Vertical and Horizontal Turbulence

The last calculation in the previous section used all the turbulence computation defaults. 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 in this section.

1. The default calculation uses the Kanthar-Clayson equations to compute the turbulence values from stability parameters estimated from the meteorological data. Within the boundary layer these 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 turbulence is a function of the friction velocity, height, and boundary layer depth. The horizontal and vertical components are explicitly predicted when using this method.


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 then run the previously saved display batch file and open the resulting graphic which shows a much narrower plume than the default calculation. The primary reason for this is that the turbulence is partitioned equally between the vertical and horizontal, while in the default calculation more turbulence goes into the horizontal component [3.0/(4.0+4.5)].



3. Another alternative is to use the meteorological model's Turbulent Kinetic Energy (TKE) field if that variable is available. In this case the turbulence components are derived from the definition of the TKE
   • E = 0.5 (u’² + v’² + w’²)
   and the relationship between the components is either defined by the Kanthar-Clayson equations or the turbulence aniosotropy factor: w’² / (u’² + v’²).

   To configure this calculation, open Menu #7 and select the Met model TKE field radio-button for vertical turbulence. Save all menus to exit, run the model, and then run the previously saved display batch file and open the resulting graphic which shows a tracer cloud with dimensions more similar to the default Kanthar-Clayson approach than the Beljaars-Holtslag.



4. The TKE field only provides information about the total turbulence and the turbulence partition between the vertical and horizontal components must be defined (the anisotropy ratio). The default is to assume a value of 0.18. To test the sensitivity of the calculation to this parameter, open Menu #7 and change the anisotropy factors from 0.18 to 0.05, an extreme change to illustrate its effect upon the calculation. Save all menus to exit, run the model, and then run the previously saved display batch file and open the resulting graphic which shows a slightly more circular plume but still very similar to the original calculation. The reason for this is that horizontal dispersion is not just a result of horizontal turbulence, but also of particles mixing to different levels with winds at different speeds and direction. An examination of the vertical mass profile shown in the MESSAGE file reveals that the 0.18 factor results in a plume that reaches 4 km rather than 2 km when the factor was 0.05.


5. Another way to approach the anisotropy 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 then run the previously saved display batch file and open the resulting graphic which shows a narrower plume using the Kanthar-Clayson, 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.