Yet your conceptual parcel, the beach ball, seems to be acted upon only by the uppermost and lowermost winds it's as if that mid-level wind has no influence. Let's say that the surface wind is westerly, the mid-level wind northerly, and the upper wind easterly. But you've also got a mid-level wind blowing perpendicular to the top and bottom layers-thus, in this case you've got directional shear due to winds backing with height. Now, if you had only the upper and lower winds, which are blowing in opposite directions, then I could understand why the axis of rotation would remain horizontal as you've shown. You've got winds veering 180 degrees with height. In example 2, the placement of the axis is not so apparent to me. The first illustration clearly portrays speed shear, and it's obvious to me why the beach ball's axis is horizontal, and perpendicular to the unidirectional flow. I have a question for you regarding your beachball illustrations. Robert, your graphics are stellar as usual. I don't grasp equations, but between the hodograph and that last table, the concept gels for me. After a while, though, one kind of wants to understand what this meters-squared-per-second-squared has to do with it. From a practical point of view, all I've needed to know for purposes of tornado forecasting is that >150 m2/s2 in the lowest kilometer of the BL is a Good Thing, with a similar threshold of 250 for 0-3 km helicity. Tim and Robert-thank you! I think my non-linear brain has grasped, or at least is coming close to grasping, the reasoning behind how helicity is expressed. So if you had a hodograph that exactly followed the edge of that box, the helicity would be 200 m^2/s^2 (twice the "area" of the box). Now say the units of the x- and y-axes are m/s (or knots as plotted on the hodograph), the "area" of a box is (10 m/s) * (10 m/s) = 100 m^2/s^2. The area of that box would be (10 m) * (10 m) = 100 m^2. Think of one of the boxes on the hodograph (doesn't matter which one), and pretend for a second that we're looking at a map and that the units of the x- and y- axes are meters (or feet or attoparsecs). It's not really an "area," though, because we're not talking about physical space, but it sort of works the same. The "area" between your two lines and the hodograph is proportional to the helicity (it's half the helicity, I think). To find helicity, you would draw lines from the origin (the point where u and v are both 0) to two other points on the hodograph. Rather than lines of direction and magnitude, this one has lines of u and v wind components, which I think helps show the idea better. I'm kind of assuming you're familiar with them and how they're plotted.
So it's kind of like area in physical space, except that "space" in this sense is velocity. One of the definitions of helicity is the area between the hodograph and the origin, which is in "velocity space," if you could call it that. One could possibly think of it is velocity (m/s) squared.
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