11.3 Statistical properties of turbulence

We start with a simple model of turbulence. It must explain why the scale size of the finest turbulence structures becomes smaller and smaller with increasing $Re$, and should allow to treat the finest details in a homogeneous way. It cannot explain why certain structures form and not others, but it describes the average flow of energy across the scale sizes of turbulence.

• Kinetic energy enters the medium on large scales, in the form of convection or friction on an obstacle (energy range).
• The energy is transferred towards smaller scale sizes over eddy fragmentation, while the Reynolds number decreases (inertia range).
• The smallest eddies have sub-critical $Re$'s, dissipate heat, and are stable (viscous range).

[Kolmogorov 1941] advanced a hypothesis for high ( $Re > 10^6-10^7$) Reynold's numbers, postulating that turbulence in the inertia range was determined only by one parameter $\epsilon$ (kinetic energy converted to heat by viscous friction per unit time and unit mass). In the viscous range, it would only depend on $\epsilon$ and the already discussed viscosity $\eta$. This model treats cases like the seemingly amorph eddies-within-eddies part in the Fig.41-6 (d),(e) (from [Feynman et al. 1964]). As we have derived in the previous section, (d) and (e) are indeed the cases to be expected in the troposphere.

The inertia range is interesting for us because it corresponds to spatial dimensions of some meters to 2-3 kilometers, i.e. the baselines of the PdBI fall into this range.

For the mathematical treatment of highly developed turbulence, one can use a formalism based on random variations.

Most ``classical'' statistics represent a given distribution of probability (binomial, Poisson, Gaussian, ...) around a most likely measurement value. For atmospheric parameters like e.g. temperature and wind velocity, we must make a more general approach: the most likely measurement values vary with time and space, which means they can be represented by non-stationary random processes. The classical average and its variation are not very useful to describe these systems.

An instrument for the characterization of non-stationary random variables are structure functions, which were first introduced by [Kolmogorov 1941]. A scalar structure function has the form given in Eq.11.5,

$$D_f(x_i,x_j) = \overline{\left (f(x_i) - f(x_j) \right )^2}$$ (11.5)

i.e. a function $f(x)$ is measured at the positions $x_i$ and $x_j$, squared and averaged over many samples to obtain a $D_f(x_i,x_j)$. When the average level of $f$ changes, the average differences between $f(x_i)$ and $f(x_j)$ stay constant.

The structure function formalism can even be used to describe vector parameters like the turbulent wind velocity, in this case one simply needs $3 \times 3$ tensors of structure functions for their description. We won't need tensors in the following discussion, however. The detailed mathematical formalism of random fields would be too much for this course (see [Tatarski 1971] for details). We will only discuss the basic concepts and their application to phase shifts.

Real atmospheric parameters are functions of time and space. For time dependency, Taylor's hypothesis of frozen turbulence has been quite successful (Eq.11.6). It states that the pattern of refractive index variations stays fixed while it is moving with the wind.

$$f(x,t+t') = f(x -V_{al} \cdot t',t)$$ (11.6)

This means that for the structure functions, one can either measure at two different sites simultaneously or measure in one place and compare different times. Time-like structure functions are often easier to determine because the sampling is continuous and instrumental effects are reduced by averaging. The velocity $V_{al}$ is also called ``Velocity aloft'' and can differ notably from measurements of a ground-based meteorological station: wind speeds increase with altitude and change direction due to the diminishing effect of ground friction.

For two measurement points which are a distance $r$ apart from each other, one finds that the structure functions of many atmospheric parameters (temperature, refractive index, absolute wind velocity, ...) obey a $r^{2/3}$ power law. This law can be derived from the theory of random fields, but the easiest way is as follows:

Consider a velocity fluctuation $\delta V_r$ (where $\delta V_r$ may be large) which occurs on a scale size $r$ and a time $t=r/\delta V_r$:

• its energy per mass unit is $\propto (\delta V_r)^2$
• The energy per mass and time: $\epsilon \propto \delta V_r^2/t= \delta V_r^3/r$
• Stationary transport of this energy from large to small scale sizes, where it is finally dissipated
• Approximately, $D_{rr}(r)=\overline{\left (V(r_l+r) - V(r_l) \right)^2}$ is dominated by eddies of size $r$, i.e. $D_{rr}(r)\approx (\delta V_r)^2$
• Therefore, the formula for intermediate scale sizes is:
  

  $$D_{rr}(r)\propto (\epsilon \cdot r)^{2/3}$$ (11.7)

  
  

For a thin layer, refractive index fluctuations and phase fluctuations are identical. This is the thin screen approximation.

$$D_{\varphi}(r) = C \cdot (\epsilon \cdot r)^{2/3}$$ (11.8)

In a thick turbulent layer, the phase front encounters multiple refractive index fluctuations and the power law index changes. This problem can be solved by analyzing the irregular refracting medium over its Fourier transform. After [Tatarski 1961], the spectral density of the function

$$D(r) = r^p$$ (11.9)

with $0<p<2$ is in the three dimensional case

$$F'(\kappa) = \frac{\Gamma (p+2)}{4 \pi^2} sin(\pi p/2) \kappa^{-(p+3)}$$ (11.10)

An important condition is that the fluctuations must have an outer limit, i.e. that the power law does not increase indefinitely. To get the phase fluctuations from the refractive index spectrum, we take

$$D_{\varphi}(r)=4 \pi \int_0^{\infty}[1-J_0(\kappa r)] F'(\kappa) \kappa d\kappa$$ (11.11)

with the Bessel function $J_0$ and finally obtain the power law for thick screen turbulence:

$$D_{\varphi}(r) \propto (\epsilon \cdot r)^{5/3}$$ (11.12)

For the phase noise ( $\Delta \varphi(r) = \sqrt{D_{\varphi}(r)}$), one has therefore to expect power laws with exponents between $1/3$ and $5/6$. The absolute scaling factor for the power law and the position of the break where the phase noise levels off depend on the observing site, and of course on the weather.

Due to the quasi-random nature of phase fluctuations, forecasts and inter/extrapolations can be considered inadequate for a phase correction system. Direct measurements of the water vapor column along the line of sight are therefore the most reliable approach.

Figure: Gradient $\Delta T_{sky}/\Delta path$ (K/mm) as a function of frequency and total precipitable water under clear sky conditions. The atmospheric model assumed an ambient temperature of 275 K, pressure 780 mbar, elevation 45$^{\circ }$, an observing frequency of 90 GHz and various amounts of water vapor. Light grey: 3mm water, middle grey: 5mm water, dark grey: 8mm water. Dash-dotted lines indicate the receiver tuning ranges of the PdBI.