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Previously, we have reported theory and experiments on the spin-up of a linearly stratified fluid contained in a sealed square cylinder with a bottom sloped at angle α (Munro & Foster, Phys. Fluids, 26, 026603, 2014). The motion is characterized by the small Rossby number, ϵ , the fractional change in the rotation rate, the Ekman number, E, and the Burger number, S, the square of the ratio of the buoyancy frequency (or Brunt-Väisälä frequency) to the base rotation rate. This paper, supplementing what has gone before, is focused on the horizontal and vertical structure of the propagating vorticity waves (Rossby waves), with alternating anti-cyclonic and cyclonic cells, which move across the tank from east to west, thereby accomplishing at least the early stages of the spin-up. We find strong dependence of the modal structure of these Rossby waves, and their speed, on the Burger number. However, what is particularly noteworthy is the observed strong north-south asymmetry of the wave field. We find that the asymmetry is not consistent with typical Rossby-wave theories (Pedlosky and Greenspan, J. Fluid Mech., 27, p. 291, 1967), but is attributable to second-order effects in the asymptotic expansion of the streamfuntion in powers of α . [ABSTRACT FROM AUTHOR]

We report experimental and theoretical results on how a fluid (homogeneous or continuously stratified) is spun up in a closed, semicircular cylinder. Experiments were performed for Rossby numbers Ro = 0.02, 0.2 and 1 (the latter corresponding to the limiting case of spin-up from rest), with the Ekman number E = O(10-5), and the Burger number (S) varied between 0 and 10. There are two key processes: Ekman pumping that drives the core flow; and the formation and breakdown of the vertical-wall boundary layers, with respective characteristic time scales t ~ E-1/2 and Ro-1. When these time scales are comparable, the observed flow is dominated by the gradual spin-up of the initial anticyclone that forms when the rotation rate is increased, which fills the container's interior; vorticity generated adjacent to the vertical walls throughout remains confined to the neighbourhood of the container's walls and corners. Conversely, when E1/2/Ro ≪ 1, the vertical-wall boundary layers rapidly break down, resulting in the formation of cyclonic vortices in the container's vertical corners, which grow and interact with the initial anticyclone, leading to the formation of a three-cell flow pattern. For Ro = 0.02, our theoretical description of the flow generally agrees well with experiments, and the computation of the eruption times for the unsteady boundary layers is consistent with the observations for both Ro = 0.02 and Ro = 0.2. [ABSTRACT FROM AUTHOR]

A linearly stratified fluid contained in a circular cylinder with a linearly sloped base, whose axis is aligned with the rotation axis, is spun-up from a rotation rate Ω-ΔΩ to Ω (with ΔΩ≪Ω) by Rossby waves propagating across the container. Experimental results presented here, however, show that if the Burger number S is not small, then that spin-up looks quite different from that reported by Pedlosky & Greenspan (J. Fluid Mech., vol. 27, 1967, pp. 291-304) for S=0. That is particularly so if the Burger number is large, since the Rossby waves are then confined to a region of height S-1/2 above the sloped base. Axial vortices, ubiquitous features even at tiny Rossby numbers of spin-up in containers with vertical corners (see van Heijst et al.Phys. Fluids A, vol. 2, 1990, pp. 150-159 and Munro & Foster Phys. Fluids, vol. 26, 2014, 026603, for example), are less prominent here, forming at locations that are not obvious a priori, but in the 'western half' of the container only, and confined to the bottom S-1/2 region. Both decay rates from friction at top and bottom walls and the propagation speed of the waves are found to increase with S as well. An asymptotic theory for Rossby numbers that are not too large shows good agreement with many features seen in the experiments. The full frequency spectrum and decay rates for these waves are discussed, again for large S, and vertical vortices are found to occur only for Rossby numbers comparable to E1/2, where E is the Ekman number. Symmetry anomalies in the observations are determined by analysis to be due to second-order corrections to the lower-wall boundary condition. [ABSTRACT FROM AUTHOR]

We present experimental and theoretical results for the adjustment of a fluid (homogeneous or linearly stratified), which is initially rotating as a solid body with angular frequency Ω - ΔΩ, to a nonlinear increase ΔΩ in the angular frequency of all bounding surfaces. The fluid is contained in a cylinder of square cross-section which is aligned centrally along the rotation axis, and we focus on the O(R0-lΩ-l) time scale, where R0 = ΔΩ/Ω is the Rossby number. The flow development is shown to be dominated by unsteady separation of a viscous sidewall layer, leading to an eruption of vorticity that becomes trapped in the four vertical corners of the container. The longer-time evolution on the standard 'spin-up' time scale, E-1/2Ω-1 (where E is the associated Ekman number), has been described in detail for this geometry by Foster & Munro (J. Fluid Mech., vol. 712, 2012, pp. 7-40), but only for small changes in the container's rotation rate (i.e. R0 « 1). In the linear case, for R0 « E1/2 « 1, there is no sidewall separation. In the present investigation we focus on the fully nonlinear problem, R0 = O(1), for which the sidewall viscous layers are Prandtl boundary layers and (somewhat unusually) periodic around the container's circumference. Some care is required in the corners of the container, but we show that the sidewall boundary layer breaks down (separates) shortly after an impulsive change in rotation rate. These theoretical boundary-layer results are compared with two-dimensional Navier-Stokes results which capture the eruption of vorticity, and these are in turn compared to laboratory observations and data. The experiments show that when the Burger number, S = (N/Ω)² (where N is the buoyancy frequency), is relatively large - corresponding to a strongly stratified fluid - the flow remains (horizontally) two-dimensional on the O(Ro-1Ω-1) time scale, and good quantitative predictions can be made by a two-dimensional theory. As S was reduced in the experiments, three-dimensional effects were observed to become important in the core of each corner vortex, on this time scale, but only after the breakdown of the sidewall layers. [ABSTRACT FROM AUTHOR]

Fluid entering the periphery of a cylindrical tank mounted on a rotating table is pumped inwards toward a central, floor drain by a potential vortex that is established in the fluid interior. We present here an asymptotic theory for small Rossby and Ekman numbers, including detailed solutions in the vortex core. Results for azimuthal velocity variation with radius agree quite well with the experiments of Andersen et al. (J. Fluid Mech., vol. 556, 2006, pp. 121–146), in spite of their free upper boundary. Modifications of the flow are presented in the instance that a short cylinder is place on the tank axis as in the work of Chen et al. (J. Fluid Mech., vol. 733, 2013, pp. 134–157). The overall flow structure found here is exactly that noted by both Andersen et al. and Chen et al. [ABSTRACT FROM AUTHOR]