This study seeks understanding of the role played by the diapycnal diffusivity in determining the transient climate evolution in a scenario with enhanced atmospheric CO$_2$ concentration. We use an Earth system Model of Intermediate Complexity (EMIC) composed of a 3D Ocean Model with idealized topography and a 2D Atmospheric Model. The model is spun up to equilibrium for three different values of the diapycnal diffusivity (small k=0.2 cm$^2$/s, standard k=0.5 cm$^2$/s and large k=1.0 cm$^2$/s) and global warming experiments are performed after the spinup. Three different climatic forcing scenarios are applied to each equilibrium state: CO$_2$ increases at a rate of 1%, 2% and 4% per year for 75 years and constant afterwards. Comparing the climate change of these experiments allows one to detect eventual non-linear behavior of the climate system induced by the different values of the diapycnal diffusivity. The major differences in the transient runs are found in the mid-high latitudes in the North Atlantic Ocean but strong non-linear behavior has not been found. For the scenarios with 1% (2%) CO$_2$ increase per year, the Meridional Overturning Circulation (MOC) slows down by about 25% (50%) of the control value after about 100 years of integration and then it recovers. The recovery is complete or almost complete (80-90% of the initial value) depending on the value of the diffusivity and the strength of the forcing scenario. The natural variability of the MOC seems to be higher for both the standard and large diffusivity models, as compared to the small diffusivity model. This is true both for the control climate and the global warming climate. For the scenario with 4% CO$_2$ increase per year the circulation shuts down for 150 years in the small diffusivity model and then recovers while in the higher diffusivity models it slows for about 4 decades to around 5 [Sv] and then recovers. In a 2XCO$_2$ global warming experiment Kamenkovich {\em et al} (Climate Dynamics, submitted) proved that the changes in the MOC in this model are mainly driven by the heat flux while the contribute of the moisture flux is negligible.