Boundary layer parameterization for Finnish regulatory dispersion models
Ari Karppinen, Sylvain M. Joffre and Pentti Vaajama
1. Description of the FMI-meteorological pre-processor
The parameterization schemes used in the dispersion models of the FMI are based on the energy flux method of van Ulden and Holtslag (1985), where the turbulent heat and momentum fluxes in the boundary layer are estimated from synoptic weather observations. The basic concepts of the parameterization are given below.
1.1 Basic scaling parameters
The first important scaling parameter is the friction velocity u* , which determines the shear production of turbulence kinetic energy at the surface. According to surface-layer similarity theory we have a relationship between the measured wind speed at height z and the friction velocity:
where zo is the roughness parameter and for the stability function we use
, for L<0 and
, for L>0
The turbulent heat flux Ho at the surface and the friction velocity determine together the second important parameter, a temperature scale
where is the air density and cp is the specific heat capacity of air.
The third important ABL scaling parameter is the Monin-Obukhov length scale L defined by
where T2 is the air temperature at the height of 2 m , and k is the von Karman constant ( k = 0.4.).
If the primary parameters to be evaluated are zo, u*, and h, all other parameters can be estimated as functions of these four
1.2 The surface heat flux
The basic parameter governing the stability and turbulence production in the boundary layer is the surface turbulent heat flux Ho . In a stationary and horizontally homogenous boundary layer the balance of the turbulent energy at the surface is given by (van Ulden & Holtslag, 1985) :
where is the latent heat flux, Q* is the net radiation flux and G is the conductive heat flux to the ground. The latent heat flux is estimated by applying the linearized saturation curve and the surface energy budget
where S is the slope of the saturation specific humidity curve, is the latent heat of vaporization for water and is an empirical temperature scale ( = 0.033 K, according to de Bruin & Holtslag, 1982).
The parameter is a function of the surface moisture (Priestley and Taylor, 1972) and it gets values in the range 0.5 -1 according to following table:
Table 1 Values of moisture parameter in the FMI- model .
Parameter S is a function of temperature : S = 51371.0 exp( 5423.0 (1/273.16 - 1/T2))/ T22 .
Using the equations given above the temperature scale (= - Ho / cpu* ) can be written as
1.3 The surface radiation budget
Equation (4) contains the net radiation flux Q* and G ,the conductive heat flux to ground, both quantities not routinely observed. The net radiation parameterization in the FMI model is basically that of van Ulden and Holtslag (1985) and unless otherwise stated all the following equations and values of parameters are based on that report.
The equation for the net radiation reads
where Qi* is the radiation flux in an isothermal atmosphere, and Q* is the deviation from this in the real atmosphere. This deviation is evaluated as a function of the difference between air temperature Tr at a reference height zr = 50 m and the surface radiation temperature To
The parameter is the Stefan-Boltzmann-constant (=5.67 10-8W m-2 K-4). The temperature difference Tr -To will be explicitly calculated later.
1.4 The incoming shortwave radiation
The net radiation for an isothermal atmosphere Qi* is evaluated from the net incoming shortwave radiation K* , the net longwave radiation Lnet =Lin-Lout and the incoming longwave radiation of the clouds Lc:
(9) Qi* = K* + Lnet + Lc
At high latitudes the net radiation at the surface correlates better with the sunshine duration than with the cloud cover . Thus, in the FMI model, K* is a function of the incoming net radiation flux at the surface RS and
(10) K* = (1-r) RS(KR)
where r is the surface albedo and Rs is a function of the observed hourly sunshine time and KR.
The clear sky radiation part (KR) is a function of the solar elevation angle
where Io is the solar constant (1372 W/m2).
The effect of clouds is incorporated in a regression equation based on the records of hourly solar radiation and sunshine amount from the Jokioinen observatory in Southern Finland. The regression equation reads
where RSS is the observed hourly sunshine time. The empirical constants refer to a ten year period (1965-74) at Jokioinen.
1.5 The outgoing and incoming longwave radiation
The outgoing longwave radiation Lup of the surface is a function of the surface radiation temperature To. In isothermal atmosphere this unkonown temperature can be replaced by temperature at reference height Tr. With this replacement the equation for Lnet reads
where is an empirical constant.
The contribution of the incoming longwave radiation from the clouds to the radiation budget is parameterized by a regression equation. The cloud black body radiation is defined through the equation
The cloud base temperature Tc is evaluated assuming an adiabatic lapse rate between the surface and cloud base:
(14b) Tc = T2 +dZc
where d = - 0.01 Co/m and Zc is the cloud base height derived from observations. The total amount of dominant clouds Cc is the sum of the amount of low and middle cloud. The contribution of high clouds to the surface radiation budget is considered negligible, especially when there are several cloud layers. It has been assumed above that in presence of low clouds the boundary layer is nearly neutrally stratified and, in contrast to the shortwave radiation, the longwave radiation part is not a function of the temperature difference Tr-To.
To account for the annual cycle in albedo, the regression method was applied separately for winter and summer. The following regression equations based upon the ten years period 1965-74 from the Jokioinen radiation records were obtained for Finnish conditions (Vaajama, 1989) :
For summer (defined here by the absence of snow cover)
(15a) Lc = 0.64 L'c
and for wintertime (ground covered by snow)
(15b) Lc = 0.56 L'c
The ground heat flux G still remains to be parameterized. This flux is evaluated as a function of the difference of the air temperature Tr at the reference height and the surface radiation temperature To
where AG is an empirical constant ( 5 Wm-2 K-1 ).
Again, there are no routine observations of the radiation temperature To. In this case the errors caused by using the screen temperature T2, would be significant. However, the value of (Q*-G) can be computed from the set of equations (4)-(6) simultaneously with equation (16) with the other boundary layer parameters. This is actually one of the main advantages of the van Ulden-Holtslag-method.
For unstable boundary layer (L < 0) we use the fact that Q* is strongly correlated with Q* :
(17) Q*= - AHQ*
where AH is an empirical heating coefficient estimated from 10 year radiation measurements at Jokioinen observatory
From equations (7),(8),(16),(17) now follows
For stable boundary layer (L<0) it follows from equations (7) and (8) that
and the temperature difference which is now strongly affected by wind speed is estimated from the potential temperature profile equation:
where z0h = 0.03 cm for the temperature roughness height for heat and zr = 50 m have been assumed.
This completes the parameterization for the difference (Q* - G) and we have now an estimate for the temperature scale as a function of the friction velocity u* and the Monin-Obukhov-length L. Together with equations (1) and (3) we can solve the 3 parameters L, u* and by iteration.
2 Comparison of the FMI-method to the Berkowicz/Prahm scheme
2.1 Net radiation estimations
FMI-method divides the net radiation to three parts: shortwave radiation from sun , blackbody radiation from clouds and ground and longwave radiation of (isothermal) atmosphere. Shortwave radiation is approximated by a regression equation which uses observed hourly sunshine time as the explaining variable in the regression model. The radiation from clouds is modelled by another regression equation , which uses the total cloudiness and cloud height as explaining parameters.
The Berkowicz/Prahm method uses two regression models (one for daytime and one for nighttime) which use the synoptic measurements of cloudiness as the most important explaining variables.
The net radiation estimates of these two schemes is illustrated in figures 1 and 2
Figure 1. Daytime net radiation estimates by FMI- and Berkowicz/Prahm preprocessor models.
Figure 2. Nighttime net radiation estimates by FMI and Berkowicz/Prahm preprocessor models.
2.2 Partitioning of the energy
The methods have one basic difference in the energy partitioning scheme. As the Berkowicz/Prahm-model (Berkowicz and Prahm,1982) evaluates the resistances ra and rc and humidity deficit in a quite complicated way the FMI-model utilizes the modified Priestly-Taylor model (van Ulden and Holtslag, 1985) which divides the evaporation in a part which is strongly correlated with (Q*-G) and a part which is not correlated with the equalibrium evaporation. That leaves us with only two empirical parameters to be evaluated (see eq. 5) in the FMI-model.
The surface and soil moisture are estimated differently in the models. The FMI-model uses synoptic weather codes and amount of rain to estimate the surface moisture and the Priestly-Taylor parameter a (see table 1). The Berkowicz/Prahm.-model uses accumulated net radiation for the measure of the soil moisture. These estimates are not directly comparable as the partition schemes and use of these moisture estimates differs in these two models.
2.3 Statistical comparison of the outputs of the two models
As the two methods differ quite a lot in the details of parametriazation it was extremely difficult to compare the models without actually doing some calculations with the models using identical synoptic data for the input of the two models.
At the end of year 1995 FMI got the code of the OML Meteorological preprocessor (Olesen and Brown, 1992) from the National Enviromental Research Institute (as a part of larger co-operation in the air-quality modelling between FMI and NERI). The basics of this OML -model are essentially identical to those used in the SMHI-model so this OML pre-processor was suitable for continuing the comparison studies and estimating the performances of the models against each other with some real synoptic data. The period chosen was 10 years from 1983 to 1993 , but the preliminary results presented here cover only the first year 1983. The meteorological data used is collected from southern Finland and the only preparation mede before using the synoptic data for the input of the preprocessors was an interpolation form 3 hour measurements to hourly values.
2.3.1 Comparison of the net radiation and turbulent heat flux
Figure3. Quantile-quantile plot of net radiation estimates of FMI and OML-models.
The figure 3 suggest strongly that in practice (in statistical sense) the estimates of net radiation by these two models give very similar results.
The plot of the cumulative distributions of turbulent heat flux estimates of these two models is presented in figure 4 , which shows that the energy partitioning schemes compared differ quite a lot, not only in the parametrisation details but also in the resulting turbulent heat flux estimates.
¨Figure 4. Quantile-quantile plot of estimates of sensible heat flux.
There is one obvious similarity in the estimates : the ratio of number of stable vs. number of unstable situations is roughle equal for these two models. In the stable side the OML-model gives consistently more negative enrgy-flux values than the FMI- model. In the unstable side the FMI-model gives larger values for turbulent heat flux than the OML-model. These results (althought based on only one year data so far) suggest that these two paramerization schemes divide the available energy differently between the latent and turbulent heat fluxes althought the basic input is identical and net radiation estimates are nearly identical.
2.3.2 Comparison of the Monin-Obukhov lengths
For the Monin -Obukhov lengths a straigthforward comparison is not very instructive. Instead the MO-lengths should be used for stablity classification and a comparison between the frequencies between these classes could be made. At this stage however such a classification is not yet made so a simple density distribution of the inverse MO-lengths produced by these two models is presented.
Figure 5. Density distribution of the inverse Monin-Obukhov lengths.
There is one important difference between these two figures: the OML-model limits the growth of 1/L in the stable side , which makes the distribution function quite symmetrical around the neutral (1/L small) area. Ther is no such a limit in the Finnish model , which produces quite a many super-stable situations. This is a very important difference for the air quality modeller as these very stable situations are often connected with air-pollution episodes. So it is of highest importance to be able to paramerisize these situations correctly.
3. Comparison with measurements
Within the COST-710 action a comparison between measurements and parametrization models is made and reported by Johansson, 1996
In FMI there are several measurement-campaigns (Walden et al., 1995, Härkönen et al., 1996) which produce data suitable for similar comparison studies. As an example of the behaviour of the FMI-model against the measurement data a comparison between the measured (mast measurements of wind at heights 2,5and 10 meters) and modelled friction velocity is given in figure 6.
Figure 6. Friction velocity: FMI parametrization model vs. local measurement.
The pre-processor of metorogical data used Finland is described and compared to a similar scheme which is in use in Sweden and Denmark.
The physics behind these pre-processors is nearly identical. The measurements used in these schemes are slightly different. Estimation procedures for net radiation differ quite a lot but the results (at least statistically) are nearly identical. The partitioning schemes vary quite a lot and also the results of parametrization differ significantly especially in the stable regime.
At the moment synoptical data only for one year has been used for the statistical analysis. After the analysis for the whole 10 year period is finished final conclusions about the performance of these two models (parametrizations) against each other can be stated.
The comparison of the paremetrization schemes against the measurements is on the way. The main difficulty in these comparison is the availability of good quality measurement data. We believe that the measurement campaigns at FMI will produce data which could be used for these comparison studies in the near future.
Berkowicz, R. and Prahm, L. P.,1982. Sensible heat flux estimated from routine meteorological data by
resistance method. J. Appl. Meteor. 21, 1845-1864 .
Karppinen, A., Kukkonen, J., Nordlund, G., Rantakrans, E., and Valkama, I., 1996. A dispersion modelling
system for urban air pollution. Finnish Meteorological Institute, Publications on Air Quality. Helsinki, 40 p. (in