Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-22T04:10:22.367Z Has data issue: false hasContentIssue false
  • English
  • Français

A probabilistic method for Navier-Stokes vortices

Part of: Incompressible inviscid fluids Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Xinyu He
Xinyu He*
Affiliation:
University of Warwick
*
∗ Postal address: Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Consider a Navier-Stokes incompressible turbulent fluid in R2. Let x(t) denote the position coordinate of a moving vortex with initial circulation Γ0 > 0 in the fluid, subject to a force F. Define x(t) as a stochastic process with continuous sample paths described by a stochastic differential equation. Assuming a suitable notion of weak rotationality, it is shown that the stochastic equation is equivalent to a linear partial differential equation for the complex function ψ, i∂ψ/∂t = [-Γ0Δ + F] ψ, where |ψ|2 = ρ(x,t), ρ being the probability density function of finding the vortex centre in position x at time t.

Keywords

Vortex flowsNavier-Stokes

MSC classification

Primary: 76B47: Vortex flows
Secondary: 60G35: Signal detection and filtering
Type
Short Communications
Information
Journal of Applied Probability , Volume 38 , Issue 4 , December 2001 , pp. 1059 - 1066
Copyright
Copyright © by the Applied Probability Trust 2001 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Chorin, A. J. (1991). Equilibrium statistics of a vortex filament with applications. Commun. Math. Phys. 141, 619–631.Google Scholar
Dritschel, D. G. (1989). Strain-induced vortex stripping. In Mathematical Aspects of Vortex Dynamics, ed. Caflisch, R. E., SIAM, Philadelphia, pp. 107–119.Google Scholar
Elworthy, K. D. (1989). Stochastic Differential Equations on Manifolds (London Math. Soc. Lecture Notes 70). Cambridge University Press.Google Scholar
Flandoli, F., and Gubinelli, M. (2001). The Gibbs ensemble of a vortex filament. To appear in Prob. Theory Relat. Fields.Google Scholar
Gliklikh, Y. E. (1996). Ordinary and Stochastic Differential Geometry as Tool for Mathematical Physics. Kluwer, Dordrecht.Google Scholar
He, X. (1996). Dispersion of a vortex trajectory in two-dimensional turbulence. Physica D 95, 163–166.Google Scholar
Marchioro, C., and Pulvirenti, P. (1982). Hydrodynamics in two dimensions and vortex theory. Commun. Math. Phys. 84, 483–503.Google Scholar
Massey, W. S. (1967). Algebraic Topology: An Introduction. Springer, New York.Google Scholar
Moffatt, H. K. (1969). The degree of knottedness of tangled vortex lines. J. Fluid Mech. 35, 117–129.Google Scholar
Mumford, D. (2000). The dawning of the age of stochasticity. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. Spec. Issue 2000, 107–125.Google Scholar
Nelson, E. (1966). Derivation of the Schrödinger equation from Newtonian Mechanics. Phys. Rev. 150, 1079–1085.Google Scholar
Nelson, E. (1988). Stochastic mechanics and random fields. In École d'Été de Probabilités de Saint-Fleur XV–XVII, 1985–1987 (Lecture Notes Math. 1362), Springer, Berlin, pp. 429–450.Google Scholar
Pullin, D. I., and Saffman, P. G. (1998). Vortex dynamics in turbulence. Ann. Rev. Fluid Mech. 30, 31–51.Google Scholar
Robert, R. (1991). Maximum entropy principle for two-dimensional Euler equations. J. Statist. Phys. 65, 531–551.Google Scholar