[1] If you're interested, the formulas for this relationship are:

[2] Doppler radars only see the component of the wind flow that is along the radial direction (i.e., in the direction the radar is pointed as it sweeps around in a circle). This document is not the place for a discussion of the characteristics of Doppler radars.

[3] For synoptic scale flows, typical vertical vorticities are on the order of 10-5 s-1, whereas horizontal vorticity values are on the order of 10-3 s-1, roughly 100 times larger.

[4] The validity of hydrostatic balance in the vicinity of severe thunderstorms and tornadoes is certainly open to question, so now the possibility exists for the horizontal gradients of vertical motion to begin to become important. Recall that these were neglected in the environmental form of the horizontal vorticity ... i.e.

Thus, a full treatment of the problem needs to consider

However, rather than doing so, consider the following argument. If it's assumed that the convection operates on the environmental vorticity, which can be closely approximated by its horizontal component and that the contribution to this environmental horizontal vorticity by horizontal gradients of vertical velocity is negligible, then we can approximate what happens simply by letting the updraft operate on the intially horizontal vortex lines.

[5] The vertical component in the real world is typically not zero, but as noted already, it's usually very small in comparison to the horizontal vorticity.