Hamiltonian dynamics of vortex lines in hydrodynamic-type systems

Кузнецов, Е. А.; Ruban, V. P.

doi:10.1134/1.567795

Cited by 47 publications

(69 citation statements)

References 6 publications

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“…The Jacobian J can take arbitrary values. From the formula (22) it is easily to get the expression for the vorticity Ω in the given point r at the instant t (compare with [2,7]):…”

Section: Solution Of the Equation (18) Yields The Mappingmentioning

confidence: 99%

“…Transition to the Lagrangian description in a new hydrodynamics is equivalent for the original Euler equations to the mixed Lagrangian-Eulerian description -the vortex line representation (VLR) [2]. Due to compressibility of a "new" fluid the collapse of vortex lines can happen as the result of breaking (or overturning) of vortex lines.…”

Section: Introductionmentioning

confidence: 99%

“…From this point of view a vortex line represents the invariant object and therefore it is natural to seek for such a transformation when this invariance is seen from the very beginning. Such type of description -the vortex line representation -was introduced in the papers [2,7] by Ruban and the author of this paper.…”

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confidence: 99%

“…Performing in (11) the transformations (7), we arrive at the equation of motion for vortex lines [2]:…”

mentioning

confidence: 99%

“…And this is possible in spite of incompressibility of both divergence-free fields, i.e., vorticity and velocity. To describe the breaking of vortex lines in the papers [2,7] it was suggested the vortex line representation -a mixed Lagrangian-Eulerian description when each vortex line is labeled by a two-dimensional marker and another parameter defines the vortex line itself.…”

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confidence: 99%

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Vortex line representation for flows of ideal and viscous fluids

Kuznetsov

2002

Jetp Lett.

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It is shown that the Euler hydrodynamics for vortical flows of an ideal fluid coincides with the equations of motion of a charged compressible fluid moving due to a self-consistent electromagnetic field. Transition to the Lagrangian description in a new hydrodynamics is equivalent for the original Euler equations to the mixed Lagrangian-Eulerian description -the vortex line representation (VLR) [2]. Due to compressibility of a "new" fluid the collapse of vortex lines can happen as the result of breaking (or overturning) of vortex lines. It is found that the Navier-Stokes equation in the vortex line representation can be reduced to the equation of the diffusive type for the Cauchy invariant with the diffusion tensor given by the metric of the VLR.PACS: 47.15.Ki, 47.32.Cc 1. Collapse as a process of a singularity formation in a finite time from the initially smooth distribution plays the very important role being considered as one of the most effective mechanisms of the energy dissipation. For hydrodynamics of incompressible fluids collapse must play also a very essential role. It is well known that appearance of singularity in gasodynamics, i.e., in compressible hydrodynamics, is connected with the phenomenon of breaking that is the physical mechanism leading to emergence of shocks. From the point of view of the classical catastrophe theory [3] this process is nothing more than the formation of folds. It is completely characterized by the mapping corresponding to transition from the Eulerian description to the Lagrangian one. Vanishing the Jacobian J of this mapping means emergence of a singularity for spatial derivatives of velocity and density of a gas. In the incompressible case breaking as intersection of trajectories of Lagrangian particles is absent because the Jacobian of the corresponding mapping is fixed, in the simplest case equal to unity. By this reason, it would seem that there were no any reasons for existence of such phenomenon at all. In spite of this fact, as it was shown in [4,5,6], breaking, however, is possible in this case also. It can happen with vortex lines. Unlike the breaking in gasodynamics, the breaking of 1

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“…The Jacobian J can take arbitrary values. From the formula (22) it is easily to get the expression for the vorticity Ω in the given point r at the instant t (compare with [2,7]):…”

Section: Solution Of the Equation (18) Yields The Mappingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

“…Performing in (11) the transformations (7), we arrive at the equation of motion for vortex lines [2]:…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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Vortex line representation for flows of ideal and viscous fluids

Kuznetsov

2002

Jetp Lett.

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Numerical evidence of breaking of vortex lines in an ideal fluid

Kuznetsov

Podvigina

Zheligovsky

Fluid Mechanics and Its Applications

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Emergence of singularity of vorticity at a single point, not related to any symmetry of the initial distribution, has been demonstrated numerically for the first time. Behavior of the maximum of vorticity near the point of collapse closely follows the dependence (t0 − t) −1 , where t0 is the time of collapse. This agrees with the interpretation of collapse in an ideal incompressible fluid as of the process of vortex lines breaking.

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Dynamics of Vortex and Magnetic Lines in Ideal Hydrodynamics and MHD

Кузнецов

Ruban

1999

Nonlinear MHD Waves and Turbulence

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Vortex line and magnetic line representations are introduced for description of flows in ideal hydrodynamics and MHD, respectively. For incompressible fluids it is shown that the equations of motion for vorticity Ω and magnetic field with the help of this transformation follow from the variational principle. By means of this representation it is possible to integrate the system of hydrodynamic type with the Hamiltonian H = |Ω|dr. It is also demonstrated that these representations allow to remove from the noncanonical Poisson brackets, defined on the space of divergence-free vector fields, degeneracy connected with the vorticity frozenness for the Euler equation and with magnetic field frozenness for ideal MHD. For MHD a new Weber type transformation is found. It is shown how this transformation can be obtained from the two-fluid model when electrons and ions can be considered as two independent fluids. The Weber type transformation for ideal MHD gives the whole Lagrangian vector invariant. When this invariant is absent this transformation coincides with the Clebsch representation analog introduced

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Hamiltonian dynamics of vortex lines in hydrodynamic-type systems

Cited by 47 publications

References 6 publications

Vortex line representation for flows of ideal and viscous fluids

Vortex line representation for flows of ideal and viscous fluids

Numerical evidence of breaking of vortex lines in an ideal fluid

Dynamics of Vortex and Magnetic Lines in Ideal Hydrodynamics and MHD

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