ABSTRACT
Spatially localized invariant solutions of plane Couette flow are organized in a snakes-and-ladders structure strikingly similar to that observed for simpler pattern-forming partial differential equations [Schneider, Gibson, and Burke, Phys. Rev. Lett. 104, 104501 (2010)PRLTAO0031-900710.1103/PhysRevLett.104.104501]. We demonstrate the mechanism by which these snaking solutions originate from well-known periodic states of the Taylor-Couette system. They are formed by a localized slug of wavy-vortex flow that emerges from a background of Taylor vortices via a modulational sideband instability. This mechanism suggests a close connection between pattern-formation theory and Navier-Stokes flow.
ABSTRACT
The formalisms of Wyld [Ann. Phys. 14, 143 (1961)] and Martin, Siggia, and Rose (MSR) [Phys. Rev. A 8, 423 (1973)] address the closure problem of a statistical treatment of homogeneous isotropic turbulence (HIT) based on techniques primarily developed for quantum field theory. In the Wyld formalism, there is a well-known double-counting problem, for which an ad hoc solution was suggested by Lee [Ann. Phys. 32, 292 (1965)]. We show how to implement this correction in a more natural way from the basic equations of the formalism. This leads to what we call the Improved Wyld-Lee Renormalized Perturbation Theory. MSR had noted that their formalism had more vertex functions than Wyld's formalism and based on this felt Wyld's formalism was incorrect. However a careful comparison of both formalisms here shows that the Wyld formalism follows a different procedure to that of the MSR formalism and so the treatment of vertex corrections appears in different ways in the two formalisms. Taking that into account, along with clarifications made to both formalisms, we find that they are equivalent and we demonstrate this up to fourth order.
