correlation coefficients....?

It depends on the length scales of the processes you are interested in. If you are looking at the correlation as a function of distance along or across the Gulf Stream, then you want the area you analyze to be small enough to be completely contained in the Gulf Stream or completely outside of the Gulf Stream. However, you don't want it to be so small that you are looking at pixel-scale noise.

You haven't provided nearly enough information to give anything like a definitive answer. In general however , you want the spatial scale of your sample area to be no larger than 1/2 the length of the smallest spatial scale you are interested in (this is the spatial Nyquist Theorem).