Conservation Laws For Vorticity Research Articles

On the Quartic Korteweg–de Vries hierarchy of nonlinear Rossby waves and its dynamics

A long-standing challenge in exploring the dynamics of large-scale atmospheric or oceanic motions is to find more appropriate theoretical models. This paper aims at the mechanisms of nonlinear Rossby waves by a new obtained nonlinear evolutionary equation hierarchy approach. The multiple scales method and weak nonlinear perturbation method are adopted based on the quasi-geostrophic potential vorticity conservation law for large scale atmospheric or oceanic motions. It is the first time that a Quartic Korteweg–de Vries model equation is derived to describe the propagation of nonlinear Rossby waves amplitude, which provides a new acceptable theoretical model for the study of atmospheric blocking phenomena. Qualitative and quantitative analysis for the new Quartic KdV model are finished to disclose the evolutionary mechanisms of nonlinear Rossby waves. The corresponding results declares that the β effect, topographic features and detuning parameter are all important factors on the nonlinear Rossby waves.

Stochastic Lagrangian dynamics of vorticity. Part 1. General theory for viscous, incompressible fluids

Abstract

Hamiltonian description of vortex systems

In the framework of 2D ideal Hydrodynamics a vortex system is defined as a smooth vorticity function having few positive local maxima and negative local minima separated by curves of zero vorticity. Invariants of such structures are discussed following from the vorticity conservation law and invertibility of Lagrangian motion. Hamiltonian formalism for vortex systems is developed by introducing new functional variables diagonalizing the original non-canonical Poisson bracket.

Multi-symplectic magnetohydrodynamics: II, addendum and erratum

A recent paper by Webb et al. (J. Plasma Phys., vol. 80, 2014, pp. 707–743) on multi-symplectic magnetohydrodynamics (MHD) using Clebsch variables in an Eulerian action principle with constraints is further extended. We relate a class of symplecticity conservation laws to a vorticity conservation law, and provide a corrected form of the Cartan–Poincaré differential form formulation of the system. We also correct some typographical errors (omissions) in Webb et al. (J. Plasma Phys., vol. 80, 2014, pp. 707–743). We show that the vorticity–symplecticity conservation law, that arises as a compatibility condition on the system, expressed in terms of the Clebsch variables is equivalent to taking the curl of the conservation form of the MHD momentum equation. We use the Cartan–Poincaré form to obtain a class of differential forms that represent the system using Cartan’s geometric theory of partial differential equations

On the generation of vorticity by force fields in rotor- and actuator flows

In most rotor design methods, the blade load is found by a blade element analysis in an iterative procedure with flow solvers like actuator disc and -line analyses as well as momentum balances. For the flow solvers the force field is the input. In most other aerodynamic analyses the force field is the output result instead of input. This is done by applying boundary conditions at the lifting surface with which the flow is solved and the pressure at the surface, so the load, is determined (only inviscid flows are considered here). Both approaches are consistent, but appear to differ with respect to the generation of vorticity. In the lifting surface approach, usually Helmoltz's laws are used to show that bound and free vorticity is conserved instead of being generated, while in the force field approach vorticity is generated instead of conserved. It is shown that both methods are consistent since sometimes Helmholtz's laws are incorrectly referred to. These laws have been derived in absence of non-conservative forces, while the surface pressure distribution is shown to be such a force field. Besides this, the question is discussed how a force field creates vorticity in an inviscid flow, since some papers consider viscosity to be necessary to generate vorticity. A literature study contradicts this, showing that in inviscid flows vorticity is generated by tangential pressure gradients or, equivalently, a non-uniform force field. This makes the Euler equation including the force field term well suited to express the generation of vorticity in characteristics of the force field. A comparison of the convection of vorticity in the wake of a disc, rotor blade and wing shows several differences. The azimuthal vorticity in the disc wake does not depend on vorticity conservation laws, in contrast to the axial and radial components. For a rotor and wing all components are governed by vorticity conservation.

산불 발화에 따른 하층 대기 순환장 변화에 관한 수치 실험

In order to clarify the impact of wildfire and its thermal forcing on atmospheric wind and temperature patterns, several numerical experiments were carried out using three dimensional atmospheric dynamic model WRF with wildfire parametrization module SFIRE. Since wind can accelerate fire spread speed, the moving speed of fireline is faster than its initial values, and the fireline tends to move the northeast, because of the wind direction and absolute vorticity conservation law associated with driving force induced by terrain. In comparison with non-fire case, the hydraulic jump that often occurs over downwind side of mountain became weak due to huge heat flux originated by surface wildfire and wind pattern over downwind side of mountain tends to vary asymmetrically with time passing. Therefore temporal variation of wind pattern should be catched to prevent the risk of widfire.

An alternative view of generalized Lagrangian mean theory

AbstractIf the variables describing wave–mean flow interactions are chosen to include a set of fluid–particle labels corresponding to the mean flow, then the generalized Lagrangian mean (GLM) theory takes the form of an ordinary classical field theory. Its only truly distinctive features then arise from the distinctive feature of fluid dynamicsas a field theory, namely, the particle-relabelling symmetry property, which corresponds by Noether’s theorem to the many vorticity conservation laws of fluid mechanics. The key feature of the formulation is thatallthe dependent variables depend on a common set of space–time coordinates. This feature permits an easy and transparent derivation of the GLM equations by use of the energy–momentum tensor formalism. The particle-relabelling symmetry property leads to the GLM potential vorticity law in which pseudo-momentum is the only wave activity term present. Thus the particle-relabelling symmetry explains the prominent importance of pseudo-momentum in GLM theory.

Numerical investigation on mechanism of multiple vortex rings formation in late boundary-layer transition

Mechanism of multiple vortex ring formation is one of key issues of late flow transition in a boundary layer. In this paper, a new mechanism is presented that the multiple ring formation is a result of shear layer instability. According to our DNS observation, a momentum deficit zone is formed above the original hairpin vortex tube due to the ejection of the streamwise hairpin vortex tube, which brings the low speed flow from the lower level to the higher level of the boundary layer through vortex rotation. Then the momentum deficit zone generates a sharp olive-shaped shear layer surrounded by the high speed flow. As the shear layer moves up, inviscid instability is developed due to the existence of inflection points (inflection surface in 3-D). Consequently, the new vortex rings are formed one by one. During the new spanwise vortex ring formation process, the spanwise vorticity is increased; however, the original streamwise hairpin vortex tube is weakened with decreasing of the streamwise vorticity. This process follows the Helmoholtz vorticity conservation law.

Conservation laws of inviscid non-isentropic compressible fluid flow in n > 1 spatial dimensions

Recent work giving a classification of kinematic and vorticity conservation laws of compressible fluid flow with barotropic equations of state (where pressure is a function only of the fluid density) in n >1 spatial dimensions is extended to general non-isentropic equations of state in which the pressure is also a function of the dynamical entropy (per unit mass) of the fluid. Two main results are obtained. First, we find that, apart from the familiar conserved integrals for mass, momentum, energy, angular momentum, Galilean momentum and volumetric entropy, additional kinematic conserved integrals arise only for non-isentropic equations of state given by a generalized form of the well-known polytropic equation of state with dimension-dependent exponent γ =1+2/ n , such that the proportionality coefficient is an arbitrary function of the entropy (per unit mass). Second, we show that the only vorticity conserved integrals consist of a circulatory entropy (which vanishes precisely when the fluid flow is irrotational) in all even dimensions. In particular, the vorticity integrals for helicity in odd dimensions and enstrophy in even dimensions are found to be no longer conserved for any non-isentropic equation of state.

Chaotic advection and nonlinear resonances in an oceanic flow above submerged obstacle

The effect of an isolated submarine obstacle on the motion of fluid particles in a periodic external flow is studied within the framework of the barotropic, quasi-geostrophic approximation on f -plane. The concept of background currents advanced by Kozlov [1995. Background currents in geophysical hydrodynamics. Izvestia, Atmos. Oceanic Phys. 31 (2), 245–250] is used to construct a dynamically consistent stream function satisfying the potential vorticity conservation law. It is shown that a system of two topographic vortices revolving about a rotation center can form in a circular external flow. Unsteady periodic perturbations, associated with either variations in the background current or deviations of the external flow from circulation, are analyzed. Unsteadiness in the external flow essentially complicates the pattern of the motion of fluid particles. Vortex-type quasi-periodic structures, identified with nonlinear resonances that form in Lagrangian equations of fluid particle advection, are examined. They either surround the stationary configuration by a vortex chain—a ringlet-like structure [ Kennelly, M.A., Evans, R.H., Joyce, T.M., 1985. Small-scale cyclones on the periphery of Gulf Stream warm-core rings. J. Geophys. Res. 90(5), 8845–8857], or they form a complex-structure multivortex domain. Asymptotic estimates and numerical modeling are used to study the distribution and widths of the nonlinear resonance domains that appear under unsteady perturbations of different types. The onset of chaotic regimes owing to the overlapping of nonlinear resonance domains is analyzed. Transport fluxes determined by chaotic advection and barriers for transport (KAM-tori) and the conditions of their existence are studied. The relation of the rotation frequency of fluid particles on their initial position (when the dependence is calculated in the undisturbed system) is shown to completely determine the main features of the pattern of Lagrangian trajectories and chaotization effects. Because of nonlinear effects, the domain involved in quasi-periodic and chaotic motions can be much larger than the domain occupied by steady topographic vortices. The results of study by Sokolovskiy et al. [1988. On the influence of an isolated submerged obstacle on a barotropic tidal flow. Geophys. Astrophys. Fluid Dyn. 88(1), 1–30] concerning the due regard on the irrotational background component as the necessary factor for the transportation of fluid particles from the vortex domain to infinity are confirmed.

Current nature of the Kuroshio in the vicinity of the Kii peninsula

The Kuroshio flows very close to Cape Shionomisaki when it takes a straight path. The detailed observations of the Kuroshio were made both on board the R/V Seisui-maru of Mie University and on board the R/V Wakayama of the Wakayama Prefectural Fisheries Experimental Station on June 11–14, 1996. It was confirmed that the current zone of the Kuroshio touches the coast and bottom slope just off Cape Shionomiaki, and that the coastal water to the east of the cape was completely separated from that to the west. The relatively high sea level difference between Kushimoto and Uragami could be caused by this separation of the coastal waters when the Kuroshio takes a straight path. This flow is rather curious, as the geostrophic flow, which has a barotropic nature and touches the bottom, would be constrained to follow bottom contours due to the vorticity conservation law. The reason why the Kuroshio leaves the bottom slope to the east of Cape Shionomisaki is attributed to the high curvature of the bottom contours there: if the current were to follow the contours, the centrifugal term in the equation of motion would become large and comparablee to the Coriolis (or pressure gradient) term, and the geostrophic balance would be destroyed. This creates a current-shadow zone just to the east of the cape. As the reason why the current zone of the Kuroshio intrudes into the coastal region to the west of the cape, it is suggested that the Kii Bifurcation Current off the southwest coast of the Kii Peninsula, which is usually found when the Kuroshio takes the straight path, has the effect of drawing the Kuroshio water into the coastal region. The sea level difference between Kushimoto and Uragami is often used to monitor the flow pattern of the Kuroshio near the Kii Peninsula. It should be noted that Uragami is located in the current shadow zone, while Kushimoto lies in the region where the offshore Kuroshio water intrudes into the coastal region. The resulting large sea level difference indicates that the Kuroshio is flowing along the straight path.

The circulation density and its role in 3D turbulence

The scaling argument applied to the vorticity cascade in 2D turbulence results in the well-known k −3 energy spectrum. Kelvin theorem for the velocity circulation generalizes the vorticity conservation law on the 3D fluids. Using the Kelvin theorem and the incompressibility of the fluid one can derive Lagrangian conservation of a quantity having the physical meaning of a circulation density. Integrated over the entire space, the powers of the circulation density form a series of the motion integrals which coincides with the vorticity series in the 2D limit. We will discuss the question of applicability of the scaling arguments to the cascades of the geometrical integrals in 3D turbulence. We will see that the cascades of all but one circulation integrals are ruled out by the reconnection kinematics. The only exception is the integral corresponding to the total volume of the vortex tubes, whose cascade corresponds to the k −3 energy spectrum. Relation of the circulation density to the Clebsch variables will be considered. We will see that the circulation density can be chosen as one of the Clebsch variables. In the case of the stationary flows, such Clebsch variables becomes an action-angle pair.

Layer-Mean Quantities, Local Conservation Laws, and Vorticity

We derive local conservation laws for layer-mean quantities in two general settings. When applied to Euler flows, the first of these settings yields well-known local conservation laws for quantities averaged between material surfaces. The second, however, leads to new local conservation laws for quantities involving the vorticity that are averaged between arbitrary surfaces. These produce the crucial vorticity conservation laws in shallow water models that admit nonhydrostatic and noncolumnar motion. Moreover, they seem to lie outside the Hamiltonian paradigm of fluid dynamics. The formalism generalizes to skew-symmetric matrix fields; applications to electromagnetism are suggested. {copyright} {ital 1997} {ital The American Physical Society}

The balance and redistribution of potential vorticity within the ocean

Abstract The balance and redistribution of potential vorticity in a turbulent, inhomogeneous ocean is examined analytically. In the first part of the paper, we derive a vorticity conservation law describing the way in which vorticity is lost and gained in a closed ocean basin. We show that the potential vorticity within the ocean can only be altered by molecular diffusion at the bounding surfaces, or by the advective input of vorticity via the turbulent velocity at the free upper surface. An imbalance between these two effects within the body of the ocean leads to a net temporal change in the total potential vorticity, which we are able to express in terms of the temporal changes in the magnitude and orientation of a vector formed by the surface velocity and density and the earth's rotation. Finally, the steady-state balance of integrated vorticity is found to be between the surface wind-stress curl and the molecular diffusion at the solid boundaries. We then consider the localized redistribution of potential vorticity in detail. In recent papers, Green (1970) and Welander (1973) have purported to show that an internal redistribution of potential vorticity can give rise to important forces in the atmosphere (Green) and the ocean (Welander). Using their recipe leads in some cases to such unphysical results as a breakdown of the laws of conservation of angular momentum. Such consequences cause us to question the validity of the argument, and lead us to a close examination' of Welander's mechanism. (Green's argument is more general and does not lend itself to such close examination.) Our investigation reveals important flaws not only in Welander's argument, but also in the classical mixing-length theory of Prandtl (1925) and of Taylor (1915) on which it is modelled. More specifically, the internal redistribution process presented in these theories is found to be incomplete since it fails to take into account the lift force exerted on individual parcels of fluid as they are transported by the turbulent motions. Inclusion of these lift forces not only permits the unification of the classical mixing-length theories, but also shows that the forces claimed by Welander do not in fact arise.