When a porous medium saturated with initially stagnant cold water around the density maximum temperature is cooled from above, the nonmonotonic density profile is developed and convection may be induced in an unstable lower layer. The initial growth rate analysis suggested that the system is initially unconditionally stable and therefore, the critical onset time exists. Here we analyze the onset of buoyancy-driven convection during the time-dependent cooling using the linear stability theory and nonlinear numerical simulation. For the linear region, the growth rates are obtained from the quasi-steady state approximation and the initial value problem approach, and they support to each other. The critical time tau(c) is found as a function of the dimensionless density maximum temperature 0(max). Based on the linear analysis, the nonlinear numerical simulation is conducted using Fourier pseudo-spectral method. From the nonlinear numerical simulation, it is found that the longer growth period is required for the smaller 0(max). (C) 2013 Elsevier Ltd. All rights reserved.