A Model for Turbulence: The modified von Karman Spectrum
The atmospheric Turbulence is modeled using the power spectral density (PSD) of the refractive index fluctuations. There are many models which can describe the PSD of $\partial \mathrm{n}$ such as the Kolmogorov, Tatarski, von Karman and modified von Karman spectrum. We will just focus here on the modified von Karman spectrum and in this case the PSD of the atmospheric turbulence looks like this [2]:
$$\Phi(\kappa) = 0.033 C^2_n \frac{\exp{\left(-\frac{\kappa^{2}}{k^{2}_{m}}\right)}}{\left(\kappa^{2}+k^{2}_{0}\right)^{\frac{11}{6}}} \space\space , \space\space\space 0\leq \kappa < \infty$$
$C^{2}_{n}$ $\space$ is the refractive index structure constant as seen on the previous slide

$k_{m}$ = $\frac{5.92}{l_0}$ $\space$ is an equivalent wavenumber related to the inner scale $\space l_{0}$

$k_{0}$ = $\frac{2\pi}{L_{0}}\space$ is an equivalent wavenumber related to the outer scale $\space L_{0}$,

$l_{0}$ and ${L_{0}}$ are respectively the inner and outer scale of the turbulent eddies

$L_{0}$ ranges generaly from $1\mathrm{m}$ to approximately $100 \mathrm{m}$, while $l_{0}$ is a few $\mathrm{mm}$

$\kappa$ is the spatial angular frequency