Similarity Solution of the Fractional Time Independent Advection-Diffusion Equation in Two Dimensions

Technical Summary: Similarity Solution of the Fractional Time-Independent Advection-Diffusion Equation in Two Dimensions

Problem Statement
The paper addresses the challenge of modeling the dispersion of pollutants, specifically sulfur dioxide (SO₂), in an unstable atmospheric environment. While the classical Advection-Diffusion Equation (ADE) is widely used, the authors note that it often fails to capture complex, anomalous diffusion behaviors found in mixed environments. To address this, the study focuses on the Fractional Advection-Diffusion Equation (FADE) in two dimensions (downwind distance $x$ and vertical height $z$). The specific problem involves solving the time-independent FADE under conditions of atmospheric instability, where both wind speed ($u$) and vertical eddy diffusivity ($k_z$) vary with height. The goal is to derive a similarity solution that accounts for these variations and to validate the model against experimental data from the Prairie-Grass experiment.

Methodology
The authors employ the similarity solution method to reduce the complexity of the partial differential equation (PDE) to an ordinary differential equation (ODE). The mathematical framework is built on the following steps:

1. Governing Equation: The study utilizes the steady-state fractional ADE:
   $$u \frac{\partial^\alpha C}{\partial x^\alpha} = \frac{\partial}{\partial z} \left( k_z \frac{\partial C}{\partial z} \right)$$
   where $0 < \alpha < 1$ represents the fractional order, $C$ is the pollutant concentration, and $u$ and $k_z$ are height-dependent wind speed and eddy diffusivity, respectively.

2. Atmospheric Parameters: Under unstable conditions, the wind speed profile $u(z)$ and eddy diffusivity $k_z(z)$ are defined using Monin-Obukhov similarity theory, incorporating friction velocity ($u_*$), roughness height ($z_0$), and the Monin-Obukhov length ($L$).

3. Similarity Transformation: The authors introduce a similarity variable $\mu = z/\delta(x)$ and a scaling function $\delta(x)$ (similarity thickness) to balance the fractional advection term with the vertical diffusion term. This reduces the PDE to a second-order ODE with constant coefficients.

   • The effective wind speed ($u_{eff}$) and effective diffusivity ($k_{eff}$) are evaluated at the stack height ($h_s$).
   • The similarity thickness is derived as $\delta(x) \propto x^{\alpha/2}$.

4. Boundary Conditions: The solution enforces zero-flux (Neumann) boundary conditions at the ground ($z=0$) and the mixing height ($z=h$). The final analytical solution is constructed using an image series method to satisfy these boundaries, resulting in a summation of Gaussian kernels.

5. Validation: The derived fractional concentrations are compared against observed SO₂ concentrations from the Prairie-Grass experiment (Barad, 1958), conducted in O'Neil, Nebraska. The comparison covers downwind distances of 50, 100, 200, 400, and 800 meters under unstable atmospheric conditions.

Key Contributions

• Analytical Derivation: The paper provides a closed-form similarity solution for the two-dimensional, time-independent fractional ADE, explicitly accounting for the vertical variation of wind speed and eddy diffusivity in unstable atmospheres.
• Fractional Order Analysis: The study evaluates the model's performance across different fractional orders ($\alpha = 0.85, 0.95,$ and the classical case $\alpha = 1$).
• Statistical Evaluation: The model's accuracy is quantified using standard statistical indices: Fraction Bias (FB), Normalized Mean Square Error (NMSE), Correlation Coefficient (COR), and the Factor of Two (FAC2) criterion.

Results
The comparison between the fractional predictions and the Prairie-Grass experimental data yields the following findings:

• Overall Agreement: For most distances, the fractional predicted concentrations fall within a factor of two of the observed values, though specific distances show deviations.
• Distance Anomaly: The model performs well at distances of 50, 100, 200, and 800 meters. However, at a distance of 400 meters, the predicted concentrations fall outside the factor-of-two range for all tested $\alpha$ values.
• Optimal Fractional Order: The results indicate that the fractional model with $\alpha = 0.85$ provides the best agreement with observed data compared to $\alpha = 0.95$ and the classical case ($\alpha = 1$).
  • At $x=50$m, the FAC2 for $\alpha=0.85$ is 0.63.
  • At $x=100$m, the FAC2 for $\alpha=0.85$ is 0.99.
  • At $x=200$m, the FAC2 for $\alpha=0.85$ is 1.5 (as reported in Table 4, exceeding the standard [0,1] range).
  • At $x=800$m, the FAC2 for $\alpha=0.85$ is 0.84, indicating that 16% of the data points fall outside the factor-of-two range.
• Statistical Performance: The statistical indices (NMSE, FB, COR) generally show better performance for the fractional cases ($\alpha < 1$) than the classical case ($\alpha = 1$), particularly at shorter distances.

Significance and Claims
The authors conclude that the numerical similarity solution yields dependable answers with a respectable level of accuracy for modeling pollutant dispersion in unstable atmospheric conditions. The study claims that the fractional approach, particularly with $\alpha = 0.85$, offers a more accurate representation of the physical diffusion processes than the classical integer-order model. The paper emphasizes that the similarity method successfully reduces computational complexity while maintaining high precision, making it a viable tool for environmental impact assessments involving SO₂ dispersion. The authors explicitly state that the results are reliable but modestly note the discrepancy at the 400m distance, suggesting that while the method is robust, specific local conditions at that distance may require further investigation.