We demonstrated in a companion paper (Watts et al., this issue) that the transient response of a globally averaged upwelling diffusion energy balance climate model depends upon three time scales: the mixed-layer thermal-response time scale, the thermocline thermal-response time scale, and the upwelling time scale. These time scales are defined and interpreted physically in that paper. The asymptotic solution for the case of linear radiative forcing was shown to depend upon the sum of the thermocline and mixed-layer time scales, while the solution for small values of time depended only upon the mixed-layer time scale. When the land fraction is included in the model, two additional time scales emerge: the land and ocean redistribution time scales. It has previously been shown by Kim et al. (1992) in a two-dimensional energy balance model that the thermal-response time scales are functions of position. The short-time solution (for the responses to both a sudden increase in radiative forcing and a linear increase in radiative forcing) is, however, accurate for only very short times (on the order of 1 year). Here we present closed-form, exact, analytical solutions for the response to both a sudden and a linear increase in radiative forcing. In addition to the mixed-layer and thermocline time scales, these depend upon the upwelling time scale. We show here that for constant mixed-layer and thermocline time scales, the response of the system is faster in intermediate stages when the upwelling time scale is longer.