Numerical models are systematically presented for time-dependent thermal convection of Newtonian fluid with strongly temperature-dependent viscosity in a two-dimensional rectangular box of aspect ratio 3 at various values of the Rayleigh number Ra-b defined with viscosity at the bottom boundary up to 1.6 x 10(8) and the viscosity contrast across the box r(eta) up to 10(8). We found that there are two different series of bifurcations that take place as r(eta) increases. One series of bifurcations causes changes in the behavior of the thermal boundary layer along the surface boundary from small-viscosity-contrast (SVC) mode, through transitional (TR) mode, to stagnant-lid (ST) mode, or from SVC mode directly to ST mode, depending on Ra-b. Another series of bifurcations causes changes in the aspect ratio of convection cells; convection with an elongated cell can take place at moderate r(eta) (10(3)-10(5.5) at Ra-b = 6 x 10(6)), while only convection of aspect ratio close to 1 takes place at small r(eta) and large r(eta). The parameter range of r(eta) and Ra-b for elongated-cell convection overlaps the parameter range for SVC and ST modes and include the entire parameter range for TR mode. In the elongated-ST regime, the lid of highly viscous fluid along the top boundary is not literally 'stagnant' but can horizontally move at a velocity high enough to induce a convection cell with aspect ratio much larger than 1. (C) 2000 Elsevier Science B.V. All rights reserved.