Skip to main content

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

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.

This is a preview of subscription content, access via your institution.

References

  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)

    Article  Google Scholar 

  2. 2

    Busse F.H.: Shear flow instabilities in rotating systems. J. Fluid Mech. 33, 577–589 (1968)

    MATH  Article  Google 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)

    MATH  Google Scholar 

  4. 4

    Eady E.A.: Long waves and cyclone waves. Tellus 1, 33–52 (1949)

    MathSciNet  Article  Google 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)

    Article  Google 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)

    Article  Google 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)

    Article  Google 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)

    MathSciNet  MATH  Article  Google 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)

    Article  Google 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)

  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)

    MATH  Article  Google 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)

    Article  Google 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)

  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). http://dx.doi.org/10.1007/s00348-009-0792-5. 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)

    Article  Google 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)

    Article  Google 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)

    Article  Google 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)

    Article  Google Scholar 

  20. 20

    Hide R., Titman C.W.: Detached shear layers in a rotating fluid. J. Fluid Mech. 29, 39–60 (1967)

    Article  Google 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)

    MathSciNet  MATH  Article  Google 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)

    MATH  Article  Google 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)

    Article  Google 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)

    MathSciNet  Article  Google Scholar 

  25. 25

    Lorenz E.N.: Simplified dynamic equations applied to the rotating-basin experiments. J. Atmos. Sci. 19, 39–51 (1962)

    Article  Google Scholar 

  26. 26

    Lu H., Miller T.: Wave dispersion in a rotating, differentially-heated fluid model. Dyn. Atmos. Oceans 27, 505–526 (1997)

    Article  Google 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)

    Article  Google 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)

    Article  Google 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)

    Article  Google 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)

    MathSciNet  Article  Google Scholar 

  33. 33

    Sitte B., Egbers C.: LDV-measurements on baroclinic waves. Phys. Chem. Earth (B) 24, 437–476 (1999)

    Article  Google 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)

    Chapter  Google Scholar 

  35. 35

    Stewartson K.: On almost rigid rotations. J. Fluid Mech. 3, 17–26 (1957)

    MathSciNet  MATH  Article  Google Scholar 

  36. 36

    Stewartson K.: On almost rigid rotations. Part 2. J. Fluid Mech. 26, 131–144 (1966)

    MATH  Article  Google 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)

    Article  Google 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)

    Article  Google 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)

    Article  Google Scholar 

  42. 42

    Williams G.P.: Baroclinic annulus waves. J. Fluid Mech. 49, 417–449 (1971)

    MATH  Article  Google 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)

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Thomas von Larcher.

Additional information

Communicated by R. Klein.

Rights and permissions

Reprints 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

Download citation

Keywords

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