Local Stable and Unstable Manifolds and Their Control in Nonautonomous Finite-Time Flows

Abstract

It is well known that stable and unstable manifolds strongly influence fluid motion in unsteady flows. These emanate from hyperbolic trajectories, with the structures moving nonautonomously in time. The local directions of emanation at each instance in time is the focus of this article. Within a nearly autonomous setting, it is shown that these time-varying directions can be characterised through the accumulated effect of velocity shear. Connections to Oseledets spaces and projection operators in exponential dichotomies are established. Availability of data for both infinite-and finite-time intervals is considered. With microfluidic flow control in mind, a methodology for manipulating these directions in any prescribed time-varying fashion by applying a local velocity shear is developed. The results are verified for both smoothly and discontinuously time-varying directions using finite-time Lyapunov exponent fields, and excellent agreement is obtained.

Meridional and Zonal Wavenumber Dependence in Potential Vorticity Flux in Rossby Waves

“…The disturbance would in general give this northward emanation a time-dependent O(ε)-variation, which is characterized by Theorem 2.2 in [40]. Using this, it turns out (calculations not shown) that the O(ε) change to this direction is zero.…”

“…Using Theorem 2.10 in [86], Theorem 2.1 in [40] or equation (2.5) in [90], the leading-order in ε location of the western (left) hyperbolic trajectory is representible as…”

“…In this case, however, the stable and unstable manifolds are attached to hyperbolic (anchoring) trajectories which are varying with time, and whose location is not explicitly known. To deal with this difficulty, recent developments to approximate the hyperbolic trajectory's location [86,90] and direction of emanation of stable/unstable manifolds [40,41] will be used to advantage. Note that hyperbolic trajectory locations are not in general given by instantaneous stagnation points (see examples by Balasuriya [41]), necessitating a more refined approach.…”

“…The formula for A kl and φ kl is exactly as is given in (37) and (38). The PV flux would, by (40), simply be (43) mutiplied by f 0 . Given the fact that the wavenumbers need not be commensurate, the flux (43) will in general not be periodic in time.…”

“…4 and 5 to have tiny, moderate, large and enormous flux amplitudes. Obtaining the manifolds numerically is nontrivial in this case since they are associated with a time-varying anchoring trajectory as opposed to a fixed point, and requires utilizing recent theoretical results [40,41]; a discussion of how these were obtained is given in Appendix C. Figure 6 captures the numerically obtained manifolds only up to a some finite length. All figures are produced by using ε = 0.02, a very small perturbation, and the A kl value associated with each (k, l) pair is given in the captions.…”