So this is only happening in the top view. I have reset the tools, deleted preferences. Uninstalled... installed. It isn't not specific to one file. It happens for everything.
That's how it's supposed to be. You don't see the green part in the top view because it's pointing towards the camera. Manipulators point towards their positive axis.
Your model is upside down
Your view is upside down, but still orthographic (use the view cube and make sure it says top)
You're not in world space (hold w, left click, make sure "world" is selected)
Y is not set as up and down. Z could be set that way (possible due to importing shit from another program).
What JMYoung said, it's suppposed to be that way. Unless of course the problem is something else that I've failed to see.
The view axis in the lower left corner of the viewport is very useful when things get confusing.
Replies
Your model is upside down
Your view is upside down, but still orthographic (use the view cube and make sure it says top)
You're not in world space (hold w, left click, make sure "world" is selected)
Y is not set as up and down. Z could be set that way (possible due to importing shit from another program).
The view axis in the lower left corner of the viewport is very useful when things get confusing.
take a peek at the bottom view, and Z will be point up like you expeceted.