The curvature of material lines in chaotic cavity flows

Abstract

Material line folding is studied in two-dimensional chaotic cavity flows. Line folding is measured by the local curvature kϭl؋lЈ/͉l͉ 3 , where l(q) is an infinitesimal vector in the tangential direction of the line, q is a coordinate along the line, and lЈ is the derivative of l with respect to q. It is shown both analytically and numerically that folding is always accompanied by compression. The vector lЈ plays a crucial role as a driving force for the stretching and folding processes. A material line is stretched when lЈ is tangential to the line and it is folded when lЈ is normal to the line. The spatial structure of the curvature field is computed numerically. The short-time structure of the curvature field is similar to the structure of unstable manifolds of periodic hyperbolic points, and closely resembles patterns observed in tracer mixing experiments and in stretching field computations. The long time structure of the field asymptotically approaches an entirely different time-independent structure. Probability density functions of curvature are independent of both time and initial conditions.

Structure of small-scale magnetic fields in the kinematic dynamo theory

“…Such anticorrelation was also noticed by Boozer and coworkers [21], who, in their theory of finite-time Lyapunov exponents for chaotic flows, found that the local Lyapunov exponent of the flow was strongly suppressed in the regions of high curvature. Finally, results very similar to those surveyed above were obtained by Muzzio and coworkers [20] for a number of 2D and 3D deterministic chaotic flows. Thus, the exact theory of the curvature statistics that we have been able to develop in the framework of the Kazantsev-Kraichnan model incorporates all of the essential features thus far observed numerically, as well as surmised in less direct theoretical ways.…”

“…The curvature statistics also appear to be largely insensitive to the type of flows considered. Thus, numerical simulations involving 3D forced [17] and 2D decaying [14] Navier-Stokes turbulence, Kraichnan's [42] random flow model [18,15], 2D and 3D deterministic but chaotic flows [20], and, finally, our own 3D forced-MHD simulations and theoretical results based on the Kazantsev-Kraichan velocity field (both compressible and incompressible), consistently reveal the same set of properties of the curvature distribution. The unbounded exponential growth of the mean-square curvature was first observed in the numerical studies of Pope and coworkers [17], who also found that, unlike the moments of the curvature itself, the moments of its logarithm tended to time-independent asymptotic values.…”

“…His results were extended and, in part, made rigorous by a number of authors [11][12][13][14][15]. Starting with the work of Pope [16], much attention was focused on the statistics of the extrinsic principal curvature of material surface elements in turbulent flows [16][17][18][19] and of the curvature of material lines in both turbulent [18,14,15] and deterministic but chaotic [20] flows. Our results on the statistics of magnetic-field-line curvature are subject to direct comparison with these earlier studies.…”

“…Therefore, besides being important for the astrophysical dynamo as outlined above, studying passive advection of the magnetic field is equivalent to studying the statistics of stretching and distortion of material lines by random flows, which is a fundamental problem in the theory of turbulence [10]. This subject has attracted considerable attention [10][11][12][13][14][15][16][17][18][19][20][21]. Our results on the statistical geometry of magnetic-field lines will have direct applicability in this area.…”

“…(20)], the moments of the curvature would grow even from such an initial state. Two distinct asymptotic regimes can be identified in the evolution of the curvature distribution.…”