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The motion of two-dimensional incompressible and viscous fluids in doubly periodic domains is often used as a numerical model to investigate the dynamics and statistical properties of two-dimensional (2d) turbulence. In the study of 2d turbulence, it is important to describe interactions of coherent vortex structures of various sizes in turbulent flows, but identifying such vortex structures accurately from complex flow patterns is not easy. In this paper, we provide a classification theory for the topological structure of particle orbits generated by two-dimensional Hamiltonian flows on a flat torus $\mathbb{T}^2$, which is a mathematical model of 2d incompressible and viscous flows. Based on this theory, we show that the global orbit structure of every Hamiltonian flow is converted into a planar tree, called {\it a partially Cyclically-Ordered rooted Tree (COT)}, and its string expression (COT representation). Applying the conversion algorithm to snapshots of two-dimensional free-decaying turbulence and enstrophy cascade turbulence, we demonstrate that the topological structure of a complex flow pattern can be represented by a simple tree and a sequence of letters by the conversion algorithm, thereby extracting coherent vortex structures successfully from the viewpoint of topology.

Data assimilation is a method of uncertainty quantification to estimate the hidden true state by updating the prediction owing to model dynamics with observation data. As a prediction model, we consider a class of nonlinear dynamical systems on Hilbert spaces including the two-dimensional Navier-Stokes equations and the Lorenz '63 and '96 equations. For nonlinear model dynamics, the ensemble Kalman filter (EnKF) is often used to approximate the mean and covariance of the probability distribution with a set of particles called an ensemble. In this paper, we consider a deterministic version of the EnKF known as the ensemble transform Kalman filter (ETKF), performing well even with limited ensemble sizes in comparison to other stochastic implementations of the EnKF. When the ETKF is applied to large-scale systems, an ad-hoc numerical technique called a covariance inflation is often employed to reduce approximation errors. Despite the practical effectiveness of the ETKF, little is theoretically known. The present study aims to establish the theoretical analysis of the ETKF. We obtain that the estimation error of the ETKF with and without the covariance inflation is bounded for any finite time. In particular, the uniform-in-time error bound is obtained when an inflation parameter is chosen appropriately, justifying the effectiveness of the covariance inflation in the ETKF., Comment: 18 pages, 0 figures

We consider the rotating and translating equilibria of open finite vortex sheets with endpoints in two-dimensional potential flows. New results are obtained concerning the stability of these equilibrium configurations which complement analogous results known for unbounded, periodic and circular vortex sheets. First, we show that the rotating and translating equilibria of finite vortex sheets are linearly unstable. However, while in the first case unstable perturbations grow exponentially fast in time, the growth of such perturbations in the second case is algebraic. In both cases the growth rates are increasing functions of the wavenumbers of the perturbations. Remarkably, these stability results are obtained entirely with analytical computations. Second, we obtain and analyze equations describing the time evolution of a straight vortex sheet in linear external fields. Third, it is demonstrated that the results concerning the linear stability analysis of the rotating sheet are consistent with the infinite-aspect-ratio limit of the stability results known for Kirchhoff's ellipse (Love 1893; Mitchell & Rossi 2008) and that the solutions we obtained accounting for the presence of external fields are also consistent with the infinite-aspect-ratio limits of the analogous solutions known for vortex patches., Comment: 24 pages, 5 figures; accepted for publication in Journal of Fluid Mechanics

This paper shows that the topological structures of particle orbits generated by a generic class of vector fields on spherical surfaces, called {\it the flow of finite type}, are in one-to-one correspondence with discrete structures such as trees/graphs and sequence 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 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.

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., Comment: 22 pages, 5 figures; accepted for publication in Physica D

In this investigation we use a simple model of the dynamics of an inviscid vortex sheet given by the Birkhoff-Rott equation to obtain fundamental insights about the potential for stabilization of shear layers using feedback control. As actuation we consider two arrays of point sinks/sources located a certain distance above and below the vortex sheet and subject to the constraint that their mass fluxes separately add up to zero. First, we demonstrate using analytical computations that the Birkhoff-Rott equation linearized around the flat-sheet configuration is in fact controllable when the number of actuator pairs is sufficiently large relative to the number of discrete degrees of freedom present in the system, a result valid for generic actuator locations. Next we design a state-based LQR stabilization strategy where the key difficulty is the numerical solution of the Riccati equation in the presence of severe ill-conditioning resulting from the properties of the Birkhoff-Rott equation and the chosen form of actuation, an issue which is overcome by performing computations with a suitably increased arithmetic precision. Analysis of the linear closed-loop system reveals exponential decay of the perturbation energy and of the corresponding actuation energy in all cases. Computations performed for the nonlinear closed-loop system demonstrate that initial perturbations of nonnegligible amplitude can be effectively stabilized when a sufficient number of actuators is used. We also thoroughly analyze the sensitivity of the closed-loop stabilization strategies to the variation of a number of key parameters. Subject to the known limitations of inviscid vortex models, our findings indicate that, in principle, it may be possible to stabilize shear layers for relatively large initial perturbations, provided the actuation has sufficiently many degrees of freedom., Comment: 38 pages, 13 figures; accepted for publication in "Journal of Fluid Mechanics"

We study numerically a Constantin-Lax-Majda-De Gregorio model generalized by Okamoto, Sakajo and Wunsch, which is a model of fluid turbulence in one dimension with an inviscid conservation law. In the presence of the viscosity and two types of the large-scale forcings, we show that turbulent cascade of the inviscid invariant, which is not limited to quadratic quantity, occurs and that properties of this model's turbulent state are related to singularity of the inviscid case by adopting standard tools of analyzing fluid turbulence., Comment: 22 pages, 22 figures

Anomalous enstrophy dissipation of incompressible flows in the inviscid limit is a significant property characterizing two-dimensional turbulence. It indicates that the investigation of non-smooth incompressible and inviscid flows contributes to the theoretical understanding of turbulent phenomena. In the preceding study, a unique global weak solution to the Euler-$\alpha$ equations, which is a regularized Euler equations, for point-vortex initial data is considered, and thereby it has been shown that, as $\alpha \rightarrow 0$, 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 or not this singular orbit can be constructed independently on the regularization method, we consider a functional generalization of the Euler-$\alpha$ equations, called the \textit{Euler-Poincar\'{e} 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-$\alpha$ equations but universal within the Euler-Poincar\'{e} models., Comment: 22 pages, 4 figures

This paper concerns feedback stabilization of point vortex equilibria above an inclined thin plate and a three-plate configuration known as the Kasper Wing in the presence of an oncoming uniform flow. The flow is assumed to be potential and is modeled by the 2D incompressible Euler equations. Actuation has the form of blowing and suction localized on the main plate and is represented in terms of a sink-source singularity, whereas measurement of pressure across the plate serves as system output. We focus on point-vortex equilibria forming a one-parameter family with locus approaching the trailing edge of the main plate and show that these equilibria are either unstable or neutrally stable. Using methods of linear control theory we find that the system dynamics linearised around these equilibria are both controllable and observable for almost all actuator and sensor locations. The design of the feedback control is based on the Linear-Quadratic-Gaussian (LQG) compensator. Computational results demonstrate the effectiveness of this control and the key finding is that Kasper Wing configurations are in general more controllable than their single plate counterparts and also exhibit larger basins of attraction under LQG feedback control. The feedback control is then applied to systems with additional perturbations added to the flow in the form of random fluctuations of the angle of attack and a vorticity shedding mechanism. Another important observation is that, in the presence of these additional perturbations, the control remains robust, provided the system does not deviate too far from its original state. Furthermore, introducing a vorticity shedding mechanism tends to enhance the effectiveness of the control. Physical interpretation is provided for the results of the controllability and observability analysis as well as the response of the feedback control to different perturbations., Comment: 42 pages, 20 figures

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 analogue (enstrophy) in the inviscid case. With a large-scale 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., Comment: 5 pages, 4 figures