Pressure waves in a vortex towards the centre?

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Is there a pressure wave in every vortex (or cyclone etc), created by the motions of the molecules of the fluid or air which make up the vortex, pressing towards the centre of the vortex, and causing the ... buoyancy-like result that: The solids are moving towards the centre of the vortex WITHOUT the molecules of the fluid or the air following the solid?

Any links saying that this actually happens, if it happens?
 

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