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 roundoff errors.
Let w_{*}
be the number of units of roundoff 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
W = w_{*} lnZ / k
Z is equal to the scale ratio
R
/ dR, i.e, the numerical value of R, the computed
length unit measured in units of roundoff error dR. Therefore
the computed numerical result W in units of R
scales with computer precision or the roundoff 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 roundoff 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 anticlockwise, 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
roundoff 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 spacetime integration of turbulent acceleration w_{*}
to give the root mean square large scale acceleration W in
fluid turbulence we have
Successive
W
values are obtained by integrating over the roundoff error structure in
the W domain. Therefore considering two successive steps
n
and
(n+1) of computation.
To begin with,
let dR = W_{n} = 1. R = SdR
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.
Table 1


































































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
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).
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_{n1} 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 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 selforganised 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
f_{e} = 1/t = 1/1.618 = 0.618
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 cauliflowerlike 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 cloudbase 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
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