This example is from measurements of a 10 cm diameter jet in air. To understand the affect of the atmosphere on astronomical images we need to review the theory of incompressible turbulence, describe the relevant mathematics, and derive the optical properties of the atmosphere. The associated power spectrum of velocity fluctuations is or where . Consider structure function which is defined as where r is the magnitude of . Thus is the mean squared fluctuation for some lag .
Power laws predominate in the mathematics of turbulence.
Since the integral of a power law is a power law it is easy to calculate power law indices. However, it is tricky to find the normalization, hence numerical coefficients are hard. Thus if then the structure function is We can write where the constant is known as the structure constant. With these results we can find the nature of the refractive index fluctuations that cause atmospheric wavefront aberrations. Temperature fluctuations in the atmosphere, represented by the bubbles in this figure, distort an incoming plane wavefront.