Abstract
We present results of a linear stability analysis of non-axisymmetric thermally driven flows in the classical model of the rotating cylindrical gap of fluid with a horizontal temperature gradient [inner (outer) sidewall cool (warm)] and a sloping bottom endwall configuration where fluid depth increases with radius. For comparison, results of a flat-bottomed endwall case study are also discussed. In both cases, the model setup has a free top surface. The analysis is carried out numerically using a Fourier–Legendre spectral element method (in azimuth and in the meridional plane, respectively) well suited to handle the axisymmetry of the fluid container. We find significant differences between the neutral stability curve for the sloping and the flat-bottomed endwall configuration. In case of a sloping bottom endwall, the wave flow regime is extended to lower rotation rates, that is, the transition curve is shifted systematically to lower Taylor numbers. Moreover, in the sloping bottom endwall case, a sharp reversal of the instability curve is found in its upper part, that is, at large temperature differences, whereas the instability line becomes almost horizontal in the flat-bottomed endwall case. The linear onset of instability is then almost independent of the rotation rate.
Similar content being viewed by others
References
Bastin M.E., Read P.L.: A laboratory study of baroclinic waves and turbulence in an internally heated, rotating fluid annulus with sloping endwalls. J. Fluid Mech. 339, 173–198 (1997)
Busse F.H.: Shear flow instabilities in rotating systems. J. Fluid Mech. 33, 577–589 (1968)
Canuto C., Hussaini M.Y., Quarteroni A., Zang T.A.: Spectral methods: evolution to complex geometries and applications to fluid dynamics scientific computation. Springer, Berlin (2007)
Eady E.A.: Long waves and cyclone waves. Tellus 1, 33–52 (1949)
Fein J.S.: An experimental study of the effects of the upper boundary condition on the thermal convection in a rotating, differentially heated cylindrical annulus of water. Geophys. Fluid Dyn. 5, 213–243 (1973)
Fein J.S., Pfeffer R.L.: An experimental study of the effects of Prandtl number on thermal convection in a rotating, differentially heated cylindrical annulus of fluid. J. Fluid Mech. 75, 81–112 (1976)
Fournier A., Bunge H.P., Hollerbach R., Vilotte J.P.: Application of the spectral element method to the axisymmetric Navier-Stokes equation. Geophys. J. Int. 156, 682–700 (2004)
Fournier A., Bunge H.P., Hollerbach R., Vilotte J.P.: A fourier-spectral element algorithm for thermal convection in rotating axisymmetric containers. J. Comput. Phys. 204, 462–489 (2005)
Fowlis W.W., Hide R.: Thermal convection in a rotating annulus of liquid: effect of viscosity on the transition between axisymmetric and non-axisymmetric flow regimes. J. Atmos. Sci. 22, 541–558 (1965)
Früh, W.G., Maubert, P., Read, P., Randriamampianina, A.: DNS of structural vacillation in the transition to geostrophic turbulence. In: Palma J., Lopes A.S. (eds.) Advances in Turbulence XI, Proceedings in Physics, vol. 117, pp. 432–434. Springer (2007)
Früh W.G., Read P.L.: Wave interactions and the transition to chaos of baroclinic waves in a thermally driven rotating annulus. Philos. Trans. R. Soc. Lond. A 355, 101–153 (1997)
Früh W.G., Read P.L.: Experiments on a barotropic rotating shear layer. part 1. instability and steady vortices. J. Fluid Mech. 83, 143–173 (1999)
Fultz, D.: Development in controlled experiments on larger scale geophysical problems. In: Advances in Geophysics, vol. 7, pp. 1–104. Academic Press (1961)
Harlander, U., Larcher, T., Wang, Y., Egbers, C.: PIV- and LDV-measurements of baroclinic wave interactions in a thermally driven rotating annulus. Experiments in Fluids, pp. 1–13 (2009). http://dx.doi.org/10.1007/s00348-009-0792-5. doi:10.1007/s00348-009-0792-5
Hide R.: An experimental study of thermal convection in a rotating fluid. Philos. Trans. R. Soc. Lond. A 250, 441–478 (1958)
Hide R.: Some laboratory experiments on free thermal convection in a rotating fluid subject to a horizontal temperature gradient and their relation to the theory of the global atmospheric circulation. In: Corby, G. (ed.) The global circulation of the atmosphere, pp. 196–221. R. Met. Office, London (1969)
Hide R., Mason P.J.: Sloping convection in a rotating fluid. Adv. Phys. 24, 47–99 (1975)
Hide R., Mason P.J.: On the transition between axisymmetric and non-axisymmetric flow in a rotating liquid annulus subject to a horizontal temperature gradient. Geophys. Astrophys. Fluid Dyn. 10, 121–156 (1978)
Hide R., Mason P.J., Plumb R.A.: Thermal convection in a rotating fluid subject to a horizontal temperature gradient: spatial and temporal characteristics of fully developed baroclinic waves. J. Atmos. Sci. 34, 930–950 (1977)
Hide R., Titman C.W.: Detached shear layers in a rotating fluid. J. Fluid Mech. 29, 39–60 (1967)
Hollerbach R.: Instabilities of the Stewartson layer. Part 1. The dependence on the sign of Ro. J. Fluid Mech. 492, 289–302 (2003)
Hollerbach R., Futterer B., More T., Egbers C.: Instabilities of the Stewartson layer part 2. Supercritical mode transitions. Theor. Comput. Fluid Dyn. 18, 197–204 (2004)
James I., Jonas P., Farnell L.: A combined laboratory and numerical study of fully developed steady baroclinic waves in a cylindrical annulus. Q. J. R. Met. Soc. 107, 51–78 (1981)
Lewis G.M., Nagata W.: Linear stability analysis for the differentially heated rotating annulus. Geophys. Astrophys. Fluid Dyn. 98, 279–299 (2004)
Lorenz E.N.: Simplified dynamic equations applied to the rotating-basin experiments. J. Atmos. Sci. 19, 39–51 (1962)
Lu H., Miller T.: Wave dispersion in a rotating, differentially-heated fluid model. Dyn. Atmos. Oceans 27, 505–526 (1997)
Marschall J., Plumb R.A.: Atmosphere, Ocean, and Climate Dynamics. Elsevier Academic Press, USA (2008)
Mason P.: Baroclinic waves in a container with sloping end walls. Philos. Trans. R. Soc. Lond. A 278, 397–445 (1975)
Miller T.L., Gall R.L.: A linear analysis of the transition curve for the baroclinic annulus. J. Atmos. Sci. 40, 2293–2303 (1983)
Pfeffer R.L., Fowlis W.W.: Wave dispersion in a rotating differentially heated cylindrical annulus of fluid. J. Atmos. Sci. 25, 361–371 (1968)
Read P.L.: Rotating annulus flows and baroclinic waves. In: Hopfinger, E. (ed.) Rotating Fluids in Geophysical and Industrial Applications, pp. 185–214. Springer, Wien-New York (1992)
Read P.L., Bell M.J., Johnson D.W., Small R.M.: Quasi-periodic and chaotic flow regimes in a thermaly-driven, rotating fluid annulus. J. Fluid Mech. 238, 599–632 (1992)
Sitte B., Egbers C.: LDV-measurements on baroclinic waves. Phys. Chem. Earth (B) 24, 437–476 (1999)
Sitte B., Egbers C.: Higher order dynamics of baroclinc waves. In: Pfister, G., Egbers, C. (eds.) Physics of Rotating Fluids, pp. 355–375. Springer, Berlin [u.a.] (2000)
Stewartson K.: On almost rigid rotations. J. Fluid Mech. 3, 17–26 (1957)
Stewartson K.: On almost rigid rotations. Part 2. J. Fluid Mech. 26, 131–144 (1966)
Travnikov V., Egbers C., Hollerbach R.: The geoflow-experiment on ISS (part II): numerical simulation. Adv. Space Res. 32(2), 181–189 (2003)
Veronis G.: On the approximation involved in transforming the equations of motion from a spherical surface onto a β-plane-plane. J. Mar. Res. 21, 110–124 (1963)
von Larcher T., Egbers C.: Experiments on transitions of baroclinic waves in a differentially heated rotating annulus. Nonlinear Process. Geophys. 12, 1033–1041 (2005)
White A.A.: The dynamics of rotating fluids: numerical modelling of annulus flows. Met. Mag. 117, 54–63 (1988)
Williams G.P.: Thermal convection in a rotating fluid annulus: part i. The basic axisymmetric flow. J. Atmos. Sci. 24, 144–161 (1967)
Williams G.P.: Baroclinic annulus waves. J. Fluid Mech. 49, 417–449 (1971)
Wordsworth, R.D., Read, P.L., Yamazaki, Y.H.: Turbulence, waves, and jets in a differentially heated rotating annulus experiment. Phys. Fluids 20, doi:10.1063/1.2990,042 (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by R. Klein.
Rights and permissions
About this article
Cite this article
von Larcher, T., Fournier, A. & Hollerbach, R. The influence of a sloping bottom endwall on the linear stability in the thermally driven baroclinic annulus with a free surface. Theor. Comput. Fluid Dyn. 27, 433–451 (2013). https://doi.org/10.1007/s00162-012-0289-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00162-012-0289-3