9.3 Turbulence Parameterizations

If not continuing from the last section, first retrieve conc_case_control.txt and conc_case_setup.txt. The calculation in the previous section used the turbulence computation defaults, the Kanthar-Clayson equations. These equations determine the way the meteorological data are processed to compute the horizontal and vertical turbulence. The method can be changed from the Configuration Setup / Concentration / Turbulence Method Menu #7. Some of these options will be explored in more detail here.

1. When the momentum and heat flux variables are available in the meteorological data file, the default calculation would use the Kanthar-Clayson equations to compute the turbulence values from stability parameters. The computation of the stability parameters from the meteorological data will be discussed in the next section. Within the boundary layer the turbulence equations have the following form:
   
   
   • w’² = 3.0 u_*²(1 – z/Z_i)^3/2
   • u’² = 4.0 u_*²(1 – z/Z_i)^3/2
   • v’² = 4.5 u_*²(1 – z/Z_i)^3/2
   
   where the turbulent velocities are a function of the friction velocity, height, and boundary layer depth. The horizontal and vertical components are explicitly predicted when using this method. When this method is defined as the default, the associated namelist variable KBLT is set to zero and no radio-button is selected for vertical turbulence in Menu #7. Undefined means HYSPLIT will select the optimum method depending upon the meteorological data. Select Kanthar-Clayson, save all menus to exit, run the model, and then create the plume graphic to confirm the result is identical to the last simulation.


2. The other turbulence computation is an older approach that follows the Beljaars-Holtslag equations parameterized in terms of the vertical diffusivity for heat:
   • K_h = k w_h z (1 - z/Z_i)²
   
   where the stability parameter w_h is a function of the friction velocity, Monin-Obukhov length, and convective velocity scale:
   • w_h = f(u_*, 1/L, w_*)
   
   The vertical diffusivity profile is converted to a turbulence value:
   • σ _w = (K_h / T_L)^0.5
   
   assuming a constant value for the Lagrangian time scale. The horizontal turbulence components are computed by assuming that they are equal to the vertical components:
   • w’² = u’² + v’²

   To configure this calculation, open Menu #7 and select the Beljaars-Holtslag radio-button for vertical turbulence. Save all menus to exit, run the model, and display the resulting graphic which shows a similar plume as the default Kanthar-Clayson calculation but with a slightly higher peak concentration.

3. Conceptually the two previous approaches are similar to the earlier (1981) equations developed by S.R. Hanna in Applications in Air Pollution Modeling, where the vertical turbulence is proportional to the height of the PBL and the friction and convective velocities. For example, in the mid-boundary layer:
   • Stable:     σ _w = 1.3 u_* (1 - z/Z_i)
   • Unstable: σ _w = 0.722 w_* (1 - z/Z_i)^0.207

   To configure this calculation, open Menu #7 and select the Hanna radio-button for vertical turbulence. Save all menus to exit, run the model, and display the resulting graphic which shows a plume between the default Kanthar-Clayson and Beljaars-Holtslag calculations in terms of peak concentration.

4. The Lagrangian time scale determines how corrrelated the turbulence is between integration time steps. Time steps less than T_L are correlated and the dispersion rate is more linear, while time steps larger than T_L are uncorrelated and the dispersive growth becomes more random. Constant values for the vertical and horizontal time scales are the default. However, Hanna developed an approach where the vertical time scale can vary in space and time based upon the σ_w and the peak energy wavelength λ, which is a function of z/Z_i and z/L.
   • T_L = 0.1 λ / σ_w

   Setting the stable vertical Lagrangian Time Scale (VSCALES) to -1 will result in the Hanna vertical time scale calculation for all stabilities. Repeating the previous calculation with Hanna turbulence and variable T_L shows a large increase in the plume concentrations. The variable Hanna T_L can be used with any turbulence method.

5. Another way to approach the anisotropy (ratio between vertical and horizontal) issue is to use a different method to compute the horizontal turbulence. The default approach is to compute the horizontal turbulence as defined in the above approaches. However, the horizontal turbulence values can be replaced by a value computed from the deformation of the velocity field:
   • K_hor = 2^-0.5 (c Χ) ² [(∂v/∂x + ∂u/∂y)² + (∂u/∂x - ∂v/∂y)²]^0.5

   To configure this calculation from the original default calculation, open Menu #7, press the Reset to set previous changes back to their defaults, and then select the velocity deformation radio-button for horizontal turbulence. Save all menus to exit, run the model, and create the graphic which shows a narrower plume using the default Kanthar-Clayson for vertical turbulence, a result more similar to the Beljaars-Holtslag calculation. In general the deformation approach should not be used for simulations where the grid spacing of the meteorological data under-samples the variations in the flow field (grid too coarse) because then the deformation provides no information about sub-grid scale turbulence.

The results shown here provide some rationale for the the default vertical and horizontal turbulence settings used in the model calculation, which has been shown to provide an excellent representation of the initial tracer cloud mixing. The turbulence options are intended to provide some flexibility when using different input meteorological data. The use of Turbulent Kinetic Energy (TKE) will be explored in a separate section.