Constant Vorticity Research Articles

Exact planetary waves and jet streams

We investigate exact nonlinear waves on surfaces locally approximating the rotating sphere for two-dimensional inviscid incompressible flow. Our first system corresponds to a $\beta$ -plane approximation at the equator, and the second to a $\gamma$ approximation, with the latter describing flow near the poles. We find exact wave solutions in the Lagrangian reference frame that cannot be written down in closed form in the Eulerian reference frame. The wave particle trajectories, contours of potential vorticity and Lagrangian mean velocity take relatively simple forms. The waves possess a non-trivial Lagrangian mean flow that depends on the amplitude of the waves and on a particle label that characterizes values of constant potential vorticity. The mean flow arises due to potential vorticity conservation on fluid particles. Solutions over the entire space are generated by assuming that the flow far from the origin is zonal and there is a region of uniform potential vorticity between this zonal flow and the waves. In the $\gamma$ approximation, a class of waves is found that, based on analogous solutions on the plane, we call Ptolemaic vortex waves. The mean flow of some of these waves, which we can describe in highly nonlinear scenarios due to the exact nature of the solutions, resembles polar jet streams. Several illustrative solutions are used as initial conditions in the fully spherical rotating Navier–Stokes equations, where integration is performed via the numerical scheme presented in Salmon & Pizzo (Atmosphere, vol. 14, issue 4, 2023, 747). The potential vorticity contours found from these numerical experiments vary between stable permanent progressive form and fully turbulent flows generated by wave breaking.

Boundary vorticity of incompressible 2D flows

For a homogeneous incompressible 2D fluid confined within a bounded Lipschitz simply connected domain, homogeneous Neumann pressure boundary conditions are equivalent to a constant boundary vorticity. We investigate the rigidity of such conditions.

Hamiltonian Birkhoff Normal Form for Gravity-Capillary Water Waves with Constant Vorticity: Almost Global Existence

We prove an almost global existence result for space periodic solutions of the 1D gravity-capillary water waves equations with constant vorticity. The result holds for any value of gravity, vorticity and depth, a full measure set of surface tensions, and any small and smooth enough initial datum. The proof demands a novel approach—that we call paradifferential Hamiltonian Birkhoff normal form for quasi-linear PDEs—in presence of resonant wave interactions: the normal form is not integrable but it preserves the Sobolev norms thanks to its Hamiltonian nature. A major difficulty is that paradifferential calculus used to prove local well posedness (as the celebrated Alinhac good unknown) breaks the Hamiltonian structure. A major achievement of this paper is to correct (possibly) unbounded paradifferential transformations to symplectic maps, up to an arbitrary degree of homogeneity. Thanks to a deep cancellation, our symplectic correctors are smoothing perturbations of the identity. Thus we are able to preserve both the paradifferential structure and the Hamiltonian nature of the equations. Such Darboux procedure is written in an abstract functional setting applicable also in other contexts.

Flow structure beneath periodic waves with constant vorticity under strong horizontal electric fields

While several articles have been written on Electrohydrodynamics (EHD) flows or flows with constant vorticity separately, little is known about the extent to which the combined effects of EHD and constant vorticity affect the flow. This study aims to shed light on this topic by investigating the combined influence of a horizontal electric field and constant vorticity on the free surface and the emergence of stagnation points. Using the Euler equations framework, we employ conformal mapping and pseudo-spectral numerical methods. Our findings reveal that increasing the electric field intensity eliminates stagnation points and smoothen the wave profile. This implies that a horizontal electric field acts as a mechanism for the elimination of stagnation points within the fluid body. Besides, we have identified regimes where three stagnation points appear on the free surface — something that cannot occur in purely gravity rotational waves.

Freak wave generation modulated by high wind and linear shear flow in finite water depth

In finite water depths, the effects of high winds and linear shear flow (LSF), encompassing both uniform flow and constant vorticity shear flow on freak wave generation are explored. A nonlinear Schrödinger equation, adjusted for high wind and LSF conditions, is derived using potential flow theory and the multiscale method. This equation accounts for the modulational instability (MI) of water waves and the evolution of freak wave amplitudes. MI analysis reveals that for waves to maintain MI, high tail winds (moving in the same direction as the wave) require less vorticity and deeper water, while adverse winds (moving in the opposite direction) necessitate more vorticity and shallower water depths compared to conditions without wind. Uniform up-flows (down-flows), positive (negative) vorticity, and high tail (adverse) winds, which inhibit (promote) wave propagation, increase (decrease) the MI growth rate and amplify (diminish) freak wave heights. It is through this MI that the generation of freak waves is either promoted or inhibited.

Stratified steady inviscid water flows with effects of surface tension and constant non-zero vorticity

AbstractIn this paper we consider steady inviscid three-dimensional stratified water flows of finite depth with a free surface and an interface. The interface plays the role of an internal wave that separates two layers of constant and different density. We study two cases separately: when the free surface and the interface are functions of one variable and when the free surface and the interface are functions of two variables. In both cases, considering effects of surface tension, we prove that the bounded solutions to the three-dimensional equations are essentially two-dimensional. More specifically, assuming that the vorticity vectors in the two layers are constant, non-vanishing and parallel to each other we prove that their third coordinate vanishes in both layers. Also we prove that the free surface, the interface, the pressure and the velocity field present no variations in the direction orthogonal to the direction of motion.

Estimating the performance of wind turbines misaligned with respect to the wind direction using a simplified CFD based approach

Wind direction in atmospheric boundary layer changes constantly. As such, the power extracted from the wind turbines in a wind farm is subjected to change. This change in wind direction impacts the wake pattern which in turn affects the power output of the downstream wind turbines. Empirical wind farm models are useful in such cases to assess the overall performance of the wind farms. However, the results provided by these models can be made more useful if fluid-structure interaction (FSI) effects are also considered. In the present work, we begin by looking at the potential flow model and the issues associated with it and used a modified model with constant vorticity to analyze the wind power available to the downstream wind turbines in a wind farm misaligned with respect to wind direction for irrotational flow. A simplified FSI based approach is used to carry out the simulations without the need of modelling the complicated wind turbine geometry.

Self-reverting vortices in chiral active matter

There is currently a strong interest in the collective behavior of chiral active particles that can propel and rotate themselves. In the presence of alignment interactions for many chiral particles, chiral self-propulsion can induce vortex patterns in the velocity fields. However, these emerging patterns are non-permanent, and do not induce global vorticity. Here we combine theoretical arguments and computer simulations to predict a so-far unknown class of collective behavior. We show that, for chiral active particles, vortices with significant dynamical coherence emerge spontaneously. They originate from the interplay between attraction interactions and chirality in the absence of alignment interactions. Depending on parameters, the vortices can either feature a constant vorticity or a vorticity that oscillates periodically in time, resulting in self-reverting vortices. Our results may guide future experiments to realize customized collective phenomena such as spontaneously rotating gears and patterns with a self-reverting order.

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.

A Sharp Version of the Benjamin and Lighthill Conjecture for Steady Waves with Vorticity

We give a complete proof of the classical Benjamin and Lighthill conjecture for arbitrary two-dimensional steady water waves with vorticity. We show that the flow force constant of an arbitrary smooth solution is bounded by the flow force constants for the corresponding conjugate laminar flows. We prove these inequalities without any assumptions on the geometry of the surface profile and put no restrictions on the wave amplitude. Furthermore, we give a complete description of all cases when the equalities can occur. In particular, that excludes the existence of one-sided bores and multi-hump solitary waves. Our conclusions are new already for Stokes waves with a constant vorticity, while the case of equalities is new even in the classical setting of irrotational waves.

Multipoint stress mixed finite element methods for linear viscoelasticity with weak symmetry

In this paper, we propose two Multipoint Stress Mixed Finite Element (MSMFE) methods for linear viscoelasticity with weak symmetry on quadrilateral grids. The methods are constructed based on the lowest order Brezzi–Douglas–Marini mixed finite element spaces for elastic and viscous stress, piecewise constant velocity and piecewise constant (linear) vorticity. A special quadrature rule is applied for local stress and vorticity elimination. This results in a positive definite cell centered velocity–vorticity or only velocity system at each time step. Unconditional energy-dissipation of the MSMFE methods is proved rigorously. The accuracy of all the numerical solutions in their nature norms are established to be first order space convergence, both for the semi-discrete and fully-discrete formulations. Numerical results validate the theory results of the proposed methods.

Global solutions of two‐dimensional inviscid and resistive magnetohydrodynamics system near an equilibrium

We investigate the global existence of classical solutions for two‐dimensional incompressible, inviscid, and resistive magnetohydrodynamics (MHD) system. Assuming that the initial magnetic field is close to an equilibrium state of constant magnetic vorticity, the global existence result could be obtained using a time‐weighted energy estimate.

Diagnosing Flavors of Tropospheric Rossby Wave Breaking and Their Associated Dynamical and Sensible Weather Features

Abstract Rossby wave breaking (RWB) can be manifested by the irreversible overturning of isentropes on constant potential vorticity (PV) surfaces. Traditionally, the type of breaking is categorized as anticyclonic (AWB) or cyclonic (CWB) and can be identified using the orientation of streamers of high potential temperature (θ) and low θ air on a PV surface. However, an examination of the differences in RWB structure and their associated tropospheric impacts within these types remains unexplored. In this study, AWB and CWB are identified from overturning isentropes on the dynamic tropopause (DT), defined as the 2 potential vorticity unit (PVU; 1 PVU = 10−6 K kg−1 m2 s−1) surface, in the ERA5 dataset during December, January, and February 1979–2019. Self-organizing maps (SOM), a machine learning method, is used to cluster the identified RWB events into archetypal patterns, or “flavors,” for each type. AWB and CWB flavors capture variations in the θ minima/maxima of each streamer and the localized meridional θ gradient (∇θ) flanking the streamers. Variations in the magnitude and position of ∇θ between flavors correspond to a diversity of jet structures leading to differences in vertical motion patterns and troposphere-deep circulations. A subset of flavors of AWB (CWB) events are associated with the development of strong surface high (low) pressure systems and the generation of extreme poleward moisture transport. For CWB, many events occurred in similar geographical regions, but the precipitation and moisture patterns were vastly different between flavors. Our findings suggest that the location, type, and severity of the tropospheric impacts from RWB are strongly dictated by RWB flavor. Significance Statement Large-scale atmospheric waves ∼15 km above Earth’s surface are responsible for the daily weather patterns that we experience. These waves can undergo wave breaking, a process that is analogous to ocean waves breaking along the seashore. Wave breaking events have been linked to extreme weather impacts at the surface including cold and heat waves, strong low pressure systems, and extreme precipitation events. Machine learning is used to identify and analyze different flavors, or patterns, of wave breaking events that result in differing surface weather impacts. Some flavors are able to generate notable channels of moisture that result in extreme high precipitation events. This is a crucial insight as forecasting of extreme weather events could be improved from this work.

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>

Rigidity of three-dimensional steady internal waves with constant vorticity governed by the [formula omitted]-plane approximation

This paper investigates the structural implications of constant vorticity for steady three-dimensional internal water waves governed by the equatorial f-plane approximation. Under the assumption that the first components of the non-zero constant vorticity vectors in the upper and lower layers are zeros, it is proved that every three-dimensional traveling internal wave with bounded velocity is either two-dimensional or a trivially three-dimensional shear flow with the pressures in both layers are identically equal.

Turbulence via intermolecular potential: Uncovering the origin

In recent works, we proposed a hypothesis, according to which turbulence in gases is created by the mean field effect of an intermolecular potential. We discovered that, in a numerically simulated inertial flow, turbulent solutions indeed spontaneously emerge from a laminar initial condition, as observed in nature and experiments. To study the origin of turbulent dynamics, in the current work we examine the equations of a two-dimensional inertial flow, linearized around a large scale constant vorticity state. Remarkably, even in this simplified setting, we find that turbulent dynamics emerge as linearly unstable fluctuations of the velocity divergence.In particular, for the linearized dynamics at a high Reynolds number, we find that, at short time scales, the coupling of the mean field potential with the large scale background vorticity creates linearly unstable and rapidly oscillating fluctuations of the divergence of velocity at inertial scales. In the asymptotic time limit, we find a persistent eigenvector, also aligned largely with the divergence of velocity, which allows these fluctuations to propagate in the form of traveling waves in the Fourier domain. Remarkably, these traveling waves decay at a constant, scale-independent exponential rate. Furthermore, it appears that the Kolmogorov scaling of the kinetic energy is produced by this persistent velocity divergence, due to a cubic relation between the physical time variable, and a pseudo-time variable, in which the dynamics become autonomous. These effects vanish when the mean field potential is removed.

General procedure for free-surface recovery from bottom pressure measurements: application to rotational overhanging waves

A novel boundary integral approach for the recovery of overhanging (or not) rotational water waves (with constant vorticity) from pressure measurements at the bottom is presented. The method is based on the Cauchy integral formula and on an Eulerian–Lagrangian formalism to accommodate overturning free surfaces. This approach eliminates the need to introduce a priori a special basis of functions, thus providing a general means of fitting the pressure data and, consequently, recovering the free surface. The effectiveness and accuracy of the method are demonstrated through numerical examples.

Nonexistence of periodic peakon and peakon for a highly nonlinear shallow-water model

In this paper, we investigate the nonexistence of the periodic peakon and peakon for a highly nonlinear shallow-water model, which has been recently derived from the full governing equations for two dimensional flow with the Coriolis effect or with constant vorticity, under a larger scaling than the Camassa-Holm (CH) one. Note that the so obtained model not only has CH-type terms, but also exhibits cubic order nonlinearities. Thus it is interesting to study how the higher power nonlinear terms affect the existence of the (periodic) peakon.

Triad resonance of flexural gravity waves in the presence of shear current with constant vorticity

This study examines the formation of triads of flexural gravity wave in a homogeneous fluid within the context of blocking dynamics due to the presence of shear current. This study will enable us to understand the distribution of wave energy on an ice-covered sea surface. New classes of triads for flexural gravity waves are introduced depending on the direction of wave propagation with following and opposing currents. The study reveals that triad formation occurs due to the interaction of flexural gravity waves irrespective of the presence of compression and current, which has not been found in the case of free surface gravity waves. In addition, at most, three triads are formed in the case of flexural gravity waves in the presence of following and opposing currents prior to the threshold of blocking. In contrast, at least three triads are formed for any frequency within the primary and secondary blocking limits for certain values of compressive force and current speed. On the other hand, 11 triads are formed in the presence of uniform current speed as well as in the case of linear shear current with constant vorticity for a certain frequency within the blocking limit for higher values of compressive force and current speed.

Fully Nonlinear Evolution of Free-Surface Waves with Constant Vorticity under Horizontal Electric Fields

This work presents the results of a direct numerical simulation of the nonlinear free surface evolution of a finite-depth fluid with a linear shear flow under the action of horizontal electric fields. The method of time-dependent conformal transformation for the description of the combined effects of the electric fields and constant vorticity is generalized for the first time. The simulation results show that strong shear flow co-directed in the wave propagation direction leads to the formation of large-amplitude surface waves, and, for some limiting vorticity value, a wave breaking process with the formation of an air bubble in the liquid is possible. The oppositely directed shear flow can cause the retrograde motion of a surface wave (wave propagation in the opposite direction to the linear wave speed). The simulations conducted taking into account the electro-hydrodynamic effects demonstrate that a high enough external horizontal electric field suppresses these strongly nonlinear processes, and the surface waves tend to preserve their shape.