Isaac Newton Maths poster 2: Maths Stirs

February 2000: Maths Stirs
(click on the image for larger version)

Hurricanes have huge potential for damage to both life and property, but are incredibly difficult to predict. If we were able to tell which direction they were likely to travel in then we would be able to take evasive action. Mathematics can in principle provide the answer to this complex problem; one of the crucial equations describes the mixing process in a swirling flow, a problem familiar at the breakfast table when we stir cream into a cup of coffee.

Cyclones arise over the deep tropical ocean by a process of convergence of airflow which intensifies the rate at which the air spins: it is like the familiar bathtub vortex, but on a very much larger scale. The pressure is low at the centre of the resulting atmospheric vortex, and so water vapour and droplets are sucked up into the vortex from the ocean surface. A complex process of interaction between the swirling air flow and the water vapour ensues. This process influences the power of the vortex and whether it will eventually turn into a tropical cyclone with devastating destructive potential: if it does, then the cyclone interacts with atmospheric winds in further complex processes that determine its path across the ocean surface. Mathematicians seek to predict this path in order to give warning of potential disaster when the cyclone hits land.

The mixing of water vapour within the vortex is a crucial part of this whole scenario. This mixing process is described mathematically in its simplest form by the famous "advection-diffusion equation" shown on the poster. In this equation, $c$ represents the concentration of water vapour. There are two processes that change $c$ :

advection (i.e.transport of the water vapour by the air flow)

diffusion (i.e.spreading out by molecular effects)

$\mathbf{u}$ represents the velocity of the swirling air flow, and the term $\mathbf{u}\mathbin{\textbf.}\nabla c$ in the equation represents the advective process. $\kappa$ measures the rate of the diffusion of water vapour in air, and the term $\kappa\nabla^2c$ represents the diffusion process.

What happens in practice is that, just as observed in the analogous "cream in coffee" experiment, advection by circulating flow generates tightly wound spirals of water vapour (or of cream and coffee) - so tight that molecular diffusion is always important no matter how small $kappa$ may be. The interaction between advection and diffusion is subtle, and can only be understood through mathematical and computational analysis of the advection-diffusion equation. In the real-life situation, there are many other complications, some of which are considered in detail in the paper by Schubert et al (1999) from which the computer simulation shown in the poster was drawn.

The type of advection-diffusion process described above occurs in many other contexts. For example, it plays a part in the process by which spiral galaxies are formed. Think of this when you drink your next cup of coffee! In both contexts, the inner regions rotate more rapidly than the outer regions, and this is why spirals are formed.

Ref. Schubert WH, Montgomery MT, Taft RK et al: Polygonal eyewalls, asymmetric eye contraction, and potential vorticity mixing in hurricanes Journal of the Atmospheric Sciences 56, 9, 1 May 1999

Links:

The UK Met Office Tropical Cyclone page, which includes images and movies of past cyclones

Tropical Cyclones - bulletins and reports from the University of Hawaii Meteorology Department

An impressive animated simulation of the formation of a spiral galaxy at Cardiff University Dept of Physics and Astronomy

Credits:

Graphic design and background fractals: A Burbanks

Simulation graphics: W Schubert, M Montgomery, R Taft, T Guinn, S Fulton, J Kossin, J Edwards.

Text: HK Moffatt

Millenium mug photo: HK Moffatt. Mug by Spode.

Website text: HK Moffatt, RE Hunt, AD Burbanks

Isaac Newton Institute for Mathematical Sciences

Posters in the London Underground