International Journal of Earth Sciences

, Volume 97, Issue 2, pp 375–384 | Cite as

Precise temperature monitoring in boreholes: evidence for oscillatory convection? Part II: theory and interpretation

  • Vladimir Cermak
  • Louise Bodri
  • Jan Safanda
Original Paper


In the previous part of this work (Cermak, Safanda and Bodri, this volume p.MMM) we have described experimental data and quantified the heterogeneity features of the microtemperature time series. The spectral analysis and the local growth of the second moment technique revealed scaling structure of all observed time series generally similar and suggested the presence of two temperature forming processes. The longer-scale part can be attributed to the heat conduction in compositional and structural heterogeneous solid rocks, further affected by various local conditions. Short-scale temperature oscillations are produced by the intra-hole fluid convection due to inherent instability of water column filling the hole. Here we present how the observational evidence is supported by the results of the computer simulations. The exact modes of intra-hole convection may be different, ranging from quasi-periodic (“quiescent”) state to close of turbulence. As demonstrated by numerical modeling and referred on laboratory experiments, at higher Rayleigh numbers the periodic character of oscillation characteristic for “quiescent” regime is superseded by stochastic features. This so called “oscillatory” convection occurs due to instability within the horizontal boundary layers between the individual convectional cells. In spite of the fact that the basic convective cell motion is maintained and convection is characterized by slow motion, the oscillatory intra-hole flow and corresponding temperature patterns exhibit typical features of turbulence. The idea of boundary layer instability as a source of stochastic temperature fluctuations could explain many distinct features of borehole temperatures that previously cannot be interpreted.


Temperature monitoring Intra-hole convection Numerical modeling Oscillatory regime 



We have greatly profited from the discussions with our colleagues when preparing this work. Special thanks are to two anonymous reviewers who have read the original text and offered valuable comments. Most of the reported studies were done under the co-operation programme between the Czech and Hungarian Academies of Sciences. The support of the Czech participation was partly provided by the Grant Agency of the Czech Republic (project GACR 205/03/0998) and the institutional expenses were funding from the research programmes Z3012916 and K3046108.


  1. Abernathey JR, Rosenberger F (1985) Time-dependent convective instabilities in a closed vertical cylinder heated from below. J Fluid Mech 160:137–154CrossRefGoogle Scholar
  2. Behzinger RP, Ahlers G (1982) Heat transport and temporal evolution of fluid flow near the Raileigh–Bénard instability in cylindrical containers. J Fluid Mech 125:219–258CrossRefGoogle Scholar
  3. Bodri L, Cermak V (2005) Multifractal analysis of temperature time series: data from boreholes in Kamchatka. Fractals 13:299–310CrossRefGoogle Scholar
  4. Bodri L, Cermak V, Yamano M (2004) Multifractal analysis of temperature time series from boreholes in Kamchatka. In: Proceedings of 4th international conference on “Fractals and Dynamic Systems in Geoscience” TU München, May 2004, Selden and Tamm, Garching (Germany), ISBN 3-923561-24-5, pp 10–14Google Scholar
  5. Busse FH, Whitehead JA (1974) Oscillatory and collective instabilities in large Prandtl number convection. J Fluid Mech 66:67–79CrossRefGoogle Scholar
  6. Caltagirone JP (1975) Thermoconvective instabilities in a horizontal porous layer. J Fluid Mech 72:269–287CrossRefGoogle Scholar
  7. Cermak V, Bodri L, Safanda J (2007) Precise temperature monitoring in boreholes: Evidence for oscillatory convection? Part II. Theory and interpretation. Int J Earth Sci. doi:  10.1007/s00531-007-0237-4
  8. Diment WH (1967) Thermal regime of a large diameter borehole: instability of the water column and comparison of air- and water-filled conditions. Geophysics 32:720–726CrossRefGoogle Scholar
  9. Frick H, Clever RM (1982) The influence of side walls on finite-amplitude convection in a layer heated from below. J Fluid Mech 111:467–480CrossRefGoogle Scholar
  10. Frick H, Müller U (1983) Oscillatory Hele–Shaw convection. J Fluid Mech 126:521–532CrossRefGoogle Scholar
  11. Gill AE, Davey A (1969) Instabilities of a buoyancy-driven system. J Fluid Mech 35:775–798CrossRefGoogle Scholar
  12. Hales AL (1937) Convection currents in geysers. Mon Not Roy Ast Soc Geophys Suppl 4:122–131Google Scholar
  13. Howard LN (1964) Convection at high Rayleigh number. In: Görtler H (ed) Proc.11th Int. Congr. Appl. Mech. Springer, München, pp 1109–1115Google Scholar
  14. Koster JN, Müller U (1982) Free convection in vertical gaps. J Fluid Mech 125:429–451CrossRefGoogle Scholar
  15. Koster JN, Müller U (1984) Oscillatory convection in vertical slots. J Fluid Mech 139:363–390CrossRefGoogle Scholar
  16. Koster JN, Ehrhard P, Müller U (1986) Nonsteady end effects in Hele-Shaw cells. Phys Rev Lett 56:1802–1804CrossRefGoogle Scholar
  17. Kvernvold O (1979) On the stability of non-linear convection in a Hele-Shaw cell. Int J Heat Mass Transfer 22:395–400CrossRefGoogle Scholar
  18. Schubert G, Straus JM (1979) Three-dimensional and multicellular steady and unsteady convection in fluid saturated porous media at high Rayleigh numbers. J Fluid Mech 94:25–38CrossRefGoogle Scholar
  19. Tritton DJ (1977) Physical fluid dynamics, Van Nostrand Reinhold Company, New York, pp 362Google Scholar
  20. Vest CM, Arpaci VS (1969) Stability of natural convection in a vertical slot. J Fluid Mech 36:1–15CrossRefGoogle Scholar
  21. Walden RW, Kolodner P, Passner A, Surko CM (1987) Heat transport by parallel-roll convection in a rectangular container. J Fluid Mech 185:205–234CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Czech Academy of SciencesPrague 1Czech Republic

Personalised recommendations