Vorticity Theorem Research Articles

On the Determination of the 3D Velocity Field in Terms of Conserved Variables in a Compressible Ocean

Explicit expressions of the 3D velocity field in terms of the conserved quantities of ideal fluid thermocline theory, namely the Bernoulli function, density, and potential vorticity, are generalised in this paper to a compressible ocean with a realistic nonlinear equation of state. The most general such expression is the ‘inactive wind’ solution, an exact nonlinear solution of the inviscid compressible Navier–Stokes equation that satisfies the continuity equation as a consequence of Ertel’s potential vorticity theorem. However, due to the non-uniqueness of the choice of the Bernoulli function, such expressions are not unique and primarily differ in the magnitude of their vertical velocity component. Due to the thermobaric nonlinearity of the equation of state, the expression for the 3D velocity field of a compressible ocean is found to resemble its ideal fluid counterpart only if constructed using the available form of the Bernoulli function, the Bernoulli equivalent of Lorenz’s available potential energy (APE). APE theory also naturally defines a quasi-material, approximately neutral density variable known as the Lorenz reference density. This density variable, in turn, defines a potential vorticity variable that is minimally affected by thermobaric production, thus providing all the necessary tools for extending most results of ideal fluid thermocline theory to a compressible ocean.

On (non-)conservative body forces, vorticity generation and energy conversion in ideal flows

Usually, the load on lifting bodies in incompressible, inviscid flow is determined by integration of the pressure on the body surface, once the flow is solved. Prandtl (Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch- Physikalische Klasse, vol. 1918, pp. 451–477) proposed an opposite method in which the body force field is the source term in the equation of motion to solve the flow. This force field method has not been used intensively but has regained importance in modern wind energy research. However, an analysis of which type of body force field generates vorticity and converts energy, and which body force field does not, is lacking. Prandtl's method is adopted here, but with the addition that the force field is allowed to be conservative or non-conservative. The relation between conservative/non-conservative body forces, vorticity generation, potential and kinetic energy and Helmholtz's vorticity theorems are derived. Similarly, the load on lifting bodies in two and three dimensions is classified as (non-)conservative, with some examples. To show that the force field method is consistent with the method where the load is output of an analysis, the expression for the Kutta–Joukowsky load and the relation between bound and trailing vorticity of a wing have been rederived using the force field method. The analysis confirms that this relation between bound and trailing vorticity is not governed by Helmholtz's theorems, as is often assumed, but by the non-conservative force field generating vorticity.

Wave-activity relation containing wave-basic flow interaction based on decomposition of general potential vorticity* *Project supported by the Strategic Pilot Science and Technology Special Program of the Chinese Academy of Sciences (Grant No. XDA17010105), the National Key Research and Development Project, China (Grant No. 2018YFC1507104), the Key Scientific and Technology Research and Development Program of Jilin Province, China (Grant No. 20180201035SF),

On the basis of the general potential vorticity theorem (GPV), a new wave–activity relation is derived in an ageostrophic and nonhydrostatic dynamic framework. When the Reynolds average is taken, the wave–activity relation shares an exchange term with the equation of the basic-state GPV. Thus, the two equations are capable of presenting the dynamic process of the wave–basic flow interaction. Unlike the E–P flux theory which can only be used in large-scale atmosphere, the corresponding derivation provides a useful tool to analyze the feedback of waves to basic states and the forcing of basic states to waves simultaneously, and it can be used in mesoscale systems, such as heavy rainfall processes. The theory was initially applied to the landfalling Typhoon Mujigae (2015) by assigning the scalar φ to the generalized potential temperature (GPT). The results showed that the newly-derived wave–activity density is able to describe the wave activities associated with strong precipitation in Typhoon Mujigae (2015), including the eyewall and spiral rainbands. However, the interaction between the basic-state vortex and mesoscale waves denoted by the exchange term between basic-state GPV and wave–activity density mainly occurs in the eyewall in Typhoon Mujigae (2015). A comparison of the exchange term with other forcing terms in the newly-derived wave–activity relation indicates that the basic state–wave interaction plays a significant role in enhancing wave activities in the high-precipitation eyewall. By a magnitude analysis of the interaction term, it is found that the strong interaction between basic-state vortex and mesoscale waves is mainly attributed to two factors: the vertical vorticity intensity of the basic-state vortex and the averaged perturbation advection of perturbation GPT which is an exchange between the basic-state GPT and perturbation GPT.

On the divergence and vorticity of vector ambit fields

This paper studies the asymptotic behaviour of the flux and circulation of a subclass of random fields within the family of 2-dimensional vector ambit fields. We show that, under proper normalization, the flux and the circulation converge stably in distribution to certain stationary random fields that are defined as line integrals of a Lévy basis. A full description of the rates of convergence and the limiting fields is given in terms of the roughness of the background driving Lévy basis and the geometry of the ambit set involved. We further discuss the connection of our results with the classical Divergence and Vorticity Theorems. Finally, we introduce a class of models that are capable to reflect stationarity, isotropy and null divergence as key properties.

A simple theory of the quasi-equilibrium regime of atmospheric circulation, using the concept of a modified potential vorticity

The application of the general Ertel’s vorticity theorem to the diagnosis of general atmospheric circulation processes is discussed. The concepts of a modified potential vorticity (PV) and the “PV of the second generation” are used; the latter quantity is obtained by substituting a PV into Ertel’s vorticity theorem. The requirement that the “second-generation PV” vanishes when the PV is properly modified leads to the condition for local compensation of the action of heat sources and frictional stresses on the Earth’s surface: namely, to the condition of surface air heating / cooling under anticyclonic / cyclonic circulation conditions. The situation with locally uncompensated anomalous wintertime near-surface air heating over the Barents Sea is discussed in the context of the current reduction in the ice cover over the Arctic, which at the global level should be compensated by the intensification of anticyclonic circulation systems over the continents and the corresponding surface air cooling.

Ertel's vorticity theorem and new flux surfaces in multi-fluid plasmas

Dedicated to Professor Harold Weitzner on the occasion of his retirement “Say to wisdom ‘you are my sister,’ and to insight ‘you are my relative.’”—Proverbs 7:4 Based on an extension to plasmas of Ertel's classical vorticity theorem in fluid dynamics, it is shown that for each species in a multi-fluid plasma there can be constructed a set of nested surfaces that have this species' fluid particles confined within them. Variational formulations for the plasma evolution and its equilibrium states are developed, based on the new surfaces and all of the dynamical conservation laws associated with them. It is shown that in the general equilibrium case, the energy principle lacks a minimum and cannot be used as a stability criterion. A limit of the variational integral yields the two-fluid Hall-magnetohydrodynamic (MHD) model. A further special limit yields MHD equilibria and can be used to approximate the equilibrium state of a Hall-MHD plasma in a perturbative way.

Vortex dynamics in $\mathbb {R}^4$R4

The vortex dynamics of Euler's equations for a constant density fluid flow in \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^4$\end{document}R4 is studied. Most of the paper focuses on singular Dirac delta distributions of the vorticity two-form ω in \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^4$\end{document}R4. These distributions are supported on two-dimensional surfaces termed membranes and are the analogs of vortex filaments in \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^3$\end{document}R3 and point vortices in \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^2$\end{document}R2. The self-induced velocity field of a membrane is shown to be unbounded and is regularized using a local induction approximation. The regularized self-induced velocity field is then shown to be proportional to the mean curvature vector field of the membrane but rotated by 90° in the plane of normals. Next, the Hamiltonian membrane model is presented. The symplectic structure for this model is derived from a general formula for vorticity distributions due to Marsden and Weinstein [“Coadjoint orbits, vortices and Clebsch variables for incompressible fluids,” Physica D 7, 305–323 (1983)10.1016/0167-2789(83)90134-3]. Finally, the dynamics of the four-form ω ∧ ω is examined. It is shown that Ertel's vorticity theorem in \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^3$\end{document}R3, for the constant density case, can be viewed as a special case of the dynamics of this four-form.

On Lagrangian and vortex-surface fields for flows with Taylor–Green and Kida–Pelz initial conditions

For a strictly inviscid barotropic flow with conservative body forces, the Helmholtz vorticity theorem shows that material or Lagrangian surfaces which are vortex surfaces at time t = 0 remain so for t > 0. In this study, a systematic methodology is developed for constructing smooth scalar fields φ(x, y, z, t = 0) for Taylor–Green and Kida–Pelz velocity fields, which, at t = 0, satisfy ω·∇φ = 0. We refer to such fields as vortex-surface fields. Then, for some constant C, iso-surfaces φ = C define vortex surfaces. It is shown that, given the vorticity, our definition of a vortex-surface field admits non-uniqueness, and this is presently resolved numerically using an optimization approach. Additionally, relations between vortex-surface fields and the classical Clebsch representation are discussed for flows with zero helicity. Equations describing the evolution of vortex-surface fields are then obtained for both inviscid and viscous incompressible flows. Both uniqueness and the distinction separating the evolution of vortex-surface fields and Lagrangian fields are discussed. By tracking φ as a Lagrangian field in slightly viscous flows, we show that the well-defined evolution of Lagrangian surfaces that are initially vortex surfaces can be a good approximation to vortex surfaces at later times prior to vortex reconnection. In the evolution of such Lagrangian fields, we observe that initially blob-like vortex surfaces are progressively stretched to sheet-like shapes so that neighbouring portions approach each other, with subsequent rolling up of structures near the interface, which reveals more information on dynamics than the iso-surfaces of vorticity magnitude. The non-local geometry in the evolution is quantified by two differential geometry properties. Rolled-up local shapes are found in the Lagrangian structures that were initially vortex surfaces close to the time of vortex reconnection. It is hypothesized that this is related to the formation of the very high vorticity regions.

Diagnosis of wave activity in a heavy‐rainfall event

Three general wave‐activity laws with an inclusion of ageostrophic winds are derived in a nonhydrostatic dynamic framework by introducing an arbitrary scalar ϕ into potential vorticity theorem. The wave activity theory may be applied to diagnoses of mesoscale weather systems. It is shown that the general wave‐activity densities cannot represent monochromatic wave but slowly varying wave train. They cannot be transported through the surface of perturbation scalar ϕe and can neither be created nor destroyed within a layer bounded by the two ϕe surfaces. The general wave‐activity law associated with perturbation vertical velocity is embodied by setting the arbitrary scalar ϕ to specific humidity, equivalent potential temperature, and virtual potential temperature, respectively. The simulation data of a heavy‐rainfall event by the ARPS model are used to calculate the three specific wave‐activity densities and to study their laws. It is shown that the three specific wave‐activity densities are closely related to the simulated rain rate. This suggests that they may serve as track for detecting precipitation. The variation of moist wave‐activity density is mainly caused by the wave‐activity flux divergence associated with the vertical component of perturbation pressure gradient force.

Quaternions and particle dynamics in the Euler fluid equations

Vorticity dynamics of the three-dimensional incompressible Euler equations are cast into a quaternionic representation governed by the Lagrangian evolution of the tetrad consisting of the growth rate and rotation rate of the vorticity. In turn, the Lagrangian evolution of this tetrad is governed by another that depends on the pressure Hessian. Together these form the basis for a direction of vorticity theorem on Lagrangian trajectories. Moreover, in this representation, fluid particles carry ortho-normal frames whose Lagrangian evolution in time are shown to be directly related to the Frenet–Serret equations for a vortex line. The frame dynamics suggest an elegant Lagrangian relation regarding the pressure Hessian tetrad. The equations for ideal MHD are similarly considered.

Material derivatives of higher dimension in geophysical fluid dynamics

The familiar operator D/Dt in fluid dynamics defines the material derivative for a fluid particle with dimension zero. In this paper we define and use macroscopic or multidimensional material derivatives D 1 /Dt, D 2 /Dt and D 3 /Dt. They are the material derivatives of infinitesimal properties of the fluid having dimensions, i. e. when particles build a line, a surface area, or a volume. Simple rules between the three operators are presented that avoid complicated calculations in fluid dynamics. For example, these operators are invariant with respect to solid rotations of coordinate systems. We rewrite a number of equations of fluid dynamics in terms of these operators and show that simple identities involving these operators already contain the structure of known vorticity theorems, especially those given by Hans Ertel. One application deals with the circulation of eddy velocities in atmospheric turbulence, showing that this circutation maybe an almost material invariant with time. Further possible applications (e.g., in electrodynamics of fluids and in radiation hydrodynamics) are also suggested.

Ertel's vorticity theorems, the particle relabelling symmetry and the energy-vorticity theory of fluid mechanics

The various vorticity theorems of ERTEL are related to a unique symmetry in the substantial or Lagrangian description of fluid mechanics. According to NOETHER'S theorem, this is the particle relabelling symmetry (PRS), also called exchange symmetry. What is missing is a classification of this symmetry for barotropic and baroclinic fluids and a transformation to the Eulerian frame, revealing the universal and practical implications of this symmetry. The paper shows that the relabelling symmetry is an infinite symmetry like the gauge symmetries in elementary particle physics. It can be parameterised by one free arbitrary function for baroclinic flows and two functions for barotropic flows. In the baroclinic case we get ERTEL'S potential enstrophy and in the barotropic case the helicity as global conserved quantities. In both cases the conservation laws are explicitly dependent upon arbitrary functions of the physical phase space. The generator of this symmetry can be also transformed to the Eulerian reference space. As a special case we can derive the generators of rotation and translation and the constraints for the related conservation laws of angular and linear momentum. In this case the arbitrary functions depend solely on the space coordinates. This surprising result is an indication that the relabelling generator is of great universal nature. Moreover, this relabelling symmetry is responsible for the closed Eulerian description of fluid mechanics and gives a theoretical explanation of the energy-vorticity theory of fluid mechanics. A dynamic state index (DSI) and a chemical state index (CSI) can be defined for the diagnosis of the weather and climate system as a practical application of this theory.

Dynamics of the atmosphere: a course in theoretical meteorology

Preface M1. Algebra of vectors M2. Vector functions M3. Differential relations M4. Coordinate transformations M5. The method of covariant differentiation M6. Integral operations M7. Introduction to the concepts of nonlinear dynamics 1. The laws of atmospheric motion 2. Scale analysis 3. The material and the local description of flow 4. Atmospheric flow fields 5. The Navier-Stokes stress tensor 6. The Helmholtz theorem 7. Kinematics of two-dimensional flow 8. Natural coordinates 9. Boundary surfaces and boundary conditions 10. Circulation and vorticity theorems 11. Turbulent systems 12. An excursion into the spectral turbulence theory 13. The atmospheric boundary-layer 14. Wave motion in the atmosphere 15. The barotropic model 16. Rossby waves 17. Inertial and dynamic stability 18. The equation of motion in general coordinate systems 19. The geographical coordinate system 20. The stereographic coordinate system 21. Orography-following coordinate systems 22. The stereographic system with a generalised vertical coordinate 23. A quasi-geostrophic baroclinic model 24. A two-level prognostic model: baroclinic instability 25. An excursion into numerical procedures 26. Modeling of atmospheric flow by spectral techniques 27. Predictability Answers to problems List of commonly used symbols References Index.

Tracers and Potential Vorticities in Ocean Dynamics

Abstract The Ertel potential vorticity theorem for stratified viscous fluids in a rotating system is analyzed herein. A set of “tracers,” that is, materially conserved scalar quantities, and the corresponding Ertel potential vorticities are used to obtain an absolute fluid velocity determination (including both horizontal and vertical components) that generalizes earlier formulations known in the literature within the framework of the beta-spiral method. Potential vorticity fields, respectively, of (i) density, (ii) potential temperature, (iii) salinity, and (iv) the latter's potential vorticities ratio are analyzed in order to infer properties of steady, or quasi-steady, nonhorizontal or slightly viscous currents. For horizontal flows, general conservative properties of a large class of tracer potential vorticities are found and discussed. These ideas are then applied to various steady cases of physical interest, such as density fronts and thermohaline currents. These arguments, together with observation...

The general relativistic and covariant form of the I. Helmholtz vorticity theorem and geophysical application

The general relativistic and covariant differential form of Helmholtz's first vorticity theorem is presented. We prove in relation with it an invariant kinematic identity which is the generalisation of the Helmholtz theorem for general continua.

The General-Relativistic and Covariant Form of the First Helmholtz Vorticity Theorem

The general relativistic and covariant differential form of Helmholtz's first vorticity theorem is presented. We prove in relation with it an invariant kinematic identity which is the generalisation of the Helmholtz theorem for general continua.

Boltzmann's Formulation of the Helmholtz Vorticity Theorem

A dual form of the general relativistic and covariant generalization of the first vorticity theorem of Helmholtz is proven. This dual form is the generalization of Boltzmann's formulation of the Helmholtz theorem.

On the General-Relativistic and Covariant Form of Integration of the Helmholtz Theorem

The covariant and relativistic generalization of the Helmholtz integral vorticity theorems and Thomson's circulation theorem are derived.

Some Aspects of General Relativistic Vortex Dynamics and Ertel's Identity

According to Erters identity and Ertel's vorticity theorem some aspects on general relativistic vortex dynamics and relationships are discussed in detail.

On Ertel’s Potential Vorticity Theorem. On the Impermeability Theorem for Potential Vorticity

Abstract Potential vorticity (PV) is usually defined as αω · gradϕ, where α is the specific volume, ω is vorticity, and ϕ is any quantity, usually a conserved one. The most common derivation of the PV theorem therefore uses the component of the vorticity equation normal to the ϕ surfaces. Since PV can also be expressed as α div(u × gradϕ) and α div(ωϕ), alternative derivations of the PV conservation law are introduced. In these derivations the PV conservation theorem is considered as the divergence of the projection (weighted by |gradϕ|) of the equation of motion onto the direction of gradϕ, or, alternately, as the divergence of a ϕ-weighted vorticity equation. The first of these interpretations is closely related to the procedure of considering every ϕ surface as a surface of constraint for the infinitesimal virtual displacements used in variational methods, and therefore it is closely related to a Hamiltonian derivation of the PV theorem. The different expressions are presented using the spatial as well...