A cell dynamical system model is developed for thundercloud electrification by consideration of microscopic domain eddy dynamical processes in the atmospheric boundary layer (ABL). This non-deterministic model based on cellular automata computational technique enables formulation of simple scale invariant governing equations for cloud electrical parameters in terms of non-dimensional steady state mass and momentum fluxes in the ABL.

The exact mechanism for the electrification of thunderclouds and the spectacular lightning phenomenon is still in dispute (1). At present there exist two models with distinctly different conceptual basis for cloud electrification. The first model, originally proposed by Elster and Geitel in 1985 (2) postulates that relative fall speeds of precipitation particles give rise to charge separation with charges of one sign residing exclusively on larger precipitation particles. Heavier particles are postulated to carry negative charges thereby accounting for the negatively charged cloud-base. This model cannot account for the origin of the cloud charges conclusively and further, ignores totally the convection currents (turbulent eddies) inside the cloud.

The second model, originally proposed by Grenet in 1947 and later in a refined form by Vonnegut in 1953(2) postulates that convection currents are solely responsible for charge separation and ignores completely the role of precipitation. According to this model, the naturally occurring vertical space charge distribution with negative excess at higher layers is redistributed by convection currents, i.e., the negative space charges from above cloud layers are brought down to cloud base in downdrafts and the positive space charges from below cloud base are transferred to cloud top regions by updrafts thereby giving rise to a positive dipole structure in the cloud.

The above two models for cloud electrification are both unrealistic since the elementary precipitation hypothesis invokes no convection while the convection hypothesis invokes no precipitation, though it is an observational fact that convection and precipitation are inseparable in clouds large enough to produce lightning. Further, both the models can account for only positive dipole cloud electrical structure while observations indicate a tripolar structure for cloud electrification, namely, in addition to the main positive dipole there is a small pocket of positive charges at cloud-base below the main negative charge center. Cloud electrification is intimately connected with the dynamics of cloud growth processes and therefore model formulations require a complete understanding of the microscopic domain cloud dynamical processes. In this paper a cell dynamical system model (3) for cloud growth and electrification is developed by consideration of microscopic domain eddy dynamical processes. The model enables formulation of scale invariant governing equations for the cloud charging processes. To begin with, a brief summary of latest developments in the modelling of dynamical systems, in particular, the concept of deterministic chaos is presented in the context of modelling the cloud electrification processes.

Mathematical models of dynamical systems, i.e. systems which evolve with time are formulated by tradition using Newtonian continuum dynamics where it is assumed that all change is continuous and the evolution equations of dynamical systems are given by a system of partial differential equations representing continuous rates of change. The partial differential equations in general do not have analytical solutions and therefore numerical solutions are obtained using digital computers having finite precision. Such digital computer realizations of continuum mathematical models for dynamical systems are inherently unrealistic because of the following reasons. (1) The nonlinear partial differential equations are sensitively dependent on initial conditions and give chaotic solutions characteristic of deterministic chaos, (2) Computer capacity related truncations in governing equations result in errors of approximations, (3) computer precision related round-off errors magnify exponentially with time the above mentioned uncertainties and give unrealistic solutions.

The sensitive dependence on initial conditions whereby deterministic mathematical models of dynamical systems give chaotic (or random) solutions is now named 'deterministic chaos' and was first identified in a mathematical model of atmospheric flow by Lorenz in 1963 (4). Later mathematical studies by other scientists revealed the existence of deterministic chaos in disparate dynamical systems (5-8). The computed trajectory of the dynamical system in the phase space comprising of the position and momenta coordinates traces out the selfsimilar fractal geometrical shape of the 'strange attractor' so named because of its strange convoluted shape being the final destination (attractor) of the trajectories. Any two initially close points in the strange attractor rapidly diverge with time and follow totally different paths though still within the strange attractor domain. Therefore the future trajectories of initially close points are unpredictable or random. The exact physical reason for the sensitive dependence on initial conditions of deterministic nonlinear partial differential equations used for modelling dynamical systems as well as the selfsimilar fractal geometry of the strange attractor design characterizing the evolution trajectory in the phase space of the dynamical system is not yet identified (9). Selfsimilarity implies scale invariance and is a manifestation of dilation symmetry whereby the shape of an object is preserved during stretching. A selfsimilar object possesses the same internal structure at all scales. Such selfsimilar objects are non-Euclidean in shape and therefore possess fractional or fractal dimension (10-12). The fractal dimension D is given by the relation D = dlnM / dlnR where M is the mass contained within a distance R from a fixed pointed in the object. The fractal dimension therefore gives in logarithmic scale the mass distribution per unit length along any direction in the extended object. When the fractal dimension is a constant for all length scales R, it indicates uniform stretching on a logarithmic scale or an inverse power-law form with constant exponent D for the mass distribution with respect to distance R from a fixed point in the object. In general objects in nature possess a multifractal structure, i.e., the fractal dimension is different for different length scales R.

Spatially extended dynamical systems exhibit long range spatial correlations exhibited as the fractal geometry of the spatial pattern and long range temporal correlations manifested as l / f noise (or flicker noise). Such long range spatial and temporal correlations which are ubiquitous to all real world dynamical systems has been identified as a signature of self-organised criticality (13) or deterministic chaos. In addition, the following characteristics generic to all natural dynamical systems may also be included as further identification of deterministic chaos. (1) the cumulative normal probability density distribution (or Gaussian distribution). (2) The logarithmic spiral geometrical form, e.g. hurricane cloud spiral pattern. (3) The icosahedral symmetry or the pentagonal dodecahedron for manifested stable shapes in nature (14). Such shapes are associated with the golden mean section, namely, (1 + Ö5)/2 and is also incorporated in the quasiperiodic Penrose tiling pattern identified as the quasicrystalline structure in condensed matter physics (15). In recent years the quasicrystalline structure is found to be more ubiquitous to the molecular level structure of matter than the periodic crystalline structure.

The computed strange attractor of dynamical systems is a mathematical artifact (16) as explained earlier and bears no relationship to the actual evolution trajectory. Further, even for a realistic mathematical model, computer precision related round-off errors introduces uncertainities in the space differentials, namely, dx, dy and dz resulting in artificial curvature to the trajectory which eventually end up as limit cycles or periodicities for sufficiently long integration time periods. Computer precision therefore plays the role of yardstick in numerical model realizations and generates selfsimilar structure for the continuum phase space trajectory, namely the strange attractor.

Recent studies show that numerical model results scale with computer precision and periodicities in numerical model results are again a function of computer precision (17-18). Computer model realizations which require long integration time periods are therefore subject to computer precision related uncertainities resulting in loss of predictability of the future state of the system. The sensitive dependence on initial conditions of the nonlinear partial differential equations may be a direct consequence of the computer precision related round off errors as shown later in Section 5.

However, such sensitive dependence on initial conditions is actually exhibited by disparate real world dynamical systems and maybe associated with information transport from the microscales to the macroscales indicated by the long term spatial and temporal correlations intrinsic to such systems. Therefore microscopic scale differences in initial conditions may contribute to appreciably different large scale space-time structure. It is important to understand the exact microscopic scale mechanisms which contribnute to the macroscale space-time evolution of the robust selfsimialr strange attractor design. It should be possible to identify a simple conceptual model which is scale invariant for the dynamical evolution of the system, i.e., a microscopic scale unit cell model which is directly applicable to the macroscale multicellular model. Such a model should enable formulation of the dynamical processes of evolution in simple mathematical formulations with analytical solutions or where the numerical solution does not require long term integration using digital computers.

In the following it is shown that computer precision is directly related to the scale invariant structure of the strange attractor and also to curvature of trajectories resulting in peridiocities in numerical output. The sensitive dependence on initial conditions of nonlinear partial differential equations which is a signature of deterministic chaos is also shown to be a direct consequence of computer precision related round-off errors.

Let w_* be the number of units of round-off error of magnitude dR each in the numerical computation (incorporating the error) comprising of dW units of magnitude R per unit. Therefore the nondimensional steady state fractional error k in each step dW of numerical computation is given by

k = w_*dR / dW.R                                                                                                     (1)

Assuming constant values for w_* and k, W is obtained by integrating Eq.(1).

Z is equal to the scale ratio R / dR, i.e, the numerical value of R, the computed length unit measured in units of round-off error dR. Therefore the computed numerical result W in units of R scales with computer precision or the round-off error length unit dR. Also, the error grows exponentially with number of unit lengths W of the numerical computer output and is responsible for the exponential divergence of two initially close points. The phase space trajectory in polar coordinates of the computed output W follows a logarithmic spiral as a direct result of the round-off errors. Computational error occurs in all three spatial dimensions (cartesian coordinates) and may equivalently be considered to occur as error length unit dR for computed length unit R in polar coordinates. dR propagates in both clockwise and anti-clockwise, logarithmic spiral curves (corresponding to appropriate dx, dy error values in cartesian coordinates) and is shown in the following to trace out the quasiperiodic Penrose tiling pattern in two dimensional phase space. The W units of numerical output is therefore the spatial average of w_* units of error distrtibution in the W domain (Eq.2). In the absence of error multiplication as visiualized above, W will trace out a structure less homogenous Euclidean object, e.g. sphere, circle, square etc. in the appropriate phase space. The selfsimilar geometrical structure of the strange attractor design traced out by W is therefore a direct consequence of inherent round-off error which multiplies exponentially with time and traces out a selfsimilar internal structure with overall logarithmic spiral design for the numerical computations W in the phase space. The numerically computed values W units of scale length R is therefore the integrated value of w_* units of error scale length dR contained in the domain of W. Analogous to the space-time integration of turbulent acceleration w_* to give the root mean square large scale acceleration W in fluid turbulence we have

                                                                                                     (3)

Successive W values are obtained by integrating over the round-off error structure in the W domain. Therefore considering two successive steps n and (n+1) of computation.

                                                                                                     (4)

To begin with, let dR = W_n = 1. R = S dR and the successive angular turning dq =dR/R. The successive values of W_n, W_n+1, R, dR and dq are computed and given in Table 1. The W values are found to follow the Fibonacci mathematical number series and therefore the spatial domain of W is made up of the quasiperiodic Penrose tiling pattern (15). The error growth dR on either side of the origin, i.e., clockwise and anticlockwise rotation generates an overall logarithmic spiral shape to W domain with quasiperiodic Penrose tiling pattern for the internal structure as shown in Fig.1. It is seen that a complete cycle of error growth structure is generated for scale ratio Z equal to t ⁵, i.e., 11.1 = R₅ / dR and corresponds to a total of 41.1 w_* units of scale length dR either in the clockwise or anticlockwise direction. Dominant error structure are generated for scale ratio Z = 11.1 and is consistent since, because of the logarithmic spiral relationship at Eq.(2). W can have measurable value for Z > 10 only. Further the fractional error k is computed to be equal to 0.382 from Eqs. (1) and (3) for dominant error structure growth, i.e., Z = 11.1 and is consistent since it is seen that for meaningful numerical computation W, the fractional error should be less than 0.5 which corresponds to Z > 10. Beck and Roepstroff (17) also find the universal constant 0.382 for the scaling relation between length of periodic orbits and computer precision in numerical experiments. Dominant coherent structures in numerical computation W evolve for scale ratio Z = 11.1 and are characterized by round-off error related quasiperiodic Penrose tiling pattern for the internal structure.

Traditional computational techniques are digital in concept, i.e., require a unit or yardstick for the computation, thereby splitting the continuum computed quantity into discrete bits and inevitably lead to approximations, i.e., round-off errors. Since computed quantity structure can be infinitesimally small in the limit, there exists no practical lower limit for the yardstick length. Therefore, numerical computations in the long run give results which scale with computer precision and also give quasiperiodic structures as shown above.

The real world dynamical systems also exhibit selfsimilar structures on all measurable scales with long range spatial and temporal correlations characteristic of self-organized criticality or deterministic chaos and further exhibit sensitive dependence on initial conditions associated with unpredictability or randomness. Analogous to the numerical computational small scale uncertainities giving rise to coherent sturctures in the computed output, the microscopic scale dynamical processes control macorscale long term evolution of real world dynamical systems as explained later in Section 8. There is a close parallelism between the numerically computed model evolution and real world dynamical system and it is seen from the arguments put forth above that the field of chaos is characterized by geometrically precise patterns which can be exactly quantified and therefore predictable to whatever short space-time extent of their life span.

In this non-deterministic computational technique the dynamical system is assumed to consist of an assembly of identical unit cells. Starting with arbitrary initial conditions the evolution of the dynamical system proceeds at successive unit length steps during unit intervals of time following arbitrary laws of interaction between adjacent cells. The cellular automata belongs to the cell dynamical system described above and does not require calculus based long term numerical integration schemes. However, the cellular automata rules for evolution do not have any physical basis. It is therefore required to incorporate the relevant physical processes into the cellular automata schemes.

The existence of deterministic chaos in atmospheric flows was proved conclusively by Lovejoy and Schertzer (20) who showed that clouds of all sizes exhibit fractal geometry and that the atmospheric eddy energy spectrum follows inverse power law of the form f ^-B where f is the frequency and B the exponent. Such long range spatial (self-similar fractal geometry) and temporal (l / f noise spectrum) correlations are signatures of self-organised criticality or deterministic chaos. Further, the cumulative probability density distribution of rainfall, rain area, cloud heights and other parameters follow the normal distribution. Also curvature is inherent to atmospheric flows as seen in the markedly spiral configuration of the hurricanes and cyclones. The full continuum of atmospheric fluctuations extend from the planetary scale of thousands of kilometers to a few millimeters and act cooperatively to give rise to the observed global weather systems. The predictability of such weather phenomena has proved to be a difficult problem since the numerical weather prediction models are based on the Navier-Stokes equations for fluid flows which consist of nonlinear partial differential equations (21) and are therefore subject to deterministic chaos or sensitive dependence on initial conditions as explained in Section 5 earlier. Further, though it is recognized that atmospheric electrification (both fair and disturbed weather) is an intrinsic part of atmospheric flows, it has not been possible to incorporate the dynamics of the electrification processes in the governing equations mainly because of a lack of conclusive knowledge of the basic physics of atmospheric electrification. In this paper a cell dynamical system model for atmospheric flows is developed which incorporates atmospheric electrification as a natural consequence of microscopic domain atmospheric eddy dynamical processes or analogous microhydrodynamical processes.

In the following, theoretical considerations similar to those developed in Section 5 for deterministic chaos in numerical model results is advanced for coherent atmospheric flow structures. In summary, the mean flow at the planetary atmospheric boundary layer (ABL) possesses an inherent upward momentum flux of frictional origin at the planetary surface. This turbulence scale upward momentum flux is progressively amplified by the exponential decrease of atmospheric density with height coupled with the buoyant energy supply by microscale fractional condensation on hygroscopic nuclei even in an unsaturated environment. The mean large scale upward momentum flux generates helical vortex roll (or large eddy) circulation in the planetary atmospheric boundary layer and is manifested as cloud rows / streets and mesoscale cloud clusters (MCC) in the global cloud cover pattern.

The space-time integrated mean of the turbulence scale vertical acceleration w_* generated by dominant eddy fluctuations of radius r give rise to large eddy acceleration W of radius R and is given by the following relation originally derived by Townsend (19).

(5)

The above relation holds good for any instantaneous values of w_* and W. The above concept of large eddy growth from turbulence scale buoyant energy generation envisagers large eddy growth in discrete length step increments dR and to r and is therefore analogous to the 'cellular automata' computational technique where cell dynamical system growth occurs in unit length steps during unit intervals of time since turbulence scale yardstick for length and time are used for measuring large eddy growth. Large eddy growth by such length scale doubling is hereby identified as the universal period doubling route to chaos eddy growth process. Therefore for turbulent eddy acceleration w_* large eddy incremental growth is dR and is associated with large eddy acceleration dW and is given by

(6)

During each length step growth dR the small scale energising perturbation W_n at the n^th instant generates the large scale perturbation W_n+1 of radius R such that from Eq.(5).

(7)

where R = S dR since successive length scale doubling give rise to R. The angular turning dq inherent to eddy circulation for each length step growth is equal to dR / R. The perturbation dR is generated by the small scale acceleration W_n and therefore dR = W_n. Starting with unit value for dR the successive W_n, W_n+1, R and dq values derived computed from Eq.7 and is given in Table 1 derived earlier in Section 5 on identical theoretical considerations. It is seen that W follows the Fibonacci mathematical number series such that R_n+1 = R_n+R_n-1 and R_n+1 / R_n is equal to the golden mean t equal to (1 + Ö 5) / 2 (= 1.618). Further, the successive large eddy values follow the geometrical progression

R_o (1+t +t ²+t ³+... where R_o is the initial value of eddy radius. Using polar coordinates the large eddy growth from primary perturbation may be depicted as in Figure 1. The primary perturbation OR_o startiang from the origin O gives rise to compensating return circulations OR₁ R_o on either side of OR_o thereby generating and large eddy radii OR₁ such that OR₁ / OR_o = t and angle R_o OR₁ = p / 5= angle R_oR₁O_. Therefore short range circulation balance requirements generate successively larger circulation patterns with precise geometry governed by the Fibonacci mathematical number series and is identified as the universal period doubling route to chaos. Such long-range non-local connections are analogous to manifestation of Berry's phase in quantum mechanics (27). It may be shown that atmospheric flow structure follows quantum mechanical laws (Mary Selvam, 1988). It is seen from Figure 1 that five such successive length step OR₂, OR₃, OR₄ and OR₅ tracing out one complete vortex roll circulation such that the scale ratio OR₅ / OR_o is equal to t ⁵ = 11.1. The envelope R₁, R₂, R₃, R₄, R₅ of the dominant large eddy (or vortex toll) is found to fit the logarithmic spiral R = R_oe^bq where b = tan a with a = p / 5 and q = p / 5 for each length step growth. The successively larger eddy radii may be subdivided again in the golden mean ratio. The internal structure of large eddy circulations is therefore made up of balanced small scale circulations tracing out the well known quasiperiodic Penrose tiling pattern identified as the quasicrystalline structure in condensed matter physics. A complete description of the atmospheric flow field is given by the quasiperiodic cycles with Fibonacci winding number in agreement with results of the others (28). The steady state non-dimensional fractional volume dilution k of the large eddy volume by environmental mixing is given by

(8)

The scale ratio for dominant large eddy growth has been shown in the above to be equal to t ⁵ = 11.1. Therefore the steady state fractional volume dilution k by eddy mixing for dominant large eddy growth as computed from Eqs. (5) and (8) is equal to 0.382. It is also seen that K 0.5 for Z < 10. Therefore, identifiable large eddy growth can occur for scale ratio Z > 10 only since for smaller scale ratios the large eddy identity is erased by environmental mixing. The above result is consistent with the earlier derived value of Z = 11.1 for self-organised dominant large eddy growth by the period doubling route to chaos growth process. The root mean square (r.m.s.) circulation speed W of the large eddy which grows from the turbulence scale at the planetary surface is obtained by integrating Eq (8) and for constant w_* and k is given as

(9)

The above equation is the well known logarithmic spiral relationship for wind profile in the surface ABL derived from conventional eddy diffusion theory, where the constant k is a constant of integration and its magnitude is obtained by observation as 0.4. The cell dynamical system model for atmospheric flows enables prediction of the logarithmic spiral profile for wind in the entire ABL and further the value of the Von Karman's constant k is obtained as equal to 0.382 as a natural consequence of environmental mixing during large eddy growth and is in agreement with observations. As seen from Figure 1 and from the concept of eddy mixing vigorous counterflow (mixing) characterizes the large eddy volume and the fractional outward mass flux of air across unit cross section for any two successive stages of eddy growth is given by

f_e is also the percolation threshold for critical phenomena, i.e., where the liquid gas mixture separates into the liquid and gas phases and in this case is associated with manifestation of coherent vortex roll structures. Clouds form in the updraft regions at the crest of large eddy circulations under favorable synoptic conditions. The cloud water condensation in the turbulent eddy fluctuations give the distinctive cauliflower-like surface granularity to the cumulus cloud. The ratio of the actual cloud liquid water content to the adiabatic liquid water content q_a is found to be less than one and has been attributed to mixing of environmental air into the cloud volume. The measured value of q / q_a at cloud-base is found to be 0.61 and is in agreement with f_e and is consistent with the observed fractal geometry to cloud shape. Incidentally, the Von Karman's constant k is equal to 1 - f_e = 0.382. The fractional upward mass flux of air of surface origin at any scale height Z is shown to be given by

(10)

(11)

Fluxes of mass, momentum from the surface ABL are therefore transported upward by the vigorous counter flow of air in intrinsic fractal structure between lower and higher levels and not by conventional eddy diffusion theory. Such a concept is analogous to superfluid turbulence in liquid helium (29).

The cell dynamical system model for cloud electrification enables to formulate simple model concepts for cloud electrification processes. In summary, the vigorous counterflow characterizing the internal structure of atmospheric large eddy circulation transport the naturally occurring negative space charges from above cloud-top regions to the cloud base and simultaneously transport the naturally occurring positive space charges from below cloud-base to the cloud top regions and thereby account for the commonly observed positive dipole structure of the average thundercloud. The small pocket of positive charges observed at the cloud-base of intense mature thunderclouds may form as a result of generation, within the cloud, of a double vortex roll circulation such that a dominant downcurrent extending from the cloud top to the cloud-base brings down positively charged precipitation particles from cloud top to form a pocket of positive charge at cloud-base. Observations show that microbursts releasing heavy precipitation and / or heavy downdrafts occur in regions of the positive pocket of charge at cloud-base. Increase in ozone levels have also been reported underneath such pockets of positive charge probably by downward transport of ozone from stratospheric levels. In summary, the cell dynamical system model for cloud electrification leads to the following conclusions. (a) Negative charges from the ionospheric level and positive charges released by corona discharge at the planetary surface are the main sources for cloud electrification, (b) Small scale nested continuum of vortex roll circulations inside the cloud are mainly responsible for the transport of charges and later attachment to precipitation particle, (c) The small scale vortices with bi-directional flow structure inside the cloud trace out the quasiperiodic Penrose tiling pattern with self selfsimilar fractal geometry manifested as the distinctive cauliflower like surface granularity to the cumulus cloud generated by cloud water condensation and dissipation respectively in adjacent small scale up and downdrafts, (d) The observed visible structure of lightning discharge has fractal geometry consistent with the concept of bi-directional electric current flow in a selfsimilar fractal network of the quasiperiodic Penrose tiling pattern, (e) The observed inverse power-law form for peak received amplitude for electromagnetic signals radiated by lightning (30) in consistent with the observed inverse power-law form for the atmospheric eddy energy structure (20) (f). The observed log-normal frequency distribution of lightning current amplitude (30) is consistent since the cloud charging currents circulate at the eddy circulation speed W which inherently follows log-normal distribution from Eq.9 since Z, the scale ratio follows normal distribution because R = S r and Z = R / r, (g) The scale invariant characteristics of atmospheric eddy dynamical processes enables to formulate simple scale invariant governing equations in terms of the non-dimensional universal functions k and f for the vertical profile of cloud electrical parameters (22-23).

Further the above described two-way charge transport mechanism from the ionosphere to the earth's surface occurs less vigorously in the fair weather regions of the globe and generates the fair weather atmospheric electric field (31). The vigorous turbulence scale counterflow of negative charges downwards and positive charges upwards extend upto the ionospheric levels and above (32) to form the 'double layers' in the auroral potential structure of the auroral curtains in the auroral oval. The beautiful shimmering curtains of auroras in the auroral oval are the visible manifestation of charge transport or energy exchange between the upper and lower atmosphere by the agency of the turbulent eddies or vortices and generates the fair weather atmospheric electric field and a corresponding horizontal component of the geomagnetic field. The structure of the charged counterflow as manifested in the visible aurorae is found to be a series of vortices between the double layers of opposite charges. High speed television cameras pointing upward at the bottom of the auroral curtain have captured images of such vortices and it has been confirmed that the auroral potential structure also accelerates positive ions upwards, indeed, such ions become at times a significant part of magnetospheric plasma (34). The mechanism of generation of the fair weather electric field and the cloud electric field is exactly the same. The direction of the electric field inside a thundercloud is the same as that of the fair weather electric field, but of a very high intensity because of vigorous charge separation by the intense small scale vertical mixing and confinement of charges within the vertical cloud volume by attachment to precipitation particles held aloft by the air currents. The thundercloud is therefore a steady state self-organized 'double layer' consisting of inherent small scale double layers and acts as an efficient generator of electrical power from the solar wind coupled ionospheric charge supply. The frequently reported horizontal travel path for lightning discharges maybe due to the adjacent positive and negative charge concentrations in the double layers of the small scale eddy circulations.

The cell dynamical system model enables formulation of simple scale invariant governing equations for cloud dynamical and electrical processes.

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