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W01 · Physics · Vortices, vectors, radar

tornadoes

A hurricane is hundreds of miles wide and lasts for days. A tornado is a few hundred yards wide, lasts minutes, and can spin twice as fast. The same physics applies, squeezed into a column you can see: a pressure drop that draws the funnel, a vector sum that decides which side hits hardest, and a radar that only sees half the wind.

LAB 1

the funnel is a pressure map

A tornado's wind is close to a Rankine vortex: the core spins like a solid disk, and outside it the speed falls off as 1/r. Spinning air needs lower pressure in the middle to hold it in its circle. The funnel cloud is simply where that pressure drop is deep enough for water vapor to condense.

side view
213 mph
300 ft 2,950 ft

How to read this: a side view. The funnel's shape is drawn from the pressure: wherever the pressure drops far enough, the cloud reaches down.

  • edges of the spinning core
  • debris, kicked up where wind tops 78 mph
  • "… across" (lower right): width of this whole picture

Try this: raise Peak wind or lower Cloud base until "Funnel reaches ground?" says Yes.

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Pressure drop at center
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EF rating for this wind
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Funnel reaches ground?
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One lap at the core edge

Cyclostrophic balance

dp/dr = ρ·v²/r ⇒ Δpcenter = ρ·vmax²

Hurricanes are wide enough that Earth's rotation matters. Tornadoes are too small and fast, so the pressure gradient alone holds the air in its circle. Half the drop happens outside the core and half inside.

At that's about 3.5 inHg, more than a tenth of an atmosphere. Each 0.1 inHg of drop lowers the condensation level by roughly 100 ft, so the cloud base sags into a funnel.

LAB 2

where the spin comes from

Tornadoes don't get their spin from Earth's rotation. When wind near the ground is slow and wind higher up is fast and turning, the layer of air between them rolls like a pencil between your palms. That's horizontal spin. A storm's updraft lifts the middle of that roll and stands its ends upright.

no updraft
70%

How to read this: the tube of rings is a roll of spinning air lying on its side. The dot running around each ring shows which way it's turning. "Cyclonic" means spinning the same way as big storms (counterclockwise here).

  • spin still lying flat
  • cyclonic leg: the one that usually makes tornadoes
  • anticyclonic leg: spinning the other way

Try this: start Updraft strength at 0 and slowly raise it.

Now

Tilting and stretching

horizontal vorticity → tilted by updraft → stretched upward

Tilting turns sideways spin into upright spin, producing a rotating updraft called a mesocyclone, a few miles wide. Stretching then narrows the column. As in the skater analogy from W00 · Hurricanes, a narrower column spins faster.

The arch makes one cyclonic leg and one anticyclonic leg. In the Northern Hemisphere, wind that turns clockwise with height favors the cyclonic side, which is why most strong tornadoes spin counterclockwise there.

LAB 3

the right side hits harder

A tornado also travels, usually –. The wind that reaches the ground is the spin plus the forward motion, added as vectors. On one side they add and on the other they partly cancel. Watch the damage path build up behind the vortex.

top view · moving east
34 mph
134 mph

How to read this: the tornado (gold circle) slides left to right and paints the strongest wind each spot felt. The colors match the EF damage ratings in the strip below the sliders, where the two lit chips are the right-side and left-side ratings. The bottom (right-hand) side comes out hotter.

  • tornado core
  • small arrows: ground-wind direction (longer = faster)
  • colored cells: damage rating, see EF strip

Try this: set Forward speed to 0 and click Clear path: the path comes out the same on both sides.

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Right of track
—
Left of track

Vector sum

ground wind = vspin + vforward

For a counterclockwise tornado moving east, the south side spins east too, so the two speeds add. The north side spins west against the motion, so they partly cancel. Facing the direction of travel, that's the right side and the left side.

The same asymmetry happens in hurricanes. Their right-front quadrant is usually the most dangerous.

Rated by damage, not by wind

Almost no anemometer survives a direct hit, so the Enhanced Fujita scale (US, 2007) estimates wind from surveyed damage to buildings and trees. The fastest wind ever measured, about 300 mph, came from a mobile Doppler radar near Bridge Creek–Moore, Oklahoma, in 1999.

LAB 4

what radar sees

Doppler radar measures only the part of the wind moving straight toward or away from the antenna. That's the dot product of the wind with the beam direction. A spinning vortex then shows up as a bright green patch beside a bright red one: the velocity couplet forecasters look for before issuing a warning. Drag the vortex around the scan.

radial velocity · drag the vortex
toward radar
away from radar
123 mph

How to read this: the radar sits just below the picture, looking up. Green is air moving toward it, red is air moving away, and brighter means faster. A tight green-red pair is the tornado's signature.

  • the tornado
  • radar beam through its center
  • faint lines and arcs: beam directions and distance rings
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Strongest inbound
—
Strongest outbound
—
Rotational velocity (Δv / 2)
—
Range · bearing

The radial component

vradial = v · r̂ = |v| cos θ

Wind moving across the beam (θ = 90°) is invisible to the radar. On the side of the vortex where air swirls toward the antenna the display is green, and where it swirls away it's red. Half the difference between them estimates the spin speed.

Flip the spin direction and the colors swap sides. A counterclockwise tornado north of the radar puts green on the left and red on the right.