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The spin-up of a linearly stratified fluid in a sliced, circular cylinder

  • R. J. Munro (a1) and M. R. Foster (a2) (a3)

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.

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Benton, E. R. & Clark, A. 1974 Spin-up. Annu. Rev. Fluid Mech. 6, 257280.
Dalziel, S. B.2006 Digiflow user guide.
Duck, P. W. & Foster, M. R. 2001 Spin-up of homogeneous and stratified fluids. Annu. Rev. Fluid Mech. 33, 231263.
Economidou, M. & Hunt, G. R. 2009 Density stratified environments: the double-tank method. Exp. Fluids 46, 453466.
Foster, M. R. & Munro, R. J. 2012 The linear spin-up of a stratified, rotating fluid in a square cylinder. J. Fluid Mech. 712, 740.
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.
Greenspan, H. P. & Howard, L. N. 1963 On a time-dependent motion of a rotating fluid. J. Fluid Mech. 17, 385404.
van Heijst, G. J. F. 1989 Spin-up phenomena in non-axisymmetric containers. J. Fluid Mech. 206, 171191.
van Heijst, G. J. F., Davies, P. A. & Davis, R. G. 1990 Spin-up in a rectangular container. Phys. Fluids A 2, 150159.
van Heijst, G. J. F., Maas, L. R. M. & Williams, C. W. M. 1994 The spin-up of fluid in a rectangular container with a sloping bottom. J. Fluid Mech. 265, 125159.
van de Konijnenberg, J. A., Naulin, V., Rasmussen, J. J., Stenum, B. & van Heijst, G. J. F. 2000 Linear spin-up in a sliced cylinder. Geophys. Astrophys. Fluid Dyn. 92, 85114.
Munk, W. H. & Carrier, G. F. 1950 The wind-driven circulation in ocean basins of various shapes. Tellus 2, 158167.
Munro, R. J. & Foster, M. R. 2014 Stratified spin-up in a sliced, square cylinder. Phys. Fluids 26, 026603.
Munro, R. J., Foster, M. R. & Davies, P. A. 2010 Instabilities in the spin-up of a rotating, stratified fluid. Phys. Fluids 22, 054108.
Munro, R. J., Hewitt, R. E. & Foster, M. R. 2015 On the formation of axial corner vortices during spin-up in a cylinder of square cross section. J. Fluid Mech. 772, 246271.
Pedlosky, J. 1965 A study of the time-dependent ocean circulation. J. Atmos. Sci. 22, 267272.
Pedlosky, J. & Greenspan, H. P. 1967 A simple laboratory model for the oceanic circulation. J. Fluid Mech. 27, 291304.
Rhines, P. 1970 Edge-, bottom-, and Rossby waves in a rotating stratified fluid. Geophys. Fluid Dyn. 1, 273302.
Sneddon, I. N. 1960 On summing infinite series involving the zeros of Bessel functions of the first kind. Proc. Glasgow Math. Assoc. 4, 144156.
Stewartson, K. 1957 On almost rigid rotations. J. Fluid Mech. 3, 1726.
Walin, G. 1969 Some aspects of time-dependent motion of a stratified rotating fluid. J. Fluid Mech. 36 (2), 289307.
Wedemeyer, E. H. 1964 The unsteady flow within a spinning cylinder. J. Fluid Mech. 20, 383399.
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