where W and w are respectively the r.m.s. circulation speeds of the large and turbulent eddies and R and r their respective radii. The buoyant production of turbulent energy by the MFC process is maximum at the crest of the large eddies and results in the warming of the large eddy volume. The turbulent eddies at the crest of the large eddies are identifiable by a microscale-capping-inversion (MCI) which rises upwards with the convective growth of the large eddy in the course of the day. This is seen as the rising inversion of the day time ABL in the echosonde records.
As the parcel of air corresponding to the large eddy rises in the stable environment of the MCI, Brunt Vaisala oscillation are generated (Mary Selvam et al., 1983a, 1984b). Thus the large eddy growth is associated with generation of a continuous spectrum of gravity (buoyancy) waves in the atmosphere. The slopes of temperature and wind spectra were theoretically shown to be equal to - 1.8 for the eddy scale ratio (Z), i.e., R/r = 10 (Mary Selvam et al., 1984c) and it is in agreement with the observed spectral slopes in the troposphere-stratosphere-ionosphere-magnetosphere (Weinstock, 1980, Dewan, 1979, Keskinen et al., 1980, Tsurutani et al., 1981, Carter and Balsley, 1982, Van Zandt, 1982).
This fractional volume
dilution of the large eddy occurs in the environment of the turbulent eddy.
The fractional volume of the large eddy which is in the environment of
the turbulent eddy where dilution occurs = r/R. Therefore, the total
fractional volume dilution k of the large eddy per second across unit cross
section can be expressed as
The value of k = 0.4 when R/r = 10 since dW = 0.25w* (Eqn.1).
In Equation (2), dW
is the increase in vertical velocity of the large eddy per second as a
result of w*. The height interval in which this incremental
change in the vertical velocity occurs is dZ which is equal to r
Using the above expressions Equation (2) can be written as follows
A normalised height Z with reference to the turbulence scale r can be defined as
Using the above expression Equation (4) can be written as follows:
The value of k is constant for a fixed value of R/r. As defined earlier k represents the fractional volume dilution rate of the large eddy by turbulent eddy fluctuations across unit cross-section on its envelope and is constant for a fixed value of the scale ratio Z.
It is well known from observations and from existing theory of eddy diffusion (Holton, 1979) that the vertical wind profile in the atmospheric ABL follows the logarithmic law which is identical to the expression shown in Equation 6. The constant k for the observed wind profile is called the Von Karman constant. The value of k as determined from observations is equal to 0.4 and has not been assigned any physical meaning in the literature.
The new theory relating to the eddy mixing to the ABL proposed in the present study enable to predict the observed logarithmic wind profile without involving any assumption as in the case of existing theories of eddy diffusion processes. Also it is shown that the Von Karman constant is associated with a specific physical process. It is the fractional volume dilution rate of the large eddy by the turbulent scale eddies for the scale ratio of 10.
Identifiable large eddies can grow in the atmospheric ABL only for scale ratios Z³10 since for smaller scale ratios the volume dilution rate by turbulent eddy mixing is more than 0.5. The convective scale eddy of radius Rc evolves from the turbulence scale eddy of radius r for scale ratio (Z), i.e., Rc/r = 10. This type of successive decadic scale range eddy mixing generates the convective, meso-, synoptic- and planetary scale eddies starting from the turbulence scale as the basic unit (Mary Selvam et al., 1984c).
Observational evidence for the tropospheric eddy chain link up with ionosphere is seen in satellite observations which indicate that increased currents at ionospheric levels are accompanied by a simultaneous increase in wind speed at lower levels. Measurements with Poker Flat radar and instruments at Alaska and with NOAA radar at Fairbanks support this contention. From the motions of chemically released ions and neutral clouds it is apparent that neutral winds in the high latitude ionosphere are driven principally by ion drag forces. Observations of infrasonic waves following sudden ionization enhancements indicates the existence of momentum transfer (Heppner, 1975). Carter and Balsley (1982) have reported correlation between short term fluctuations in the wind field near the mesopause (~ 83 - 90 km) and the intensity variations of the auroral electrojet (~110-115 km)
Weinstock (1981) has shown that the energy dissipation rate e of turbulence in the stable free atmosphere can be expressed by the following relation:
is the variance of the vertical velocity and NB the Brunt-Vaisala frequency. Observations indicate that the above relation for e holds good for the atmospheric PBL up to the stratosphere. The new theory of atmospheric eddy mixing can be used to derive the relation as follows:
The dominant eddies in the atmospheric PBL are in the Brunt-Vaisala eddies since they are generated and sustained by the microscale-fractional-condensation process in the turbulent eddies.
Let WB = root mean square value of the vertical velocity of the large eddy, i.e., for one complete cycle of the large eddy oscillation. Let NB = Brunt-Vaisala frequency associated with the large eddy.
Thus e, the dissipation rate per second of the vertical velocity variance associated with the Brunt-Vaisala eddy is given as follows:
The value of k = 0.4 for size ratio Z = 10. Thus the predicted relation for e at Equation 8 is identical to the observed relation for e at Equation (7) providing proof for the existence of a vertical eddy chain with convective scale as the basic unit.
ia* = w* s*
s*= net positive space charge density in the surface layer.
As the large eddy grows
there is upward transport of net positive space charge from surface layers
the concentration decreasing with height due to dilution of the large eddy
volume by turbulent mixing as explained earlier.
The fractional mass flux f of the surface air in the vertical can be derived as follows: Across unit cross-section of the large eddy surface at normalised height Z the ratio of the upward mass flux of air to the upward mass flux at surface level = W/w*. This excess upward mass flux occurs in the environment of the turbulent eddy and hence the fractional volume of the large eddy associated with this dilution = r/R = 1/Z. Hence the fractional mass flux f of the surface air at normalised height Z is given as
Earlier (Equation 6) it was shown that
the wind profile of the large eddy is logarithmic with height.
Under steady state conditions a fraction f of surface air will be found at normalised height Z.
The atmospheric nuclei
and thus space charge concentration originate from the surface layers.
Thus the net positive space charge concentration
The atmospheric electric field F at any level is given as
where sis the net positive space charge density at the level. F and s are expected to decrease with height according to the f distribution (Eqn.9). The vertical profile of f is shown in Fig.3 and it is similar to the observed F and nuclei profile (Imyanitov and Chubarina, 1967) in the atmospheric (Fig.4). The value of f has been computed assuming that the dominant turbulent eddy radius (r) is equal to 100m and lm respectively above and below the lifting condensation level (LCL).
The aerosol current at any level Z is given by
ia = s* f Z w* f Z
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