Piecewise Constant Vorticity Research Articles

Solitary solutions to the steady Euler equations with piecewise constant vorticity in a channel

We consider a two-dimensional, two-layer, incompressible, steady flow, with vorticity which is constant in each layer, in an infinite channel with rigid walls. The velocity is continuous across the interface, there is no surface tension or difference in density between the two layers, and the flow is inviscid. Unlike in previous studies, we consider solutions which are localised perturbations rather than periodic or quasi-periodic perturbations of a background shear flow. We rigorously construct a curve of exact solutions and give the leading order terms in an asymptotic expansion. We also give a thorough qualitative description of the fluid particle paths, which can include stagnation points, critical layers, and streamlines which meet the boundary.

Discontinuous differential equation for modelling the Antarctic Circumpolar Current

<p>In this paper, we were concerned with the existence of the solution related to the discontinuous differential equation, which corresponded to the stratification phenomenon in the Antarctic Circumpolar Current (ACC). By considering the piecewise vorticity function, we demonstrated the existence of solution corresponding to the discontinuous differential equation using Green's function, fixed point theory, and topological degree theory. This primarily included cases with piecewise constant vorticity, piecewise linear vorticity, and piecewise nonlinear vorticity. Additionally, we provided some examples to verify our results.</p>

Superharmonic and triadic resonances in a horizontally oscillated stably stratified square cavity

The response to harmonic horizontal oscillations of a stably stratified fluid-filled two-dimensional square container is examined as the forcing amplitude is increased. For the studied forcing frequency, the response flow at very small forcing amplitudes is a synchronous periodic flow with piecewise-constant vorticity in regions delineated by the characteristics emanating from the corners of the container, regularized by viscosity. The second temporal harmonic of the forced response flow resonantly excites an intrinsic mode of the stratified container, whose magnitude grows as the square of the forcing amplitude. Above a critical forcing amplitude, a sequence of pairs of other container modes are excited via triadic resonances with the second-harmonic-driven mode. The flows are computed from the Navier–Stokes–Boussinesq equations and the ensuing dynamics is analysed using Fourier techniques, providing a comprehensive picture of the transition to internal wave turbulence.

On rotational flows with discontinuous vorticity beneath steady water waves near stagnation

Abstract

On the determination of the number of positive and negative polynomial zeros and their isolation

Abstract A novel method with two variations is proposed with which the number of positive and negative zeros of a polynomial with real coefficients and degree n can be restricted with significantly better determinacy than that provided by the Descartes rule of signs. The method also allows the isolation of the zeros of the polynomial quite successfully, and the determined root bounds are significantly narrower than the Cauchy and the Lagrange bounds. The method relies on solving equations of degree smaller than that of the given polynomial. One can determine analytically the exact number of positive and negative zeros of a polynomial of degree up to and including five and also fully isolate the zeros of the polynomial analytically, and with one of the variations of the method, one can analytically approach polynomials of degree up to and including nine by solving equations of degree not more than four. For polynomials of higher degree, either of the two variations of the method should be applied recursively. Numerous examples are given. Presented is the full classification of the roots of the cubic equation, together with their isolation intervals – especially important for various scientific models for which the coefficients of the equation might be functions of the model parameters. An application of the method to a quartic equation with variable coefficients (resulting from the study of wave–current interactions in the physically realistic model of azimuthal two-dimensional non-viscous flows with piecewise constant vorticity in a two-layer fluid with a flat bed and a free surface) has been demonstrated. A step-by-step algorithm for each version of the method has also been presented.

Resonances for water waves over flows with piecewise constant vorticity

We investigate three and four-wave resonances of capillary–gravity water waves arising as the free surface of water flows exhibiting piecewise constant vorticity. More precisely, our type of flow has a jump in the vorticity distribution that separates a rotational layer at the top (commonly generated by wind-shear) from another rotational layer adjacent to the sea-bed, region that accommodates strong sheared currents. Instrumental in deriving our findings is the dispersion relation for such flows having a jump in the distribution of the vorticity. In general, for rotational flows of non-constant vorticity, the dispersion relation is very intricate. However, we show that a disentanglement occurs in the case of capillary–gravity water waves for which the ratio “thickness of the near-surface vortical layer/the wavelength of the surface wave” is sufficiently large. More precisely, we find explicitly two solutions, λ0 and λ1 that represent the (relative) surface wave speed. We then confirm analytically that λ1 gives rise to three-wave resonances for capillary–gravity water waves with wavelengths not exceeding 2 cm. However, for significant wavelength range, we establish that λ1 does not lead to four-wave resonances. In contrast to the previous conclusion λ0 does not bring about three-wave resonances, but is able to generate four-wave resonances.

Equatorial Wave\u2013Current Interactions

We study the nonlinear equations of motion for equatorial wave–current interactions in the physically realistic setting of azimuthal two-dimensional inviscid flows with piecewise constant vorticity in a two-layer fluid with a flat bed and a free surface. We derive a Hamiltonian formulation for the nonlinear governing equations that is adequate for structure-preserving perturbations, at the linear and at the nonlinear level. Linear theory reveals some important features of the dynamics, highlighting differences between the short- and long-wave regimes. The fact that ocean energy is concentrated in the long-wave propagation modes motivates the pursuit of in-depth nonlinear analysis in the long-wave regime. In particular, specific weakly nonlinear long-wave regimes capture the wave-breaking phenomenon while others are structure-enhancing since therein the dynamics is described by an integrable Hamiltonian system whose solitary-wave solutions are solitons.

Dispersion relations for steady periodic water waves of fixed mean-depth with two rotational layers

The aim of this paper is to obtain the dispersion relation for small-amplitude periodic travelling water waves propagating over a flat bed with a specified mean depth under the presence of a discontinuous piecewise constant vorticity. An analysis of the dispersion relation for a model with two rotational layers each having a non-zero constant vorticity is presented. Moreover, we present a stability result for the bifurcation inducing laminar flow solutions.

Periodic equatorial water flows from a Hamiltonian perspective

The main result of this paper is a Hamiltonian formulation of the nonlinear governing equations for geophysical periodic stratified water flows in the equatorial f-plane approximation allowing for piecewise constant vorticity.

Eroding dipoles and vorticity growth for Euler flows in : axisymmetric flow without swirl

A review of analyses based upon anti-parallel vortex structures suggests that structurally stable dipoles with eroding circulation may offer a path to the study of vorticity growth in solutions of Euler’s equations in $\mathbb{R}^{3}$. We examine here the possible formation of such a structure in axisymmetric flow without swirl, leading to maximal growth of vorticity as $t^{4/3}$. Our study suggests that the optimizing flow giving the $t^{4/3}$ growth mimics an exact solution of Euler’s equations representing an eroding toroidal vortex dipole which locally conserves kinetic energy. The dipole cross-section is a perturbation of the classical Sadovskii dipole having piecewise constant vorticity, which breaks the symmetry of closed streamlines. The structure of this perturbed Sadovskii dipole is analysed asymptotically at large times, and its predicted properties are verified numerically. We also show numerically that if mirror symmetry of the dipole is not imposed but axial symmetry maintained, an instability leads to breakup into smaller vortical structures.

Hamiltonian Formulation for Wave-Current Interactions in Stratified Rotational Flows

We show that the Hamiltonian framework permits an elegant formulation of the nonlinear governing equations for the coupling between internal and surface waves in stratified water flows with piecewise constant vorticity.

A Hamiltonian approach to wave-current interactions in two-layer fluids

We provide a Hamiltonian formulation for the governing equations describing the two-dimensional nonlinear interaction between coupled surface waves, internal waves, and an underlying current with piecewise constant vorticity, in a two-layered fluid overlying a flat bed. This Hamiltonian structure is a starting point for the derivation of simpler models, which can be obtained systematically by expanding the Hamiltonian in dimensionless parameters. These enable an in-depth study of the coupling between the surface and internal waves, and how both these wave systems interact with the background current.

Optimal Excitation of Asymmetric Perturbations on an Axisymmetric Barotropic Vortex: A Linear Singular-Value Analysis

Abstract Singular vectors on a barotropic circular vortex consisting of three regions of piecewise-constant vorticity are investigated under the L2 norm to reveal the shape and growth rate of possible perturbations that may contribute to the formation of the hierarchal structure seen in natural and artificial vortices. Here, the analytical form of the singular values and the forward and backward singular vectors are derived for comparison with the corresponding eigenvalues and vectors. The results indicate no notable difference between the growing eigenvector and the singular vectors for Michalke and Timme’s profile, which is characterized by a ring of high vorticity. In contrast, for Syōno’s profile, which incorporates a negative vorticity region between a solidly rotating core and potentially rotating surroundings, the pattern of the forward singular vector giving the maximum growth in a specified time interval differs crucially from that of the growing eigenvector: whereas the radial peak of the streamfunction exists solely at the maximum-wind radius of the basic vortex in the growing eigenvector, it shifts outward in the growing forward singular vector. This implies the injection of considerable energy into the outer part of the basic vortex, rather than in the inner ring of the highest wind, facilitating the formation of hierarchal structure concentrated around the core region of the vortex.

Dispersion relations for gravity water flows with two rotational layers

We derive the dispersion relation for periodic traveling water waves propagating at the surface of water possessing a layer of constant non-zero vorticity γ1 adjacent to the free surface above another rotational layer of vorticity γ2 which is adjacent to the flat bed. As a by-product we give necessary and sufficient condition for local bifurcation in the frame-work of piecewise constant vorticity. Moreover, we give estimates on the speed at the free surface of the bifurcating laminar flows. These estimates involve only the vorticity γ1, the mean depth of water d and the depth at which the jump in vorticity occurs. A stability result for certain laminar flows is also presented.

Existence of capillary-gravity water waves with piecewise constant vorticity

In this paper we construct small-amplitude periodic capillary-gravity water waves with a piecewise constant vorticity distribution. They describe water waves traveling on superposed linearly sheared currents that have different vorticities. This is achieved by associating to the height function formulation of the water wave problem a diffraction problem where we impose suitable transmission conditions on each line where the vorticity function has a jump. The solutions of the diffraction problem, found by using local bifurcation theory, are the desired solutions of the hydrodynamical problem.

Dispersion relations for periodic water waves with surface tension and discontinuous vorticity

We derive the dispersion relation for water waves with surface tension and having a piecewise constant vorticity distribution. More precisely, we consider here two scenarios; the first one is that of a flow with constant non-zero vorticity adjacent to the flat bed while above this layer of vorticity we assume the flow to be irrotational. The second type of flow has a layer of non-vanishing vorticity adjacent to the free surface and is irrotational below.

Baroclinic multipole formation from heton interaction

In a two-layer quasi-geostrophic model, the interaction between two opposite-signed hetons (baroclinic vortex pairs) is studied analytically and numerically, for singular and finite-area vortices.For point vortices, using trilinear coordinates, it is shown that the possible evolutions depend on the deformation radius Rd: for large Rd, the layers decouple, vortices pair in each layer and their trajectories are open; for medium Rd, the exchange of opposite-sign partners between layers becomes possible; for small Rd, two other regimes appear: one where hetons remain unaltered during their evolution but follow open trajectories, and one where hetons occupy only a bounded subdomain of space at all times. Conditions for invariant co-rotation of the heton pair are derived and analyzed.Then, the nonlinear evolutions of finite-area heton pairs, with piecewise-constant vorticity, are computed with contour dynamics. When the central cyclonic vortex is initially aligned vertically, a transition occurs between three nonlinear regimes as layer coupling increases: for weak coupling, the vortices pair horizontally and drift away in opposite directions; for moderate layer coupling, the core vortex splits into two parts, one of which remains as a tilted columnar vortex at the center; for stronger layer coupling, each anticyclone pairs with part of the cyclone in each layer, thus forming an L-shaped dipole, a new coherent structure of two-layer flows. When the initial distance between the central and satellite vortices is increased, the velocity shear at the center decreases and the central vortex remains vertically aligned, thus forming a Z-shaped tripole, also a newly observed vortex compound. Such tripoles also compete with oscillating states, in which the core vortex periodically aligns and tilts, a regime observed when layer coupling is moderate and as vortices become closer in each layer. This Z-shaped tripole forms for various values of stratification and of initial distances between vortices, and is therefore a robust vortex compound in two-layer quasi-geostrophic flows.

Interaction of shear flows of an ideal incompressible fluid in a channel

The problem of the decay of an arbitrary discontinuity for the equations describing plane-parallel shear flows of an ideal fluid in a narrow channel is considered. The class of particular solutions corresponding to fluid flows with piecewise constant vorticity is studied. In this class, the existence of self-similar solutions describing all possible unsteady wave configurations resulting from the nonlinear interaction of the specified shear flows is established.

Unsteady interaction of uniformly vortex flows

The problem of the decay of an arbitrary discontinuity (the Riemann problem) for the system of equations describing vortex plane-parallel flows of an ideal incompressible liquid with a free boundary is studied in a long-wave approximation. A class of particular solutions that correspond to flows with piecewise-constant vorticity is considered. Under certain restrictions on the initial data of the problem, it is proved that this class contains self-similar solutions that describe the propagation of strong and weak discontinuities and the simple waves resulting from the nonlinear interaction of the specified vortex flows. An algorithm for determining the type of resulting wave configurations from initial data is proposed. It extends the known approaches of the theory of one-dimensional gas flows to the case of substantially two-dimensional flows.

Contour Dynamics for the Euler Equations in Two Dimensions

We present a contour dynamics algorithm for the Euler equations of fluid dynamics in two dimensions. This is applied to regions of piecewise-constant vorticity withinfinite-area-vortexregions (FAVRs). Essentially, this reduces the dimensionality by one and we are treating the interaction of closed polygonal contours whose nodes are advected by the total fluid motion computed self-consistently. A leapfrog centered scheme is used for temporal advancement. Computer simulation results are given for two and four like-signed interacting FAVRs. In all cases wavelike surface deformations are observed. If the distance between FAVRs is comparable to their extent (“diameter”), these surface deformations are large. They play an essential role in the observed coalescence of FAVRs.