Non-zero Vorticity Research Articles

Long-range ordering of topological excitations in a two-dimensional superfluid far from equilibrium

We study the relaxation of a 2D ultracold Bose-gas from a nonequilibrium initial state containing vortex excitations in experimentally realizable square and rectangular traps. We show that the subsystem of vortex gas excitations results in the spontaneous emergence of a coherent superfluid flow with a non-zero coarse-grained vorticity field. The streamfunction of this emergent quasi-classical 2D flow is governed by a Boltzmann-Poisson equation. This equation reveals that maximum entropy states of a neutral vortex gas that describe the spectral condensation of energy can be classified into types of flow depending on whether or not the flow spontaneously acquires angular momentum. Numerical simulations of a neutral point vortex model and a Bose gas governed by the 2D Gross-Pitaevskii equation in a square reveal that a large scale monopole flow field with net angular momentum emerges that is consistent with predictions of the Boltzmann-Poisson equation. The results allow us to characterise the spectral energy condensate in a 2D quantum fluid that bears striking similarity with similar flows observed in experiments of 2D classical turbulence. By deforming the square into a rectangular region, the resulting maximum entropy state switches to a dipolar flow field with zero net angular momentum.

Lagrangian theory for cosmic structure formation with vorticity: Newtonian and post-Friedmann approximations

We study the nonlinear gravitational dynamics of a universe filled with a pressureless fluid and a cosmological constant $\Lambda$ in the context of Newtonian gravity, and in the relativistic post-Friedmann approach proposed in paper I [I. Milillo et al., Phys. Rev. D 92, 023519 (2015).]. The post-Friedmann approximation scheme is based on the $1/c$ expansion of the space-time metric and the energy-momentum tensor, and includes nonlinear Newtonian cosmology. Here we establish the nonlinear post-Friedmann framework in the Lagrangian-coordinates approach for structure formation. For this we first identify a Lagrangian gauge which is suitable for incorporating nonzero vorticity. We analyze our results in two limits: at the leading order we recover the fully nonlinear Newtonian cosmological equations in the Lagrangian formulation, and we provide a space-time metric consistent from the perspective of general relativity. We then linearize our expressions and recover the relativistic results at first order in cosmological perturbation theory. Therefore, the introduced approximation scheme provides a unified treatment for the two leading-order regimes, from the small scales described by Newtonian gravity to the large linear scale, where first-order relativistic cosmological perturbation theory gives a very good description of structure formation.

Symmetry Breaking Drift of Particles Settling in Homogeneous Shear Turbulence

We investigate the influence of shear on the gravitational settling of heavy inertial particles in homogeneous shear turbulence (HST). In addition to the well-known enhanced settling velocity, observed for heavy inertial particles in homogeneous isotropic turbulence (HIT), a horizontal drift velocity is also observed in the shearing direction due to the presence of a nonzero mean vorticity (introducing symmetry breaking due to the mean shear). This drift velocity is due to the combination of shear, gravity, and turbulence, and all three of these elements are needed for this effect to occur. We extend the mechanism responsible for the enhanced settling velocity in HIT to the case of HST. Two separate regimes are observed, characterized by positive or negative drift velocity, depending on the particle settling velocity.

Weakly nonlinear convective regimes in a horizontal fluid layer with poorly conducting boundaries and temperature-dependent viscosity

In this work, the influence of temperature dependence of viscosity on the weakly nonlinear regimes in a horizontal fluid layer with poorly conducting boundaries is studied. A multi-scale method is used to derive the amplitude equations, which are investigated analytically and numerically. The stability of roll patterns, square cells, hexagonal cells and quasi-periodic dodecagonal structures is investigated analytically. As a numerical method we use one of modifications of the spectral method. It has been found that only hexagonal and square patterns maintain stability, the hexagonal pattern being always stable for sufficiently large values of the Prandtl number. The calculations have shown that there is the region of co-existence of hexagonal and square patterns for a small range of the parameter, which is responsible for temperature dependence of viscosity. The hexagonal cells are always excited subcritically, whereas the square cells can be excited both in a subcritical and supercritical manner. Our investigations at small values of the Prandtl number have also revealed the instability, which is associated with nonzero vertical vorticity.

Vorticity addition method

It is shown that in the general case it is not possible to propose the Lagrangian viewpoint on the vorticity evolution, which would be unique for the entire flow, using the existing analogs of the Helmholtz theorems. This is related to the fact that, as distinct from the Helmholtz theorem oneself, these analogs are valid only for nonzero vorticity zones. New analogs of the Helmholtz theorems are proposed for the general case of flows (from incompressible fluid to viscous gas). They describe the vorticity evolution at all the points including the points of nonzero vorticity.

A Geometric Interpretation of Eddy Reynolds Stresses in Barotropic Ocean Jets

AbstractBarotropic eddy fluxes are analyzed through a geometric decomposition of the eddy stress tensor. Specifically, the geometry of the eddy variance ellipse, a two-dimensional visualization of the stress tensor describing the mean eddy shape and tilt, is used to elucidate eddy propagation and eddy feedback on the mean flow. Linear shear and jet profiles are analyzed and theoretical results are compared against fully nonlinear simulations. For flows with zero planetary vorticity gradient, analytic solutions for the eddy ellipse tilt and anisotropy are obtained that provide a direct relationship between the eddy tilt and the phase difference of a normal-mode solution. This allows a straightforward interpretation of the eddy–mean flow interaction in terms of classical stability theory: the initially unstable jet gives rise to eddies that are tilted “against the shear” and extract energy from the mean flow; once the jet stabilizes, eddies become tilted “with the shear” and return their energy to the mean flow. For a nonzero planetary vorticity gradient, ray-tracing theory is used to predict ellipse geometry and its impact on eddy propagation within a jet. An analytic solution for the eddy tilt is found for a Rossby wave on a constant background shear. The ray-tracing results broadly agree with the eddy tilt diagnosed from a fully nonlinear simulation.

On the well-posedness of relativistic viscous fluids with non-zero vorticity

We study the problem of coupling Einstein’s equations to a physically well-motivated relativistic modification of the Navier-Stokes equations. Under a technical condition for the vorticity, we prove existence and uniqueness in a suitable Gevrey class if the fluid’s dynamic velocity has vanishing divergence, and show that the solutions enjoy the finite propagation speed property.

On the dynamics of internal waves interacting with the equatorial undercurrent

The interaction of the nonlinear internal waves with a nonuniform current with a specific form, characteristic for the equatorial undercurrent, is studied. The current has no vorticity in the layer, where the internal wave motion takes place. We show that the nonzero vorticity that might be occuring in other layers of the current does not affect the wave motion. The equations of motion are formulated as a Hamiltonian system.

Consistent crystal plasticity kinematics and linearization for the implicit finite element method

Purpose – The purpose of this paper is to describe several novel techniques for implementing a crystal plasticity (CP) material model in a large deformation, implicit finite element framework. Design/methodology/approach – Starting from the key kinematic assumptions of CP, the presentation develops the necessary CP correction terms to several common objective stress rates and the consistent linearization of the stress update algorithm. Connections to models for slip system hardening are isolated from these processes. Findings – A kinematically consistent implementation is found to require a correction to the stress update to include plastic vorticity developed by slip deformation in polycrystals. A simpler, more direct form for the algorithmic tangent is described. Several numerical examples demonstrate the capabilities and computational efficiency of the formulation. Research limitations/implications – The implementation assumes isotropic slip system hardening. With simple modifications, the described approach extends readily to anisotropic coupled or uncoupled hardening functions. Practical implications – The modular formulation and implementation support streamlined development of new models for slip system hardening without modifications of the stress update and algorithmic tangent computations. This implementation is available in the open-source code WARP3D. Originality/value – In the process of developing the CP formulation, this work realized the need for corrections to the Green-Naghdi and Jaumann objective stress rates to account properly for non-zero plastic vorticity. The paper describes fully the consistent linearization of the stress update algorithm and details a new scheme to implement the model with improved efficiency.

ON THE MECHANISM OF THE FORMATION OF MAGNETOHYDRODYNAMIC VORTICES IN THE SOLAR PLASMA

Based on the magnetohydrodynamic (MHD) equations for an incompressible conductive viscous fluid, the possible mechanism of the formation of giant MHD vortices recently discovered in the solar atmosphere (chromosphere) is analyzed. It is assumed that these vortices arise in the regions of the solar surface (photosphere) with ascending flows of hot plasma that arrives from the inner regions of the Sun as a result of thermal convection and is accelerated upward under the action of the chromospheric plasma pressure gradient. It is shown that, under the assumption of plasma incompressibility and flow continuity, the ascending plasma flows induce converging radial plasma flows, which create the convective and Coriolis nonlinear hydrodynamic forces due to the nonzero initial vorticity of the chromospheric plasma caused by Sun’s rotation. The combined action of these two forces leads to the exponential acceleration of the solid-body rotation of plasma inside the ascending flow, thereby creating a vortex that generates an axial magnetic field, in agreement with astrophysical observations.

Geometric Decomposition of Eddy Feedbacks in Barotropic Systems

AbstractIn oceanic and atmospheric flows, the eddy vorticity flux divergence—denoted “F” herein—emerges as a key dynamical quantity, capturing the average effect of fluctuations on the time-mean circulation. For a barotropic system, F is derived from the horizontal velocity covariance matrix, which itself can be represented geometrically in terms of the so-called variance ellipse. This study proves that F may be decomposed into two different components, with distinct geometric interpretations. The first arises from variations in variance ellipse orientation, and the second arises from variations in the kinetic energy of the anisotropic part of the velocity fluctuations, which can be seen as a function of variance ellipse size and shape. Application of the divergence theorem shows that F integrated over a closed region is explained entirely by separate variations in these two quantities around the region periphery. A further decomposition into four terms shows that only four specific spatial patterns of ellipse variability can give rise to a nonzero eddy vorticity flux divergence. The geometric decomposition offers a new tool for the study of eddy–mean flow interactions, as is illustrated with application to an unstable eastward jet on a beta plane.

Hamiltonian approach to the modeling of internal geophysical waves with vorticity

We examine a simplified model of internal geophysical waves in a rotational 2-dimensional water-wave system, under the influence of Coriolis forces and with gravitationally induced waves. The system consists of a lower medium, bound underneath by an impermeable flat bed, and an upper lid. The two media have a free common interface. Both media have constant density and constant (non-zero) vorticity. By examining the governing equations of the system we calculate the Hamiltonian of the system in terms of its conjugate variables and perform a variable transformation to show that it has canonical Hamiltonian structure. We then linearize the system, determine the equations of motion of the linearized system and calculate the dispersion relation. Finally, limiting cases are examined to recover irrotational and single medium systems as well as an infinite two media system.

Hamiltonian formulation of 2 bounded immiscible media with constant non-zero vorticities and a common interface

We examine a 2-dimensional water-wave system, with gravitationally induced waves, consisting of a lower medium bound underneath by an impermeable flat bed and an upper medium bound above by an impermeable lid such that the 2 media have a free common interface. Both media have constant density and constant (non-zero) vorticity. By examining the governing equations of the system we calculate the Hamiltonian of the system in terms of its conjugate variables and perform a variable transformation to show that it has canonical Hamiltonian structure.

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.

On the mechanism of the formation of magnetohydrodynamic vortices in the solar plasma

Based on the magnetohydrodynamic (MHD) equations for an incompressible conductive viscous fluid, the possible mechanism of the formation of giant MHD vortices recently discovered in the solar atmosphere (chromosphere) is analyzed. It is assumed that these vortices arise in the regions of the solar surface (photosphere) with ascending flows of hot plasma that arrives from the inner regions of the Sun as a result of thermal convection and is accelerated upward under the action of the chromospheric plasma pressure gradient. It is shown that, under the assumption of plasma incompressibility and flow continuity, the ascending plasma flows induce converging radial plasma flows, which create the convective and Coriolis nonlinear hydrodynamic forces due to the nonzero initial vorticity of the chromospheric plasma caused by Sun’s rotation. The combined action of these two forces leads to the exponential acceleration of the solid-body rotation of plasma inside the ascending flow, thereby creating a vortex that generates an axial magnetic field, in agreement with astrophysical observations.

Dispersion relations for rotational gravity water flows having two jumps in the vorticity distribution

We derive the dispersion relation for periodic traveling waves propagating at the surface of water with a layer of constant non-zero vorticity situated between two layers of irrotational flow. Due to the complicated nature of the dispersion relation – a fourth order algebraic equation with intricate coefficients – we also give an estimate of a very simple form involving only the levels at which the vorticity has jumps. Our formula generalizes a corresponding one from [5].

Non-existence of three-dimensional travelling water waves with constant non-zero vorticity

AbstractWe prove that there are no three-dimensional bounded travelling gravity waves with constant non-zero vorticity on water of finite depth. The result also holds for gravity–capillary waves under a certain condition on the pressure at the surface, which is satisfied by sufficiently small waves. The proof relies on unique continuation arguments and Liouville-type results for elliptic equations.

Three dimensional steady subsonic Euler flows in bounded nozzles

The existence and uniqueness of three dimensional steady subsonic Euler flows in rectangular nozzles were obtained when prescribing normal component of momentum at both the entrance and exit. If, in addition, the normal component of the voriticity and the variation of Bernoulli's function at the entrance are both zero, then there exists a unique subsonic potential flow when the magnitude of the normal component of the momentum is less than a critical number. As the magnitude of the normal component of the momentum approaches the critical number, the associated flows converge to a subsonic–sonic flow. Furthermore, when the normal component of vorticity and the variation of Bernoulli function are both small, the existence and uniqueness of subsonic Euler flows with non-zero vorticity are established. The proof of these results is based on a new formulation for the Euler system, a priori estimate for nonlinear elliptic equations with nonlinear boundary conditions, detailed study for a linear div–curl system, and delicate estimate for the transport equations.

Exact analytical solutions of second order conformal hydrodynamics

We present some exact solutions of relativistic second-order hydrodynamic equations in theories with conformal symmetry. Starting from a spherically expanding solution in ideal hydrodynamics, we take into account general conformal second-order corrections, and construct, for the first time, fully analytical axisymmetric exact solutions including the case with nonzero vorticity. These solutions are time-reversible despite having a nonvanishing shear stress tensor and provide a useful quantitative measure of the second-order effects in relativistic hydrodynamics.

Transient growth of droplet instabilities in a stream

Droplet deformation is the first stage of all aerodynamically induced-breakups, considerably affecting the characteristics of the atomization. In the present study, using an adaptive volume of fluid method, two and three-dimensional direct numerical simulations have been performed to understand droplet deformation. A high Reynolds number and a range of relatively high Weber numbers are chosen, addressing the shear breakup of droplets in a stream. The study is focused on the initiation and growth of instabilities over the droplet. The role of Kelvin-Helmholtz and Rayleigh-Taylor instabilities in wave formation and azimuthal transverse modulation are shown and the obtained results for the most amplified wave-numbers are compared with instability theories for zero and non-zero vorticity layers. The present results for the most amplified wave-numbers and deformation topologies are in good agreement with the previous experimental results.