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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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