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.

    Numerous studies indicate correlation between geomagnetic field variations and tropospheric weather activity ( Herman and Goldberg, 1978; Gribbins, 1981; Courtillot et al., 1982; Kalinin and Rozanova, 1984; Taylor, 1986 ). However, the exact physical mechanism for the observed coupling between meteorological and geomagnetic phenomena is not yet identified. In this paper a gravity wave feedback mechanism for the troposphere-ionosphere coupling is proposed.

    Vertical mass exchange in the troposphere-ionosphere-magnetosphere takes place by means of an eddy continuum. Any perturbation in the troposphere would be transmitted to ionosphere and vice versa. A global perturbation in the ionosphere, as the one caused by solar variability, is transmitted to troposphere influencing weather systems/geomagnetic/atmospheric electrification processes ( Mary Selvam et al., 1984; Poonam Sikka et al., 1984 ).
    The atmospheric boundary layer ( ABL ) contains large eddies or helical vortex  roll circulations as seen from well organized cloud rows, cloud streets and meso-scale cloud clusters ( MCC ) in global satellite cloud cover pictures ( Eymard, 1985 ) . It is however not clear how such long-lived circulations exist in the dissipative turbulent planetary boundary layer flows. In the following it is shown that the buoyant production of energy by micro scale fractional condensation ( MFC )  in turbulent eddies is responsible for the sustenance and growth of large eddies in the atmospheric boundary layer ( ABL ) (Mary Selvam and Murty, 1985; Mary Selvam, 1986; Mary Selvam and Murty, 1987; Mary Selvam, 1987). The circulation speed of the large eddy is related to that of the turbulent eddy according to the following relation ( Townsend, 1956) .
 
 

     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

From Eqs. (2) and (3)

The steady state fractional upward mass flux of surface air is dependent only on the dominant turbulent eddy radius.
 
 

    The eddy energy power spectrum is conventionally plotted as ln E versus lnn where E is the eddy energy and n its frequency.

The spectral slope S of the scale invariant eddy energy continuum is given as

    The kinetic energy KE per unit mass of an eddy of frequency n in the hierarchical eddy continuum is shown to be equal to Hnwhere H is the spin angular momentum of unit mass of the largest eddy in the hierarchy. The circulation speed W_p of the largest
eddy in the continuum is equal to the integrated mean of all the inherent turbulent eddy circulations. Let W_p be the mean circulation speed or the zero level about which all the smaller frequency fluctuations occur.

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.

    Natural phenomena posses energy/geometrical structure which is found to follow statistical normal distribution characteristics. Such normal distributions conventionally attributed to random chance, are in reality the deterministic laws of the unified eddy dynamical processes, and therefore represent the implicit order in natural phenomena as shown in the following. Let P represent the probability of occurrence in the medium of bi-directional energy flux with characteristics of a particular large eddy radius R .

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.
    The period doubling sequence for eddy growth gives

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.

    The statistical distribution characteristics of natural phenomena commonly follow normal distribution associated conventionally with random chance. The normal distribution is characterized by the moment coefficient of skewness equal to zero, signifying symmetry. The moment coefficient of kurtosis is equal to 3 and represents the intermittency of turbulence. In the following it is shown that the universal period doubling route to chaos growth phenomena in nature gives rise to the observed statistical normal distribution parameters as a natural consequence. The period doubling route to growth is initiated and sustained by the turbulent ( fine scale ) eddy acceleration w_*  which then propagates by the inherent property of inertia of the medium. Therefore, the statistical parameters mean, variance, skewness and kurtosis of the perturbation field in the medium is given by w_* , w_*² , w_*³ , and  w_*⁴  respectively. By analogy, the perturbation speed w_*  ( motion ) per second of the medium sustained by its inertia represents the mass, w_*²  the acceleration ( or force ), w_*³  the momentum ( or potential energy ) and w_*⁴  the spin angular momentum since an eddy motion has an inherent curvature to its trajectory. Because the eddy motion is inherently symmetric with bi-directional energy flow, the skewness factor w_*³  is equal to zero for one complete eddy circulation thereby satisfying the law of conservation of momentum. The moment coefficient of kurtosis which represents the intermittency of turbulence is shown in the following to be equal to 3 as a natural consequence of the growth phenomenon by period doubling route to chaos.

From Eq.3

(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 .

    Therefore moment 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 .

    The universal period doubling route to chaos has been studied extensively by mathematicians. The basic example with the potential to display the main features of the erratic behaviour is the Julia model ( Delbourgo, 1986 ) given below.

The above non-linear model represents the population values of the parameter X_n  at different time periods of X  for small  X  . Feigenbaum's ( 1980 ) research showed that the two universal constants 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.
    From Eq.(1) the function a may be defined as

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.

    In the following it is shown that the atmospheric electric field and geomagnetic field variations are manifestations of the vertical mass exchange process between the lower troposphere and ionosphere. The vertical mass exchange on the sunlit hemisphere gives rise to upward transport of surface air. The nuclei in the surface air layers contain a net positive space charge and thus there is an aerosol current i_a* = w_* s_* where sis the 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 net positive space charge concentration sat any level can be expressed in terms of the surface concentration s_*   as follows

The aerosol current at any level z is given as

Thus the aerosol current i_a  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.

    The scale invariant self-similar atmospheric eddy continuum generated by the universal period doubling route to chaos growth process exists as a unified whole , spanning the total planetary atmospheric boundary layer from the planetary surface to the outermost limits of the magnetosphere and is in constant two-way communication for energy flow from solar and other planetary/stellar atmospheres. The fine structure signal of the unified atmospheric eddy continuum is manifested as the luminescence phenomena of the auroral oval, where field-aligned currents in discrete auroral arcs indicated bi-directional eddy enegy/charge flow. Field aligned currents in discrete auroral arcs are visible manifestations of atmospheric vertical mixing coupled to solar wind pumping of negative space charges downwards accompanied by simultaneous upward transport of terrestrial positive ions. Therefore, the aerosol currents originating from the planetary surface extend into the magnetosphere in the discrete auroral arcs and couple to the solar wind energy pump accounting for the observed close association between solar wind dynamic parameters , discrete auroras and geomagnetic micropulsations ( Feldstein and Galperin, 1985 ) . Therefore ground based monitoring of geomagnetic micropulsations will indicate the extra-terrestrial energy flow rate into the planetary atmosphere. The above theory proposed for magnetosphere - ionosphere - troposphere energy coupling is consistent with satellite observations of the close association between magnetic field aligned currents, parallel electric fields, inverted V structures and plasma pressure inhomogeneities in the magnetosphere and related geomagnetic micropulsations ( Lundin and Evans, 1985; Chmyrev et al., 1985 ; Stasiewicz, 1985; Bosqued et al., 1986).

    The universal period doubling route to chaos is the growth phenomena of a dynamic unified scale invariant atmospheric eddy continuum from turbulence scale at the planetary surface with inherent steady state ordered energy flow between the component eddies and the continuum. The bi-directional energy flow in the planetary eddy continuum is manifested as various tropospheric, ionospheric and magnetospheric phenomena in a continuous chain of individual perturbation events in a space-time continuum which is super symmetric being the fusion of the individual component eddy symmetries. The bi-directional energy flow in the unified planetary atmospheric eddy continuum is the coupling mechanism of upper atmosphere to the lower atmosphere and occurs in the form of a vertical global aerosol current and the accompanying geomagnetic H component. Since the time scale for the downward transport of extra-terrestrial energy input is in the order of several days, global geomagnetic field variations can serve as precursor signals to weather/climate change. The solar modulation of magnetospheric - ionospheric  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.

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