Non-zero Vorticity Research Articles

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.

Mathematical Model of Gases Flux in Boundary Layer

Analytical dynamic model of the turbulent flux in the three-layer boundary system is presented. Turbulence is described as a presence of the non-zero vorticity. The generalized advection-diffusion-reaction equation is derived for an arbitrary number of components in the flux. The fluxes in the layers are objects for matching requirements on the boundaries between the layers. Different types of transport mechanisms are dominant on the different levels of the layers.

Vorticity in Green House Gases Flux Model in Boundary Layer

Mathematical model of the turbulent flux in the three-layer boundary system is presented. Turbulence is described as a presence of the nonzero vorticity. Generalized advection-diffusion-reaction equation is derived for arbitrary number components in the flux. The fluxes in the layers are objects for matching requirements on the boundaries between the layers.

COSMIC VORTICITY AND THE ORIGIN HALO SPINS

In the standard model of cosmology, structure emerges out of non-rotational flow and the angular momentum of collapsing halos is induced by tidal torques. The growth of halo angular momentum in the linear and quasi-linear phases is associated with a shear, curl-free, flow and it is well described within the linear framework of tidal torque theory (TTT). However, TTT is rendered irrelevant as haloes approach turn around and virialization. At that stage the flow field around halos has non-zero vorticity. Using a cosmological simulation, we have examined the importance of the curl of the velocity field (vorticity) in determining halo spin, finding a strong alignment between the two. We have also examined the alignment of vorticity with the principle axes of the shear tensor, finding that it tends to be perpendicular to the axis along which material is collapsing fastest (e1). This behavior is independent of halo masses and cosmic web environment. Our results agree with previous findings on the tendency of halo spin to be perpendicular to e1, and of the spin of (simulated) halos and (observed) galaxies to be aligned with the large-scale structure. Our results imply that angular momentum growth proceeds in two distinct phases. In the first phase angular momentum emerges out of a shear, curl-free, potential flow, as described by TTT. In the second phase, in which haloes approach virialization, the angular momentum emerges out of a vortical flow and halo spin becomes strongly aligned with the vorticity of the ambient flow field.

A nonlinear Schrödinger equation for water waves on finite depth with constant vorticity

A nonlinear Schrödinger equation for the envelope of two dimensional surface water waves on finite depth with non-zero constant vorticity is derived, and the influence of this constant vorticity on the well-known stability properties of weakly nonlinear wave packets is studied. It is demonstrated that vorticity modifies significantly the modulational instability properties of weakly nonlinear plane waves, namely the growth rate and bandwidth. At third order, we have shown the importance of the nonlinear coupling between the mean flow induced by the modulation and the vorticity. Furthermore, it is shown that these plane wave solutions may be linearly stable to modulational instability for an opposite shear current independently of the dimensionless parameter kh, where k and h are the carrier wavenumber and depth, respectively.

On the wave breaking phenomena for the generalized periodic two-component Dullin-Gottwald-Holm system

Considered herein is the generalized two-component periodic Dullin-Gottwald-Holm system, which can be derived from the Euler equation with nonzero constant vorticity in shallow water waves moving over a linear shear flow. The precise blow-up scenarios of strong solutions and several results of blow-up solutions with certain initial profiles are described in detail. The exact blow-up rates are also determined. Finally, a sufficient condition for global solutions is established.

Dispersion relations for steady periodic water waves with an isolated layer of vorticity at the surface

In this article we obtain the dispersion relation for small-amplitude steady periodic water waves, which propagate over a flat bed with a specified and fixed mean depth, and where the underlying flow has a discontinuous vorticity distribution. This discontinuity takes the form of an isolated layer of constant non-zero vorticity at the surface of the flow, with purely irrotational flow beneath this layer.

The second variation of the stream function energy of water waves with vorticity

We compute the second variation of the stream function energy of two-dimensional steady free surface gravity water waves with vorticity in the stream function formulation. We prove that for nonpositive vorticity the second variation of the stream function energy at extreme waves with Stokes corner asymptotics cannot be nonnegative in any small neighbourhood of a given isolated stagnation point. The particular form of our second variation suggests however the possibility that certain singularities in the case of nonzero vorticity might be constructible as minimizers of the stream function energy.

Eastern boundary drainage of the North Atlantic subtropical gyre

The eastern boundary of the North Atlantic subtropical gyre (NASG) is an upwelling favorable region characterized by a mean southward flow. The Canary Upwelling Current (CUC) feeds from the interior ocean and flows south along the continental slope off NW Africa, effectively providing the eastern boundary condition for the NASG. We follow a joint approach using slope and deep-ocean data together with process-oriented modeling to investigate the characteristics and seasonal variability of the interior–coastal ocean connection, focusing on how much NASG interior water drains along the continental slope. First, the compiled sets of data show that interior central waters flow permanently between Madeira and the Iberian Peninsula at a rate of 2.5 ± 0.6 Sv (1 Sv = 106 m3 s-1 109 km s-1), with most of it reaching the slope and shelf regions north of the Canary Islands (1.5 ± 0.7 Sv). Most of the water entering the African slope and shelf regions escapes south between the easternmost Canary Islands and the African coast: In 18 out of 22 monthly realizations, the flow was southward (−0.9 ± 0.4 Sv) although an intense flow reversal occurred usually around November (1.7 ± 0.9 Sv), probably as the result of a late fall intensification of the CUC north of the Canary Islands followed by instability and offshore flow diversion. Secondly, we explore how the eastern boundary drainage may be specified in a process-oriented one-layer quasigeostrophic numerical model. Non-zero normal flow and constant potential vorticity are alternative eastern boundary conditions, consistent with the idea of anticyclonic vorticity induced at the boundary by coastal jets. These boundary conditions cause interior water to exit the domain at the boundary, as if recirculating through the coastal ocean, and induce substantial modifications to the shape of the eastern NASG. The best model estimate for the annual mean eastward flow north of Madeira is 3.9 Sv and at the boundary is 3.3 Sv. The water exiting at the boundary splits with 1 Sv flowing into the Strait of Gibraltar and the remaining 2.3 Sv continuing south along the coastal ocean until the latitude of Cape Ghir. The model also displays significant wind-induced seasonal variability, with a maximum connection between the interior and coastal oceans taking place in autumn and winter, in qualitative agreement with the observations.

The nonlinear future stability of the FLRW family of solutions to the Euler–Einstein system with a positive cosmological constant

In this article, we study small perturbations of the family of Friedmann–Lemaître–Robertson–Walker cosmological background solutions to the 1 + 3 dimensional Euler–Einstein system with a positive cosmological constant. These background solutions describe an initially uniform quiet fluid of positive energy density evolving in a spacetime undergoing accelerated expansion. Our nonlinear analysis shows that under the equation of state \({p = c^2_s \rho}\) , \({0 < c^2_s < 1/3}\) , the background solutions are globally future-stable. In particular, we prove that the perturbed spacetime solutions, which have the topological structure \({[0,\infty) \times \mathbb{T}^3}\) , are future-causally geodesically complete. These results are extensions of previous results derived by the author in a collaboration with I. Rodnianski, in which the fluid was assumed to be irrotational. Our novel analysis of a fluid with non-zero vorticity is based on the use of suitably defined energy currents.

Models for the formation of a critical layer in water wave propagation

A theory is presented which provides a model for the appearance of critical layers within the flow below a water wave. The wave propagates over constant depth, with constant (non-zero) vorticity. The mechanism described here involves adjusting the surface-pressure boundary condition; two models are discussed. In the first, the pressure at the surface is controlled (mimicking the movement of a low-pressure region associated with a storm) so that the speed and development of the pressure region ensure the appearance of a critical layer. In the second, the pressure boundary condition is allowed to accommodate the reduction of pressure with altitude, although the effects have to be greatly enhanced for this mechanism to produce a critical layer. These two problems are analysed using formal parameter asymptotics. In the second problem, this leads to a Korteweg-de Vries equation for the surface wave, and then the evolution of appropriate solutions of this equation gives rise to the appearance of a critical layer near the bottom; the corresponding problem at the surface can be formulated but not completely resolved. The appearance of a stagnation point and then a critical layer, either at the surface or the bottom, are discussed; the nature of the flow, and the corresponding streamlines are obtained and some typical flow fields are depicted.

Vorticity forces on an impulsively started finite plate

AbstractIn this study, we consider various contributions to the forces on an impulsively started finite plate from the perspective of a diagnostic vorticity force theory. The wing plate has an aspect ratio (AR) between 1 and 3, and is placed at low and high angles of attack ($\ensuremath{\alpha} \leq $ and ${\gt }2{0}^{\ensuremath{\circ} } $), while the Reynolds number is either 100 or 300. The theory enables us to quantify the contributions to the forces exerted on the plate in terms of all of the fluid elements with non-zero vorticity, such as in the tip vortices (TiVs), leading- and trailing-edge vortices (LEV and TEV) as well on the plate surface. This line of force analysis has been pursued for two-dimensional flow in our previous studies. In contrast to the pressure force analysis (PFA), the vorticity force analysis (VFA) reveals new salient features in its applications to three-dimensional flow by examining sectional force contributions along the spanwise direction. In particular, at a large aspect ratio ($\mathit{AR}= 3$), the force distributions of PFA and VFA show close agreements with each other in the middle sections, while at a lower aspect ratio ($\mathit{AR}= 1$), the force distribution of PFA is substantially larger than that of VFA in most of the sections. The difference is compensated for by the contributions partly by the edge sections and mainly by the vortices in the outer regions. Further investigation is made fruitful by decomposing the vorticity into the spanwise (longitudinal) component (the only one in two-dimensional flow) and the other two orthogonal (transverse) components. The relative importance of the force contributions credited to the transverse components in the entire flow regions as well as in the two outer regions signifies the three-dimensional nature of the flow over a finite plate. The interplay between the LEV and the TiVs at various time stages is shown to play a key role in distinguishing the force contributions for the plate with a smaller aspect ratio and that with a larger aspect ratio. The present VFA provides a better perspective for flow control by relating the forces directly to the various sources of vorticity (or vortex structures) on or near the wing plate.

DISPERSION RELATIONS FOR STEADY PERIODIC WATER WAVES OF FIXED MEAN-DEPTH WITH AN ISOLATED BOTTOM VORTICITY LAYER

In this paper we obtain the dispersion relations for small-amplitude steady periodic water waves, which propagate over a flat bed with a specified mean depth, and which exhibit discontinuous vorticity. We take as a model an isolated layer of constant nonzero vorticity adjacent to the flat bed, with irrotational flow above the layer.

Two-dimensionality of gravity water flows of constant nonzero vorticity beneath a surface wave train

We show that a free surface water flow of constant nonzero vorticity beneath a wave train and above a flat bed has to be two-dimensional and the vorticity must have only one nonzero component which points in the horizontal direction orthogonal to the direction of wave propagation. The obtained results are of relevance to studies of resonant wave train interactions in flows of constant nonzero vorticity: in contrast to irrotational flows, all wave trains have to propagate in the same direction. An important practical aspect of these considerations lies in that wave trains of constant vorticity model the interaction of swell with tidal currents, negative vorticity being appropriate for the flood current and positive vorticity for the ebb current.

Global solutions of the Navier-Stokes equations in a uniformly rotating space

We consider the Cauchy problem for the Navier-Stokes system of equations in a three-dimensional space rotating uniformly about the vertical axis with the periodicity condition with respect to the spatial variables. Studying this problem is based on expanding given and sought vector functions in Fourier series in terms of the eigenfunctions of the curl and Stokes operators. Using the Galerkin method, we reduce the problem to the Cauchy problem for the system of ordinary differential equations, which has a simple explicit form in the basis under consideration. Its linear part is diagonal, which allows writing explicit solutions of the linear Stokes-Sobolev system, to which fluid flows with a nonzero vorticity correspond. Based on the study of the nonlinear interaction of vortical flows, we find an approach that we can use to obtain families of explicit global solutions of the nonlinear problem.

Nonquasineutral current equilibria as elementary structures of plasma dynamics

A study is made of the fundamental features of current filaments with a nonzero electron vorticity Ωe ≡ B − (c/e) ▿ × pee ≠ 0 and the corresponding Lagrangian invariant Ie. Such current structures can exist on spatial scales of up to ωpi−1. It is shown that the dissipative stage of the plasma evolution and the violation of Thomson’s theorem on vorticity conservation in an electron fluid are of fundamental importance for the onset of electron current structures. A key role of the screening of electric and magnetic fields at distances on the order of the magnetic Debye radius rB = B/(4πene)—the main property of such current structures in a Hall medium with σB/(enec) ≫ 1—is stressed. Since the minimum size of a vortex structure is the London length c/ωpe, the structures under consideration correspond to the condition rB > c/ωpe or B2 > 4πnemec2, which leads to strong charge separation in the filament and relativistic electron drift. It is demonstrated that the specific energy content in current structures is high at a filament current of 10–15 kA: from 100 J/cm3 at a plasma density of 1014 cm−3 (the parameters of a lightning leader) to 107J/cm3 for a fully ionized atmospheric-pressure air. Estimates are presented showing that the Earth’s ionosphere, circumsolar space, and interstellar space are all Hall media in which current vortex structures can occur. A localized cylindrical equilibrium with a magnetic field reversal is constructed—an equilibrium that correlates with the magnetic structures observed in intergalactic space. It is shown that a magnetized plasma can be studied by using evolutionary equations for the electron and ion Lagrangian invariants Ie and Ii. An investigation is carried out of the evolution of a current-carrying plasma in a cylinder with a strong external magnetic field and with a longitudinal electron current turned on in the initial stage—an object that can serve as the simplest electrodynamic model of a tokamak. In this case, it is assumed that the plasma conductivity is low in the initial stage and then increases substantially with time. Based on the conservation of the integral momentum of the charged particles and electromagnetic field in a plasma cylinder within a perfectly conducting wall impenetrable by particles, arguments are presented in support of the generation of a radial electric field in a plasma cylinder and the production of drift ion fluxes along the cylinder axis. A hypothesis is proposed that the ionized intergalactic gas expands under the action of electromagnetic forces.

Glider observations of kinematics in a Gulf of Alaska eddy

The depth‐radial structure of a Gulf of Alaska (GoA) eddy is investigated using observations from an autonomous winged buoyancy‐driven glider, drifters, and satellite altimeters. The eddy formed when Sitka and Yakutat eddies merged in June 2005. Sitka and Yakutat eddies form in the northeastern GoA and travel westward carrying anomalous energy, temperature, and chemical properties. The glider crossed the eddy seven times from mid‐August to late October 2005. A temperature maximum, at 130 m depth near the eddy's center, is indicative of coastal water. The strongest azimuthal current is 0.35 m s−1, occurring at 270 m depth and 17.5 km radius. The eddy's Rossby (Burger) number is 0.16 (0.47). Using a model with nonzero constant potential vorticity anomaly only within each depth's radius of strongest current, a function is fit to azimuthal current. The fit explains a large percentage of the current's radial variance, with a squared correlation coefficient of ≥0.9 at the depths of strongest current (270 m) and weakest stratification (360 m). Temporal trends in azimuthal volume transport rate and depth mean potential vorticity are not observed during the glider survey.

Linear dynamics of Eady waves in the presence of horizontal shear

The problem of the instability of the flow of a stratified rotating fluid with constant vertical and horizontal shears is investigated within the framework of a quasi-geostrophic approximation. It is shown that the horizontal shear, when taken into account, leads to a qualitative change in the dynamics of Eady waves, i.e., wave solutions with zero potential vorticity. The main salient feature is related to the effect of the temporary exponential growth of the unstable waves, i.e., the growth effect in a finite time interval. This effect is manifested by an alternation of the stages of a smooth oscillating behavior (in time) with an exponential (explosive) growth of finite duration. A kinematic interpretation of the effect of the temporary exponential growth is suggested which is associated with the passage of a time-dependent wave disturbance vector across the domain of the exponential instability existing in the absence of horizontal shear. Along with the dynamics of individual Eady waves, the generation process of these waves—caused by an initial disturbance defined by one spatial Fourier harmonic—is also investigated. It is shown that this process is accompanied by the formation of nonmodal waves, with time-varying horizontal and vertical wavenumbers and nonzero potential vorticity. The interaction of the nonmodal wave with the background flow leads to an algebraic growth of the Eady wave at the initial cyclogenesis stage.

Effect of temperature on the vorticity field of an evolving water droplet

Vorticity in liquid droplets and the impact of temperature on the evolution of dripping droplets have received little attention in the published literature. Here, an investigation is conducted to determine the vorticity field within a droplet as it evolves from the onset to the breakup point and to show the impact of temperature on the vorticity field. The work is performed both experimentally and computationally. It was observed that the water leaving the nozzle grows into a cylindrical shape, forms a throat, contracts near the throat, and finally breaks up and forms an independent nearly spherical droplet. Presence of a flow recirculation within the droplet was noticed. Vorticity iso-surfaces were circumferentially asymmetric, and indicated a three-dimensional droplet behavior with rotational (non-zero vorticity) effects. Temperature enhanced the vorticity-related asymmetry and affected the pressure and velocity within the droplet.

Swimming in an inviscid fluid

We present a set of equations governing the motion of a body due to prescribed shape changes in an inviscid, planar fluid with nonzero vorticity. The derived equations, when neglecting vorticity, reduce to the model developed in Kanso et al. (J Nonlinear Sci 15:255–289, 2005) for swimming in potential flow, and are also consistent with the models developed in Borisov et al. (J Math Phys 48:1–9, 2007), Kanso and Oskouei (J Fluid Mech 800:77–94, 2008), Shasikanth et al. (Phys Fluids 14(3):1214–1227, 2002) for a rigid body interacting dynamically with point vortices. The effects of cyclic shape changes and the presence of vorticity on the locomotion of a submerged body are discussed through examples.