where W and w* are respectively the r.m.s (root mean square) circulation speeds of the large and turbulent eddies and R and r are their respective radii.
The total fractional volume dilution rate of the large eddy by vertical mixing across unit cross-section is derived from Eq.(1) (ref.1) and is given as follows:
where w*is the increase in vertical velocity per second of the turbulent eddy due to MFC process and dW is the corresponding increase in vertical velocity of large eddy.
The fractional volume dilution rate k is equal to 0.4 for a scale ratio ( z ) i.e., R/r =10 . Identifiable large eddies can exist in the atmosphere only for scale ratios more than 10 since, for smaller scale ratios the fractional volume dilution rate k becomes more than half. Thus atmospheric eddies of various scales, i.e., convective, meso-, synoptic and planetary scale eddies are generated by successive decadic scale range eddy mixing process starting from the basic turbulence scale (ref.2).
From Eq.(2) the following logarithmic wind profile relationship for the ABL is obtained (ref. 1).
The steady state fractional upward mass flux f of surface air at any height z can be derived using Eq.(3) and is given by the following expression (ref. 1).
where f represents the steady state fractional volume of surface air at any level z . Since atmospheric aerosols originate from surface, the vertical profile of mass and number concentration of aerosols follow the f distribution.
The model predicted aerosol vertical distribution are computed using Eq.(4) and are shown in Fig. 1. The model predicted profile closely resemble the observed profiles reported by other investigators (ref. 3). The peaks in the aerosol concentration at 1 km (lifting condensation level) and at about 10-15 km (stratosphere) identify the MCI at the crests of the convective and meso-scale eddies respectively. Earlier it was shown that the scale ratios for the convective and meso-scale eddies are respectively 10 and 100 with respect to the turbulence scale. Thus for the turbulent eddy of radius 100m, the MCI's for the convective and meso-scale eddies occur at 1 km and 10 km respectively.
The vertical mass exchange mechanism
f distribution for the steady state vertical
transport of aerosols at higher levels. Thus aerosol injection into the
stratosphere by volcanic eruptions gives rise to the enhanced peaks in
the regions of MCI in the stratosphere and other higher levels determined
by the radius of the dominant turbulent eddy at that level.
The time T taken for the steady state aerosol concentration f to be established at the normalised height z is equal to the time taken for the large eddy to grow to the height z and is computed using the following relation.
The vertical dispersion rate of of aerosols/pollutants from known sources (e.g., volcanic eruptions, industrial emissions ) can be computed using the relation for f and T (Eqs. 4 and 5).
W = w*f z
The aerosols are held in suspension by the eddy vertical velocity perturbations. Thus the suspended aerosol mass concentration m at any level z will be directly related to the vertical velocity perturbation W at z , i.e., W=mg where g is the acceleration due to gravity. Eq.6 can be expressed as follows:
m = m*f z
The number concentration of aerosol decreases with height according to the f distribution which can be expressed as follows:
N = N* f
r = r*z1/3
The mean aeosol size increases with height according to the cube root of z . As the large edy grows in the vertical, the aerosol size spectrum extends towards larger sizes while the total number concentration decreases with height according to the f distribution. The variable f can be expressed in terms of the incremental growth dW of the large eddy across unit crosssection on its surface as follows. Let dN denote the number concentration of aerosols in the ascending volume dW of the large eddy.
dN = N* f dW
i.e., f = (1/ N* )( dN/ dW)
From the logarithmic wind profile relationship (Eq.3) it follows that
In steady state atmospheric conditions the aerosol number concentration at any level is determind by the upward and downward transports each of which follow the f distribution. Hence the total number concentration of aerosols follow the f 2 distribution. With the aeosol source at the surface the aerosol size spectrum in the steady state atmospheric conditions can be expressed as follows:
A graph of ln(f2/ln3) ) versus lnr is shown in Fig.2. The computed aerosol size spectrum is identical to the Junge aerosol size spectrum. The aerosol number concentration initially increases with size for small sizes and for further size increase, the aerosol number concentration decreases steeply. The slope of the spectrum ranges from -2 to -4.8, in the decreasing number concentration size range.
Figure 2: computed aerosol size spectrum
The particles which scatter visible light occur normally in the size interval where the spectral slope is -3. This gives rise to the normal blue colour of the sky. However in the regions of high humidity or in the presence of volcanic dust clouds, the minimum particulate size r*(Eq.10) increases and thus the spectral slope become less than 3 and sometimes shifts to the peak region of the spectrum for the visible light scattering particles size range. This results in anomalous turbidities with whitish sky colour.
2. A. Mary Selvam, A. S. R.Murty and Bh. V. Ramana Murty, Role of frictional turbulence in the evolution of cloud systems, Preprint volume, 9th International Cloud Physics Conference, Tallinn, USSR, (1984).
3. E.C.Junge, Air chemistry and radioactivity,
Academic Press, London, 1963, pp.382.