Stuart-type vortices on a rotating sphere

Stuart vortices are among the few known smooth explicit solutions of the planar Euler equations with a nonlinear vorticity, and they have a counterpart for inviscid flow on the surface of a fixed sphere. By means of a perturbative approach we adapt the method used to investigate Stuart vortices on a fixed sphere to provide insight into some large-scale shallow-water flows on a rotating sphere that model the dynamics of ocean gyres.

Strato-hyperbolic instability: a new mechanism of instability in stably stratified vortices

The stability of stably stratified vortices is studied by local stability analysis. Three base flows that possess hyperbolic stagnation points are considered: the two-dimensional (2-D) Taylor–Green vortices, the Stuart vortices and the Lamb–Chaplygin dipole. It is shown that the elliptic instability is stabilized by stratification; it is completely stabilized for the 2-D Taylor–Green vortices, while it remains and merges into hyperbolic instability near the boundary or the heteroclinic streamlines connecting the hyperbolic stagnation points for the Stuart vortices and the Lamb–Chaplygin dipole. More importantly, a new instability caused by hyperbolic instability near the hyperbolic stagnation points and phase shift by the internal gravity waves is found; it is named the strato-hyperbolic instability; the underlying mechanism is parametric resonance as unstable band structures appear in contours of the growth rate. A simplified model explains the mechanism and the resonance curves. The growth rate of the strato-hyperbolic instability is comparable to that of the elliptic instability for the 2-D Taylor–Green vortices, while it is smaller for the Stuart vortices and the Lamb–Chaplygin dipole. For the Lamb–Chaplygin dipole, the tripolar instability is found to merge with the strato-hyperbolic instability as stratification becomes strong. The modal stability analysis is also performed for the 2-D Taylor–Green vortices. It is shown that global modes of the strato-hyperbolic instability exist; the structure of an unstable eigenmode is in good agreement with the results obtained by local stability analysis. The strato-hyperbolic mode becomes dominant depending on the parameter values.

Effects of an axial flow on the centrifugal, elliptic and hyperbolic instabilities in Stuart vortices

Linear stability of the Stuart vortices in the presence of an axial flow is studied. The local stability equations derived by Lifschitz & Hameiri (Phys. Fluids A, vol. 3 (11), 1991, pp. 2644–2651) are rewritten for a three-component (3C) two-dimensional (2D) base flow represented by a 2D streamfunction and an axial velocity that is a function of the streamfunction. We show that the local perturbations that describe an eigenmode of the flow should have wavevectors that are periodic upon their evolution around helical flow trajectories that are themselves periodic once projected on a plane perpendicular to the axial direction. Integrating the amplitude equations around periodic trajectories for wavevectors that are also periodic, it is found that the elliptic and hyperbolic instabilities, which are present without the axial velocity, disappear beyond a threshold value for the axial velocity strength. Furthermore, a threshold axial velocity strength, above which a new centrifugal instability branch is present, is identified. A heuristic criterion, which reduces to the Leibovich & Stewartson criterion in the limit of an axisymmetric vortex, for centrifugal instability in a non-axisymmetric vortex with an axial flow is then proposed. The new criterion, upon comparison with the numerical solutions of the local stability equations, is shown to describe the onset of centrifugal instability (and the corresponding growth rate) very accurately.

Critical Layer Analysis of Stuart Vortices in a Plane Jet

Asymptotic techniques are used to model quasi-steady-state vortices in the plane (Bickley) inviscid jet. A nonlinear critical layer analysis is used to find a family of steady-state finite amplitude two-dimensional vortices which are based on the Stuart vortex.

On Kelvin–Stuart vortices in a viscous fluid

When one contemplates the one-parameter family of steady inviscid shear flows discovered by J. T. Stuart in 1967, an obvious thought is that these flows resemble a row of vortices diffusing in a viscous fluid, with the parameter playing the role of a reversed time. In this paper, we ask how close this resemblance is. Accordingly, the paper begins to explore Navier–Stokes solutions having as initial condition the classical, irrotational flow due to a row of point vortices. However, since we seek explicit answers, such exploration seems possible only in two relatively easy cases: that of small time and arbitrary Reynolds number and that of small Reynolds number and arbitrary time.

Zonal approach to centrifugal, elliptic and hyperbolic instabilities in Stuart vortices with external rotation

The stability analysis of a street of Stuart vortices in a rotating frame is performed by integrating the Kelvin–Townsend equations along the mean flow trajectories, using the geometrical optics technique (Lifschitz & Hameiri 1991) for short-wave perturbations. A parallel is drawn between the formulations of this zonal approach and that of rapid distortion theory, better known to the turbulence community. The results presented confirm those obtained by the standard stability analysis based on normal-mode decomposition: depending on the rotation parameter and the oblique mode considered, three unstable zones are identified, related to the centrifugal, elliptic and hyperbolic instabilities, as observed for Taylor–Green cells (Sipp et al. 1999). Anticyclonic rotation is shown to destabilize Stuart vortices through a combination of the elliptical and centrifugal instability mechanisms, depending on the ratio of its rate to the structure core vorticity. Available stability criteria are discussed in the general case of two-dimensional rotating flows, in relation to their streamline topology and the values of the local Rossby number or vorticity.

Three-dimensional destabilization of Stuart vortices: the influence of rotation and ellipticity

We investigate the influence of the ellipticity of a columnar vortex in a rotating environment on its linear stability to three-dimensional perturbations. As a model of the basic-state vorticity distribution, we employ the Stuart steady-state solution of the Euler equations. In the presence of background rotation, an anticyclonic vortex column is shown to be strongly destabilized to three-dimensional perturbations when background rotation is weak, while rapid rotation strongly stabilizes both anticyclonic and cyclonic columns, as might be expected on the basis of the Taylor–Proudman theorem. We demonstrate that there exist three distinct forms of three-dimensional instability to which strong anticyclonic vortices are subject. One form consists of a Coriolis force modified form of the ‘elliptical’ instability, which is dominant for vortex columns whose cross-sections are strongly elliptical. This mode was recently discussed by Potylitsin & Peltier (1998) and Leblanc & Cambon (1998). The second form of instability may be understood to constitute a three-dimensional inertial (centrifugal) mode, which becomes the dominant mechanism of instability as the ellipticity of the vortex column decreases. Also evident in the Stuart model of the vorticity distribution is a third ‘hyperbolic’ mode of instability that is focused on the stagnation point that exists between adjacent vortex cores. Although this short-wavelength cross-stream mode is much less important in the spectrum of the Stuart model than it is in the case of a true homogeneous mixing layer, it nevertheless does exist even though its presence has remained undetected in most previous analyses of the stability of the Stuart solution.