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Waves are mainly generated by wind via the transfer of wind energy to the water through friction. When the wind subsides, the waves transition into swells and eventually dissipate. Friction plays a crucial role in the generation and dissipation of waves. Numerous wave theories have been developed based on the assumption of inviscid flow, but these theories are inadequate in explaining the transformation of waves into swells. This study addressed these limitations by analytically deriving general solutions to the Navier–Stokes equations. By expressing the velocity field as the product of a solution to the Helmholtz equation and a time-dependent univariate function, the Navier–Stokes equations are decomposed into an ordinary differential equation and the Euler equations, which are solved using tensor calculus. This paper provides solutions for viscous flow with shear currents when applied to the water wave problem. These solutions were validated through their application to the vorticity equation. The decay modulus of water waves was compared with experimental data, showing a significant degree of concordance. In contrast to other wave theories, this study clarified the process through which waves evolve into swells.

This paper is aimed at eliciting consistency conditions for the existence of unsteady incompressible axisymmetric swirling viscous Beltrami vortices and explicitly constructing solutions that obey the conditions as well as the Navier–Stokes equations. By Beltrami flow, it is meant that vorticity, i.e., the curl of velocity, is proportional to velocity at any local point in space and time. The consistency conditions are derived for the proportionality coefficient, the velocity field and external force. The coefficient, whose dimension is of [length−1], is either constant or nonconstant. In the former case, the well-known exact nondivergent three-dimensional unsteady vortex solutions are obtained by solving the evolution equations for the stream function directly. In the latter case, the consistency conditions are given by nonlinear equations of the stream function, one of which corresponds to the Bragg–Hawthorne equation for steady inviscid flow. Solutions of a novel type are found by numerically solving the nonlinear constraint equation at a fixed time. Time dependence is recovered by taking advantage of the linearity of the evolution equation of the stream function. The proportionality coefficient is found to depend on space and time. A phenomenon of partial restoration of the broken scaling invariance is observed at short distances.

We explored here the case of three-dimensional non-stationary flows of helical type for the incompressible couple stress fluid with given Bernoulli-function in the whole space (the Cauchy problem). In our presentation, the case of non-stationary helical flows with constant coefficient of proportionality α between velocity and the curl field of flow is investigated. In the given analysis for this given type of couple stress fluid flows, an absolutely novel class of exact solutions in theoretical hydrodynamics is illuminated. Conditions for the existence of the exact solution for the aforementioned type of flows were obtained, for which non-stationary helical flow with invariant Bernoulli-function satisfying to the Laplace equation was considered. The spatial and time-dependent parts of the pressure field of the fluid flow should be determined via Bernoulli-function if components of the velocity of the flow are already obtained. Analytical and numerical findings are outlined, including outstanding graphical presentations of various types of constructed solutions, in order to elucidate dynamic snapshots that show the timely development of the topological behavior of said solutions. [ABSTRACT FROM AUTHOR]

This paper is aimed at eliciting consistency conditions for the existence of unsteady incompressible axisymmetric swirling viscous Beltrami vortices and explicitly constructing solutions that obey the conditions as well as the Navier–Stokes equations. By Beltrami flow, it is meant that vorticity, i.e., the curl of velocity, is proportional to velocity at any local point in space and time. The consistency conditions are derived for the proportionality coefficient, the velocity field and external force. The coefficient, whose dimension is of [length−1], is either constant or nonconstant. In the former case, the well-known exact nondivergent three-dimensional unsteady vortex solutions are obtained by solving the evolution equations for the stream function directly. In the latter case, the consistency conditions are given by nonlinear equations of the stream function, one of which corresponds to the Bragg–Hawthorne equation for steady inviscid flow. Solutions of a novel type are found by numerically solving the nonlinear constraint equation at a fixed time. Time dependence is recovered by taking advantage of the linearity of the evolution equation of the stream function. The proportionality coefficient is found to depend on space and time. A phenomenon of partial restoration of the broken scaling invariance is observed at short distances. [ABSTRACT FROM AUTHOR]

We explored here the case of three-dimensional non-stationary flows of helical type for the incompressible couple stress fluid with given Bernoulli-function in the whole space (the Cauchy problem). In our presentation, the case of non-stationary helical flows with constant coefficient of proportionality α between velocity and the curl field of flow is investigated. In the given analysis for this given type of couple stress fluid flows, an absolutely novel class of exact solutions in theoretical hydrodynamics is illuminated. Conditions for the existence of the exact solution for the aforementioned type of flows were obtained, for which non-stationary helical flow with invariant Bernoulli-function satisfying to the Laplace equation was considered. The spatial and time-dependent parts of the pressure field of the fluid flow should be determined via Bernoulli-function if components of the velocity of the flow are already obtained. Analytical and numerical findings are outlined, including outstanding graphical presentations of various types of constructed solutions, in order to elucidate dynamic snapshots that show the timely development of the topological behavior of said solutions.

The arrangements of invariant tori that resemble rod packings with cubic symmetries are considered in three-dimensional solenoidal vector fields. To find them systematically, vector fields whose components are represented in the form of multiple Fourier series with finite terms are classified using magnetic groups. The maximal magnetic group compatible with each arrangement is specified on the assumption that the cores of the nested invariant tori are straight and located on the lines corresponding to the central axes of the rods packed. Desired rod-packing arrangements are demonstrated by selecting vector fields whose magnetic groups are the maximal ones and by drawing their integral curves that twine around invariant tori. In the demonstration of chiral arrangements, Beltrami flows (or force-free fields in plasma physics), which have the strongest chirality of all solenoidal vector fields satisfying the same vector Helmholtz equation, are used. As by-products, several chain-like arrangements of closed invariant tori were found. One of the chains consists of knotted invariant tori. In all vector fields (chiral or achiral) selected for the demonstration, the volume percentages of ordered regions formed by invariant tori in a unit cell were roughly measured with the aid of a supervised machine learning technique. [ABSTRACT FROM AUTHOR]

Recently, the Nash-style convex integration has been becoming the main scheme for the mathematical study of turbulence, and the main building block of it has been either Beltrami flow (finite mode) or Mikado flow (compactly supported in the physical side). On the other hand, in physics, it is observed that turbulence is composed of a hierarchy of scale-by-scale vortex stretching. Thus our mathematical motivation in this study is to find another type of building blocks accompanied by vortex stretching and scale locality (possibly finitely many Fourier modes). In this paper, we give a complete list of solutions to the 3D Euler equations with finitely many Fourier modes, which is an extension of the corresponding 2D result by Elgindi et al. (Comm Math Phys 355(1): 145–159, 2017). In particular, we show that there are no 3D Euler flows with finitely many Fourier modes, except for stationary 2D-like flows and Beltrami flows. We also discuss the case when viscosity and Coriolis effect are present. [ABSTRACT FROM AUTHOR]

Can every physical system simulate any Turing machine? This is a classical problem that is intimately connected with the undecidability of certain physical phenomena. Concerning fluid flows, Moore [C. Moore, Nonlinearity 4, 199 (1991)] asked if hydrodynamics is capable of performing computations. More recently, Tao launched a program based on the Turing completeness of the Euler equations to address the blow-up problem in the Navier-Stokes equations. In this direction, the undecidability of some physical systems has been studied in recent years, from the quantum gap problem to quantum-field theories. To the best of our knowledge, the existence of undecidable particle paths of threedimensional fluid flows has remained an elusive open problem since Moore's works in the early 1990s. In this article, we construct a Turing complete stationary Euler flow on a Riemannian S3 and speculate on its implications concerning Tao's approach to the blow-up problem in the Navier-Stokes equations. [ABSTRACT FROM AUTHOR]

Magnetic groups are applied to three‐dimensional fluid flows with chirality, which are called Beltrami flows (or force‐free fields in plasma physics). First, six Beltrami flows are derived so that their symmetries and antisymmetries are described by six different hexagonal magnetic groups. The general Wyckoff positions are used to derive the flows. Special Wyckoff positions are shown to be useful for finding the zero points of the flows. Tube‐like surfaces called invariant tori are observed to interlace and form various crystal‐like structures when streamlines winding around the surfaces are numerically plotted. Next, two simpler hexagonal Beltrami flows are derived, and their zero points and invariant tori are studied. Some families of the invariant tori have arrangements similar to those observed in materials science. [ABSTRACT FROM AUTHOR]

In this paper, a new framework for iterative speckle noise reduction in polarimetric synthetic aperture radar (SAR) data is introduced. Speckle is inherent to all coherent imaging systems and affects SAR imagery in the form of strong intensity variations in pixels with similar backscattering coefficients. This makes the interpretation of SAR data in several applications a difficult task. The proposed framework includes a preprocessing step capable of dealing with noise correlation usually found in single-look data. The general filtering approach is based on the Beltrami flow for denoising manifolds or images painted on manifolds. The principal contribution of this work is to adapt this approach to deal with covariance or coherency matrices instead of optical imagery. The evaluation presented suggests that this approach allows for good spatial and radiometric preservation compared to other state-of-the-art methods. Experiments are performed on the basis of synthetic and real-world experimental data. The validation of the proposed framework is accomplished using two refined error performance measures and the well-known effective number of looks measured. The source code of a parallel implementation of the proposed framework is released under the MPL 2.0 (https://www.mozilla.org/en-US/MPL/2.0/) alongside this paper. [ABSTRACT FROM AUTHOR]

Abstract The Arnold–Beltrami–Childress (ABC) flow is the best-known example of a Beltrami flow (or a force-free field in plasma physics) periodic in three orthogonal directions. When its three parameters are equalised, its set of streamlines has the same crystallographic symmetry as a model of a blue phase (BP) of cholesteric liquid crystals. This commonality appears in the similarity between the arrangements of invariant tori in the ABC flow and of double-twist cylinders in the BP model. In this paper, three Beltrami flows whose sets of streamlines have the same crystallographic symmetries as three other BP models are derived by using cubic space groups. The first and the third Beltrami flows have invariant tori arranged similarly to double-twist cylinders in two of the BP models. The second Beltrami flow has invariant tori whose arrangement has been neither theoretically nor experimentally known in liquid crystals. Heteroclinic orbits in the three flows are also studied. [ABSTRACT FROM AUTHOR]

In this note, we obtain uniqueness results for Beltrami flow in both bounded and unbounded domain with nonempty boundary by establishing an elementary but useful formula involving operators div and curl. We also use this formula to deal with Maxwell and Stokes eigenvalue problems. [ABSTRACT FROM AUTHOR]

In this paper, we establish the global regularity of classical solutions for the two-dimensional MHD equations with only velocity diffusion for a class of large initial data. Both the initial velocity and magnetic field can be arbitrarily large in H s . [ABSTRACT FROM AUTHOR]

This paper presents a novel solution to a two-dimensional Beltrami flow through hexagonal channel of a dusty fluid evaluation. The uniqueness of the solution is focused on characterizing the flow behavior of fluid and dust phases by considering the effect of Reynolds number and permeability of porous medium. The governing Navier–Stokes equations are transformed into dimensionless equations and solved using Laplace transform, Fourier transform and Crank–Nicolson techniques. A detailed comparison of analytical and numerical solutions are presented and results are interpreted through performance analysis graphs. Finally, the skin friction at the boundaries are calculated to understand the phase of dusty fluid in the porous medium. [ABSTRACT FROM AUTHOR]

Beltrami flows with crystallographic cubic symmetry were demonstrated numerically to have knots or links of invariant tori, which are not only stream tubes but also vortex tubes. The cubic symmetry is associated with the ABC flow with its three parameters equalised. The knots formed by invariant tori are the trefoil knot, the granny knot, true-lovers' knot, etc. The linked invariant tori are shown to form a variety of chainlike structures. • Invariant tori in Beltrami flows with cubic symmetry are considered. • Knotted invariant tori of the trefoil knot type etc. are demonstrated. • A variety of chainlike structures of linked invariant tori are shown. [ABSTRACT FROM AUTHOR]

In this paper, we establish the global existence of smooth solutions of the three-dimensional MHD system for a class of large initial data. Both the initial velocity and magnetic field can be arbitrarily large in the critical norm. [ABSTRACT FROM AUTHOR]

The Beltrami flow is an efficient nonlinear filter, that was shown to be effective for color image processing. The corresponding anisotropic diffusion operator strongly couples the spectral components. Usually, this flow is implemented by explicit schemes, that are stable only for very small time steps and therefore require many iterations. In this paper we introduce a semi-implicit Crank-Nicolson scheme based on locally one-dimensional (LOD)/additive operator splitting (AOS) for implementing the anisotropic Beltrami operator. The mixed spatial derivatives are treated explicitly, while the non-mixed derivatives are approximated in an implicit manner. In case of constant coefficients, the LOD splitting scheme is proven to be unconditionally stable. Numerical experiments indicate that the proposed scheme is also stable in more general settings. Stability, accuracy, and efficiency of the splitting schemes are tested in applications such as the Beltrami-based scale-space, Beltrami denoising and Beltrami deblurring. In order to further accelerate the convergence of the numerical scheme, the reduced rank extrapolation (RRE) vector extrapolation technique is employed. [ABSTRACT FROM AUTHOR]

Based on expansion terms of the Beltrami-flow type, we use multiscale methods to effectively construct an asymptotic expansion at large Reynolds numbers R for the long-wavelength perturbation of the nonstationary anisotropic helical solution of the force-free Navier-Stokes equation (the Trkal solution). We prove that the systematic asymptotic procedure can be implemented only in the case where the scaling parameter is R. Projections of quasistationary large-scale streamlines on a plane orthogonal to the anisotropy direction turn out to be the gradient lines of the energy density determined by the initial conditions for two modulated anisotropic Beltrami flows (modulated as a result of scaling) with the same eigenvalues of the curl operator. The three-dimensional streamlines and the curl lines, not coinciding, fill invariant vorticity tubes inside which the velocity and vorticity vectors are collinear up to terms of the order of 1/ R. [ABSTRACT FROM AUTHOR]

In this paper, inverse solutions are obtained for the class of 2D steady incompressible couple stress fluid flows. This class consists of flows for which the vorticity distribution is given by ▽2 ψ= ψ+ ƒ( x,y). The solutions are obtained by applying the inverse method which makes certain hypotheses regarding the form of the velocity field and pressure but without making any regarding the boundaries of the domain occupied by the fluid. Inverse solutions are derived for three different forms of ƒ( x,y). [ABSTRACT FROM AUTHOR]

An analytical study of Beltrami flow of viscous dusty fluid between two parallel plates has been studied. The flow is due to influence of movement of plates. Flow analysis is carried out using differential geometry techniques and exact solutions of the problem are obtained using Laplace Transform technique also which are discussed with the help of graphs drawn for different values of Reynolds number. Further the expressions for skin-friction are obtained at the boundaries. [ABSTRACT FROM AUTHOR]

In many instances, numerical integration of space-scale PDEs is the most time consuming operation of image processing. This is because the scale step is limited by conditional stability of explicit schemes. In this work, we introduce the unconditionally stable semi-implicit linearized difference scheme that is fashioned after additive operator split (AOS), for Beltrami and the subjective surface computation. The Beltrami flow is one of the most effective denoising algorithms in image processing. For gray-level images, we show that the flow equation can be arranged in an advection-diffusion form, revealing the edge-enhancing properties of this flow. This also suggests the application of AOS method for faster convergence. The subjective surface deals with constructing a perceptually meaningful interpretation from partial image data by mimicking the human visual system. However, initialization of the surface is critical for the final result and its main drawbacks are very slow convergence and the huge number of iterations required. In this paper, we first show that the governing equation for the subjective surface flow can be rearranged in an AOS implementation, providing a near real-time solution to the shape completion problem in 2D and 3D. Then, we devise a new initialization paradigm where we first "condition" the viewpoint surface using the Fast-Marching algorithm. We compare the original method with our new algorithm on several examples of real 3D medical images, thus revealing the improvement achieved. [ABSTRACT FROM AUTHOR]

One kind of tensor denotation of nth Beltrami axisymmetric and nonaxisymmetric spherical vortices, their classification and symmetries were discussed. Chaotic phenomena will occur in the dynamic system of the nonaxisymmetric Beltrami spherical vortices. From these aspects, it is shown that the tensor denotation has more meaningful characters and nonaxisymmetric Beltrami shperical vortices are various and very complex. [ABSTRACT FROM AUTHOR]