The pseudo-singularity, as viewed from the domain of the inertial range,
looks as if a genuine singularity, whose actual form is now to
be identified.
We start with checking if the local isotropy in the classical sense has physical reality. The isotropy assumption is equivalent to

(36)

which is the alternative expression of Robertson's theorem[5]. Substitution of (36) into equation of continuity (33) gives

Obviously, this is a source/ sink flow, corresponding to the velocity
potential

(37)

If is positive it represents a sink flow with mass flux
vanishing at the origin. Reminding that the
dissipation range
is the `black hole' of turbulent
kinetic energy losing the amount by converted into heat every second,
we see that no mass is supposed to be lost. Thus solution (36) for
spherical isotropy is ruled out.
Prospective singularity now to take over can be sought within potential
flow regime as follows: It is obvious that operator
commutes with Laplacean operator, therefore

(38)

represent a group of potential flows. In particular

(39)

representing a dipole aligned with its axis parallel to direction
is seen to meet the purpose. In fact a dipole is made up of a
pair of sink/source of equal strength, so no mass flux is lost
at this spot. Thus simplest possible candidate for the
pseudo-singularity is axially isotropic.
The following observation from direct numerical simulation[6]
helps us draw a more precise picture of our pseudo-singularity: In the
physical space the dissipation occurs only within a confined volume of
`elementary particles' of rod shape, randomly dispersed in turbulent
medium. They all have a shape like a worm with radius of
several Kolmogorov lengths and several ten times of it lengthwise (
), rotating round their own axes. Their lifetime is
about
. So to say, they are like firefly worms
illuminating light for their lifetime as small as of the order of far
subseconds.
This picture when mapped onto our eddy space is such that two spinning
dipoles having opposite sign and rotation are placed at

(40)

respectively, where the axis of symmetry is axis. The direction j
= 1 is the direction of motion of turbulence-generating body,
representing the only vector prescribing the fluid motion. The flow
field induced by the pair of spinning dipoles

(41)

with
and
standing for velocities induced by dipoles and line vortices, placed at points (40), respectively:

(42)

(43)

where
is the unit vector designating axis. Expression
(42) is direct consequence of (37) and (39), and that for (43) is
Biot-Savart's law for a line vortex with infinitesimal length directing
-axis and with circulation . For example, the actual form of
rewritten in axially isotropic form
with
is,

(44)

Note that there is no contribution from
to .
For this expression approaches to

(45)

which is a quadrupole field, as it should be expected.
Streamlines are shown in Fig.1 of the fictitious flow generated by a
pair of dipoles given by the potential

(46)

This axi-symmetric flow can also be represented using streamfuntion
as

(47)

from which we have

(48)

The flow pattern :const. shows quadrupole-like structure at far
field(
) toward which the longitudinal vortices are
stretched streamwise and then getting thicker. On the returning path to the dipole
core they are chopped off and trim the aspect ratio, getting into the
dissipation region. This picture may serve the qualitative description
of what is actually observed (Fig.2).
Expression (45) gives us estimate for the outer boundary of the locally
homogeneous region. That is also the inner boundary of inhomogeneous region where the first condition of inequality (32) ceases to hold;

(49)

Let this boundary be defined by , then we have
from (45), therefore condition (49) is replaced with a more
precise one

(50)

where is the characteristic velocity corresponding to . They
supplement Kolmogorov formula (27) as

(51)

On the opposite side of the inertial subrange
,
, is estimated from (44) as

(52)

We note that the dimensional analysis developed in Sec.4 still holds by
replacing
with of (52), for instance,

(53)

Then, by eliminating
and from (50) through (53)
we have

(54)

The relationship between and
or is yet to be
reconsidered. At present no consensus formula is available for explicit
parameter dependence of worm size .
A recent observation by
numerical experiments[8] is that is of the order of the
Taylor microscale (
; r.m.s. of the velocity fluctuation ), according to which the outer boundary of the inertial
range is

(55)

The following expression may serve an easier estimate for the size of
the inertial subrange: