arXiv:physics/0305052v2 [physics.flu-dyn] 28 May 2003Submitted to Phys. Rev. Lett.Linear Stability and Subcritical Turbulence in Rotating Shear FlowsPierre-Yves Longaretti∗Laboratoire d’Astrophysique de Grenoble, BP 53X, F-38041, Grenoble, France(Dated: 03/20/03)The relation between rotating plane Couette and Taylor-Couette flows is clarified. Experimentaldata are used to quantify the behavior of the minimum Reynolds number for subcritical turbulenceas a function of rotation and curvature. This last dependence is understood through a phenomeno-logical analysis, which also allows us to relate the subcritical turbulent transport efficiency to thetransition Reynolds number. This implies that the Coriolis force reduces the efficiency of subcriticalturbulent transport with respect to nonrotating flows, and resolves an ongoing controversy.PACS numbers: 47.10.+g; 47.20.Ft; 47.27.Ak; 47.27.PaShear flows constitute one of the prototypical type ofhydrodynamical flows. Furthermore, they are commonlyfound in various instances, e.g. geophysical and astro-physical contexts. This makes the understanding of theirproperties an important issue from both a fundamentaland a practical point of view. Most prominently, char-acterizing turbulence and turbulent transport in suchflows is critically needed, as astrophysical and geophysi-cal shear flows are usually fully turbulent because of thelarge scales involved.A large body of experimental evidence has been col-lected on Taylor-Couette flows in the linearly unstableregime (see, e.g., Ref. [1] and references therein). Muchless is known about the linearly stable, or subcritical,regime [2, 3, 4]. Plane Couette flows (rotating or not)are more difficult to realize experimentally, and have beenless extensively studied [5, 6, 7].On the theoretical side, intense efforts have been de-voted to the understanding of turbulence spectral andstatistical properties (see, e.g., [8] for an introduction tothe subject); also, mostly for practical purposes, rathercomplex turbulent transport models have been developed[9]. Unfortunately, in spite of these remarkable successes,some basic properties such as magnitude of the criticalReynolds number of fully developed turbulence, or itsdependence on rotation, are not yet understood. Worse,even the most sophisticated Reynolds stress closure mod-els fail to account for the existence of subcritical turbu-lence in the presence of a stabilizing rotation.Some of these shortcomings are addressed in thepresent investigation, mostly through a phenomenolog-ical analysis of the effects of rotation and curvature inrotating plane Couette flows and Taylor-Couette flows.This provides us with an understanding of some of thecharacteristic features of subcritical shear turbulence,and, most importantly from a practical point of view, es-tablishes a relation between the efficiency of subcriticalturbulent transport and the magnitude of the minimumturbulent Reynolds number. The emphasis on subcriti-cal turbulence follows for the following reasons. First, ananalysis of angular momentum transport suggests thatboth linearly stable and unstable fully turbulent Taylor-Couette flows are controlled by similar nonlinear physics[2, 10]. Secondly, it turns out that subcritical shear flowsare easier to analyze from a phenomenological point ofview, and they are in any case important in themselves.Let us first reexamine the connection between rotat-ing plane Couette flows and Taylor-Couette flows. TheNavier-Stokes equation for rotating plane Couette flowsreads, in the rotating frame∂w∂t+ w.∇w = −∇P∗ρ− 2? × w + ν?w.(1)In this equation, P∗represents the sum of the inertialterm and of the gas pressure term, as usual when con-sidering incompressible flows. Such flows are character-ized by two dimensionless numbers, the Reynolds numberRe = ?V?L/ν ∼ |w.∇w/ν?w|, and a rotation (inverseRossby) number R?= ?2??L/?V ∼ |2? × w/w.∇w|,which is a global measure of the rotation parameter de-fined by S ≡ −2?/(dV/dy). In these definitions, the xaxis is assumed to lie in the streamwise direction, they axis in the shearwise direction, and the z axis in thespanwise direction (rotation axis); V(y) is the mean flow(along x). The rotation number is positive (resp. neg-ative) (? = ±1) depending on whether the rotation iscyclonic (resp. anticyclonic).Stability limits for inviscid rotating plane Couetteflows can be determined through a displaced particleanalysis [11, 12]. Indeed, although the total work of theCoriolis force vanishes, the work ofthe Coriolis force com-ponent in the streamwise direction during a fluid particledisplacement in the shearwise direction does not. Com-paring the resulting post-displacement streamwise veloc-ity to the equilibrium one implies that the flow is linearlyunstable if −1 < S < 0, and linearly stable otherwise, asthe squared oscillation frequency offluid particles is givenby (dV/dy)2S(S + 1).The Navier-Stokes equation for Taylor-Couette flowsis most meaningfully compared to that of rotating planeCouette flows when it is dispelled in a frame rotatingwith some characteristic rotation velocity of the flow ?o(e.g., the average angular velocity of the two cylinders),
