ENERGY-RELEASING PROCESSES AND STABILITY OF THERMALLY DRIVEN MOTIONS IN A ROTATING FLUID

Abstract

The effects of the various physical factors on the motions produced by differential heating in a rotating fluid are examined by solving the simultaneous hydrodynamic and thermodynamic equations. It is shown that both the rotation Ω and static stability s tend to inhibit the motion by increasing the resistance of the fluid. These resistances are proportional, respectively, to 4Ω2d?2 and gsa?2 where d and a are the vertical and horizontal scales of the motion, and g is the acceleration due to gravity. Therefore, the type of the motion produced depends very much on the parameter S = gsd2(2Ωa)?2. It is also shown that there are two different mechanisms through which the available potential energy produced by the heating is transformed into kinetic energy. One mechanism accomplishes this through the vertical motion associated with the meridional circulation ? about the zonal axis; another mechanism acts through the circulation ? about the north-south axis. The efficiency of the second mechanism is proportional to the rotation Ω and also proportional to the wave number l around the latitude circle, while the efficiency of the first mechanism is independent of these quantities. The stability of the motion is studied first by expanding the complete solutions of the high-order partial differential equation governing the viscous flow into double Fourier series, and then also by obtaining analytic solutions of a simplified version of the differential equation. It is shown that, at lower rotation rates, symmetric convection is the most favored motion in the sense that its maintenance requires the lowest radial temperature contrast. However, the mean temperature contrast required to maintain the symmetric convection increases with Ω2 at higher rotation rates, while that required to maintain a wave disturbance first decreases with increasing Ω, and then increases with Ω at very high rotation rates. The motion therefore breaks up into waves at the higher rotation rates. The critical radial temperature contrasts and the Rossby number RoT obtained from the theory agree roughly with the measured values in experiments performed by Fultz. These thermally driven disturbances produce a poleward and an upward transport of heat and a down-ward transport of zonal momentum in the zone of positive vertical shear. The maximum upward heat transfer occurs at the level z = d/2, and therefore has a cooling effect in the lower layers and a heating effect in the upper layers. This heat transfer is in the direction of producing and maintaining a stable stratification.