The pattern of local daily fluctuations of climate fields such as temperatures and geopotential heights is not stable and hard to predict. Surprisingly, we find that the observed relations between such fluctuations in different geographical regions yields a very robust network pattern that remains highly stable during time. Using a new systematic methodology we track the origins of the network stability. It is found that about half of this network stability is due to the spatial 2D embedding of the network, and half is due to physical coupling between climate in different locations. We also find that around the equator, the contribution of the physical coupling is significantly less pronounced compared to off-equatorial regimes. Finally, we show that there is a gradual monotonic modification of the network pattern as a function of altitude difference.