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The effect of spherical shell geometry on rapidly-rotating thermal convection is studied in a suite of high resolution three-dimensional numerical simulations. The geometry is characterized by the radius ratio, ¼ r i =r o , where r i is the inner shell radius, and r o is the outer shell radius. In this study, is varied over the broad range 0.10 to 0.92 in calculations of Boussinesq rotating convection subject to isothermal, rigid boundary conditions. Simulations are performed at Prandtl number Pr ¼ 1 and for Ekman numbers E ¼ 10 À3 , 3 Â 10 À4 and 10 À4 . Near the onset of convection, the flow takes the form of rolls aligned parallel to the rotation axis and situated adjacent to the inner shell equator. The dimensionless azimuthal wavelength, c , of the rolls is found to be independent of the shell geometry, only varying with the Ekman number. The critical wave number, m c , of the columnar rolls increases in direct proportion to the inner boundary circumference. For our simulations the critical Rayleigh number Ra c at which convection first occurs varies in proportion to E À1:16 ; a result that is consistent with previous work on rotating convection. Furthermore, we find that Ra c is a complex function of . We obtain the relation Ra c E 1:16 ¼ 0:21= 2 þ 22:4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 À Þð1 þ Þ p , which adequately fits all our results. In supercritical convection calculations the flows form quasi-geostrophic sheet-like structures that are elongated in the radial direction, stretching from the inner boundary toward the outer boundary.

We assume universal values for the color-singlet ( 1 ) and color-octet ( 8 ) nonfactorization parameters in B decays. Two sets of color-favored processes and one set of color-suppressed processes were used to give quantitative estimates of these parameters. It has been found ͑by calculating the branching ratios for a large number of Cabibbo-favored B decays͒ that the values 1 ( 0 )ϭϪ0.06Ϯ0.03 and 8 ( 0 )ϭ0.12Ϯ0.02 improve significantly the predictions of the factorization model. ͓S0556-2821͑99͒01205-9͔

Using NLL values for Wilson coefficients and including the contributions from the penguin diagrams, we estimate the amount of nonfactorization in two-body hadronic B decays. Also, we investigate the model dependence of the nonfactorization parameters by performing the calculation using different models for the form factors. The results support the universality of nonfactorizable contributions in both Cabibbo-favored and Cabibbo-suppressed B decays.13.25.Hw, 14.40.Nd

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