I hope you guys can realize how much anguish it causes me to potentially contest the understanding and actions of people in this community that I greatly respect, people who have been sharing their knowledge decades before I was even around. This stuff keeps me up at night. I could just keep quiet and not get involved, but I believe that there are literally lives on the line here. It would be immoral not to say something.

I want to make sure I get this right before I publish a video on this. So I went back and redid everything I had for the evolution of the storm, analyzed the velocity data again, and replotted the storm's track.
Chasers in the area were focused on an area of higher velocity, north of the ongoing (yet unknown by most) EF2. There is a rotation signature here:

However, if you look at the orientation of the inbounds and outbounds you can see there is a convergence signature here too:

The inbounds and outbounds aren't oriented parallel to the radar beam, and this region starts out nearly perpendicular to the radial indicating strong convergence.
I plotted the orientation of the inbounds and outbounds on KTWX storm relative velocity for thirty or so scans from 22:28 to 23:11 UTC, about 20 minutes prior to the start of the impacting EF2 to a few minutes into the start of the EF4.

There are a couple patterns that really stand out here:
A. There was a continuous, nearly linear track of rotation from southwest to northeast that existed well before the tornadoes, and both tornadoes closely followed this linear track.
B. Prior to each tornado there was an area of strong convergence to the north of this rotation track line. The convergence signature became a mix of rotation and convergence signature, and then mostly focused rotation as it appears to get "pulled in" or merges into this southwest to northeast track line. The tornadoes develop at about the time of these mergers.
My interpretation of what this means:
A is the track of the main supercell updraft or primary mesocyclone, and the supercell is cycling along this line.
B represents an inflow surge into the next supercell cycle, likely convergence on an inflow band, that is curling (yes with rotation) into A. Once this converging and rotating inflow surge becomes rooted to the updraft represented by A, a significant tornado spins up.
It's interesting to see that there was a velocity couplet due west of the start of the EF2 track, just like there were couplets west of the Lawrence EF4. It's as if the circulation is arriving on the inflow band, and the tornado is hooking left around the back rim of the mesocyclone as it's developing before settling into the center and being carried off to the northeast by the parent updraft.
Calling this the track of the primary, tornadic mesocyclone can be misleading or even dangerous from a storm chasing perspective.
It makes it appear as if this is a distinctly different part of the storm, largely disconnected from what is happening with the EF2. SLT and other chasers are actively trying to stay south of this track line, which turned out to be a critical mistake. However, without the track of the EF2 here, it looks as if this would be the correct thing to do. It's important to remember that the velocity is not necessarily showing the tornado producing structure of the storm, but instead the winds flowing into and around the storm. Dangerous misinterpretations of the radar are possible here, which is why chasers shouldn't be relying o such data for their safety.
Instead, I think it's better to think of the primary tornadic mesocyclone track like this, even if there are distinctly different mesocyclone cycles on this track:

Why? Because there is not only a continuous track of rotation along this line, but there is continuous physical structure along this line. The chaser can see this visually in the form of the RFD gust front and Bear's Cage region, which is maintained along this line. That visual identification is most important here in chase making decisions.
It's also readily apparent on the reflectivity. The end of the HP hook exhibits a commahead circulation, the head of the comma I'm interpreting as roughly corresponding to the supercell's main updraft, the tail of the comma as the flanking line updrafts. The storm maintains this commahead along the above linear track. Both tornadoes appear to be rooted to the center of it where the commahead curls in on itself:
SLT is no longer south of the mesocyclone in the above velocity plot. They've crossed its path, and have effectively made it into the storm's inflow notch, before turning around and crossing it again. To their northwest there are areas of strong converging rear flank downdraft (purple arrow) and surging inflow (yellow arrow).
Focused rotation arriving along the inflow band into the rear flank gust front is not present yet as SLT is making the decision to turn around. There is instead a broad area of rotation and convergence to the west corresponding with a surge on the storm's inflow band. This storm was like a breathing entity moving down a set of tracks. It took a deep breath before each tornado.

So what does all of this nitty gritty detail matter in what is likely just a subjective interpretation. Do I expect chasers to come up with this in the heat of the moment on the chase? NO
Do I expect chasers to note that readily identifiable tornado producing structure is moving in a mostly straight line northeast? YES
That big RFD core/Bear's Cage is a train coming down the tracks. Get out of its way.
Thanks to Jon for correcting me here. I totally misremembered the plots he posted on his blog. Before Jon responded here, I had made my own plots of the high velocity areas to the north of the EF2 and SLT, and came up with a similar track direction for this developing part of the storm referred to as the "new" or "primary meso" (white line).On a meteorological level, at no point did I indicate that any mesocyclones in the area southwest of Lawrence on May 28 were moving "southeastward" (not sure where Skip got that impression). On NWS Topeka radar images and from the tornado tracks I surveyed (even walking into some wooded areas), the two mesocyclones involved were both moving toward the east-northeast.
There's also been some assertion that the large EF3/EF4 tornado on May 28 came from the same mesocyclone that crossed into Douglas County KS southeast of Overbrook, and that the mesocyclone visible west of Lone Star Lake around 6:00 pm CDT was only "inflow" to the occluded meso wrapped up in rain to its south. Storm-relative velocity and other images below show that's not the case, where _new_ inflow velocities, convergence, and meso formation are evident _north_ of the original meso, causing the original meso (and small tornado) to interact and merge with the new and intensifying meso.

I want to make sure I get this right before I publish a video on this. So I went back and redid everything I had for the evolution of the storm, analyzed the velocity data again, and replotted the storm's track.
Chasers in the area were focused on an area of higher velocity, north of the ongoing (yet unknown by most) EF2. There is a rotation signature here:

However, if you look at the orientation of the inbounds and outbounds you can see there is a convergence signature here too:

The inbounds and outbounds aren't oriented parallel to the radar beam, and this region starts out nearly perpendicular to the radial indicating strong convergence.
I plotted the orientation of the inbounds and outbounds on KTWX storm relative velocity for thirty or so scans from 22:28 to 23:11 UTC, about 20 minutes prior to the start of the impacting EF2 to a few minutes into the start of the EF4.

There are a couple patterns that really stand out here:
A. There was a continuous, nearly linear track of rotation from southwest to northeast that existed well before the tornadoes, and both tornadoes closely followed this linear track.
B. Prior to each tornado there was an area of strong convergence to the north of this rotation track line. The convergence signature became a mix of rotation and convergence signature, and then mostly focused rotation as it appears to get "pulled in" or merges into this southwest to northeast track line. The tornadoes develop at about the time of these mergers.
My interpretation of what this means:
A is the track of the main supercell updraft or primary mesocyclone, and the supercell is cycling along this line.
B represents an inflow surge into the next supercell cycle, likely convergence on an inflow band, that is curling (yes with rotation) into A. Once this converging and rotating inflow surge becomes rooted to the updraft represented by A, a significant tornado spins up.
It's interesting to see that there was a velocity couplet due west of the start of the EF2 track, just like there were couplets west of the Lawrence EF4. It's as if the circulation is arriving on the inflow band, and the tornado is hooking left around the back rim of the mesocyclone as it's developing before settling into the center and being carried off to the northeast by the parent updraft.
Calling this the track of the primary, tornadic mesocyclone can be misleading or even dangerous from a storm chasing perspective.

It makes it appear as if this is a distinctly different part of the storm, largely disconnected from what is happening with the EF2. SLT and other chasers are actively trying to stay south of this track line, which turned out to be a critical mistake. However, without the track of the EF2 here, it looks as if this would be the correct thing to do. It's important to remember that the velocity is not necessarily showing the tornado producing structure of the storm, but instead the winds flowing into and around the storm. Dangerous misinterpretations of the radar are possible here, which is why chasers shouldn't be relying o such data for their safety.
Instead, I think it's better to think of the primary tornadic mesocyclone track like this, even if there are distinctly different mesocyclone cycles on this track:

Why? Because there is not only a continuous track of rotation along this line, but there is continuous physical structure along this line. The chaser can see this visually in the form of the RFD gust front and Bear's Cage region, which is maintained along this line. That visual identification is most important here in chase making decisions.
It's also readily apparent on the reflectivity. The end of the HP hook exhibits a commahead circulation, the head of the comma I'm interpreting as roughly corresponding to the supercell's main updraft, the tail of the comma as the flanking line updrafts. The storm maintains this commahead along the above linear track. Both tornadoes appear to be rooted to the center of it where the commahead curls in on itself:
SLT is no longer south of the mesocyclone in the above velocity plot. They've crossed its path, and have effectively made it into the storm's inflow notch, before turning around and crossing it again. To their northwest there are areas of strong converging rear flank downdraft (purple arrow) and surging inflow (yellow arrow).
Focused rotation arriving along the inflow band into the rear flank gust front is not present yet as SLT is making the decision to turn around. There is instead a broad area of rotation and convergence to the west corresponding with a surge on the storm's inflow band. This storm was like a breathing entity moving down a set of tracks. It took a deep breath before each tornado.

So what does all of this nitty gritty detail matter in what is likely just a subjective interpretation. Do I expect chasers to come up with this in the heat of the moment on the chase? NO
Do I expect chasers to note that readily identifiable tornado producing structure is moving in a mostly straight line northeast? YES
That big RFD core/Bear's Cage is a train coming down the tracks. Get out of its way.
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