# American Institute of Mathematical Sciences

ISSN:
1531-3492

eISSN:
1553-524X

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## Discrete & Continuous Dynamical Systems - B

February 2005 , Volume 5 , Issue 1

A special issue on Recent Advances in Vortex Dynamics and Turbulence

Guest Editors: Chjan C. Lim and Ka Kit Tung

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2005, 5(1): i-i doi: 10.3934/dcdsb.2005.5.1i +[Abstract](1801) +[PDF](27.2KB)
Abstract:
As the subject of vortex dynamics and its applications to two-dimensional fluid flows mature, we have witnessed an explosion in the number of research works in the field. It is the aim of this special issue to collate some of this recent advance and at the same time point to several new directions. One of these new directions is the re-entry of equilibrium statistical mechanics into the field. Many years after the classical papers of Onsager, Kraichnan, Leith, Montgomery, Lundgren, Pointin and Chorin, we are at a point where the Kraichnan, Batchelor and Leith energy-enstrophy theories in two-dimensional turbulence have been studied from new analytical and numerical points of views. A second emerging direction is in the use of a particular type of large-scale scientific computing in vortex statistics, namely Monte-Carlo simulations of vortex gas in the plane and sphere which explore an extended range of parameter values such as temperature and chemical potentials.
2005, 5(1): 1-14 doi: 10.3934/dcdsb.2005.5.1 +[Abstract](1514) +[PDF](403.0KB)
Abstract:
This paper presents the statistical equilibrium distributions of single-species vortex gas and cylindrical electron plasmas on the unbounded plane obtained by Monte Carlo simulations. We present detailed numerical evidence that at high values of $\beta >0$ and $\mu >0$, where $\beta$ is the inverse temperature and $\mu$ is the Lagrange multiplier associated with the conservation of the moment of vorticity, the equilibrium vortex gas distribution is centered about a regular crystalline distribution with very low variance. This equilibrium crystalline structure has the form of several concentric nearly regular polygons within a bounding circle of radius $R.$ When $\beta$ ~ $O(1)$, the mean vortex distributions have nearly uniform vortex density inside a circular disk of radius $R.$ In all the simulations, the radius $R=\sqrt{\beta \Omega /(2\mu )}$ where $\Omega$ is the total vorticity of the point vortex gas or number of identical point charges. Using a continuous vorticity density model and assuming that the equilibrium distribution is a uniform one within a bounding circle of radius $R$, we show that the most probable value of $R$ scales with inverse temperature $\beta >0$ and chemical potential $\mu >0$ as in $R=\sqrt{\beta \Omega /(2\mu )}.$
2005, 5(1): 15-33 doi: 10.3934/dcdsb.2005.5.15 +[Abstract](1797) +[PDF](211.4KB)
Abstract:
A new generalization of the Poincaré-Birkhoff fixed point theorem applying to small perturbations of finite-dimensional, completely integrable Hamiltonian systems is formulated and proved. The motivation for this theorem is an extension of some recent results of Blackmore and Knio on the dynamics of three coaxial vortex rings in an ideal fluid. In particular, it is proved using KAM theory and this new fixed point theorem that if $n>3$ coaxial rings all having vortex strengths of the same sign are initially in certain positions sufficiently close to one another in a three-dimensional ideal fluid environment, their motion with respect to the center of vorticity exhibits invariant $(n-1)$-dimensional tori comprised of quasiperiodic orbits together with interspersed periodic trajectories.
2005, 5(1): 35-50 doi: 10.3934/dcdsb.2005.5.35 +[Abstract](1639) +[PDF](236.2KB)
Abstract:
The paper studies the system of a rigid body interacting dynamically with point vortices in a perfect fluid. For arbitrary value of vortex strengths and circulation around the cylinder the system is shown to be Hamiltonian (the corresponding Poisson bracket structure is rather complicated). We also reduced the number of degrees of freedom of the system by two using the reduction by symmetry technique and performed a thorough qualitative analysis of the integrable system of a cylinder interacting with one vortex.
2005, 5(1): 51-66 doi: 10.3934/dcdsb.2005.5.51 +[Abstract](1681) +[PDF](183.7KB)
Abstract:
We study the existence of periodic solutions in the neighbourhood of symmetric (partially) elliptic equilibria in purely reversible Hamiltonian vector fields. These are Hamiltonian vector fields with an involutory reversing symmetry $R$. We contrast the cases where $R$ acts symplectically and anti-symplectically.
In case $R$ acts anti-symplectically, generically purely imaginary eigenvalues are isolated, and the equilibrium is contained in a local two-dimensional invariant manifold containing symmetric periodic solutions encircling the equilibrium point.
In case $R$ acts symplectically, generically purely imaginary eigenvalues are doubly degenerate, and the equilibrium is contained in two two-dimensional invariant manifolds containing nonsymmetric periodic solutions encircling the equilibrium point. In addition, there exists a three-dimensional invariant surface containing a two-parameter family of symmetric periodic solutions.
2005, 5(1): 67-78 doi: 10.3934/dcdsb.2005.5.67 +[Abstract](1721) +[PDF](242.4KB)
Abstract:
Analysis of energy spectra and fluxes of 2D forced incompressible turbulence in the energy range reveals marked departures from the $-5/3$ law and the idea of spectral locality. Departures from the locality could also be diagnosed in the enstrophy interval, and in the energy range of beta-plane turbulence.
2005, 5(1): 79-102 doi: 10.3934/dcdsb.2005.5.79 +[Abstract](1761) +[PDF](224.9KB)
Abstract:
The Kraichnan-Leith-Batchelor scenario of a dual cascade, consisting of an upscale pure energy cascade and a downscale pure enstrophy cascade, is an idealization valid only in an infi nite domain in the limit of in finite Reynolds number. In realistic situations there are double cascades of energy and enstrophy located both upscale and downscale of injection, as long as there are cascades. We outline the statistical theory governing the double cascades and predict the form of the energy spectrum. We show that in general the twin conservation of energy and enstrophy imply the presence of two constant fluxes in each inertial range. This gives rise to a more complicated energy spectrum, which cannot be predicted using dimensional arguments as in the classical theory.
2005, 5(1): 103-124 doi: 10.3934/dcdsb.2005.5.103 +[Abstract](1682) +[PDF](198.6KB)
Abstract:
This paper is concerned with three interrelated issues on our proposal of double cascades intended to serve as a more realistic theory of two-dimensional turbulence. We begin by examining the approach to the KLB limit. We present improved proofs of the result by Fjortoft. We also explain why in that limit the subleading downscale energy cascade and upscale enstrophy cascade are hidden in the energy spectrum. Then we review the experimental evidence from numerical simulations concerning the realizability of the energy and enstrophy cascade. The inverse energy cascade is found to be affected by the presense of a particular solution, and the downscale enstrophy cascade is not robust. In particular, while it is possible to have either the upscale range or the downscale range with suitable choice of dissipations, the dual cascade of KLB does not appear to be realizable, not even approximately. Finally, we amplify the hypothesis that the energy spectrum of the atmosphere reflects a combined downscale cascade of energy and enstrophy. The possibility of the downscale helicity cascade is also considered.
2005, 5(1): 125-136 doi: 10.3934/dcdsb.2005.5.125 +[Abstract](2056) +[PDF](779.0KB)
Abstract:
The point-vortex system on the surface of the sphere is examined by Monte Carlo methods. The statistical equilibria found in the system when it is constrained to keep circulation zero (but without other explicit constraints on site values) are found to be self-regulating in a sense. While site strengths will grow without bound as the number of sweeps increases, the Dirichlet quotient, the ratio of enstrophy to energy, is found to converge rapidly to a finite nonzero value. This unlimited growth in site values remains controlled. The dependences of this quotient on the temperature and on the mesh size are examined.
2005, 5(1): 137-152 doi: 10.3934/dcdsb.2005.5.137 +[Abstract](1773) +[PDF](476.0KB)
Abstract:
The Hamiltonian system governing $N$-interacting particles constrained to lie on a closed planar curve are derived. The problem is formulated in detail for the case of logarithmic (point-vortex) interactions. We show that when the curve is circular with radius $R$, the system is completely integrable for all particle strengths $\Gamma _ \beta$, with particle $\Gamma _ \beta$ moving with frequency $\omega _ \beta = (\Gamma - \Gamma _ \beta )/4 \pi R^2$, where $\Gamma = \sum^{N}_{\alpha=1} \Gamma _ \alpha$ is the sum of the strengths of all the particles. When all the particles have equal strength, they move periodically around the circle keeping their relative distances fixed. When not all the strengths are equal, two or more of the particles collide in finite time. The diffusion of a neutral particle (i.e. the problem of 1D mixing) is examined. On a circular curve, a neutral particle moves uniformly with frequency $\Gamma / 4 \pi R^2$. When the curve is not perfectly circular, for example when given a sinusoidal perturbation, or when the particles move on concentric circles with different radii, the particle dynamics is considerably more complex, as shown numerically from an examination of power spectra and collision diagrams. Thus, the circular constraint appears to be special in that it induces completely integrable dynamics.
2005, 5(1): 153-174 doi: 10.3934/dcdsb.2005.5.153 +[Abstract](1666) +[PDF](1070.0KB)
Abstract:
Incompressible, homogeneous magnetohydrodynamic (MHD) turbulence consists of fluctuating vorticity and magnetic fi elds, which are represented in terms of their Fourier coefficients. Here, a set of fi ve Fourier spectral transform method numerical simulations of two-dimensional (2-D) MHD turbulence on a $512^2$ grid is described. Each simulation is a numerically realized dynamical system consisting of Fourier modes associated with wave vectors $\mathbf{k}$, with integer components, such that $k = |\mathbf{k}| \le k_{max}$. The simulation set consists of one ideal (non-dissipative) case and four real (dissipative) cases. All fi ve runs had equivalent initial conditions. The dimensions of the dynamical systems associated with these cases are the numbers of independent real and imaginary parts of the Fourier modes. The ideal simulation has a dimension of $366104$, while each real simulation has a dimension of $411712$. The real runs vary in magnetic Prandtl number $P_M$, with $P_M \in {0.1, 0.25, 1, 4}$. In the results presented here, all runs have been taken to a simulation time of $t = 25$. Although ideal and real Fourier spectra are quite di fferent at high $k$, they are similar at low values of $k$. Their low $k$ behavior indicates the existence of broken symmetry and coherent structure in real MHD turbulence, similar to what exists in ideal MHD turbulence. The value of $P_M$ strongly affects the ratio of kinetic to magnetic energy and energy dissipation (which is mostly ohmic). The relevance of these results to 3-D Navier-Stokes and MHD turbulence is discussed.

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