Is The Taylor-Proudman Theorem Exact In Unbounded Domains? Case Study Of The Three-Dimensional Stability Of A Vortex Pair In A Rapidly Rotating Fluid

Owing to the Taylor-Proudman theorem, it is generally believed that rotating flows should have two-dimensional dynamics for rapid background rotation. Yet, we show that two infinitely long counter-rotating vertical vortices always remain unstable to three-dimensional perturbations even for large rotation rate about the vertical axis. The dominant instability consists of quasi-antisymmetric displacements of the two vortices. Its growth rate is independent of the rotation rate when it is sufficiently large, while the most amplified vertical wavelength scales like the rotation rate. Direct numerical simulations show that the instability leads ultimately to a full three-dimensional breakdown of the vortex pair. In other words, a two-dimensional vortex pair will spontaneously develop three-dimensional variations even in the limit of infinite rotation rate but the wavelength will tend to infinity. This implies that the Taylor-Proudman theorem can be strictly valid only in vertically bounded flows. The scaling for the wavelength is next generalized by showing that the typical vertical scale in rapidly rotating unbounded flows is L-v similar to L- (h)/Ro, where L-h is the horizontal scale and Ro the Rossby number. This scaling law is shown to derive from a self-similarity of the Navier-Stokes equations for Ro << 1. The resulting reduced equations are identical to those obtained first by Julien et al. (Theor. Comput. Fluid Dyn., vol. 11, issue 3-4, 1998, pp. 251-261) and Nazarenko & Schckochihin (J. Fluid Mech., vol. 677, 2011, pp. 134-153) by means of multiscale analyses in the cases of rotating convection or turbulence. The self-similarity demonstrated herein suggests that these reduced equations are valid for any rapidly rotating unbounded flow.