Solution of the Advection-Diffusion Equation in Two Dimensions using the Deposition Velocity

Technical Summary: Solution of the Advection-Diffusion Equation in Two Dimensions using the Deposition Velocity

Problem Statement
This study addresses the analytical solution of the two-dimensional steady-state advection-diffusion equation (ADE) under unstable atmospheric conditions. The primary objective is to model the dispersion of pollutants, specifically focusing on crosswind-integrated normalized concentrations at the centerline. The model incorporates a deposition velocity boundary condition at the ground level and utilizes power-law functions for both vertical eddy diffusivity and wind speed as functions of vertical height. The study seeks to validate this analytical approach against observed sulfur hexafluoride (SF6) concentration data from the Copenhagen field experiment in Denmark.

Methodology
The authors employ the separation of variables technique to solve the governing ADE. The mathematical framework is defined as follows:

• Governing Equation: The two-dimensional ADE is formulated where wind speed $u(z)$ and vertical eddy diffusivity $k_z(z)$ are expressed as power laws of vertical height ($z^p$ and $z^q$, respectively).
• Boundary Conditions:
  1. Top Boundary (Mixing Height $h$): Zero flux condition ($k_z \frac{\partial C}{\partial z} = 0$).
  2. Ground Boundary ($z=0$): A deposition flux condition where the flux equals the product of the deposition velocity ($V_d$) and the concentration ($k_z \frac{\partial C}{\partial z} = -V_d C$).
  3. Source Term: A continuous point source at stack height $h_s$ represented by a Dirac delta function.
• Analytical Derivation:
  • The equation is separated into spatial components $X(x)$ and $Z(z)$.
  • The vertical component leads to a Sturm-Liouville type differential equation with variable coefficients.
  • By assuming specific values for the exponents (e.g., $q=0.85$) and utilizing a variable transformation, the equation is reduced to a generalized Bessel equation.
  • The solution for the vertical concentration profile is derived using Bessel functions of the first and second kind, which are simplified to trigonometric forms (sine and cosine) for half-order Bessel functions.
  • Eigenvalues and eigenfunctions are determined by applying the boundary conditions, leading to a series solution for the concentration $C_y(x, z)$.
  • Coefficients are calculated using orthogonality conditions derived from the wind speed profile.

Key Contributions

• Analytical Framework: The paper presents a closed-form analytical solution for the 2D ADE under unstable conditions, explicitly incorporating a deposition velocity boundary condition, which is often a complex factor in dispersion modeling.
• Power-Law Parameterization: The model utilizes power-law representations for wind speed and eddy diffusivity, allowing for flexibility across different stability classes (defined by exponents $p$ and $q$).
• Validation: The analytical model is rigorously tested against the well-known Copenhagen SF6 dataset, covering various stability classes (A, B, C, D) and downwind distances.

Results
The study compares the predicted crosswind-integrated normalized concentrations with observed SF6 concentrations from the Copenhagen experiment.

• Data Comparison: Table 2 and Figure 1 present a direct comparison of observed versus predicted concentrations across multiple runs (Run 1 through Run 9) at varying downwind distances.
• Statistical Performance: The model's performance is evaluated using standard statistical metrics (Hanna, 1989):
  • Normalized Mean Square Error (NMSE): 0.07 (indicating low error).
  • Fractional Bias (FB): 0.003 (indicating negligible bias).
  • Correlation Coefficient (COR): 0.82.
  • Fraction within a Factor of Two (FAC2): 0.997.
• Visual Agreement: Figure 1 illustrates a high degree of agreement between the observed and predicted concentration profiles. Figure 2 confirms that the predicted values fall within a factor of two of the observed data.

Significance and Claims
The authors claim that the proposed analytical model achieves a "high degree of closed agreement" with the observed field data. Specifically, the paper states that the predicted concentrations achieved 100% agreement with the observed concentrations at the centerline, as evidenced by the FAC2 value of 0.997 and the near-zero bias. The study concludes that the separation method, combined with power-law parameterizations for wind and diffusivity, provides a robust analytical tool for predicting pollutant dispersion under unstable atmospheric conditions without the need for complex numerical simulations. The work reinforces the utility of transform-based and separation-based analytical solutions in capturing the essential physics of pollutant transport in unstable boundary layers.