Streamwise vorticity is the component of horizontal spin aligned parallel to the storm-relative wind. It is the piece of environmental rotation that becomes cyclonic vertical spin as soon as an updraft tilts it upward, without a phase lag between updraft and rotation.
The physics of two vorticity components
Any horizontal vortex tube in the inflow can be decomposed into two pieces. The part aligned with the storm-relative flow is streamwise vorticity. The part perpendicular to the storm-relative flow is crosswise vorticity. Both are pure rotation in the horizontal, but they behave very differently once an updraft lifts them.
Davies-Jones showed in 1984 that when an updraft tilts a streamwise vortex tube, the resulting vertical vorticity maximum sits inside the updraft itself, so rotation and ascent are co-located from the start. When it tilts a crosswise vortex tube, the vertical vorticity maximum splits into a positive cell on one flank and a negative cell on the other, with the updraft centered on neither.
Why supercells care
A rotating updraft is much more efficient at intensifying and persisting than an updraft that only carries rotation on its flanks. Co-located spin and ascent set up a dynamic pressure minimum inside the updraft that draws air upward through nonlinear forcing, which reinforces the updraft and helps the storm deviate to the right of the mean wind.
In practice, the environment usually contains both components. What matters for supercell organization is that a meaningful fraction of the inflow vorticity be streamwise, so the updraft is directly rotating rather than just adjacent to rotating flanks.
How it shows up on a hodograph
Streamwise vorticity is highest when the storm-relative wind at low levels is nearly perpendicular to the vertical wind shear vector at the same level. That is the geometric statement behind the critical angle concept. On a hodograph, this looks like a curved, elongated shape with the storm motion dot placed such that the arrow from the dot to a low-level wind point runs nearly at a right angle to the local shear vector.
This is also what storm-relative helicity is measuring. SRH is proportional to the streamwise vorticity present in the inflow layer times the storm-relative wind speed, integrated with height, and geometrically it equals twice the area swept out between the hodograph and the storm motion.
Important limits
Streamwise vorticity is a description of the environment as the storm sees it, not a description of the storm itself. Once a supercell matures, it modifies its own low-level wind field, generates baroclinic vorticity along its outflow, and creates local streamwise flow that can dominate over the ambient environment. Idealized modeling and the VORTEX2 field program both show that near-ground vorticity in mature supercells owes as much to storm-generated processes as to what the storm inhaled from the environment.
The parameter also inherits the storm-motion sensitivity of every storm-relative quantity. Recompute for a different motion and the split between streamwise and crosswise shifts. Treat the streamwise picture from a proximity sounding as a strong indication of what an idealized supercell in that environment would tap, then adjust for what the actual storm is doing on radar.