Kolmogorov turbulence

Kolmogorov theory in words

Kolmogorov turbulence theory explains:

How energy is transferred from larger to smaller eddies
How much energy is contained by eddies of a given size
How much energy is dissipated by eddies of each size

The idea of eddies:

An eddy is a region of turbulent motion of size l and velocity u(l) that are moderately coherent. Therefore, an eddy also has a time scale τ(l) = l/u(l). Turbulence can be considered to consist of eddies of different sizes.

Energy Transfer:

Kolmogorov theory assumes that energy is added to system at largest scales “outer scale” L_o. These large scales are unstable and break up, transferring their energy to somewhat smaller eddies. These smaller eddies undergo a similar break-up process and transfer their energy to yet smaller structures "eddies". This energy cascade process takes place in the "Inertial range" and continues until the Reynolds number Re(l) =u(l)*l/ν (where ν is the kinematic viscosity = μ/ρ (m²/s), μ is the dynamic viscosity of the fluid (Pa·s or N·s/m² or kg/(m·s), and ρ is the density of the fluid (kg/m³)) is sufficiently small that the eddy motion becomes stable, and molecular viscosity is effective in dissipating the kinetic energy. At these small scales, called "inner scale" l_o, the kinetic energy of turbulence is converted into heat. L_o ranges from 10’s to 100’s of meters; l_o is a few mm.

Kolmogorov turbulence (dimensional analysis):

Kolmogorov presented his original paper and his well known -5/3 power law from a pure dimensional analysis.

Dimensional analysis:

Given that:
u(l) the eddie's velocity.
ε the energy dissipation rate per unit mas.
ν the viscosity.
l_o the inner scale.
l the local spatial scale.
Energy/mass = u(l)²/2 ∝ u(l)²
Energy dissipation rate per unit mass is then given by:


			
			

			\epsilon \sim   \frac{u(l)^2}{\tau}  \sim  \frac{u(l)^2}{l/u(l)} \sim \frac{u(l)^3}{l}, \; or \; u(l) = (\epsilon l)^{1/3}

Energy is then u(l)² ∼ ε^2/3 l^2/3.

The 1D power spectrum of velocity fluctuations is then given by:


			
			

			\Phi(\kappa) \Delta (\kappa) \propto   u(l)^2  \propto  \epsilon^{2/3} \kappa^{-2/3}

where κ = 2π/l. The 1D power spectrum can then be written as:


			
			

			\Phi(\kappa) \propto \kappa^{-5/3}

The 3D power spectrum is related the 1D spectrum by:


			
			

			\Phi^{3D}(\kappa) = - \frac{d\Phi}{d\kappa}*\frac{1}{2\pi\kappa}

Therefore the 3D spectrum can be then written as:


			
			

			\Phi^{3D}(\kappa) \propto \kappa^{-11/3}

u(l) the velocity.
optical turbulence (OT) is a very small-scale phenomena that is caused by the fluctuations in the index of refraction of the air. In turns, the fluctuations in refractive index lead to distortion of the phase of wave-front of the light propagating through the atmosphere. The latter is assumet to be a random process with zero mean <Φ> = 0, and is often described in terms of the structure function defined as¹:


			
			

			D_\phi (\overrightarrow{r}) = <(\phi(\overrightarrow{r'}) - \phi(\overrightarrow{r} + \overrightarrow{r'}))^2>

Under the statistically homogeneous and isotropic turbulence assumption, the structure function can be written as:


			
			

			D_\phi(R) = \left\{
			\begin{array}{c l}      
    									C_n^2 \, R^of v{2/3} & l_0  \ll R \ll L_o\\
    									C_n^2 \, l_o^{-4/3}\,  R^2 &  R \ll l_o
											\end{array}\right.

Where C_n² is defined as the refractive index structure parameter, considered the most critical parameter along the propagation path. Furthermore, using the Kolmogorov model of the turbulence distortions, the phase structure functionthe structure fucntion can be written as²:


			
			

			D_\phi (\overrightarrow{r})=  6.88  \, (\frac{|\overrightarrow{r}|}{r_o})^{5/3}

where


			
			

			r_o^{-5/3} = 0.185 \, \frac{4 \pi}{k^2} \frac{1}{A}

and


			
			

			r_0^{-5/3} = \left\{
			\begin{array}{c l}      
    									{\int_0^{L} \, C_{n}^{2}(z) dz} & (plane \, wave) \\
    									{\int_0^{L} \, C_{n}^{2}(z) (\frac{L-z}{L})^{5/3} dz} & (spherical \, wave)
											\end{array}\right.

r_o represents the maximum size of a telescope which can just operate at the diffraction limit. Any larger and resolution will be seeing-limited. The integration is perfromed from the pupil plane at z = 0 to z = L, the light source location. For a constant value of C_n² through out the turbulent region, the spherical and plane wave definition of r_o are related by:


			
			

			r_o (spherical \, wave) = (\frac{8}{3})^{3/5} \, \, r_o (plane \, wave)