution lead to stationary vortex patterns with 4 n ̂ point vortices of identical strengths, n ̂ ∈ N. The quantized point vortices are stationary in the sense that they are equilibria of a "one-way interaction" model where the evolution of point vortices is subject to the continuous background vorticity, while the background vorticity distribution is not affected by the velocity field induced by the point vortices. By choosing loxodromic functions continuously dependent on a parameter and taking appropriate limits with respect to this parameter, we show that there are solutions with inhomogeneous point vortex strengths, in which the exponential part of the background vorticity disappears. The point vortices are always located at the innermost and outermost rings of the torus owing to the curvature effects. The topological features of the streamlines are found to change as the modulus of the torus changes. [ABSTRACT FROM AUTHOR]

This study proposes an algorithm detecting atmospheric blocking by extracting topological features of geopotential height data at 500 hPa. The algorithm uses topological flow data analysis (TFDA) providing a unique symbolic representation, named the partially cyclically ordered rooted tree (COT) representation, and a discrete graph structure, called a Reeb graph, to each structurally stable Hamiltonian vector field based on the mathematical theory of topological classifications for streamline patterns. It recognizes blocking events more simply and effectively using fewer meteorological parameters than conventional algorithms. Furthermore, the algorithm can determine morphological types of blocking events, an Omega shape or a dipole pattern, whereas no effective algorithm has been available so far. The identified blocking events and their morphological types are consistent with synopticians' subjective judgments. [ABSTRACT FROM AUTHOR]

Quasi-stationary states consisting of localized spots in a reaction-diffusion system are considered on the surface of a torus with major radius R and minor radius r. Under the assumption that these localized spots persist stably, the evolution equation of the spot cores is derived analytically based on the higher-order matched asymptotic expansion with the analytic expression of the Green's function of the Laplace-Beltrami operator on the toroidal surface. Owing to the analytic representation, one can investigate the existence of equilibria with a single spot, two spots, and the ring configuration where N localized spots are equally spaced along a latitudinal line with mathematical rigor. We show that localized spots at the innermost/outermost locations of the torus are equilibria for any aspect ratio α = R/r. In addition, we find that there exists a range of the aspect ratio in which localized spots stay at a special location of the torus. The theoretical results and the linear stability of these spot equilibria are confirmed by solving the nonlinear evolution of the Brusselator reaction-diffusion model by numerical means. We also compare the spot dynamics with the point vortex dynamics, which is another model of spot structures. [ABSTRACT FROM AUTHOR]

Vortex crystals are equilibrium states of point vortices whose relative configuration is unchanged throughout the evolution. They are examples of stationary point configurations subject to a logarithmic particle interaction energy, which give rise to phenomenological models of pattern formations in incompressible fluids, superconductors, superfluids and Bose-Einstein condensates. In this paper, we consider vortex crystals rotating at a constant speed in the latitudinal direction on the surface of a torus. The problem of finding vortex crystals is formulated as a linear null equation AΓ =0 for a non-normal matrix A whose entities are derived from the locations of point vortices, and a vector Γ consisting of the strengths of point vortices and the latitudinal speed of rotation. Point configurations of vortex crystals are obtained numerically through the singular value decomposition by prescribing their locations and/or by moving them randomly so that the matrix A becomes rank deficient. Their strengths are taken from the null space corresponding to the zero singular values. The toroidal surface has a non-constant curvature and a handle structure, which are geometrically different from the plane and the spherical surface where vortex crystals have been constructed in the preceding studies. We find new vortex crystals that are associated with these toroidal geometry: (i) a polygonal arrangement of point vortices around the line of longitude; (ii) multiple latitudinal polygonal ring configurations of point vortices that are evenly arranged around the handle; and (iii) point configurations along helical curves corresponding to the fundamental group of the toroidal surface. We observe the strengths of point vortices and the behaviour of their distribution as the number of point vortices gets larger. Their linear stability is also examined. [ABSTRACT FROM AUTHOR]

A steady solution of the incompressible Euler equation on a toroidal surface TR,r of major radius R and minor radius r is provided. Its streamfunction is represented by an exact solution to the modified Liouville equation, ∇²T R,r ψ =c edψ + (8/d)κ, where ∇²T R,r and κ denote the Laplace–Beltrami operator and the Gauss curvature of the toroidal surface respectively, and c, d are real parameters with cd<0. This is a generalization of the flows with smooth vorticity distributions owing to Stuart (Stuart 1967 J. Fluid Mech. 29, 417–440. (doi:10.1017/S002211206 7000941)) in the plane and Crowdy (Crowdy 2004 J. Fluid Mech. 498, 381–402. (doi:10.1017/S002211200 3007043)) on the spherical surface. The flow consists of two point vortices at the innermost and the outermost points of the toroidal surface on the same line of a longitude, and a smooth vorticity distribution centred at their antipodal position. Since the surface of a torus has non-constant curvature and a handle structure that are different geometric features from the plane and the spherical surface, we focus on how these geometric properties of the torus affect the topological flow structures along with the change of the aspect ratio α =R/r. A comparison with the Stuart vortex on the flat torus is also made. [ABSTRACT FROM AUTHOR]

Anomalous enstrophy dissipation of incompressible ows in the inviscid limit is a significant property characterizing two-dimensional turbulence. It indicates that the investigation of nonsmooth incompressible and inviscid ows contributes to the theoretical understanding of turbulent phenomena. In the preceding study [J. Nonlinear Sci., 26 (2016), pp. 1525-1570], a unique global weak solution to the Euler-α equations, which is a regularized Euler equation, for point-vortex initial data is considered, and thereby it has been shown that, as α→ 0, the evolution of three point vortices converges to a self-similar collapsing orbit dissipating the enstrophy in the sense of distributions at the critical time. In the present paper, to elucidate whether this singular orbit can be constructed independently on the regularization method, we consider a functional generalization of the Euler-α equations, called the Euler-Poincaré models, in which the incompressible velocity field is dispersively regularized by a smoothing function. We provide a sufficient condition for the existence of the singular orbit, which is applicable to many smoothing functions. As examples, we confirm that the condition is satisfied with the Gaussian regularization and the vortex blob regularization that are both utilized in the numerical scheme solving the Euler equations. Consequently, the enstrophy dissipation via the collapse of three point vortices is a generic phenomenon that is not specific to the Euler-α equations but universal within the Euler-Poincaré models. [ABSTRACT FROM AUTHOR]

We formulate the N point vortex problem in a doubly-periodic rectangle with a constant background vorticity using the hydrodynamic Green's function. We derive an explicit formula for the hydrodynamic Green's function and the velocities of the point vortices in terms of the Schottky–Klein prime function using a conformal mapping approach. The sum of vortex strengths can be arbitrary, and when it is zero we recover previous known results including the integrability of the two and three-vortex problems. A non-zero sum of vortex strengths leads to a constant background vorticity. We derive a Hamiltonian structure for the equations and show that the two-vortex problem is integrable, and classify all possible vortex motions. In addition, for general N , we obtain several fixed lattice equilibrium configurations including single-layered and double-layered equilibria. In the latter case, we also obtain lattice configurations with defects consisting of point vortices with inhomogeneous strengths. We find equilibria arranged in doubly-periodic rectangular and parallelogram lattices consisting of N = 1 , 2 , 3 , 4 vortices per fundamental lattice. • Explicit formula for hydrodynamic Green's function in a doubly-periodic domain. • Hamiltonian equations for point vortices in a rectangular fundamental domain. • Constant background vorticity and non-zero sum of point vortex strengths. • Conformal mapping approach in a complex plane using the Schottky–Klein prime function. • New equilibria with inhomogeneous vortex strengths and lattice structure with defects. [ABSTRACT FROM AUTHOR]

A flow of 2D incompressible and inviscid fluid is an example of a Hamiltonian vector field, where its Hamiltonian corresponds to the stream function whose level curves are called streamlines. A 2D Hamiltonian vector field is said to be structurally stable when the topological structure of streamlines is unchanged under any small perturbations of the vector field. In the present paper we show that the streamline topology of every structurally stable Hamiltonian vector field is in one-to-one correspondence with a labelled and directed plane tree and its associated symbolic expression called a regular expression. Consequently, we can characterize all streamline topologies with their corresponding plane trees and regular expressions uniquely. The present theory of tree representations is combinatorial; it brings us a new compression algorithm converting a large amount of streamline plots obtained by laboratory experiments and numerical simulations into a small set of simple symbolic data of regular expressions, which is amenable to a big data analysis for streamline patterns. Conversion to tree structures and their associated regular expressions is easily performed and it is flexibly applicable not only to incompressible flows but also to any physical phenomena described by Hamiltonian vector fields. We also demonstrate how the tree representation is applied to describe variations of streamline topologies for incompressible flows. [ABSTRACT FROM AUTHOR]

We carry out the linear stability analysis of a polygonal ring configuration of N point vortices, called an N-ring, along the line of latitude θ0 on a torus with the aspect ratio α. Deriving a criterion for the stability depending on the parameters N, θ0 and α, we reveal how the aspect ratio α contributes to the stability of the N-ring. While the N-ring necessarily becomes unstable when N is sufficiently large for fixed α, the stability is closely associated with the geometric property of the torus for variable α; for low aspect ratio α∼1, N=7 is a critical number determining the stability of the N-ring when it is located along a certain range of latitudes, which is an analogous result to those in a plane and on a sphere. On the other hand, the stability is determined by the sign of curvature for high aspect ratio α≫1. That is to say, the N-ring is neutrally stable if it is located on the inner side of the toroidal surface with a negative curvature, while the N-ring on its outer side with a positive curvature is unstable. Furthermore, based on the linear stability analysis, we describe nonlinear evolution of the N-ring when it becomes unstable. It is difficult to deal with this problem, since the evolution equation of the N point vortices is formulated as a Hamiltonian system with N degrees of freedom, which is in general non-integrable. Thus, we reduce the Hamiltonian system to a simple integrable system by introducing a cyclic symmetry. Owing to this reduction, we successfully find some periodic orbits in the reduced system, whose local bifurcations and global transitions for variable α are characterized in terms of the fundamental group of the torus. [ABSTRACT FROM AUTHOR]

We study the stability of a barotropic vortex strip on a rotating sphere, as a simple model of jet streams. The flow is approximated by a piecewise-continuous vorticity distribution by zonal bands of uniform vorticity. The linear stability analysis shows that the vortex strip becomes stable as the strip widens or the rotation speed increases. When the vorticity constants in the upper and the lower regions of the vortex strip have the same positive value, the inner flow region of the vortex strip becomes the most unstable. However, when the upper and the lower vorticity constants in the polar regions have different signs, a complex pattern of instability is found, depending on the wavenumber of perturbations, and interestingly, a boundary far away from the vortex strip can be unstable. We also compute the nonlinear evolution of the vortex strip on the rotating sphere and compare with the linear stability analysis. When the width of the vortex strip is small, we observe a good agreement in the growth rate of perturbation at an early time, and the eigenvector corresponding to the unstable eigenvalue coincides with the most unstable part of the flow. We demonstrate that a large structure of rolling-up vortex cores appears in the vortex strip after a long-time evolution. Furthermore, the geophysical relevance of the model to jet streams of Jupiter, Saturn and Earth is examined. [ABSTRACT FROM AUTHOR]

Dissipation of enstrophy in 2D incompressible flows in the zero viscous limit is considered to play a significant role in the emergence of the inertial range corresponding to the forward enstrophy cascade in the energy spectrum of 2D turbulent flows. However, since smooth solutions of the 2D incompressible Euler equations conserve the enstrophy, we need to consider non-smooth inviscid and incompressible flows so that the enstrophy dissipates. Moreover, it is physically uncertain what kind of a flow evolution gives rise to such an anomalous enstrophy dissipation. In this paper, in order to acquire an insight about the singular phenomenon mathematically as well as physically, we consider a dispersive regularization of the 2D Euler equations, known as the Euler- $$\alpha $$ equations, for the initial vorticity distributions whose support consists of three points, i.e., three $$\alpha $$ -point vortices, and take the $$\alpha \rightarrow 0$$ limit of its global solutions. We prove with mathematical rigor that, under a certain condition on their vortex strengths, the limit solution becomes a self-similar evolution collapsing to a point followed by the expansion from the collapse point to infinity for a wide range of initial configurations of point vortices. We also find that the enstrophy always dissipates in the sense of distributions at the collapse time. This indicates that the triple collapse is a mechanism for the anomalous enstrophy dissipation in non-smooth inviscid and incompressible flows. Furthermore, it is an interesting example elucidating the emergence of the irreversibility of time in a Hamiltonian dynamical system. [ABSTRACT FROM AUTHOR]

Owing to non-constant curvature and a handle structure, it is not easy to imagine intuitively how flows with vortex structures evolve on a toroidal surface compared with those in a plane, on a sphere and a flat torus. In order to cultivate an insight into vortex interactions on this manifold, we derive the evolution equation for N-point vortices from Green's function associated with the Laplace-Beltrami operator there, and we then formulate it as a Hamiltonian dynamical system with the help of the symplectic geometry and the uniformization theorem. Based on this Hamiltonian formulation, we show that the 2-vortex problem is integrable. We also investigate the point vortex equilibria and the motion of two-point vortices with the strengths of the same magnitude as one of the fundamental vortex interactions. As a result, we find some characteristic interactions between point vortices on the torus. In particular, two identical point vortices can be locally repulsive under a certain circumstance. [ABSTRACT FROM AUTHOR]

We consider Hamiltonian vector fields with a dipole singularity satisfying the slip boundary condition in two-dimensional multiply connected domains. One example of such Hamiltonian vector fields is an incompressible and inviscid flow in exterior multiply connected domains with a uniform flow, whose Hamiltonian is called the stream function. Here, we are concerned with topological structures of the level sets of the Hamiltonian, which we call streamlines by analogy from incompressible fluid flows. Classification of structurally stable streamline patterns has been considered in Yokoyama and Sakajo (2013), where a procedure to assign a unique sequence of words, called the maximal word, to these patterns is proposed. Thanks to this procedure, we can identify every streamline pattern with its representing sequence of words up to topological equivalence. In the present paper, based on the theory of word representations, we propose a combinatorial method to provide a list of possible transient structurally unstable streamline patterns between two different structurally stable patterns by simply comparing their maximal word representations without specifying any Hamiltonian. Although this method cannot deal with topological streamline changes induced by bifurcations, it reveals the existence of many non-trivial global transitions in a generic sense. We also demonstrate how the present theory is applied to fluid flow problems with vortex structures. [ABSTRACT FROM AUTHOR]

A i B 475 b , 20180666 (26 April 2019; Published online 3 April 2019) (doi:10.1098/rspa.2018.0666) The constant HT ht in the modified Liouville equation was incorrect. Hence, the exact solution satisfies the modified Liouville equation, HT ht for HT ht with HT ht , where HT ht and HT ht denote the Laplace-Beltrami operator and the Gauss curvature of the toroidal surface HT ht . [Extracted from the article]

The purpose of this study was to gain a deeper understanding of the postpartum experiences of older primiparas at the time of their postpartum hospital stay. Participants were Japanese primiparas aged 35-44 years who gave birth, without abnormalities for either the mother or child. Data were obtained using a semi-structured interview with 22 participants. Thirteen themes emerged through content analysis. We found that there were distinctive experiences, which included 'the health and growth of the baby: relief and continued anxiety', 'awareness of strengths and of concerns of primiparas', 'meaning of the pregnancy and childbirth experience' and 'developing one's own child-care approach'. Findings suggest that nursing care during the post-delivery hospital stay should use the unique strengths of older primiparas to enable their smooth transition to maternal role attainment. [ABSTRACT FROM AUTHOR]

We study numerically the long-time evolution of the surface quasi-geostrophic equation with generalized viscosity of the form (-Δ)α, where global regularity has been proved mathematically for the subcritical parameter range α ≥ 1/2. Even in the supercritical range, we have found numerically that smooth evolution persists, but with a slow damping in the long run. A subtle balance between nonlinear and dissipative terms is observed therein. Notably, qualitative behaviours of the analytic properties of the solution do not change in the super and subcritical ranges, suggesting the current theoretical boundary α = 1/2 is of a technical nature. [ABSTRACT FROM AUTHOR]

Abstract: The paper provides symmetric fixed configurations of point vortices in multiply connected domains in the unit circle with many circular obstacles. When the circular domain is invariant with respect to rotation around the origin by a degree of , a regular -polygonal ring configuration of identical point vortices becomes a fixed equilibrium. On the other hand, when we assume a special symmetry, called the folding symmetry, on the circular domain, we find a fixed equilibrium in which point vortices with the positive unit strength and point vortices with the negative unit strength are arranged alternately at the vertices of a -polygon. We also investigate the stability of these fixed equilibria and their bifurcation for a special circular domain with the rotational symmetry as well as the folding symmetry. Furthermore, we discuss fixed equilibria in non-circular multiply connected domains with the same symmetries. We give sufficient conditions for the conformal mappings, by which fixed equilibria in the circular domains are mapped to those in the general multiply connected domains. Some examples of such conformal mappings are also provided. [Copyright &y& Elsevier]

We answer the question of whether optimal packings of circles on a sphere are equilibrium solutions to the logarithmic particle interaction problem for values of N =3-12 and 24, the only values of N for which the optimal packing problem (also known as the Tammes problem) has rigorously known solutions. We also address the cases N =13-23 where optimal packing solutions have been computed, but not proven analytically. As in Jamaloodeen & Newton (Jamaloodeen & Newton 2006 Proc. R. Soc. Lond. Ser. A 462, 3277-3299. (doi:10.1098/rspa.2006.1731)), a logarithmic, or point vortex equilibrium is determined by formulating the problem as the one in linear algebra, AG =0, where A is a N(N - 1)/2 × N non-normal configuration matrix obtained by requiring that all interparticle distances remain constant. If A has a kernel, the strength vector G ? RN is then determined as a right-singular vector associated with the zero singular value, or a vector that lies in the nullspace of A where the kernel is multi-dimensional. First we determine if the known optimal packing solution for a given value of N has a configuration matrix A with a non-empty nullspace. The answer is yes for N =3-9, 11-14, 16 and no for N =10, 15, 17-24. We then determine a basis set for the nullspace of A associated with the optimally packed state, answer the question of whether N-equal strength particles, G =(1, 1, 1, …, 1)T, form an equilibrium for this configuration, and describe what is special about the icosahedral configuration from this point of view. We also find new equilibria by implementing two versions of a random walk algorithm. First, we cluster sub-groups of particles into patterns during the packing process, and 'grow' a packed state using a version of the 'yin-yang' algorithm of Longuet-Higgins (Longuet-Higgins 2008 Proc. R. Soc. A (doi:10.1098/rspa.2008.0219)). We also implement a version of our 'Brownian ratchet' algorithm to find new equilibria near the optimally packed state for N =10, 15, 17-24. [ABSTRACT FROM AUTHOR]

We study numerically the long-time evolution of the surface quasi-geostrophic equation with generalized viscosity of the form ([?]^)a, where global regularity has been proved mathematically for the subcritical parameter range a [?] 1/2. Even in the supercritical range, we have found numerically that smooth evolution persists, but with a very slow and oscillatory damping in the long run. A subtle balance between nonlinear and dissipative terms is observed therein. Notably, qualitative behaviours of the analytic properties of the solution do not change in the super and subcritical ranges, suggesting the current theoretical boundary a = 1/2 is of technical nature. [ABSTRACT FROM AUTHOR]

This is a review article of recent research developments on the motion of a polygonal ring configuration of vortex structures with singular vorticity distributions in incompressible and inviscid flows on a non-rotating sphere. Numerical computation of a single vortex sheet reveals that the Kelvin-Helmholtz instability gives rise to the formation of a polygonal ring arrangement of rolling-up spirals. An application of methods of Hamiltonian dynamics to the N-vortex problem on the sphere shows that the motion of the ring configuration of homogeneous point vortices, which is a simple model for the rolling-up spirals, becomes chaotic after a long time evolution. Some remarks on an extension of the present research and a generic non-self-similar collapse are also provided. [ABSTRACT FROM AUTHOR]

The paper gives the equation of motion for N point vortices in a bounded planar multiply connected domain inside the unit circle that contains many circular obstacles, called the circular domain. The velocity field induced by the point vortices is described in terms of the Schottky–Klein prime function associated with the circular domain. The explicit representation of the equation enables us not only to solve the Euler equations through the point-vortex approximation numerically, but also to investigate the interactions between localized vortex structures in the circular domain. As an application of the equation, we consider the motion of two point vortices with unit strength and of opposite signs. When the multiply connected domain is symmetric with respect to the real axis, the motion of the two point vortices is reduced to that of a single point vortex in a multiply connected semicircle, which we investigate in detail. [ABSTRACT FROM AUTHOR]

Abstract: A fast and accurate algorithm to compute interactions between point vortices and between vortex blobs on a sphere is proposed. It is an extension of the fast tree-code algorithm developed by Draghicescu for the vortex method in the plane. When we choose numerical parameters in the fast algorithm suitably, the computational cost of is reduced to and the approximation error decreases like when , as demonstrated in the present article. We also apply the fast method to long-time evolution of two vortex sheets on the sphere to see the efficiency. A key point is to describe the equation of motion for the points in the three-dimensional Cartesian coordinates. [Copyright &y& Elsevier]

We describe a Brownian ratchet scheme that we use to calculate relative equilibrium configurations of N point vortices of mixed strength on the surface of a unit sphere. We formulate it as a problem in linear algebra, AΓ=0, where A is a N(N−1)/2×N non-normal configuration matrix obtained by requiring that all inter-vortical distances on the sphere remain constant and Γ∈RN is the (unit) vector of vortex strengths that must lie in the null space of A. Existence of an equilibrium is expressed by the condition det(ATA)=0, while uniqueness follows if Rank(A)=N−1. The singular value decomposition of A is used to calculate an optimal basis set for the null space, yielding all values of the vortex strengths for which the configuration is an equilibrium and allowing us to decompose the equilibrium configuration into basis components. To home in on an equilibrium, we allow the point vortices to undergo a random walk on the sphere and, after each step, we compute the smallest singular value of the configuration matrix, keeping the new arrangement only if it decreases. When the smallest singular value drops below a predetermined convergence threshold, the existence criterion is satisfied and an equilibrium configuration is achieved. We then find a basis set for the null space of A, and hence the vortex strengths, by calculating the right singular vectors corresponding to the singular values that are zero. We show a gallery of examples of equilibria with one-dimensional null spaces obtained by this method. Then, using an unbiased ensemble of 1000 relative equilibria for each value N=4→10, we discuss some general features of the statistically averaged quantities, such as the Shannon entropy (using all of the normalized singular values) and Frobenius norm, centre-of-vorticity vector and Hamiltonian energy. [ABSTRACT FROM AUTHOR]

We study the motion of N point vortices with N∈ℕ on a sphere in the presence of fixed pole vortices, which are governed by a Hamiltonian dynamical system with N degrees of freedom. Special attention is paid to the evolution of their polygonal ring configuration called the N -ring, in which they are equally spaced along a line of latitude of the sphere. When the number of the point vortices is N=5 n or 6 n with n∈ℕ, the system is reduced to a two-degree-of-freedom Hamiltonian with some saddle-center equilibria, one of which corresponds to the unstable N-ring. Using a Melnikov-type method applicable to two-degree-of-freedom Hamiltonian systems with saddle-center equilibria and a numerical method to compute stable and unstable manifolds, we show numerically that there exist transverse homoclinic orbits to unstable periodic orbits in the neighborhood of the saddle-centers and hence chaotic motions occur. Especially, the evolution of the unstable N-ring is shown to be chaotic. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with complicated behavior in the vortex problem on a sphere, which is reduced to three-degree-of-freedom Hamiltonian systems. In the reduced Hamiltonians, the polygonal ring configuration of the point vortices becomes a saddle-center equilibrium which has two hyperbolic and four center directions in some parameter regions. Near the saddle center, there exists a normally hyperbolic, locally invariant manifold including a Cantor set of whiskered tori. For we numerically compute the stable and unstable manifolds of the locally invariant manifold with the assistance of the center manifold technique, and show that they intersect transversely and complicated dynamics may occur. Direct numerical simulations are also given to demonstrate our numerical analysis. [Copyright &y& Elsevier]

We study the evolution of N-point vortices in ring formation embedded in a background flowfield that initially corresponds to solid-body rotation on a sphere. The evolution of the point vortices is tracked numerically as an embedded dynamical system along with the M contours which separate strips of constant vorticity. The full system is a discretization of the Euler equations for incompressible flow on a rotating spherical shell, hence a ‘barotropic’ model of the one-layer atmosphere. We describe how the coupling creates a mechanism by which energy is exchanged between the ring and the background, which ultimately serves to break up the structure. When the centre-of-vorticity vector associated with the ring is initially misaligned with the axis of rotation of the background field, it sets up the propagation of Rossby waves around the sphere which move retrograde to the solid-body rotation. These waves pass energy to the ring (in the case when the solid-body field and the ring initially co-rotate) or extract energy from the ring (when the solid-body field and the ring initially counter-rotate), hence the Hamiltonian and the centre-of-vorticity vector associated with the N-point vortices are no longer conserved as they are for the one-way coupled model described by Newton & Shokraneh. In the first case, energy is transferred to the ring, the length of the centre-of-vorticity vector increases, while its tip spirals in a clockwise manner towards the North Pole. The ring stays relatively intact for short times, but ultimately breaks-up on a longer time-scale. In the latter case, energy is extracted from the ring, the length of the centre-of-vorticity vector decreases while its tip spirals towards the North Pole and the ring loses its coherence more quickly than in the co-rotating case. The special case where the ring is initially oriented so that its centre-of-vorticity vector is perpendicular to the axis of rotation is also examined as it shows how the coupling to the background field breaks this symmetry. In this case, both the length of the centre-of-vorticity vector and the Hamiltonian energy of the ring achieve a local maximum at roughly the same time. [ABSTRACT FROM AUTHOR]

We consider the motion of N vortex points on sphere, called the N-vortex problem, which is a Hamiltonian dynamical system. The three-vortex problem is integrable and its motion has already been resolved. On the other hand, when the moment of vorticity vector, which consists of weighed sums of the vortex positions, is zero at the initial moment, the four-vortex problem is integrable, but it has not been investigated yet. The present paper gives a description of the integrable four-vortex problem with the reduction method to a three-vortex problem used by Aref and Stremler. Moreover, we examine whether the vortex points collide self-similarly in finite time. The four-vortex collapse is proved to be impossible. We consider if it is possible for not all but part of the vortex points to collapse self-similarly. Moreover, we discuss the topological structure of periodic orbits obtained in the present problem. [ABSTRACT FROM AUTHOR]

Abstract: We are concerned with the system of vortex points on a sphere with two fixed vortex points at the poles. This article gives a reduction method of the system to invariant dynamical systems when all the vortex points have the same strength. It is carried out by considering the invariant property of the system with respect to the shift and pole reversal transformations, for which the polygonal ring configuration of the vortex points at the line of latitude, called the “-ring”, remains unchanged. We prove that there exists a -dimensional invariant dynamical system reduced by the -shift transformation for an arbitrary factor of . The -shift invariant system is equivalent to the -vortex-points system generated by the averaged Hamiltonian with the modified pole vortices. It is also shown that the system can be reduced by the pole reversal transformation when the pole vortices are identical. Since the reduced dynamical systems are defined in the linear space spanned by the eigenvectors given in the linear stability analysis for the -ring, we obtain the inclusion relation among the invariant reduced dynamical systems. This allows us to decompose the system of a large number of vortex points into a collection of invariant reduced subsystems. [Copyright &y& Elsevier]

We consider the particle mixing in the plane by two vortex points appearing one after the other, called the blinking vortex system. Mathematical and numerical studies of the system reveal that the chaotic particle mixing, i.e., the chaotic advection, is observed due to the homoclinic chaos, but the mixing region is restricted locally in the neighborhood of the vortex points. The present article shows that it is possible to realize a global and efficient chaotic advection in the blinking vortex system with the help of the Thurston–Nielsen theory, which classifies periodic orbits for homeomorphisms in the plane into three types: periodic, reducible, and pseudo-Anosov (pA). It is mathematically shown that periodic orbits of pA type generate a complicated dynamics, which is called topological chaos. We show that the combination of the local chaotic mixing due to the topological chaos and the dipole-like return orbits realize an efficient and global particle mixing in the blinking vortex system. [ABSTRACT FROM AUTHOR]

Abstract: We study the motion of a polygonal ring consists of identical vortex points that are equally spaced at a line of latitude on a sphere with vortex points fixed at the both poles. First, we calculate explicitly all the eigenvalues and the eigenvectors corresponding to them for the linearized problem, from which we consider the stability of the polygonal vortex ring in the presence of the pole vortices. Next, when the number of the vortex points is even in particular, the equations of the vortex points are reduced to those for a pair of two vortex points by assuming a special symmetry. Studying the reduced system mathematically and numerically, we describe an universal transition of global periodic motion of the perturbed polygonal ring. Moreover, we also discuss the stability of the periodic motion. [Copyright &y& Elsevier]

We consider the motion of a vortex sheet on the surface of a unit sphere in the presence of point vortices fixed on north and south poles. Analytic and numerical research revealed that a vortex sheet in two-dimensional space has the following three properties. First, the vortex sheet is linearly unstable due to Kelvin–Helmholtz instability. Second, the curvature of the vortex sheet diverges in finite time. Last, the vortex sheet evolves into a rolling-up doubly branched spiral, when the equation of motion is regularized by the vortex method. The purpose of this paper is to investigate how the curvature of the sphere and the presence of the pole vortices affect these three properties mathematically and numerically. We show that some low spectra of disturbance become linearly stable due to the pole vortices and thus the singularity formation tends to be delayed. On the other hand, however, the vortex sheet, which is regularized by the vortex method, acquires complex structure of many rolling-up spirals. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

We consider an axisymmetric, swirling vortex sheet in an inviscid and incompressible flow. Caflisch et al. pointed out that the vortex sheet acquired a singularity in finite time, but the property of the singularity was not revealed. In the present paper we show convincing numerical evidences of the singularity formation by applying the same numerical methods as what was used in the study of a two-dimensional (2D) vortex sheet. We find that the radial and axial components of the axisymmetric vortex sheet behave like the 2D singularity that has been observed in many vortex-sheet motions, while the azimuthal component of the sheet behaves differently. Furthermore, the singularity appears along the vortex line and the first derivative of the vortex sheet strength forms a cusp, while the known singularities are associated with the curvature along curves perpendicular to the vortex lines and the sheet strength has a cusp. [ABSTRACT FROM AUTHOR]

The theme issue surveys recent advances at the intersection between Applied and Computational Complex Analysis (ACCA) and the study of different nonlinear phenomena. While complex analysis has traditionally played an important role in the development of certain areas of physics, such as fluid and solid mechanics, in recent years there has been a resurgence of interest in new applications of complex analysis to many different areas. This renaissance has been largely driven by new development in complex analysis which include, but are not limited to, connections with classical, harmonic and asymptotic analysis, new methods based on integral transforms and conformal maps, as well as progress in the Wiener–Hopf and Riemann–Hilbert problems, These advances have been buttressed by parallel development of computational methods reflecting the fact that complex analysis has also come to play an important role in numerical analysis. In fluid mechanics, vortex dynamics continues to drive many new applications of complex analysis. New connections have been discovered between the Riemann–Hilbert problem and integrable hierarchies of partial differential equations (PDEs), Painlevé transcendents as well as problems in statistical mechanics. Rational approximations, orthogonal polynomials, tracking complex singularities in PDEs and conformal mappings have created new opportunities in the study of nonlinear problems. [ABSTRACT FROM AUTHOR]

A nearly full-length cDNA clone for catalase (pCAS01) was obtained through immunological screening of eDNA expression library constructed from size-fractionated poly(A)-rich RNA of wounded sweet potato tuberous roots by Escherichia coli expression vector-primed cDNA synthesis. Two additional catalase cDNA clones (pCAS10 and pCAS13), which contained eDNA inserts slightly longer than that of pCAS01 at their 5'-termini, were identified by colony hybridization of another eDNA library. Those three catalase cDNAs contained primary structures not identical, but closely related, to one another based on their restriction enzyme and RNase cleavage mapping analyses, suggesting that microheterogeneity exists in catalase mRNAs. The eDNA insert of pCAS13 carried the entire catalase coding capacity, since the RNA transcribed in vitro from the eDNA under the SP6 phage promoter directed the synthesis of a catalase polypeptide in the wheat germ in vitro translation assay. The nucleotide sequencing of these catalase cDNAs indicated that 1900-base catalase mRNA contained a coding region of 1476 bases. The amino acid sequence of sweet potato catalase deduced from the nucleotide sequence was 35 amino acids shorter than rat liver catalase [Furuta, S., Hayashi, H., Hijikata, M., Miyazawa, S., Osumi, T. & Hashimoto, T. (1986) Proc. Natl Acad. Sci. USA 83, 313-317]. Although these two sequences showed only 38% homology, the sequences around the amino acid residues implicated in catalytic function, heme ligand or heine contact had been well conserved during evolution. [ABSTRACT FROM AUTHOR]

We consider relative equilibrium solutions of the two-dimensional Euler equations in which the vorticity is concentrated on a union of finite-length vortex sheets. Using methods of complex analysis, more specifically the theory of the Riemann–Hilbert problem, a general approach is proposed to find such equilibria which consists of two steps: first, one finds a geometric configuration of vortex sheets ensuring that the corresponding circulation density is real-valued and also vanishes at all sheet endpoints such that the induced velocity field is well-defined; then, the circulation density is determined by evaluating a certain integral formula. As an illustration of this approach, we construct a family of rotating equilibria involving different numbers of straight vortex sheets rotating about a common center of rotation and with endpoints at the vertices of a regular polygon. This equilibrium generalizes the well-known solution involving single rotating vortex sheet. With the geometry of the configuration specified analytically, the corresponding circulation densities are obtained in terms of a integral expression which in some cases lends itself to an explicit evaluation. It is argued that as the number of sheets in the equilibrium configuration increases to infinity, the equilibrium converges in a certain distributional sense to a hollow vortex bounded by a constant-intensity vortex sheet, which is also a known equilibrium solution of the two-dimensional Euler equations. • Developed a novel approach to find relative equilibria of vortex sheets. • Found a new family of rotating equilibria of vortex sheets. • The new family generalizes a known solution and has interesting properties. [ABSTRACT FROM AUTHOR]

Aim: We evaluated the effectiveness of a nursing intervention program for Chinese women who are giving birth in Japan to reduce cross‐cultural stressors during the postpartum period and prevent postpartum depressive symptoms. Methods: A prospective, before‐and‐after study incorporating a longitudinal mixed‐method design was conducted. Thirty‐eight participants underwent this program from August 2016 to July 2017. The program comprised a maternity class, conversation cards, and a social‐network group. Data collection was initially performed using questionnaires administered in the third trimester (T1)—before the intervention—to obtain participants' basic information, stress levels, depressive symptoms, and cognitive appraisals. Then, stress levels, depressive symptoms, and social support were measured during hospitalization after having given birth (T2) and during the first month postpartum (T3). Finally, through semi‐structured interviews, cognitive appraisal, coping, stress, social support, participants' evaluations of the intervention were determined. Results: Post‐intervention, all participants showed positive cognitive appraisals, although eight also showed some negative appraisals. At T3, 36 participants did not report experiencing stress owing to cross‐cultural stressors. Furthermore, post‐intervention, participants who returned scores that were suggestive of depression remained identical to that at pre‐intervention (21.1%). Among the eight participants who showed postpartum depressive symptoms during T3, seven did not report experiencing cross‐cultural stressors, but did report encountering maternity stressors. Conclusion: The nursing intervention program may be effective for preventing postpartum depressive symptoms in Chinese women who give birth in Japan. Since this was a pre‐post study in which one group was measured pre‐intervention and again post‐intervention, we did not register in a publicly assessible database. [ABSTRACT FROM AUTHOR]

As a minimal mathematical model generating cascade analogous to that of the Navier-Stokes turbulence in the inertial range, we propose a one-dimensional partial-differential-equation model that conserves the integral of the squared vorticity analog (enstrophy) in the inviscid case. With a large-scale random forcing and small viscosity, we find numerically that the model exhibits the enstrophy cascade, the broad energy spectrum with a sizable correction to the dimensional-analysis prediction, peculiar intermittency, and self-similarity in the dynamical system structure. [ABSTRACT FROM AUTHOR]

This paper details a generalized method to, within some numerical tolerance, compute vortex equilibria in the presence of many obstacles. Given a conformal mapping between a pre-image circular domain and the physical domain, stationary conditions for the point vortices and the desired Kutta conditions are constructed and then solved using a Brownian Ratchets scheme. The method is applied to a Kasper Wing (three plate) configuration and results are compared to those of a single plate. The lift experienced by the main plate is seen to be sensitive to the placement of the two additional plates but their presence can be force enhancing for certain configurations. [ABSTRACT FROM AUTHOR]

This study considers the flow of incompressible and inviscid fluid in two-dimensional multiply connected domains. For such flows, encoding algorithms to assign a unique sequence of words to any structurally stable streamline topology based on the theory presented by Yokoyama and Sakajo (2013 Proc. R. Soc. A 469 20120558) are proposed. As an application, we utilize the algorithms to characterize the evolution of an incompressible and viscid flow around a flat plate inclined to the uniform flow in terms of the change of the word representations for their instantaneous streamline topologies. [ABSTRACT FROM AUTHOR]

Let us consider incompressible and inviscid flows in two-dimensional domains with multiple obstacles. The instantaneous velocity field becomes a Hamiltonian vector field defined from the stream function, and it is topologically characterized by the streamline pattern that corresponds to the contour plot of the stream function. The present paper provides us with a procedure to construct structurally stable streamline patterns generated by finitely many point vortices in the presence of the uniform flow. Starting from some basic structurally stable streamline patterns in a disc of low genus, i.e. a disc with a small number of holes, we repeat some fundamental operations that append a streamline pattern by increasing one genus to them. Owing to the inductive procedure, one can assign a sequence of operations as a representing word to each structurally stable streamline pattern. We also give the canonical expression for the word representation, which allows us to make a catalogue of all possible structurally stable streamline patterns in a combinatorial manner. As an example, we show all streamline patterns in the discs of genus 1 and 2. [ABSTRACT FROM AUTHOR]

Aim To examine the relationship between physical symptoms and depressive symptoms among new mothers during the first 6 months postpartum. Methods Prospective cohort study design was used in this study. Participants were recruited in 13 Japanese hospitals between 2012 and 2013. A total of 2709 women (response rate of those invited 71.9%) provided longitudinal data using self-report questionnaires at 5 time points: during the hospital stay and at 1, 2, 4, and 6 months postpartum. Depressive symptoms were measured using the Japanese version of the Edinburgh Postnatal Depression Scale. Also, 37 physical symptoms were measured using a yes-or-no checklist. To examine the association between physical symptoms and depressive symptoms, logistic regression analysis was used. Results There was a significant positive association between physical symptoms and depressive symptoms during the first 6 months postpartum. Fifteen physical symptoms were associated with significantly increased odds of depression at 1, 2, 4, and 6 months post-partum. These symptoms included: loss of appetite, tiredness/languor, and palpitation/shortness of breath. Conclusion Multiple physical symptoms after childbirth were associated with depressive symptomatology. Health professionals should assess for postpartum depression when puerperal women report multiple physical symptoms as independent complaints. [ABSTRACT FROM AUTHOR]

Background: Depressive symptoms are common in postpartum women. The present study aimed to describe changes in the prevalence of depressive symptoms during the first 6 months postpartum, and their association with maternal age and parity.Methods: A prospective cohort study was conducted with 3769 women at 13 hospitals in Japan. Depressive symptoms were measured during hospital stay and at 1, 2, 4, and 6 months postpartum, using the Japanese version of the Edinburgh Postnatal Depression Scale (EPDS). The effects of maternal age and parity were assessed by comparing four groups: younger primiparas (aged <35 years); older primiparas (≥35 years); younger multiparas (<35 years); and older multiparas (≥35 years). Data were analyzed using a mixed between/within-subjects analysis of variance, Cochran's Q tests, and chi-square tests.Results: Mean EPDS scores significantly decreased from 1 to 2 months postpartum in all groups. The proportion of women with EPDS scores ≥9 significantly decreased during the same period for primiparas but not for multiparas. Primiparas also had significantly higher EPDS scores than multiparas during hospital stay and at 1 month postpartum.Limitations: As we used convenience sampling, our study sample was not fully representative of Japanese mothers. This study was also limited by our focus on the postpartum period.Conclusions: The first month postpartum represented peak prevalence for depressive symptoms. Primiparity was a risk factor for depressive symptoms only during the first month postpartum. Healthcare professionals should be sensitive to postpartum duration and parity when monitoring depressive symptoms. [ABSTRACT FROM AUTHOR]

This study examined the psychometric properties of a 13-item fatigue scale for postpartum mothers. Japanese mothers ( n = 2026) from a cohort study completed questionnaires (e.g. fatigue scale, Japanese version of the Edinburgh Postnatal Depression Scale, demographics) during their hospital stay after childbirth (baseline) and at 1, 2, 4 and 6 months postpartum. Initial factor analysis of baseline data revealed that the fatigue scale had three factors or subscales (physical, emotional and cognitive). Within-group analysis across each measurement time revealed the same three-factor structure with acceptable fit. Between-group analysis also showed longitudinal factorial invariance across time. The fatigue subscales had acceptable divergent and convergent validities with the depression scale. The subscale scores differed significantly based on participant background. The Japanese Fatigue Scale is a concise and informative tool for assessing aspects of fatigue in clinical settings and in the community. [ABSTRACT FROM AUTHOR]

The purpose of this mixed-method design study was to examine factors contributing to depression among immigrant Chinese women (primipara and multipara) ( n = 22) delivering a child for the first time in Japan. Data were obtained just after hospital discharge by using the Edinburgh Postnatal Depression Scale (EPDS), the Social Support Scale, a new scale to measure cross-cultural stressors in the postpartum setting and a visual analogue scale for stress and a demographic survey. The average EPDS score was 9.0 (SD ± 3.7) at 1-3 weeks postpartum; yet, more than half of the subjects ( n = 12; 54.5%) were high risk for depression (EPDS ≥ 10). Low household income and primiparous status were associated with depression scores. New mothers with depression also reported more general stress and more cross-cultural stress in the postpartum setting, although social support appeared to mediate cross-cultural stressors. Semi-structured interviews were held with two immigrant women at high risk for depression; these new mothers described additional stress because they could not follow Zuoyuezi, an important postpartum Chinese tradition, in the Japanese hospital. These findings suggest that immigrant Chinese women are at higher risk for postpartum depression when they give birth for the first time in Japan. [ABSTRACT FROM AUTHOR]

Aim: Post‐partum fatigue is a serious concern for mothers worldwide. The aim of this study was to identify age‐specific determinants of self‐perceived fatigue after childbirth among Japanese primiparous women. Methods: A multicenter prospective cohort study was conducted in Japan. The degree of fatigue was assessed using the Postnatal Accumulated Fatigue Scale during the hospital stay after childbirth. Associated factors were stratified into two age groups (<35 or ≥35 years old); data were analyzed using stepwise multiple regression. Results: In the younger group, significant determinants of post‐partum fatigue included: anticipated difficulties in child‐rearing after discharge; perception of inadequate sleep conditions; dissatisfaction with childbirth; shorter sleep and meal times; longer duration between feeding and infant bedtime; unemployment; and having hyperemesis gravidarum. In the older group, significant determinants of post‐partum fatigue included: anticipated difficulties in child‐rearing after discharge; perception of inadequate sleep conditions; shorter meal times; formula feeding; and dissatisfaction with childbirth. Conclusion: Primiparous mothers who anticipated new life with their infants to be difficult and who had an irregular lifestyle tended to report higher levels of fatigue during the post‐partum hospital stay. Nurses and midwives should assess maternal sleep and eating conditions, providing support to ensure adequate maternal rest. Special efforts should be made to reduce anxiety about new life with infants among younger primiparous women who had experienced hyperemesis gravidarum. [ABSTRACT FROM AUTHOR]

This study aimed to describe the postpartum experiences of older Japanese primiparas during the first month after childbirth. The participants were 21 primiparous women over 35 years of age who gave birth to a healthy child at three urban hospitals. Data were collected from July 2011 to April 2012 through a semi-structured interview about postpartum experiences after discharge. Data were analysed using content analysis. We obtained Institutional Review Boards' approval and written informed consent from all participants prior to study initiation. Thirteen themes of postpartum experiences were extracted. The findings revealed that the mothers' experiences varied greatly and were highly personal. Also, participants described common experiences, unique child-care support needs and the strengths of older primiparas. Understanding the postpartum experiences of older primiparas assists health-care professionals to identify better ways to provide appropriate support. [ABSTRACT FROM AUTHOR]