This paper extends the stochastic DSICE model of Cai et al. (Cai et al. 2015a, 2015b) to include the case of spatial transport of heat and moisture from the Equator to the Poles. This well-known and important phenomenon in climate science has been neglected in popular IAM’s, e.g. RICE and DICE (Nordhaus 2010, 2013). Spatial transport leads to another well-known phenomenon in climate science called polar amplification where a one degree increase in the global yearly mean temperature anomaly causes a more than one degree increase of the yearly mean temperature anomaly in the high latitudes (Langen and Alexeev 2007). This extension allows us to compare the optimal paths of key quantities like the Social Cost of Carbon (SCC), emissions, abatement, and damages and their uncertainty bands when heat and moisture transport are neglected as in the received literature on IAMs to when this important phenomenon documented by climate science is included. We view our paper as a first step towards adding additional aspects of climate dynamics like heat and moisture transport across latitudes and polar amplification to the existing literature on IAMs.