It’s not so much “vortexing downward” as it is “being flung away from the center”.
Stirring up the water makes it want to pull away from the center of rotation. But it’s being pressed inward by all the surrounding water. This pressure increases with depth, so the deeper you go, the more pressure the vortex needs to exert to move water out of the way. The result is a tapered funnel shape to the vortex — the water near the top is under very little pressure, so a little centripital force can open up the vortex a decent radius, but as you go deeper, that same force gets less effective as the pressure force increases.
There’s probably a lot more to say with things like surface tension that affect the exact shapes of very small scale vorticies, like the ones spun up by a small bathtub or sink drain, but I expect this explanation to hold at larger scales, like an aggressively stirred up pitcher or a flushing toilet bowl.
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