Numerical calculations (1) -temperature and stream function

 Figure 3: Fully developed convection in the case of L=6. (a) Ra=104. (b) Ra=105. (c) Ra=106. The upper and lower panels of each figure illustrate the temperature field and the stream function, respectively. *** MOVIE *** Ra=104 : Ra=105 : Ra=106

Figure 3 shows distributions of temperature and stream function for evolved convection for different values of the Rayleigh number in the case of L=6.

When the Rayleigh number is Ra=104, many convection cells exist beneath the plate. The convective pattern becomes steady after an initial evolutionary phase. The aspect ratio of the convection cells is about 1.5, which is independent of the horizontal size of the plate. At the margins of the plate lies a convection cell with upwelling beneath the plate and downwelling outside it.

For a Rayleigh number is Ra=105, the flow pattern is unsteady even after convection has fully developed. Small warm plumes are transported to the center of the plate and cold plumes to the outside. After convection has evolved, we observe a convection cell of large horizontal scale extending from the center of the plate to the outside. This large-scale flow seems to be stable and less time-dependent compared with the smaller-scale plumes. The horizontal size of the large-scale flow is the same as that of the plate and increases as the plate gets wider. For all plate sizes considered here, a strong upwelling is produced under the center of the plate.

If the Rayleigh number is increased to Ra=106, the entire fluid layer is mixed by vigorous unsteady flow. Similar to the case of Ra=105, small warm plumes are transported to the center of the plate and cold plumes to the outside. Once again, the large-scale flow seems to be stable and less time-dependent compared with the behavior of the smaller-scale plumes. Eventually, large scale convection cells extending from the center of the plate to the outside emerge and a stable upwelling appears under the center of the plate.

 Numerical calculations (1) -temperature and stream function