1999 Fiscal Year Final Research Report Summary
Studies of Turbulence as a Nonlinear Science
| Project/Area Number |
08404006
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| Research Category |
Grant-in-Aid for Scientific Research (A)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Nagoya University |
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Principal Investigator |
KIMURA Yoshifumi Nagoya University, Graduate School of Mathematics, Professor, 大学院・多元数理科学研究科, 教授 (70169944)
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| Co-Investigator(Kenkyū-buntansha) |
NAWA Hayato Graduate School of Mathematics, Associate Professor, 大学院・多元数理科学研究科, 助教授 (90218066)
OBATA Nobuaki Graduate School of Mathematics, Associate Professor, 大学院・多元数理科学研究科, 助教授 (10169360)
OHTA Hiroshi Graduate School of Mathematics, Associate Professor, 大学院・多元数理科学研究科, 助教授 (50223839)
KIDA Shigeo National Institute for Fusion Science, Professor, 核融合科学研究所, 教授 (70093234)
NAITO Hisashi Graduate School of Mathematics, Associate Professor, 大学院・多元数理科学研究科, 助教授 (40211411)
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| Project Period (FY) |
1996 – 1999
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| Keywords | turbulence / fluid mechanics / vortex motion / diffusion / geophysical flows / Navier-Strokes equations / Hamiltonian dynamics / Group theory |
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| Research Abstract |
(1) Vortex Formation and Diffusion in Rotating Stratified Turbulence : Rotation and Stratification are two fundamental physical mechanisms in atmospheres and oceans. Studies of vortex formation and diffusion in such flows play a vital role in improving our understanding of geophysical and astrophysical turbulent phenomena. We examine results of direct numerical simulation of the Navier-Stokes equations with the effects of rotation and stably stratification. Both rotation and stratifications tend to make flows two dimensional (or two components), but we verified that they work in different ways in generating vertical structures. As for diffusion, we observed that stratification suppresses the particle migration in the vertical direction and so does rotation, and that the linear growth in time for single particle dispersion persists in the horizontal direction under strong rotation and/or stratification.
(2) Vortex Motion on Surfaces with Constant Curvature : Vortex motion on two- dimensi
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onal Riemannian surfaces with constant curvature is formulated. By way of the stereographic projection, the relation and difference between the vortex motion on a sphere and on a hyperbolic plane can be clearly analyzed. The Hamiltonian formalism is presented for the motion of point vortices on a sphere and a hyperbolic plane. As an example of analytic solutions, the motion of a vortex pair (dipole) is considered. It is shown that a dipole draws a geodesic curve as its trajectory on both surfaces.
(3) Evolution of decaying two-dimensional turbulence and self similarity : We examine the consequences of self-similarity of the energy spectrum of two-dimensional decaying turbulence, and conclude that traditional closures are consistent with this principle only if the regions of space contributing significantly to energy and enstrophy transfer comprise an ever diminishing region of space as time proceeds from the initial time of Gaussian chaos.
(4) Axisymmetrization process for a non-uniform elliptic vortex : The axisymmetrization of a 2D non-uniform elliptic vortex is studied in terms of the growth of palinstrophy, the squared of the vorticity gradient. First, it is pointed out that the equation for the palinstrophy growth, if written in terms of the strain rate tensor, has a similar form to that of enstrophy growth in 3D - the vortex-stretching equation. Then palinstrophy production is analyzed particularly for non-uniform elliptic vortices. It is shown analytically and verified numerically that a non-uniform elliptic vortex in general has quadrupole structure for the palinstrophy production, and that in the positive production regions, vortex filaments are ejected following the gradient enhancement process for vorticity.
(5) Pressure distribution for random Gaussian Velocities : Pressure distributions for random Gaussian velocities are studied both analytically and numerically. Arguments based on rotation symmetry allow to clarify the analytical structure of the characteristic function of pressure and to find the power of all its singularities, which in turn, allows to obtain the exact form of the PDF tail, including both exponent and power pre-exponent factors. For the narrow velocity spectrum (velocity restricted to a shell in k-space), the characteristic function is found explicitly, generating pressure cummulants of all orders. Less
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