If we look at a two-dimensional sphere and draw a continuous vector field on it, you can compare this with a tennis ball whose hairs you comb in a continuous way. Two points on the ball will have an undefined direction of the hair. If you comb them from up-down, or from the right to the left, or whatever way you try to comb it, two points arise where there is no direction of the balls hair.
The wind field in the atmosphere is a continuous vector field (which can be compared with the directions of the hair on the ball). The difference is that the atmosphere in which the wind field is situated is three-dimensional. But I can't imagine that this one extra degree of freedom makes any difference for the same two points.
So my question is: are there in the atmosphere of the earth (at least?) two points in which the wind field is not defined, and does this mean the same as saying that there are two positions in the atmosphere where there is no wind?