Stability Computation Method

The base calculation from the previous sections used the turbulence computation defaults. One of these defaults was to compute the atmospheric stability using the model predicted fluxes of heat (H) and momentum (M). The stability estimate, expressed as a Monin-Obukhov length, determines the magnitude of the turbulence associated with that stability. Most of the meteorological data files contain estimates of the fluxes, however if these fields are missing there is an option to use the vertical profile of wind and temperature to estimate the stability.

1. The default computation uses the momentum flux to the surface (M) to compute the friction velocity, the sensible heat flux (H) to compute the friction temperature, and the mixed layer depth to compute the convective velocity scale. Using these three values, the height normalized Monin-Obukhov (z/L) stability parameter is computed.
   • u_* = (∣M∣ / ρ)^0.5
   • t_* = -H (ρ C_p u_*)^-1
   • w_* = ∣ g u_* t_* Z_i T^-1∣^1/3
   • z/L = Z₂k g t_*(u_*²T₂)^-1


2. When no fluxes are provided by the meteorological model, z/L can be estimated from the wind and temperature profiles by first computing the bulk Richardson Number from the two lowest data levels and the z/L can be computed using several different functional forms that depend upon the stability.
   • R_b = g Δθ ΔZ {θ₁₂ [(Δu)²+(Δv)²]}^-1
   • z/L = f(R_b)

   The plume prediction should be recalculated using the wind and temperature profile by opening Configuration Setup / Concentration / Turbulence Method Menu #7, press Reset to set previous changes back to their defaults, and then select the Computed from U/T profile radio-button for the boundary layer stability section. This calculation will take considerably longer than the default calculation because there is greater puff splitting in this calculation. When the run has completed, execute the display batch file and open the resulting graphic which shows a much longer and slightly wider plume with correspondingly lower concentrations. Examination of the MESSAGE shows the tracer to have mixed much higher into regions of faster winds causing the puff splitting without the opportunity for puffs to merge.

   In this case using the wind and temperature profiles to compute stability caused a significantly different calculation result. Stability from profiles are instantaneous values valid at the model time step, while flux derived stabilities are based upon time-averaged heat and momentum fluxes. There is no single right answer that applies to all data files but it depends upon the frequency of the meteorological data and the time duration of the flux integral.

3. There is one other option in this section. Normally the vertical turbulence profile varies with height, approaching zero near the ground, increasing to a peak in the mid-boundary layer, then decreasing again to the top of the mixed layer. This is the default approach. However, the height varying profile can be replaced by a single boundary layer average value. This approach would only be used when trying to configure the model to match the Gaussian analytic solution, where the vertical diffusivity is constant with height. The effect of this change can be compared with the default (press Reset first) by selecting the Menu #7 radio-button Replaced by PBL average, save, run the model, display script, and then the resulting graphic shows a very similar result to the default calculation.

The results shown in this section suggest that there may be considerable sensitivity to the selection of how the stability is computed from the meteorological data. They key is understanding the sources of the data and their limitations. For instance, hourly fields generated by a mesoscale meteorological model may show more similarity between the flux and profile calculation that computations performed using regional or global model output that may only be available at 3- or 6-h intervals.