7.2 Air Concentration Equations

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This section provides a quick review of the dispersion equations for particles and puffs and how the air concentration is computed using each approach. The equations shown here are used to illustrate the calculation and the HYSPLIT Technical Memorandum (ARL-224) should be referenced for a more complete description.

  1. Turbulence Definitions:

    • U(t) = U'(t) + Ubar
    • U'(t) = U(t) - Ubar
    • σu = (1/n ΣU'(t)2)0.5

  2. The 3D particle calculation is composed of two steps. The first consists of the advection step, essentially the trajectory calculation component using the mean wind field from the input meteorological data file. In the second step, the mean position is adjusted by the turbulent wind.

    • Pfinal(t+Δt) = Pmean(t+Δt) + U'(t+Δt) Δt

  3. The horizontal turbulent velocity components at the current time U'(t+Δt) are computed from the turbulent velocity components at the previous time U'(t), an auto-correlation coefficient (R) that depends upon the time step, the Lagrangian time scale (TLi - see the section on turbulence options for more detail), and a computer generated random component (U"). An exponential velocity auto-correlation is assumed.

    • U'(t+Δt) = R(Δt) U'(t) + U" ( 1-R(Δt)2)0.5
    • R(Δt) = exp ( -Δt / TLi)

  4. The Gaussian random component U" comes from the computer generated random number (λ) and the standard deviation of the velocity σu, a statistical value that is estimated from the meteorological data fields (e.g. wind speed - U and stability - σθ wind direction fluctuation).

    • U" = σu λ
    • tan σθ = σu / U

  5. The same standard deviation of the velocity component (σu) used to compute particle dispersion is also used to compute the horizontal puff growth rate.

    • dσh/dt = σu

  6. Air concentrations are computed by summing the contribution of each 3D particle's mass (m) each time step that the particle resides in the grid cell and dividing the sum by the cell's volume, where x, y, and z define the grid cell's dimensions.

    • Δc = m (Δx Δy Δz)-1

  7. The incremental concentration contribution for a Top-Hat and Gaussian puff is computed at each grid cell center point when that point lies within the puff radius. Puffs must intersect the point to contribute to the concentration summation while a 3D particle can be anywhere within the grid cell. This means that puffs may pass between grid cell center points when the plume radius is small or the concentration grid resolution is too coarse. In this situation the volume is a combination of the puff radius and cell height.

    • {Top-Hat} Δc = m (π r2 Δz)-1
    • {Gaussian} Δc = m (2 π σh2Δz)-1 exp(-0.5 x2/σh2)

Only a simplified version of the air concentration computational details are shown here. Symbols in bold are vectors and a form of these equations are applied in each velocity component direction. Furthermore, the vertical particle dispersion equation has an additional term (not shown here) to compensate for the accumulation of particles in low-wind and low-turbulence regions near the ground surface.