Reynolds stresses and mean fields generated by pure waves: applications to shear flows and convection in a rotating shell

Abstract

A general reformulation of the Reynolds stresses created by two-dimensional waves breaking a translational or a rotational invariance is described. This reformulation emphasizes the importance of a geometrical factor: the slope of the separatrices of the wave flow. Its physical relevance is illustrated by two model systems: waves destabilizing open shear flows; and thermal Rossby waves in spherical shell convection with rotation. In the case of shear-flow waves, a new expression of the Reynolds-Orr amplification mechanism is obtained, and a good understanding of the form of the mean pressure and velocity fields created by weakly nonlinear waves is gained. In the case of thermal Rossby waves, results of a three-dimensional code using no-slip boundary conditions are presented in the nonlinear regime, and compared with those of a two-dimensional quasi-geostrophic model. A semi-quantitative agreement is obtained on the flow amplitudes, but discrepancies are observed concerning the nonlinear frequency shifts. With the quasi-geostrophic model we also revisit a geometrical formula proposed by Zhang to interpret the form of the zonal flow created by the waves, and explore the very low Ekman-number regime. A change in the nature of the wave bifurcation, from supercritical to subcritical, is found. IntroductionIn hydrodynamic stability theory and turbulence modelling, it is natural and customary to separate the velocity field into a mean flow V and a fluctuating part v. In the Navier-Stokes equation for V , the nonlinear term expressing the feedback of the fluctuating flow onto the mean flow is usually written as the divergence of the Reynolds stress tensorwhere the angle brackets indicate a suitable averaging. A good understanding or modelling of τ is therefore required to explain the form of the mean flow, and other mean properties of the flow, such as the flow rate and head losses, in the case of an open system for instance. The tensor τ is also quite important for energy since in purely hydrodynamical systems its contraction with the mean strain rate tensor is the only possible source of growth of the fluctuating kinetic energy, as shown in a landmark paper by Reynolds (1895) Busse (1983) in the case of a fluctuating field corresponding to a columnar quasi-two-dimensional wave. He noted in his § 3 (see also his figure 3), a link between the variations of the 'phase function' of the wave and the relevant cross-diagonal Reynolds stress, which was revisited by Zhang (1992). The reformulation proposed by Zhang for the Reynolds stress, however, is limited to a special form of the streamfunction. In the somewhat more general case of a two-dimensional, x, y, fluctuating field, Pedlosky (1987) established ( § 7.3, p. 502) a link between the product v x v y that controls the most important Reynolds stress, i.e. the cross-diagonal stress τ xy , and the slope of the streamlines of v. Pedlosky offered no simple formula for the average v x v y , however.The primary aim of this paper is to complement these pioneering works by propo...

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