Piacsek 



Eden and Piacsek (1968) for the case of small rotation rates. It may be shown 

 that the reversals in the boundary layer temperature gradient are related to 

 the curvature of the vertical velocity profile, such that its points of inflection 

 occur where the temperature equals the horizontally averaged temperature at 

 that height, i.e., where the "relative buoyancy" vanishes. For increasing rota- 

 tion, the Coriolis force deflects any radial motion into zonal motion and sets 

 up a vertical pressure gradient opposing the buoyancy force, so that the con- 

 vective flux decreases in the boundary layer and fluid particles eject sooner. 

 The net result is greater warming and cooling by conduction near the cylindri- 

 cal walls, and a shrinking of the isotherms to a "bundle." So far, no satisfactory 

 explanation has been found for the exponential behavior of the temperature with 

 height, nor for the peculiar dependence of the convective heat transfer on rota- 

 tion; for .1 < n < .9 rad/sec the quantity N - 1 is found to be -logCl/fi^^^), 

 N being the Nusselt number (Eden and Piacsek, 1968, Piacsek, 1968), whereas 

 for 1.3 < fi < 2.1 rad/sec it is found to be -1/il (Williams, 1967). The large 

 ratio of d^/d^ is attributed to the different entrainment rates into the cold 

 and hot boundary layers. 



The strong boundary layer seen on the bottom surface is due to the squeez- 

 ing of the radial motion out of the core region by the rotation to boundaries 

 where friction enables the fluid to convert zonal into radial motion again. This 

 layer is similar to the Ekman layer found near the top of wind-driven ocean 

 currents, and to those found during spin-up time near a rotation disc. For a 

 discussion of these layers, the reader is referred to Barcilon (1964) and 

 Mclntyre (1967). 



B. Convection in a Semi-Infinite Fluid Cooled from Above: 

 Penetrative Convection 



This problem considers the convection currents that arise in unstable fluid 

 layers that are bounded below by either positively or neutrally stable layers. 

 In the former case, the stable layers are penetrated to a certain extent by the 

 rising or descending thermal columns in the unstable regions, but they them- 

 selves remain stable, on the whole. In the latter case, the convection currents 

 will sooner or later involve all of the accessible fluid volume. 



Many phenomena in nature exhibit a similar process — atmospheric thermals 

 and cumulus towers impinging on stably stratified layers above, including inver- 

 sions and the tropopause; evaporation-driven ocean currents penetrating into 

 lower regions stably stratified by solar radiation, or seasonal cooling effects 

 reaching down to the thermocline; convection in the sun and stars in layers 

 where radiation causes a superadiabatic temperature gradient, bounded both 

 below and above by stable layers. Often the penetration currents are coupled 

 to larger -scale general circulations, and their mutual interaction is of great 

 interest to geo- and astrophysicists. 



Ball (1954) and Ewing (1960) have studied the difference between the radia- 

 tion temperature of the ocean's surface and the temperature of the water below 

 the surface. Ball has found a difference of -'.25°C in the top cm or so of the 

 surface layer; Ewing and McAlister found -^.60 °C in about 15 cm. In addition, 



762 



