The influence of a sloping bottom endwall on the linear stability in the thermally driven baroclinic annulus with a free surface

  • Thomas von Larcher
  • Alexandre Fournier
  • Rainer Hollerbach
Original Article


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.


Linear stability analysis Baroclinic instability Fourier–Legendre spectral element code Sloping bottom endwall Thermally driven rotating flows 


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  1. 1.
    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)CrossRefGoogle Scholar
  2. 2.
    Busse F.H.: Shear flow instabilities in rotating systems. J. Fluid Mech. 33, 577–589 (1968)MATHCrossRefGoogle Scholar
  3. 3.
    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)MATHGoogle Scholar
  4. 4.
    Eady E.A.: Long waves and cyclone waves. Tellus 1, 33–52 (1949)MathSciNetCrossRefGoogle Scholar
  5. 5.
    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)CrossRefGoogle Scholar
  6. 6.
    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)CrossRefGoogle Scholar
  7. 7.
    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)CrossRefGoogle Scholar
  8. 8.
    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)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    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)CrossRefGoogle Scholar
  10. 10.
    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)Google Scholar
  11. 11.
    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)MATHCrossRefGoogle Scholar
  12. 12.
    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)CrossRefGoogle Scholar
  13. 13.
    Fultz, D.: Development in controlled experiments on larger scale geophysical problems. In: Advances in Geophysics, vol. 7, pp. 1–104. Academic Press (1961)Google Scholar
  14. 14.
    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). doi: 10.1007/s00348-009-0792-5
  15. 15.
    Hide R.: An experimental study of thermal convection in a rotating fluid. Philos. Trans. R. Soc. Lond. A 250, 441–478 (1958)CrossRefGoogle Scholar
  16. 16.
    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)Google Scholar
  17. 17.
    Hide R., Mason P.J.: Sloping convection in a rotating fluid. Adv. Phys. 24, 47–99 (1975)CrossRefGoogle Scholar
  18. 18.
    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)CrossRefGoogle Scholar
  19. 19.
    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)CrossRefGoogle Scholar
  20. 20.
    Hide R., Titman C.W.: Detached shear layers in a rotating fluid. J. Fluid Mech. 29, 39–60 (1967)CrossRefGoogle Scholar
  21. 21.
    Hollerbach R.: Instabilities of the Stewartson layer. Part 1. The dependence on the sign of Ro. J. Fluid Mech. 492, 289–302 (2003)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    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)MATHCrossRefGoogle Scholar
  23. 23.
    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)CrossRefGoogle Scholar
  24. 24.
    Lewis G.M., Nagata W.: Linear stability analysis for the differentially heated rotating annulus. Geophys. Astrophys. Fluid Dyn. 98, 279–299 (2004)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Lorenz E.N.: Simplified dynamic equations applied to the rotating-basin experiments. J. Atmos. Sci. 19, 39–51 (1962)CrossRefGoogle Scholar
  26. 26.
    Lu H., Miller T.: Wave dispersion in a rotating, differentially-heated fluid model. Dyn. Atmos. Oceans 27, 505–526 (1997)CrossRefGoogle Scholar
  27. 27.
    Marschall J., Plumb R.A.: Atmosphere, Ocean, and Climate Dynamics. Elsevier Academic Press, USA (2008)Google Scholar
  28. 28.
    Mason P.: Baroclinic waves in a container with sloping end walls. Philos. Trans. R. Soc. Lond. A 278, 397–445 (1975)CrossRefGoogle Scholar
  29. 29.
    Miller T.L., Gall R.L.: A linear analysis of the transition curve for the baroclinic annulus. J. Atmos. Sci. 40, 2293–2303 (1983)CrossRefGoogle Scholar
  30. 30.
    Pfeffer R.L., Fowlis W.W.: Wave dispersion in a rotating differentially heated cylindrical annulus of fluid. J. Atmos. Sci. 25, 361–371 (1968)CrossRefGoogle Scholar
  31. 31.
    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)Google Scholar
  32. 32.
    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)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Sitte B., Egbers C.: LDV-measurements on baroclinic waves. Phys. Chem. Earth (B) 24, 437–476 (1999)CrossRefGoogle Scholar
  34. 34.
    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)CrossRefGoogle Scholar
  35. 35.
    Stewartson K.: On almost rigid rotations. J. Fluid Mech. 3, 17–26 (1957)MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Stewartson K.: On almost rigid rotations. Part 2. J. Fluid Mech. 26, 131–144 (1966)MATHCrossRefGoogle Scholar
  37. 37.
    Travnikov V., Egbers C., Hollerbach R.: The geoflow-experiment on ISS (part II): numerical simulation. Adv. Space Res. 32(2), 181–189 (2003)CrossRefGoogle Scholar
  38. 38.
    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)Google Scholar
  39. 39.
    von Larcher T., Egbers C.: Experiments on transitions of baroclinic waves in a differentially heated rotating annulus. Nonlinear Process. Geophys. 12, 1033–1041 (2005)CrossRefGoogle Scholar
  40. 40.
    White A.A.: The dynamics of rotating fluids: numerical modelling of annulus flows. Met. Mag. 117, 54–63 (1988)Google Scholar
  41. 41.
    Williams G.P.: Thermal convection in a rotating fluid annulus: part i. The basic axisymmetric flow. J. Atmos. Sci. 24, 144–161 (1967)CrossRefGoogle Scholar
  42. 42.
    Williams G.P.: Baroclinic annulus waves. J. Fluid Mech. 49, 417–449 (1971)MATHCrossRefGoogle Scholar
  43. 43.
    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)

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Thomas von Larcher
    • 1
  • Alexandre Fournier
    • 2
  • Rainer Hollerbach
    • 3
    • 4
  1. 1.Institute for MathematicsFreie Universität BerlinBerlinGermany
  2. 2.Geomagnetism, Institut de Physique du Globe de Paris, Sorbonne Paris-CitéUniv Paris Diderot, UMR 7154 CNRSParisFrance
  3. 3.Institute of GeophysicsETH ZurichZurichSwitzerland
  4. 4.Department of Applied MathematicsUniversity of LeedsLeedsUK

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