Observations show that geomagnetic field lines follow closely the atmospheric circulation patterns (Gribbins, New Scientist, 5 Feb., 350-353, 1981). and that geomagnetic field variations are precursors to climate change ( Courtillot et al., Nature 297, 386-387, 1982). Therefore close monitoring of the local and global geomagnetic field variations by satellite systems will assist prediction of hurricane tracks, intensification and also long-term weather trends. The exact mechanism for the observed close relationship between global geomagnetic field and the tropospheric weather patterns is not clear. In this paper a universal theory of atmospheric eddy dynamics is presented which shows that the global geomagnetic field, atmospheric electric field and weather systems are manifestations of a semi permanent scale invariant hierarchical atmospheric eddy continuum. The scale invariant energy structure for the atmospheric eddy continuum has been documented and discussed ( Lovejoy and Schertzer, Bull. Amer. Meteorol. Soc., 67, 21-32, 1986). In summary, quantitative equations are derived to show that the full continuum of atmospheric eddies exist as a unified whole and originate from buoyant energy supply from frictional turbulence at the planetary surface ( Mary Selvam et al., Proc. VII Int'l. Conf. Atmos. Elec., June 3-8, 1984, Albany, N.Y., 154-159). Large eddy growth occurs from turbulence scale by the universal period doubling route to chaos ( Fairbairn, Phys. Bull. 37, 300-302, 1986). The turbulent eddies are carried upwards on the large eddy envelopes and vertical mixing occurs by the turbulent eddy fluctuations resulting in downward transport of negative space charges from from higher levels and simultaneous upward transport of positive space charges from surface levels. The eddy circulations therefore generate a large scale vertical aerosol current which is of the correct sign and magnitude to generate the horizontal component of the geomagnetic field. Therefore, atmospheric circulation patterns leave signature on the geomagnetic field lines whose global variations can be easily monitored by satellite borne sensors and thus assist in weather and climate prediction.
where W and w* are respectively the root mean square ( r.m.s ) circulation speeds of the large and turbulent eddies and R and r their respective radii. The production of buoyant energy ( turbulent scale ) is maximum at the crest of the large eddy system. The turbulent eddies at the crest of the large eddies are identifiable by a micro scale 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 PBL ( planetary boundary layer ) in echosonde records. A conceptual model of the large and turbulent eddies in the ABL ( atmospheric boundary layer ) is shown in Figure 1.
Figure 1 : Conceptual model of large and turbulent eddies in the ABL
The turbulent eddy fluctuations mix overlying environmental air into the growing large eddy volume and the fractional volume dilution rate k of the total large eddy volume across unit cross section on its envelope is equal to
where w* is the unidirectional turbulent eddy acceleration and dW the corresponding acceleration of the large eddy circulation ( Mary Selvam and Murty, 1985) during the large eddy incremental length step growth dR equal to r . The variable k is greater than 0.5 for z less than 10. Therefore organized large eddy growth can occur for scale ratio z greater than or equal to 10 . Therefore a hierarchical scale invariant selfsimilar eddy continuum with semi permanent dominant eddies at successive decadic scale range intervals is generated by the self-organised period doubling route to chaos growth process. The large eddy circulation speed is obtained by integrating Eq.2 for large eddy growth from the turbulence scale energy pump at the planetary surface and is given as
k = 0.4 for z = 10. This is the well known logarithmic wind profile relation in the ABL ( Holton, 1979 ) and k is designated as the Von Karman's constant and its value as determined from observations is equal to 0.4 ( Hogstrom, 1985 ). The region of chaos is the dynamic growth region of large eddy by turbulence scale energy pumping and the nested vortex hierarchical continuum energy structure is manifested as the strange attractor design. The particles in the region of chaos follow laws analogous to Kepler's third law of planetary motion since
The rising large eddy gets progressively diluted by vertical mixing due to the turbulent eddy fluctuations and a fraction f of surface air which reaches the normalized height z is given by ( Mary Selvam and Murty, 1985 ).
From Eqs. (2) and (3)
W = w* f z
The steady state fractional upward mass flux of surface air is dependent only on the dominant turbulent eddy radius.
The spectral slope S of the scale invariant eddy energy continuum is given as
S = ln E / ln z
= -2 for large z
Therefore the universal period doubling route to chaos eddy growth mechanism gives rise to an eddy energy continuum spectral slope equal to -2 . The universal scale invariant -2 power spectrum for eddy energy has been observed in the atmospheric boundary layer turbulence ( Van Zandt, 1982 ).
Therefore 2 R = Wp / n
H is equal to the product of the momentum of unit mass of planetary scale eddy and its radius and therefore represents the spin angular momentum of unit mass of planetary scale eddy about the eddy center. Therefore the kinetic energy of unit mass of any component eddy of frequency n of the scale invariant continuum is equal to Hn which is analogous to quantum mechanical laws for the electron energy levels in the subatomic space.
W/w* is distributed normally since W is the integrated mean of w* over the large eddy volume. Similarly, since large eddy energy content is equal to the sum of all its individual component eddy energies and therefore the energy E of any eddy of radius R in the eddy continuum expressed as a fraction of the energy content of the largest eddy in the hierarchy will represent the cumulative normal probability distribution. The eddy continuum energy spectrum is therefore the same as the cumulative normal probability distribution plotted on a log-log scale. The eddy energy spectral slope is derived from the cumulative normal distribution curve as follows.
r = (dR/R) = 1/2
The standard deviation s with a cumulative probability of occurrence equal to dR/(R+dR) = 35% . The cumulative normal probability distribution also gives 32% probability at one standard deviation in close agreement with the statistical parameters generated by the period doubling sequence. Further, the slope of the log-log plot of the cumulative normal probability curve at one standard deviation is equal to -1.8 in agreement with the computed (see Eq.6 ) slope of -2 for the eddy energy spectra.
(dW)4 / w*4 represents the statistical moment coefficient of kurtosis . Organized eddy growth occurs for scale ratio equal to 10 and identifies the large eddy on whose envelope period doubling growth process occurs. Therefore for a dominant eddy
(dz/z) = 2 for one length
step growth by period doubling process since z = dz + dz
coefficient of kurtosis is given as
In other words, period doubling phenomena result in a threefold increase in the spin angular momentum of the large eddy for each period doubling sequence. This result is consistent since period doubling at constant pump frequency involves eddy length step growth dR on either side of the turbulent eddy length dR .
Xn+1 = F( Xn ) = L Xn (1 - Xn )
The above non-linear
model represents the population values of the parameter Xn
at different time periods of X for small
X . Feigenbaum's ( 1980 ) research showed that the two universal
a = -2.5029 and d = 4.6692
are independent of the details of the non-linear equation for the period
doubling sequences where a and d
denote the successive spacing ratios of X and L
respectively for adjoining period doublings.
The universal constants a and d assume different numerical values for period tripling, quadrupling, etc., and the appropriate values computed by Delbourgo ( 1986 ) show that the relation 3d = 2a2 has a much wider validity.
The physical concept of large eddy growth by the period doubling process enables to derive the universal constants a and d and their mutual relationship as function inherent to the scale invariant eddy energy structure as follows.
a is therefore equal to 1/k from Eq.(3) where k is the Von Karman 's constant representing the non dimensional steady state fractional volume dilution rate of large eddy by turbulent eddy fluctuations across unit cross-section on the large eddy envelope. Therefore a represents the non dimensional total fractional mass dispersion rate and a2 represents the fractional energy flux into the environment.
Let d represent the ratio of the spin angular momenta for the total mass of the large and turbulent eddies.
Therefore 2a2 = 3d from Eqs.(8) and (9). The variable 2a2 represents the total eddy energy flux into the environment in the bi-directional eddy energy flow and 3d represents the threefold increase in spin angular momentum generation in the large eddy during the period doubling sequence growth. In an earlier section it was shown that the period doubling sequence is associated with a three-fold increase in the spin angular momentum of the resulting large eddy and accounts for the moment coefficient of kurtosis of the normal distribution. Therefore the above equation relating the universal constants is a statement of the law of conservation of energy, that is, the period doubling growth process generates a three fold increase in the spin angular momentum of the resulting large eddy and propagates outward as the total large eddy energy flow in the medium. The property of inertia enables propagation of turbulence scale perturbation in the medium by release of the latent energy potential of the medium. An illustrative example is the buoyant energy generation by water vapour condensation in the updraft regions in the atmospheric boundary layer.
s =s* f
F = 4 ps
Figure 2 : The computed vertical profile of F and s
Figure 3 : The observed vertical profile of condensation nuclei sandelectric field F
The aerosol current at any level z is given as
ia = (s* f z) x ( w* f z ) = ia* f 2 z2
Thus the aerosol current ia produced by the vertical mass exchange generates the observed atmospheric electric field. The conventional air earth conduction current ( Chalmers, 1967 ) cannot discharge the atmospheric electric field thus produced since the dynamic charge transport by the vertical mass exchange process is faster than the ion mobilities by more than one order of magnitude. The convective scale aerosol current can be computed from Eq.(12) and shown to be 1000 times larger and in opposite direction to the conventional air-earth conduction current. The vertical aerosol currents are of the right order of magnitude and direction as those of the vertical current postulated to exist in the atmosphere by Bauer (1920) and Schmidt (1924) in their hypothesis for explaining the variations in the H component of the geomagnetic field. The aerosol currents occur over convective scale, that is, one square kilometer and thus were not detected by conventional spot observations. The universal period doubling route to chaos growth process generates scale invariant atmospheric eddy continuum circulations extending from the planetary surface to the magnetospheric levels and above manifested in the geomagnetic field observations.
Observational evidence for the tropospheric eddy continuum extension into the 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 with NOAA radar at Fairbanks support this contention.
The solar wind energy coupling in the terrestrial magnetosphere is indicated by the geomagnetic micropulsations and therefore also signal the continuous solar wind energy supply modulation of magnetosphere - ionosphere processes is well established and are therefore reflected in the tropospheric weather phenomena at the lower levels of the atmospheric eddy continuum which is a two-way energy flow channel between the lower and upper atmospheres. Therefore extra-terrestrial trigger of tropospheric weather changes can be forecast from the precursor signal from geomagnetic field variations.
Bosqued, J., C. Maurel, J. A. Sauvaud, R. A. Kovrazhkin and Yu. I. Galperin, 1986 ; Observations of auroral electron inverted V structures by the Aureol - 3 Satellite. Planet. Space Sci., 34, 255 - 269.
Chalmers, J. A., 1967 : Atmospheric Electricity, 2nd Ed., Oxford, Pergamon Press, pp. 515.
Chayrev, V. M., V. N. Oraevsky, S. V. Bilicheko, N. V. Isaev, G. A. Stanev, D. K. Teodosiev and S. I. Shkolnikova, 1985 : The fine structure of intensive small scale electric and magnetic fields in the high latitude ionosphere as observed by intercosmos-Bulgaria 1300 Satellite. Planet. Space Sci., 33 , 1383 - 1388 .
Courtillot, V., J. L. Le Mouel, J. Ducruix and A. Cazenave, 1982 : Geomagnetic secular variation as a precursor of climate change. Nature , 297 , 386 - 287 .
Delbourgo, R., 1986 : Universal facets of chaotic processes. ASPAP NEWS , 1 , 7 - 11.
Eymard, L., 1985 : Convective organization in a tropical boundary layer. An interpretation of doppler radar observations using Asia's model. J. Atmos. Sci., 42 ,2844 -2855 .
Feigenbaum, M. J., 1980 : Los Alamos Science , 1 , 4 -27 .
Feldstein, Y. I., and Yu. I. Galperin, 1985 : The auroral luminosity structure in the high latitude upper atmosphere : Its dynamics and relationship to the large-scale structure of the Earth's magnetosphere. Rev. Geophys ., 23 , 217 - 276 .
Gribbins, J., 1981 : Geomagnetism and climate. New Scientist , 5 Feb ., 350-354 .
Herman, J. R. and R. S. Goldberg , 1978 : Sun , Weather and Climate . NASA SP, 426, pp. 360 .
Hogstrom, U., 1985 : Von Karman's constant in atmospheric boundary layer now revevaluated. J. Atmos. Sci., 42 , 263 - 270 .
Holton, J. R., 1979 : An introduction to dynamic meteorology . Academic Press, New York, pp. 39.
Imyanitov, I. M., and E. V. Chubarina, 1967 : Electricity of the free atmosphere . Israel Program for Scientific Translations, Jerusalem, pp. 212 .
Kalinin, Yu. D., and T. S. Rozanova, 1984 : Geomagnetic moment, irregular variations in the length of days and polar variations of latitude. Geomagnetism and Aeronomy , 24 , 90 - 92 .
Lundin, R. and D. S. Evans, 1985 : Boundary layer plasmas as a source for high latitude early afternoon auroral arcs. Plant. Space Sci., 32 , 1389 - 1406 .
Mary Selvam, A., A .S. R. Murty, G. K. Manohar, S. S. Kandalgaonkar and Bh. V. Ramana Murty, 1984 : A new mechanism for the maintenance of fair weather electric field and cloud electrification. Proc. VII International Conference on atmospheric electricity , June 3 - 8, Albany, New York, 154 - 159 . http://xxx.lanl.gov/abs/physics/9910006
Mary Selvam, A. and A .S. R. Murty, 1985 : Numerical simulation of warm rain process. Proc. IV WMO Sci. Weather Modification , Honolulu, Hawaii, USA, 12 -14 Aug., 503 - 506. http://xxx.lanl.gov/abs/physics/9911021
Mary Selvam, A.,1986 : A gravity wave feedback mechanism for the evolution of meso-scale cloud clusters ( MCC ) . Proc. Int'l. Conf. Monsoon and Meso-scale Meteorol. with MSROC, 4 - 7 Nov., Taipei, Taiwan, 84 - 89.
Mary Selvam, A. and A .S. R. Murty, 1987 : Simulation of urban effects on cloud physical parameters. Proc. International Conference on Energy Transformations and Interactions with Small and Meso-scale Atmospheric Processes , 2 - 6 March, Switzerland. http://xxx.lanl.gov/html/physics/0006031
Poonam Sikka, A. Mary
Selvam, A. S. Ramachandra Murty and Bh. V. Ramana Murty, 1984 : Possible
solar influence on atmospheric electric field. Proc. VII International
Conference on atmospheric electricity , June 3 - 8, Albany, New York,
148 - 153 . http://xxx.lanl.gov/abs/chao-dyn/9806014