Fall 2007 9 Misc

Lightning is known to release a large amount of energy in a form of a short burst. Let W be the energy output per unit length of the lightning and f be the dominant acoustic frequency of the thunder it emits.

(a) Use dimensional analysis to express W in terms of f and physical parameters of the surrounding air, e.g., the speed of sound, density, etc.
(b) Under typical conditions, the thunder is heard at f = 100 Hz, the speed of sound in air is v = 343m/s, and the length of the lightning is ~ 1 km.
Estimate the total energy produced by such a lightning and compare it with the energy release of one ton of TNT explosive, 4.6 × 109 J.

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(a) A sudden release of a large energy along the track of the lightning creates an initially rapidly expanding cylinder of a superhot gas.
Remembering that the thermal velocity of molecules coincides with the speed of sound v up to a coefficient, we conclude that the expansion of a very hot gas is necessarily supersonic. Hence, it creates a shock wave. Since the energy is delivered as a short burst rather than continuously, the expanding gas cools down, slows down, and the shock eventually becomes subsonic. At that moment the sound waves can run ahead of it and be heard as thunder.

This description suggests that the most important here are the inertial and sound propagation characteristics of air, i.e., mass density $\rho$ and the speed of sound v. Temperature, diffusion coefficient, viscosity, etc, are irrelevant because the process is far from equilibrium. Ambient air pressure P does not add anything either because $P \sim \rho v^2$. Hence, we expect $W = W(f, \rho, v)$.

We try the scaling form

(1)
\begin{align} W = cf^{\alpha}\rho^{\beta} v^{\gamma} \end{align}

where $c, \alpha, \beta, \mbox{ and } \gamma$ are some dimensionless numbers. The requirement of W to have correct units fixes the last three as follows:

(2)
\begin{align} W = c \rho v^4 / f^2 \end{align}

and therefore:

(3)
\begin{align} W \left [ J/ m \right ] = c \frac{\rho\left[ kg/m^3 \right ] \times \left ( v \left[ m/s \right ] \right )^4}{\left ( f \left[ Hz \right ]\right ) ^ 2} \end{align}

Hence, the lower the frequency of the thunder, the higher must be the energy of the lightning. Although it may seem counterintuitive, it can be understood based on the argument that the wavelength of the thunder is set by the radius of the supersonic core around the lightning. Obviously, this radius increases with W.

(b) For a crude estimate, we can drop the unknown numerical coefficient c. Using the suggested numbers of f = 100 Hz, v = 343m/s, and also $\rho$ = 1.2 kg/m3, we get W = 1.7 × 106 J/m. Hence, the total energy released by the lightning is ~ 1.7 × 109 J or ~ 0.4 ton of TNT.