Atmospheric Turbulence Explained

Weak, Moderate, and Strong Atmospheric Turbulence

This is the natural follow-up after weak turbulence. Below is an intuitive explanation of moderate and strong turbulence, connected to standard equations and reduced to engineer-friendly models.

1. Physical Picture: From Weak to Strong Turbulence

Weak Turbulence

• Small refractive-index fluctuations
• Phase distortions dominate
• Intensity fluctuates mildly
• Log-normal model works

Moderate Turbulence

• Both small-scale and large-scale eddies matter
• Strong scintillation
• Deep fades start appearing
• Single log-normal model fails

Strong Turbulence

• Multiple scattering effects
• Beam breakup and saturation
• Intensity fluctuates wildly
• Negative exponential behavior emerges

One Gaussian is no longer enough.

2. Why the Gamma–Gamma Model Is Used

Intensity fluctuations arise from two independent physical effects:

• Large-scale turbulence (slow, beam-wide fluctuations)
• Small-scale turbulence (fast, speckle-like fluctuations)

Each effect is modeled as a Gamma random variable, and the received intensity is their product:

I = I_L · I_S

The product of two Gamma random variables results in the Gamma–Gamma distribution.

3. Gamma–Gamma Probability Density Function

f_I(I) =
2(αβ)^((α+β)/2) / [Γ(α)Γ(β)]
· I^((α+β)/2 − 1)
· K_(α−β)(2√(αβI))
      

• Γ(·): Gamma function
• K_ν(·): Modified Bessel function
• α: large-scale turbulence parameter
• β: small-scale turbulence parameter

4. Meaning of α and β

α = [exp(0.49σ_R^2 / (1 + 1.11σ_R^(12/5))^(7/6)) − 1]^(-1)

β = [exp(0.51σ_R^2 / (1 + 0.69σ_R^(12/5))^(5/6)) − 1]^(-1)
      

σ_R^2 is the Rytov variance, which controls turbulence strength.

• Large α, β → weak turbulence
• Small α, β → strong turbulence

5. Turbulence Regime Classification

Turbulence	Rytov Variance	Model
Weak	σ_R^2 < 1	Log-normal
Moderate	σ_R^2 ≈ 1	Gamma–Gamma
Strong	σ_R^2 ≫ 1	Negative exponential

6. MATLAB Simulations

Weak Turbulence (Log-Normal)

N = 1e6;
sigmaR2 = 0.3;

X = sqrt(sigmaR2) * randn(N,1) - sigmaR2/2;
I_weak = exp(X);

histogram(I_weak,200,'Normalization','pdf');

Moderate Turbulence (Gamma–Gamma)

sigmaR2 = 1;

alpha = (exp((0.49*sigmaR2)/(1+1.11*sigmaR2^(12/5))^(7/6))-1)^(-1);
beta  = (exp((0.51*sigmaR2)/(1+0.69*sigmaR2^(12/5))^(5/6))-1)^(-1);

IL = gamrnd(alpha,1/alpha,N,1);
IS = gamrnd(beta,1/beta,N,1);

I_mod = IL .* IS;

Strong Turbulence (Exponential)

I_strong = exprnd(1,N,1);

7. Summary

Weak turbulence is log-normal, moderate turbulence is Gamma–Gamma, and strong turbulence reduces to a negative exponential model.

Underwater Optical Channel Turbulence (\(\sigma_{\ln}\))

In an underwater wireless optical communication (UWOC) system, the optical signal propagates through water that is not perfectly uniform. Variations in temperature, salinity, and small-scale currents cause fluctuations in the refractive index, which produce fading in the received signal.

This fading is often modeled as lognormal turbulence, especially for weak to moderate turbulence conditions.

1. Lognormal Fading Model

Let the received optical power after path loss and geometric coupling be \(P_0\). The received power affected by turbulence is:

\[ P_r = P_0 \cdot G \]

where \(G\) is a random multiplicative gain representing turbulence. In the lognormal model:

\[ G = e^{X}, \quad X \sim \mathcal{N}(\mu, \sigma_{\ln}^2) \]

Here:

• \(\mu\) is chosen to ensure unit mean of \(G\):
  \[ \mathbb{E}[G] = e^{\mu + \frac{\sigma_{\ln}^2}{2}} = 1 \quad \Rightarrow \quad \mu = -\frac{\sigma_{\ln}^2}{2} \]
• \(\sigma_{\ln}\) is the standard deviation of log-amplitude fluctuations, controlling the turbulence strength.

2. Statistical Properties

Given this model:

1. Mean: \(\mathbb{E}[G] = 1\)
2. Variance: \(\text{Var}[G] = e^{\sigma_{\ln}^2} - 1\)
3. Probability density function (PDF) of \(P_r\):
   \[ f_{P_r}(p) = \frac{1}{p \sigma_{\ln} \sqrt{2 \pi}} \exp\Bigg[ -\frac{(\ln p - \mu)^2}{2\sigma_{\ln}^2} \Bigg], \quad p>0 \]

This shows the received optical power fluctuates around the mean \(P_0\), with the spread controlled by \(\sigma_{\ln}\).

3. Implementation in MATLAB

mu = -0.5 * (cfg.ch.sigma_ln^2);
gT = exp(mu + cfg.ch.sigma_ln*randn(1, nSyms));
gainSym = gainSym .* gT;  % multiplicative turbulence per symbol

Explanation:

• randn(1, nSyms) generates independent Gaussian variables \(X \sim \mathcal{N}(0,1)\).
• Multiplying by σ_ln scales fluctuations to the desired turbulence strength.
• Adding \(\mu = -\sigma_{\ln}^2/2\) ensures mean gain = 1.
• exp(...) converts the Gaussian variable into a lognormal gain.

4. Physical Interpretation

Parameter	Physical Meaning
\(\sigma_{\ln}\)	Strength of turbulence (higher → stronger intensity fluctuations)
\(\mu = -\sigma_{\ln}^2/2\)	Ensures mean received power remains unchanged
\(G = e^X\)	Multiplicative fading of optical power
\(P_r = P_0 \cdot G\)	Received optical power including turbulence

Analogy: Imagine a flashlight beam underwater. If the water is perfectly calm (\(\sigma_{\ln} = 0\)), the beam is steady. If the water has turbulence (\(\sigma_{\ln} > 0\)), the beam flickers randomly around the average intensity. The bigger the \(\sigma_{\ln}\), the stronger the flicker.

5. Typical Values

• Weak turbulence: \(\sigma_{\ln} \approx 0.1\)
• Moderate turbulence: \(\sigma_{\ln} \approx 0.2 - 0.3\)
• Strong turbulence: \(\sigma_{\ln} \gtrsim 0.4\)

These values are consistent with experimental UWOC studies for clear to moderately turbid water.