Figure 10 compares velocity amplitudes with respect to Rayleigh number and the horizontal scale of the convection cell. The region below the line corresponds to u_h >u,u'. Under these conditions we interpret the convection induced by the horizontal temperature difference to overcome the intrinsic convection induced by the vertical temperature difference, and convection cells with corresponding horizontal scales L_x can emerge. On the other hand, the region above the line corresponds to u_h <u,u', meaning that the convection induced by the horizontal temperature difference cannot overcome the intrinsic convection induced by the vertical temperature difference, and convection cells of horizontal scale L_x cannot emerge. Therefore, this curve indicates the maximum horizontal scale of the convection cells.

From Figure 10, the maximum horizontal scale of convection cells which satisfies u_h >u,u' is found to be L_x=2 in the case of Ra=10⁴, L_x=10 in the case of Ra=10⁵, and L_x=30 in the case of Ra=10⁶. In other words, convection cells whose aspect ratios are less than 2, 10 and 30 can emerge in the cases of Ra=10⁴, 10⁵ and 10⁶, respectively.

By substituting the horizontal size of the plate for L_x, we can determine whether convection cells of similar horizontal scales to the plate can emerge or not. When the plate width is located below the curve, a convection cell of the same horizontal scale as the plate’s can emerge. Conversely, when the plate width plots above the curve, the horizontal scale of a convection cell will be less than that of the plate, as its maximum horizontal scale is limited by the line. In this case, the upwelling will emerge near the margins of the plate rather than at its center.

Figure 10: The maximum aspect ratio of the convection cells. For given values of the Rayleigh number, convection cells whose horizontal sizes are smaller than the values indicated by the line can overcome the intrinsic convection induced by the vertical temperature difference. a=1.5 is adopted for the estimation using equation (10).