Billow clouds, shocked streams & shedding eddies

I flew from Bangalore to Delhi on Tuesday. The flight was early in the day, at 6, and so I had the wonderful opportunity to watch a sunrise from above a sea of clouds. One very beautiful sight was the presence of uniquely shaped ones, styled like the waves in Hokusai’s The Great Wave off Kanagawa.

'The Great Wave off Kanagawa'
Photo: Wikimedia Commons

I recalled having seen them in Tuticorin sometime in late 2011, but I couldn’t remember what they were called. Their vortex-like upper tips had me confuse them briefly with a Karman vortex street. Thankfully, one Google search led to another and I came upon the answer: billow clouds.

A photograph of billow clouds.
A photograph of billow clouds. Photo: wunderground.com

Billow clouds, I re-learnt, are the result of what’s called a Kelvin-Helmholtz instability: When two fluids of different densities and sharing a surface are moving parallel to each other, the surface becomes unstable if their relative velocity reaches a certain threshold.

When there’s talk of fluids, surface tension is likely to be involved. Fortunately, that’s what the relative velocity component takes care of. However, “surface tension is not relevant on atmospheric scales,” said Dr. Rajaram Nityananda, of IISER, Pune.

More interestingly, subtle variations on the Kelvin-Helmholtz instability give rise to more complex shapes, and even more complex titles. For example, if the lighter fluid is pushing against the heavier fluid, a Rayleigh-Taylor instability* results. A memorable manifestation of this is the mushroom cloud that forms after a powerful nuclear explosion, where cooler air is pushing into the debris rising upward.

A mushroom cloud rising from the Castle Romeo nuclear test, 1954.
Image: Wikimedia Commons

If you sent a If you sent a shockwave through two parallely flowing fluids, you’d get the Richtmyer-Meshkov instability. The shockwave will cause both fluids to accelerate and waver, the extent of which builds up over time. If the heavier accelerates into the lighter one, it pushes through as spikes. If the lighter accelerates into the heavier one, it produces bubbles. Eventually, the instability builds up until the two fluids are mixed.

Simulation of a shockwave-induced Richtmyer-Meshkov instability.
Simulation of a shockwave-induced RM instability. Image: Wikimedia Commons

This could be leveraged in the working of jet engines. A parallel flow of fuel and oxygen could be destabilized using a shockwave so the fuel is broken up into finer droplets that are easier to combust.

At last, we come to my “phenomenological” favorite (not that there’s a list): the Karman vortex street. Instead of there being two fluids, imagine just the one, in whose path a blunt obstacle is placed. When it meets the obstacle, the fluid is split into two swirling streams. If the fluid was flowing fast enough, given the shape of the obstacle, the streams reconcile their paths after crossing the obstacle by forming vortices – sometimes a street of them.

Notice the gradual onset of instability until the 49th second. Karman vortices are evidently not hard to find as many satellite images of winds blowing past small islands have shown.

Image: http://disc.sci.gsfc.nasa.gov/
Image: http://disc.sci.gsfc.nasa.gov/

These effects are as astounding as the foundational principles are elegant. If simple disturbances on one and two streams are responsible for a variety of designs, imagine what the depthless roster of fluid dynamics will have to offer.