Which expression represents the centerline concentration along the downwind axis at height z?

Which expression represents the centerline concentration along the downwind axis at height z?

• A. C_centerline(z) = Q/(4π u σ_y σ_z) exp(- (x^2+y^2)/(2 σ^2)).
• B. C_centerline(z) = Q/(√(2π) u σ_y) exp(- (z-H)^2/(2 σ_z^2)).
• C. C_centerline(z) = Q/(2π u σ_y σ_z) [exp(-(z-H)^2/(2 σ_z^2)) + exp(-(z+H)^2/(2 σ_z^2))].
• D. C_centerline(z) = Q/(2π u σ_y σ_z) [exp(- (z)^2/(2 σ_z^2))].

Answer: (C)

The important idea is that along the downwind axis the crosswind distance is zero (y = 0), so the crosswind spreading factor does not reduce the concentration there. For a continuous point source near the ground, the ground surface is represented by an image source located at a height of -H, which means you add the contributions from both the real source at height H and its image at -H. This yields two vertical-Gaussian terms, one centered at z = H and one at z = -H, that together describe how concentration varies with height above the ground along the plume centerline. The overall scaling uses Q divided by 2π u σ_y σ_z, reflecting the downwind and crosswind/vertical dispersion terms. Combining these ideas gives C_centerline(z) = Q/(2π u σ_y σ_z) [ exp(- (z - H)^2/(2 σ_z^2)) + exp(- (z + H)^2/(2 σ_z^2)) ]. This accounts for both ground reflection and evaluating exactly along the centerline where y = 0.

The important idea is that along the downwind axis the crosswind distance is zero (y = 0), so the crosswind spreading factor does not reduce the concentration there. For a continuous point source near the ground, the ground surface is represented by an image source located at a height of -H, which means you add the contributions from both the real source at height H and its image at -H. This yields two vertical-Gaussian terms, one centered at z = H and one at z = -H, that together describe how concentration varies with height above the ground along the plume centerline. The overall scaling uses Q divided by 2π u σ_y σ_z, reflecting the downwind and crosswind/vertical dispersion terms. Combining these ideas gives C_centerline(z) = Q/(2π u σ_y σ_z) [ exp(- (z - H)^2/(2 σ_z^2)) + exp(- (z + H)^2/(2 σ_z^2)) ]. This accounts for both ground reflection and evaluating exactly along the centerline where y = 0.