FIG 1. Conceptual model of large and turbulent eddies in the planetary ABL. The mean air flow at the planetary surface carries the signature of the fine scale features of the planetary surface topography as turbulent fluctuations with a net upward momentum flux. This persistent upward momentum flux of surface frictional origin generates large-eddy (or vortex-roll) circulations, which carry upward the turbulent eddies as internal circulations. Progressive upward growth of a large eddy occurs because of buoyant energy generation in turbulent fluctuations as a result of the latent heat of condensation of atmospheric water vapour on suspended hygroscopic nuclei such as common salt particles. The latent heat of condensation generated by the turbulent eddies forms a distinct warm envelope or a microscale capping inversion layer at the crest of the large-eddy circulations as shown in the upper part of the figure. The lower part of the figure shows the progressive upward growth of the large eddy from the turbulence scale at the planetary surface to a height R and is seen as the rising inversion of the daytime atmospheric boundary layer. The turbulent fluctuations at the crest of the growing large-eddy mix overlying environmental air into the large-eddy volume, i.e., there is a two-stream flow of warm air upward and cold air downward analogous to superfluid turbulence in liquid helium (see ref. 79). The convective growth of a large eddy in the atmospheric boundary layer therefore occurs by vigorous counter flow of air in turbulent fluctuations (see also Fig. 4), which releases stored buoyant energy in the medium of propagation, e.g., latent heat of condensation of atmospheric water vapour. Such a picture of atmospheric convection is different from the traditional (see ref. 78) concept of atmospheric eddy growth by diffusion, i.e., analogous to the molecular level momentum transfer by collision.
The generation of turbulent buoyant energy
by the microscale fractional condensation 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 that rises upward with the convective growth of the
large eddy during the course of the day. This is seen as the rising inversion
of the daytime planetary boundary layer in echosonde and radiosonde records
and has been identified as the entrainment zone (62) where mixing with
the environment occurs.
Townsend (63) has investigated
the structure and dynamics of large-eddy formations in turbulent shear
flows and has shown that large eddies of appreciable intensity form as
a chance configuration of turbulent motion as illustrated in the following
example. Consider a large eddy of radius R that forms in
a field of isotropic turbulence with turbulence length and velocity scales
2r
and w*
, respectively. The dominant turbulent eddy radius is therefore equal to
w*
. The mean square circulation C2 at any instant
around a circulation path of large-eddy radius R is
given by
= 2(2pR)w*(2r
w*)
= 8pR
w*2r
where w*
is tangential to the path elements ds and the motions in
sufficiently separated parts of the flow are statistically independent.
The mean-square velocity of circulation W 2 in
the large eddy of radius
R is given by
The above equation enables
us to compute the instantaneous acceleration dW for a large-eddy
of radius
R generated by the spatial integration of the inherent
dominant turbulence-scale vertical acceleration w*
of length scale 2r . The large-eddy growth from turbulence
scale fluctuations may be visualized as follows. The large-eddy domain
is defined by the overall envelope of the turbulent fluctuations, and incremental
growth of the large-eddy occurs in discrete length steps equal to the turbulent
outward displacement of air parcels. Such a concept outlined above for
large-eddy growth from turbulence scale buoyant energy generation envisages
large-eddy growth in discrete length step increments dR equal
to
r and is therefore analogous to the cellular automata
computational technique (see Sect. 4) where cell dynamical system growth
occurs in unit length steps during unit intervals of time, the turbulence
scale yardsticks for length and time being used for measuring large-eddy
growth. A continuous spectrum of progressively larger eddies are thus generated
in the ABL. Large-eddy growth by such successive length scale doubling
is hereby identified as the universal period doubling route to the chaotic
eddy growth process. Therefore, the turbulent eddy acceleration w*
generates, at any instant, the large-eddy incremental growth dR
associated with large-eddy incremental acceleration dW
as given by [1] as
Equation [1] signifies a two-way ordered energy (kinetic energy) flow between the smaller and larger scales and [1] is therefore identified as the statement of the law of conservation of energy for the universal period doubling route for chaos eddy growth processes in atmospheric flows. Figure 2 shows the concept of the universal period doubling route for chaotic eddy growth process by the self-sustaining process of ordered energy feedback between the larger and smaller scales, the smaller scales forming the internal circulations of the larger scales.
FIG 2. Physical concept of the universal period doubling route to chaotic eddy growth process by the self-sustaining process of ordered energy feedback between the larger and smaller scales, the smaller scales forming the internal circulations of the larger scales. The figure shows a uniform distribution of dominant turbulent scale eddies of length scale 2r . Large-eddy circulations such as ABCD form as coherent structures sustained by the enclosed turbulent eddies. The r.m.s. circulation speed of the large eddy is equal to the spatially integrated mean of the r.m.s. circulation speeds of the enclosed turbulent eddies. Such a concept envisages large-eddy growth in unit length step increments during unit intervals of time with turbulence-scale yardsticks for length and time, and is therefore analogous to the cellular automata computational technique. The growth of the large-eddy by successive period doubling, namely, discrete length step increments equal to the turbulence length scale is identified as the physics of the universal period doubling route to chaos eddy growth process.
Atmospheric boundary
layer flows, therefore, generate, as a natural consequence of surface friction,
persistent microscopic domain turbulent fluctuations that amplify and propagate
upward and outward spontaneously as a result of the buoyant energy supply
from the latent heat of condensation of atmospheric water vapour on suspended
hygroscopic nuclei in the upward fluctuations of air parcels. The evolution
of the macroscale atmospheric eddy continuum structure occurs in successive
microscopic fluctuation length steps in the ABL and therefore has
a self-similar scale-invariant
fractal geometrical structure by
concept and also according to [1]. Equation [1] is therefore identified
as the universal algorithm that defines the space-time continuum evolution
of the atmospheric eddy energy structure (strange attractor). Such a concept
of the autonomous growth of the atmospheric eddy continuum with ordered
energy flow between the scales is analogous to the 'bootstrap' theory
of Chew (64), the theory of implicate order envisaged by Bohm (65),
and Prigogine's concept of the spontaneous emergence of order through
a process of self-organization (65).
The turbulent eddy
circulation speed and radius increase with the progressive growth of the
large eddy as given in [1]. The successively larger turbulent fluctuations,
which form the internal structure of the growing large eddy, may be computed
from [1] as
During each length step growth dR , the small-scale energizing perturbation Wn at the nth instant generates the large-scale perturbation Wn+1 of radius R where R = S1n dR since successive length-scale doubling gives rise to R . Equation [3] may be written in terms of the successive turbulent circulation speeds Wn and Wn+1 as
[4]
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 Wn at any instant n and therefore dR = Wn . Starting with the unit value for dR the successive Wn, Wn+1 , R , and dq values are computed from [4] and are given in Table 1.
|
|
|
|
|
|
|
|
2 3.254 5.239 8.425 13.546 21.780 35.019 56.305 90.530 |
1.254 1.985 3.186 5.121 8.234 13.239 21.286 34.225 55.029 |
1.254 1.985 3.186 5.121 8.234 13.239 21.286 34.225 55.029 |
0.627 0.610 0.608 0.608 0.608 0.608 0.608 0.608 0.608 |
1.985 3.186 5.121 8.234 13.239 21.286 34.225 55.029 88.479 |
1.627 2.237 2.845 3.453 4.061 4.669 5.277 5.885 6.493 |
It is seen that the succesive values of the circulation speed W and radius R of the growing turbulent eddy follow the Fibonacci mathematical number series such that Rn+1 = Rn + Rn-1 and Rn+1/Rn is equal to the golden mean t , which is equal to [(1 +sqrt(5))/2] ~ (1.618). Further, the successive W and R values form the geometrical progression R0(1 + t + t2 + t3 + t4 + ....) where R0 is the initial value of the turbulent eddy radius.
(c)
FIG. 3. The internal structure of large-eddy circulations. (a) Turbulent eddy growth from primary perturbation ORo starting from the origin O gives rise to compensating return circulations OR1R2 on either side of ORo , thereby generating the large eddy radius OR1 such that OR1/ORo = t and RoOR1 = p/5 = RoR1O. Five such successive length step growths give successively increasing radii OR1 , OR2 , OR3 , OR4 and OR5 tracing out one complete vortex-roll circulation such that the scale ratio OR5/ORo is equal to t5 = 11.1. The envelope R1R2R3R4R5 of a dominant large eddy (or vortex roll) is found to fit the logarithmic spiral R = Ro ebq where Ro = ORo , b= tan a with a the crossing angle equal to p/5, and the angular turning q for each length step growth is equal to p/5. (b) The logarithmic spiral R = Ro ebq is drawn as OAB for clarity. The successively larger eddy radii may be subdivided again in the golden mean ratio. (c) The internal structure of large-eddy circulations is, therefore, made up of balanced small-scale circulations, which trace out the well-known quasiperiodic Penrose tiling pattern identified as the quasi-crystalline structure in condensed matter physics. 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 signature of the universal period doubling route to chaos in atmospheric flows.
Turbulent eddy growth
from primary perturbation
ORo starting from the origin
O (Fig. 3) gives rise to compensating return circulations OR1R2
on either side of
ORo , thereby generating the large
eddy radius OR1 such that
OR1/ORo
= t
and RoOR1 = p/5
= RoR1O.
Therefore, short-range circulation balance requirements generate successively
larger circulation patterns with precise geometry that is governed by the
Fibonacci mathematical number series, which is identified as a signature
of the universal period doubling route to chaos in fluid flows, in particular
atmospheric flows. It is seen from Fig. 3 that five such successive length
step growths give successively increasing radii OR1 ,
OR2
, OR3 , OR4 and
OR5
tracing out one complete vortex-roll circulation such that the scale ratio
OR5/ORo
is equal to t5
= 11.1. The envelope R1R2R3R4R5
(Fig. 3) of a dominant large eddy (or vortex roll) is found to fit the
logarithmic spiral R = Ro ebq
where Ro = ORo , b= tan a
with a
the crossing angle equal to p/5,
and the angular turning q
for each length step growth is equal to p/5.
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 quasi-crystalline
structure in condensed matter physics. A complete description of the atmospheric
flow field is given by the quasi-periodic cycles with Fibonacci
winding numbers. The self-organized large-eddy growth dynamics, therefore,
spontaneously generate an internal structure with the fivefold symmetry
of the dodecahedron, which is referred to as the icosahedral symmetry,
e.g., the geodesic dome devised by Buckminster Fuller. Incidentally,
the pentagonal dodecahedron is, after the helix, nature's second favourite
structure (67). Recently, a carbon macromolecule C60 , formed
by condensation from a carbon vapour jet, was found to exhibit the icosahedral
symmetry of the closed soccer ball and has been named Buckminsterfullerene
or footballene (68, 69). It may be noted that it has not been possible
to create such C60 Buckminsterfullerene molecules
by traditional chemical reaction methods. Such a quasi-crystalline structure
has recently been identified in numerical simulation of fluid flows (70).
The time period of
large-eddy circulation made up of internal circulations with the Fibonacci
winding number is arrived at as follows. Assuming turbulence-scale yardsticks
for length and time, the primary turbulence-scale perturbation generates
successively larger perturbations with the Fibonacci winding number
on either side of the initial perturbation. Therefore, the large-eddy time
period T is directly proportional to the total circulation
path traversed on any one side, and is given in terms of the turbulence
scale time period t as
[5] T = t [ 2 (1 + t +t2 + t3+ t4 ) + t5 ] = 43.74 t
Therefore, the large-eddy circulation time period is also related to the geometrical structure of the flow pattern.
r : R = r : t5r : t10 r : t15 r : t20 r
The limit cycles or
dominant periodicities in atmospheric flows (71), possibly originating
from solar-powered primary oscillations, are given in the following. (i)
The 40- to 50-day oscillation in the atmospheric general circulation and
the quasi-five yearly ENSO phenomena (49) may possibly arise from
diurnal surface heating. (ii) The 40- to 50-year cycle in climate
may be a direct consequence of the annual solar cycle (summer and winter
oscillation). (iii) The quasi-biennial oscillation (QBO)
in the tropical stratospheric wind flows may arise as a result of the semidiurnal
pressure oscillation. (iv) The 22-year cycle in weather patterns
associated with the solar sunspot cycle may be related to the newly identified
5-min oscillations of the sun's atmosphere (72). The growth of large eddies
by energy pumping at smaller scales, namely the diurnal surface heating,
the semidiurnal pressure oscillation, and the annual summer-winter cycles
as cited above is analogous to the generation of chaos in optical emissions
triggered by a laser pump (73). Recent barometer data on the planet Mars,
whose tenuous atmosphere magnifies atmospheric oscillations, reveal oscillations
with periods very close to 1.5 Martian days preceding episodes of global
dust storms (74), which indicates a possible cause and effect mechanism
as given in [6]. The identification of limit cycles in atmospheric flows
is possible by means of the continuous periodogram analysis of long-term
high-resolution surface pressure data and this will help long-term prediction
of regional atmospheric flow pattern (75).
As seen from Fig. 3
and from the concept of eddy growth, vigorous counter flow (mixing) characterizes
the large-eddy volume. the steady-state fractional volume dilution k
of the large-eddy volume by environmental mixing is given by
Earlier it was shown that the successive eddy length step growths generate the angular turning dq of the large-eddy radius R given by dR/R, which is a constant equal to 1/t where t is the golden mean. Further, the successive values of the r.m.s. circulation speed W and the corresponding radius R of the large eddy follow the Fibonacci mathematical number series. Therefore, the value of k , the steady state volume dilution of the large eddy by the turbulent eddy fluctuations for each length step growth of the large eddy, is found from [7] to be
k = 1/t2= 0.382
Since the steady-state
fractional volume dilution of the large eddy by inherent turbulent eddy
fluctuations during successive length step increments is equal to 0.382,
i.e., less than half, the overall Euclidean geometrical shape of
the large eddy is retained as manifested in the cloud billows, which resemble
spheres.
The fractional outward
mass flux of air across a unit cross section for any two successive steps
of eddy growth is given by
fc = 1/t = 0.618
fc
is therefore equal to the percolation threshold for critical phenomena,
i.e., where the liquid-gas mixture separates into the liquid and gas phases
with the formation of self-similar fractal structures (76) and in
the case of atmosphereric flows this is associated with the manifestation
of coherent vortex-roll structures. The ratio of the actual (observed)
cloud liquid water content q to the adiabatic liquid water
content qa , i.e., without mixing with the environment,
is found to be less than one and has been attributed to the mixing
of the environmental air into the cloud volume. The measured value of q/qa
at the cloud base is found to be 0.61 (77) in agreement with that
predicted by
fc above; it is also consistent with
the observed fractal geometry of cloud shape.
The vigorous counterflow
of air (mainly vertically) in turbulent eddy fluctuations characterizes
the internal structure of the growing large eddy. The turbulent eddies
carried upward by the growing large eddy are amplified to form 'cloud-top
gravity oscillations' and are manifested as the distictive cauliflower-like
surface granularity of the cumulus cloud growing in the large-eddy updraft
regions under favourable conditions of moisture supply (Fig. 4). Therefore,
atmospheric convection and the associated mass, heat, and momentum transport
in the ABL occur by the vigorous counterflow of air in intrinsic
fractal structures and not by eddy diffusion processes postulated
by the conventional theories of atmospheric convection (78). Such a concept
of atmospheric convection is analogous to superfluid turbulence in liquid
helium (79).
FIG. 4. Cloud structure in the ABL. The turbulent eddies carried upward by the growing large eddy (see Fig. 1) are amplified to form cloud-top gravity (buoyancy) oscillations and are manifested as the distinctive cauliflower-like surface granularity of the cumulus cloud growing in the large-eddy updraft regions under favourable conditions of moisture supply in the environment. The fractal or broken cloud structure is a direct result of cloud water condensation and evaporation, respectively, in updrafts and downdrafts of the innumerable microscale turbulent eddy fluctuations in the cloud volume. Therefore, atmospheric convection and the associated mass, heat, and momentum transport in the ABL occur by the vigorous counterflow of air in intrinsic fractal structures and not by eddy diffusion processes postulated by the conventional theories of atmospheric convection (see Fig. 1).
The above equation is the well-known logarithmic spiral relationship for wind profile in the surface ABL derived from conventional eddy diffusion theory (78) where k is a constant of integration and its magnitude is obtained from observations as 0.4 (80). The logarithmic wind-profile relationship is consistent with the overall logarithmic flow structure pattern of the quasi- periodic Penrose tiling pattern that is traced by atmospheric flows as was deduced earlier and shown in Fig. 3. The cell dynamical system model for atmospheric flows enables us to predict the logarithmic spiral profile for the wind for the entire ABL . 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 dominant large-eddy growth, and is in agreement with observations. Von Karman's constant is therefore identified as the universal constant for deterministic chaos in the real world dynamical system of atmospheric flows. The predicted logarithmic spiral trajectory for ABL flows is seen markedly in the hurricane spiral pattern. Such coherent helicity is intrinsic to atmospheric flows (41).
Wp = 2pnRp
From [1],
Furthermore,
KE = pHn = (1/2)Hw
H is equal to the product of the momentum of the planetary scale eddy and its radius and therefore represents the angular momentum of the planetary scale eddy about the eddy centre. Therefore, the KE of any component eddy of frequency n of the scale invariant eddy continuum is equal to pHn . Further, since the large eddy is but the sum total of the smaller scales, the large eddy energy content is equal to the sum of all its individual component eddy energies and therefore the KE energy distribution is normal and the KE 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 density distribution. The eddy continuum energy spectrum is therefore the same as as the cumulative normal probability density distribution plotted on a log-log scale and the eddy energy probability density distribution is equal to the square of the eddy amplitude. Therefore, the atmospheric eddy continuum energy structure follows laws that are similar to quantum mechanical laws without exhibiting any of the apparent inconsistencies of the quantum mechanical laws for subatomic dynamics as illustrated in Fig. 5 and explained in the following.
FIG. 5. Quantum mechanical analogy with macroscale phenomena of atmospheric flows. The upper part of the figure illustrates the concept of wave-particle duality as physically consistent in the common place observed phenomena of the formation of clouds in a row as a natural consequence of cloud formation and dissipation, respectively, in the updrafts and downdrafts of vortex roll circulations in the ABL. The lower part of the figure illustrates the concept of non-locality by analogy with instantaneous transfer of energy from effort to load in a pulley and as also inferred by the physically consistent phenomena of instantaneous circulation balance in the atmospheric vortex-roll circulations with alternating balanced high- and low-pressure areas.
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