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Simple theoretical and experimental study of convection with some geophysical applications and analogies

Published online by Cambridge University Press:  19 April 2006

G. S. Golitsyn
G. S. Golitsyn
Affiliation:
Institute of Atmospheric Physics, Academy of Sciences of U.S.S.R., Moscow 109017
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Abstract

A plane horizontal layer of a fluid with depth d is considered into which heat is introduced. Within the Boussinesq approximation an exact expression is obtained for the efficiency of convection γ in transforming the rate of heat supplied into the generation of kinetic energy. It agrees with results of numerical and laboratory experiments whose data can be used to calculate this value. In laboratory experiments γ is usually of order 10−8 to 10−6. Using this quantity estimates are obtained for the r.m.s. velocity of convective motions $\overline{u}$ and their time scale $\tau = d/\overline{u}$ for a regime where viscosity is important. These estimates agree with the results of a number of previous numerical experiments over a wide range of Rayleigh, R, and Prandtl numbers. Dimensionless convection equations normalized by these scales show the existence of thermal boundary layers and of almost isothermal regions within the bulk of the fluid. From this, the main regimes of heat transfer follow immediately: the Nusselt number N ∼ (R − Rcr)¼ for moderate R and $N \sim R^{\frac{1}{3}}$ for sufficiently large R.

A number of simple experiments have been carried out to measure $\overline{u}$ and τ for convection in water. Their results confirm the theoretical dependences of $\overline{u}$ and τ on external parameters and show that a smooth transition region exists from the viscous regime of convection to the more fully-developed turbulent one. The latter regime is considered by a scaling analysis. The results are compared with the author's measurements and other experimental data.

Similarly density convection is considered which arises by the separation of a medium into light and heavy fractions. Differences and analogies with the thermal convection are established. Elementary experiments confirm qualitatively the predicted dependences for $\overline{u}$ and τ. Applications of the results obtained are briefly discussed for studies of heat and mass transfer in the ocean and of convection in the Earth's mantle.

In the last section some general properties are considered for various forced flows, convection, turbulence and some types of atmospheric circulation, that allow one to formulate a ‘rule’ of the fastest response, which asserts that the kinetic energy of a fluid system is of order of the supplied power times the fastest relaxation time which the system possesses.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 95 , Issue 3 , 11 December 1979 , pp. 567 - 608
Copyright
© 1979 Cambridge University Press

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References

Artyushkov, E. V. 1968 Gravitational convection in the Earth's interior. Izv. Acad. Sci. USSR, Physics of the Earth 9, 3–17.Google Scholar
Barenblatt, G. I. 1976 Self-similarity, similitude and intermediate asymptotics. Izv. Radio-phys. 19, 902–931.Google Scholar
Barenblatt, G. I. 1978 Similitude, Self-Similarity, Intermediate Asymptotics. Theory and Application to Geophysical Fluid Dynamics. Leningrad: Hydrometeorol.
Barenblatt, G. I., Entov, V. M. & Ryzhik, V. M. 1972 The Theory of Non-Stationary Filtration of liquid and Gas. Moscow: Nedra.
Booker, J. R. 1976 Thermal convection with strongly temperature dependent, viscosity. J. Fluid Mech. 76, 481–496.Google Scholar
Bourangulov, N. I. & Zilitinkevich, S. S. 1976 On the generation of kinetic energy of atmospheric circulation on a slowly rotating planet. Dokl. Akad. Nauk S.S.S.R. 227, 1315–1318.Google Scholar
Carter, N. L. 1976 Steady flow of rocks. Rev. Geophys. Space Phys. 14, 301–360.Google Scholar
Clever, R. M. & Busse, F. H. 1974 Transition to time dependent convection. J. Fluid Mech. 65, 625–645.Google Scholar
Deardorff, J. W. & Willis, G. E. 1967 Investigation of turbulent thermal convection between horizontal plates. J. Fluid Mech. 28, 675–704.Google Scholar
Foster, T. D. 1968 Haline convection induced by the freezing of sea water. J. Geophys. Res. 73, 1933–1938.Google Scholar
Foster, T. D. 1971 Intermittent convection. Geophys. Fluid Dyn. 2, 201–217.Google Scholar
Garron, A. M. & Goldstein, R. J. 1973 Velocity and heat transfer measurements in thermal convection. Phys. Fluids 16, 1818–1825.Google Scholar
Ginzburg, A. I. & Fedorov, K. N. 1978 Thermal state of boundary layer of cooling water at transition from the free convection to the forced one. Izv. Atmos. Oceanic Phys. 14, 778–785.Google Scholar
Ginzburg, A. I., Golitsyn, G. S. & Fedorov, K. N. 1979 Measurements of convection time scale in water at its cooling from the surface. Izv. Atmos. Oceanic Phys. 15, 333–335.Google Scholar
Ginzburg, A. I., Zatsepin, D. G. & Fedorov, K. N. 1977 Fine structure of boundary layer in water near separation surface water-air. Izv. Atmos. Oceanic Phys. 13, 1268–1277.Google Scholar
Golitsyn, G. S. 1970 A similarity approach to general circulations of planetary atmospheres. Icarus 13, 1–24.Google Scholar
Golitsyn, G. S. 1973 Introduction to the Dynamics of Planetary Atmospheres. Hydromoteorol. Publ. House, Leningrad. (English translation: 1974, NASA TT F-15, 627, Washington, D.D.)
Golitsyn, G. S. 1977a Towards a theory of convection in the upper mantle. Dokl. Akad. Nauk S.S.S.R. 234, 552–555.Google Scholar
Golitsyn, G. S. 1977b Estimates of efficiency and velocities of convection in laboratory experiments on general circulation modeling. Izv. Atmos. Oceanic Phys. 13, 891–896.Google Scholar
Golitsyn, G. S. 1978 Energetics of convection of viscous fluid. Dokl. Akad. Nauk. S.S.S.R. 240, 1054–1057.Google Scholar
Herring, J. R. 1963 Investigations of problems in thermal convection. J. Atmos. Sci. 20, 325–338.Google Scholar
Herring, J. R. 1964 Investigations of problems in thermal convection: rigid boundaries. J. Atmos. Sci. 21, 277–290.Google Scholar
Herring, J. R. 1966 Some analytic results in the theory of thermal convection. J. Atmos. Sci. 23, 672–677.Google Scholar
Hewitt, J. M., Mckenzie, D. P. & Weiss, N. O. 1975 Dissipative heating of convective flows. J. Fluid Mech. 68, 721–738.Google Scholar
Hide, R. 1977 Towards a theory of irregular variations in the length of the day and coremantle coupling. Phil. Trans. Roy. Soc. A 284, 547–554.Google Scholar
Houston, M. H. & Debremaecker, J.-CL. 1975 Numerical models of convection in the upper mantle. J. Geophys. Res. 80, 742–751.Google Scholar
Howard, L. N. 1966 Convection at high Rayleigh number. Proc. 11th Int. Congr. Appl. Mech. Münich, pp. 1109–1115.Google Scholar
Jakob, M. 1949 Heat Transfer, vol. 1. New York: Wiley.
Katsaros, K. B., Liu, W. T., Businger, J. A. & Tillman, J. A. 1977 Heat transport and thermal structure in the interfacial boundary layer measured in an open tank of water in turbulent free convection. J. Fluid Mech. 83, 311–335.Google Scholar
Keondjan, V. P. & Monin, A. S. 1975 A model of gravitational differention of interior of planets. Dokl. Akad. Nauk S.S.S.R. 220, 825–828.
Keondjan, V. P. & Monin, A. S. 1977 On the wandering of poles due to continental drift. Dokl. Akad. Nauk S.S.S.R. 233, 316–319.Google Scholar
Kolmogorov, A. N. 1941 Local structure of turbulence in an incompressible fluid at very high Reynolds number. Dokl. Akad. Nauk S.S.S.R. 30, 299–302.Google Scholar
Koschmieder, E. L. & Pallas, S. G. 1974 Heat transfer through a shallow, horizontal convecting fluid layer. Int. J. Heat Mass Transfer 17, 991–1002.Google Scholar
Kraichnan, R. H. 1962 Turbulent thermal convection at arbitrary Prandtl number. Phys. Fluids 5, 1374–1389.Google Scholar
Kutateladze, S. S. 1970 The foundations of the theory of heat exchange. Novosibirsk: Science.
Landau, L. D. & Lifshits, E. M. 1954 Mechanics of Continuous Media. Moscow: Gostekhteorizdat. (English transl. 1959 Fluid Mechanics. Pergamon.)
Leovy, C. B. & Pollack, J. B. 1973 A first look at the general circulation on Titan. Icarus 19, 195–201.Google Scholar
Lipps, F. B. 1976 Numerical simulation of three-dimensional Bénard convection in air. J. Fluid Mech. 75, 113–148.Google Scholar
Lipps, F. B. & Somerville, R. C. J. 1971 Dynamics of variable wavelength in finite amplitude Bénard convection. Phys. Fluids 14, 759–765.Google Scholar
Lliboutry, L. 1972 The driving mechanism, its source of energy and its solution studied with a three-layer model. J. Geophys Res. 77, 3759–3770.Google Scholar
Malkus, W. V. R. 1954 The heat transport and spectrum of thermal turbulence. Proc. Roy. Soc. A 225, 196–212.Google Scholar
Mckenzie, D. P., Roberts, J. M. & Weiss, N. O. 1974 Convection in the Earth's mantle: towards a numerical simulation. J. Fluid Mech. 62, 465–538.Google Scholar
Monin, A. S. & Keondjan, V. P. 1976 On the evolution of planets of terrestrial group. Gerlands Beitr. Geophys. 85, (H3), 169–174.Google Scholar
Monin, A. S. & Yaglom, A. M. 1965 Statistical Hydromechanics, Pt. 1, Moscow: Nauka (English transl. 1971, MIT Press.)
Oboukhov, A. M. 1960 On the structure of temperature and velocity fields in the conditions of free convection. Izv. Acad. Sci. USSR, Geophys. Ser. no. 9, 1392–1396.Google Scholar
Prandtl, L. 1932 Meteorologische Anwendungen der Strömungslehre. Beitr. Phys. fr. Atmosph. 19, 188–202.Google Scholar
Rossby, H. T. 1969 A study of Bénard convection with and without rotation. J. Fluid Mech. 36, 309–335.Google Scholar
Schlüter, A., Lortz, D. & Busse, F. 1965 On the stability of steady finite amplitude convection. J. Fluid Mech. 23, 129–144.Google Scholar
Sorokhtin, O. G. 1974 Global Evolution of the Earth. Moscow: Nauka.
Thompson, P. D. 1967 A statistical approach to the study of turbulent convection. In Thermal Convection: A Colloquium. NCAR-TN-24, 177–195.
Turcotte, D. L. & Oxburgh, E. R. 1967 Finite amplitude convection cells and continental drift. J. Fluid Mech. 28, 29–42.Google Scholar
Veronis, G. 1966 Large amplitude Bénard convection. J. Fluid Mech. 26, 49–68.Google Scholar
Williams, G. P. 1967 Thermal convection in a rotating fluid annulus: Part 1. The basic flow experiment. J. Atmos. Sci. 24, 144–161.Google Scholar
Williams, G. P. 1971 Baroclinic annulus waves. J. Fluid Mech. 49, 417–449.Google Scholar
Willis, G. E., Deardorff, J. W. & Somerville, R. C. J. 1972 Roll-diameter dependence in Rayleigh convection and its effect upon the heat flux. J. Fluid Mech. 54, 351–367.Google Scholar