Abstract
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 $\unicode[STIX]{x1D6FA}-\unicode[STIX]{x0394}\unicode[STIX]{x1D6FA}$ to $\unicode[STIX]{x1D6FA}$ (with $\unicode[STIX]{x0394}\unicode[STIX]{x1D6FA}\ll \unicode[STIX]{x1D6FA}$) 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 $E^{1/2}$, 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.
Highlights
In a pioneering paper, Pedlosky & Greenspan (1967) examined how a rotating, homogeneous fluid is spun up inside a sliced cylinder—a closed, vertical cylinder with its base plane inclined at an angle α to the horizontal
In this paper we will consider how the spin-up mechanism and formation of Rossby waves in a sliced cylinder are affected by the introduction of a linear density stratification, a problem that has remained unresolved since publication of Pedlosky and Greenspan’s classical paper
What was surprising is that the actual spin-up time is shorter in the presence of the sloping bottom boundary: if we take the bottom slope, α, to be large enough so that α ≫ E1/2, where the Ekman number is defined by E ≡ ν/ΩL2 (L denoting the container’s width), we find that the spin-up time scale is somewhat smaller, namely (ν Ω)1/2
Summary
Pedlosky & Greenspan (1967) examined how a rotating, homogeneous fluid is spun up inside a sliced cylinder—a closed, vertical cylinder with its base plane inclined at an angle α to the horizontal (see figure 1). In all of these studies, intense vortices with axes parallel to the rotation axis are seen to form in the fluid’s interior, which are as a result of unsteady separation of the sidewall boundary layers, a mechanism that was studied in detail for the case of a square cylinder by Munro et al (2015) These vortices are in stark contrast to the long-slope propagating vorticity waves that form in the sliced cylinder, which are a result of vortex stretching. If H is the container depth and ν the kinematic viscosity, we determined that, after a few ‘spin-up times’, multiples of H/(νΩ)1/2, the Rossby waves have decayed, and the interior is partially spun-up, with precisely the same azimuthal velocity variation with radius and height as in the flat-bottom case (Foster & Munro 2012) In both cases, the eruption of the Ekman layers, at the vertical sidewalls, play a crucial role in the process.
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