A&A 491, 671680 (2008)
DOI: 10.1051/00046361:200809848
A threedimensional magnetohydrodynamic model
of planetary nebula jets, knots, and filaments
K. H. Tsui
Instituto de Física  Universidade Federal Fluminense,
Campus da Praia Vermelha, Av. General Milton Tavares de Souza s/n
Gragoatá, 24.210346, Niterói, Rio de Janeiro, Brasil
Received 26 March 2008 / Accepted 4 September 2008
Abstract
Context. Aims. A selforganizational view of planetary nebulae, driven by global conservation properties of the magnetohydrodynamic (MHD) plasma, is presented. Selfsimilarity with a selfconsistent evolution function is used, as a method and as a model for selforganized states, to solve the timedependent MHD equations in the gravitational field of a central star.
Methods. The selfsimilar configurations are constructed on the basis of a limited radially expanding MHD plasma sphere, which could be associated with the ejected hydrogen layer of the AGB star due to its strong pulsations.
Results. Although the plasma expansion velocity is isotropically radial, driven by selforganization, the interactions between the conducting plasma and the magnetic fields steer the MHD system to a highly inhomogeneous spatial distribution. This spatial redistribution involves the focusing of magnetic energy, plasma thermal energy, and plasma momentum, generating collimated jets, pointsymmetric knots, and circulating filaments.
Key words: magnetohydrodynamics (MHD)  methods: analytical  ISM: planetary nebulae: general
A star with a mass of up to approximately eight solar masses
evolves along the main sequence of the HR
(temperatureluminosity) diagram to a stage
where the accelerated fusion of the hydrogen layer,
immediately above the helium core, causes the star to expand.
The expansion is faster than the energy production rate,
such that the expansion cools the star,
moving it to the red giant branch and launching a slow wind.
The expanding outer layers of the star are convective,
and they bring the material in the fusing region below
to the surface by turbulence.
As the helium core ignites and expands,
it pushes the hydrogen layer up and slows down
the hydrogen fusion rate.
The total energy generation decreases,
and the star contracts in size
and heats up in surface temperature,
migrating it to the horizontal branch.
As nuclear fusions continue,
carbon and oxygen become the core material,
surrounded by an inner helium layer
and an outer hydrogen layer,
plus a highly convective envelope,
bringing the star to the asymptotic giant branch (AGB).
The energy output is shifted to long wavelength
emission towards the cool side of the HR diagram.
Helium burning is quite sensitive to temperature,
which causes burning instabilities
and large star pulsations.
The pulsations can become so strong
that they give the outer layers of the star
enough energy to eject matter as a fast wind,
forming a planetary nebula.
The remaining core of the star cools down
and becomes a small but dense white drawf.
Due to the rotational velocity of the star,
we believe that strong magnetic fields are present in the star,
especially in the helium and hydrogen layers,
and in the outer convective envelope,
and such fields have been actually detected in central stars
(Jordan et al. 2005).
Considering a rigidly rotating star
and selfgravity of the distributed stellar mass,
our preliminary research indicates
that magnetic field lines in the star are wound
in a torus, in layers.
These magnetic field lines could be exposed
during the pulsation phase,
and released to the fast wind.
The initial configuration of the field lines
need not stay the same
as the fast wind expands in space,
because of its interaction with the conducting plasma
and its transformation through dissipative processes.
As observational techniques and instruments become more sophisticated,
the morphologies of planetary nebulae
have become increasingly complex.
As a result, the early pioneering model
of interacting stellar winds for spherical planetary nebulae
(Dyson & de Vries 1972; Kwok et al. 1978)
has evolved to generalized interacting stellar winds
to account for axisymmetric bipolar planetary nebulae
(Morris 1981; Kahn & West 1985;
Mellema 1997; Zhang & Kwok 1998).
The bipolar features could be reproduced
by the presence of a dense equatorial cloud,
supposedly generated by the slow wind
(Kahn & West 1985; Mellema et al. 1991;
Mellema & Frank 1997; Dijkstra & Speck 2006),
which has been imaged by high resolution instruments.
Besides the bipolar features, images have also revealed
pointsymmetric knots and highly collimated jets
(Miranda & Solf 1992; Lopez et al 1993; Balick et al. 1993).
Under the generalized interacting stellar wind model,
the possibility of the AGB star having a secondary close
companion makes it possible to have an accretion disk
around the secondary star to launch the jets
(Morris 1981; Soker & Livio 1994; Mastrodemos & Morris 1999).
Nevertheless, these collimated jets and pointsymmetric knots
have led to the alternative model of magnetized windblown bubbles
where an embedded azimuthal magnetic field
in the fast stellar wind is responsable for collimated jets
(Pascoli 1993; Chevalier & Luo 1994; GarciaSegura 1997;
Matt et al. 2000; Gardiner & Frank 2001).
Pointsymmetric knots can be reproduced
with magnetized windblown bubbles
by considering jets that precess about the polar axis
generated by a tilted rotating star
or a binary system at the center (GarciaSegura & Lopez 2000).
In particular, using the thin shell approximation of Giuliani (1982),
Chevalier & Luo (1994) developed a selfsimilar model
of planetary nebulae with a preselected
constant radially expanding velocity.
Observations over time have indicated
that the winds have such extraordinarily high momentum and energy
that they cannot be accounted for by the radiation pressure
of the AGB star (Bujarrabal et al. 2001).
These observational results appear to favor
the magnetized windblown bubbles scenario
where jets could be locally magnetically driven
(GarciaSegura et al. 2005)
provided that there is a strong enough toroidal magnetic field.
To explain the morphologies of planetary nebulae,
current models attribute dynamic structures
to the central star, or binary,
to reproduce nebula morphologies in a deterministic way.
To account for the highly collimating jets,
pointsymmetric knots, and spatially resolved filaments,
we consider timedependent full MHD equations
and take an indirect form to describe the morphologies,
through selforganizational processes via turbulent dissipation.
From an analytic approach,
the end configurations are disconnected from the initial conditions
of the central engine, that are lost in the chaotic processes,
although numerically we can always simulate over the turbulence,
given enough computing power and time.
In order to solve for the end configurations of the
timedependent MHD equations,
we look for solutions of reduced dimensions
in a selfsimilar form and
with selfconsistent evolution in time.
Selfsimilarity between time and one spatial coordinate
lowers the dimensions of the solution,
and takes the MHD system through a specific time trajectory.
We believe that this kind of selfsimilar solution,
with selfconsistent time evolution,
is able to represent stable selforganized structures
(Low 1982a,b, 1984a,b).
This selfsimilar approach relies on the global conservation
properties of MHD plasma to reach selforganized configurations,
regardless of the initial conditions.
This approach is hypothetical only,
however we recall that Earth's magnetosphere
is a stationary object in space with respect to
the moving frame of Earth about the Sun.
The magnetic field lines are stationary in such a frame.
Nevertheless, near the terrestrial surface,
the field lines corotate with Earth.
In between, there is a transition layer
that joins the stationary outer part
to the corotating inner part of the magnetosphere.
This is an example of dissipative processes
in driving the end configuration,
and where the initial conditions are lost in the process.
The same can be said for the heliosphere.
Therefore, on the scale of planetary nebulae,
it is likely that the details of the AGB star initial conditions
could be lost through dissipation.
Selfsimilarity, as a method, can be applied in different ways.
In the case of disk winds
(Blandford & Payne 1982; Pelletier & Pudritz 1992),
steady state MHD equations are solved for selfsimilar
solutions in space between two spatial labels
in cylindrical coordinates to describe the transport
of plasma from the equatorial disk to the polar axes.
This amounts to a deterministic cause and effect
approach through selfsimilar solutions in space.
Under a specific magnetic field configuration of the disk,
plasma transport with a continuous velocity field
generates spatial singularities in the governing equation.
In the case of the stellar envelope,
Lou and his collaborators have treated an
aggregating fluid under its selfgravitational field
with a similarity variable x=r(t)/at and a preselected evolution function at,
where a is the sound speed
(Lou & Shen 2004; Bian & Lou 2005; Lou & Wang 2006; Lou & Gao 2006).
Because of the linear dependence on time,
this similarity variable x refers to a reference frame moving at speed a,
which is different from the radial flow velocity.
For this reason, the convective derivative remains explicit
in the x representation.
As a result, this approach amounts to finding
the plasma structures in an adequate moving frame
in the Eulerian x fluid description.
Although an evolution function with a constant
radial velocity for a spherically symmetric expanding plasma
appears reasonable from a kinematic point of view,
it is not true in most cases for plasmas.
This is because the energy density is shared
by the magnetic field, the kinetic energy of the plasma,
and the thermal energy of the plasma.
Consequently, the radial plasma velocity
could slow down or speed up,
depending on the interactions
between the magnetic field and the conducting plasma.
Without selfconsistent time evolution of the system,
this particular approach amounts to an application
of selfsimilarity as a method.
The driving force for selforganization
relies on the MHD global conservation properties
through the quadratic invariants in the absence of dissipation
(Hasegawa 1985; Biskamp 1993; Zhu et al. 1995;
Yoshida & Mahajan 2002; Kondoh et al. 2004).
These invariants are the total energy density,
which includes magnetic energy, plasma kinetic energy,
and plasma thermal energy,
and the magnetic helicity,
which is the scalar product
of the vector potential and the magnetic field.
In the case of incompressible fluids,
there is also the crosshelicity,
which is the scalar product
of the plasma velocity and the magnetic field.
In the presence of dissipation,
these invariants could undergo constrained changes.
These constrained variations could change one MHD configuration
to another topologically accessible configuration of lower energy.
These configurations are isolated configurations in the ideal case.
The continuous topological transformation
from one configuration to another
is only possible through dissipative paths,
such as magnetic reconnection.
Mathematically speaking, this evolution amounts
to the application of the variational principle on the total
energy density under the constraint of global magnetic
helicity conservation (Hasegawa 1985).
The exact processes of dissipation
in taking the system to selforganization
need not be specified under the variational description.
Tsui (2008) has described the equatorial plasma torus
of planetary nebulae from this selforganizational view,
by using temporally selfsimilar MHD solutions
with selfconsistent evolution functions
in spherical coordinates
with radially expanding plasma flow,
as a method and as a model.
For the present case, we follow the same apporach
but remove the axisymmetric assumption
to seek threedimensional selfsimilar solutions
to model jets, knots, and filaments.
Earlier works on temporally selfsimilar MHD
with selfconsistent evolution functions
include onedimensional (Osherovich et al. 1993, 1995)
and twodimensional (Tsui & Tavares 2005) cylindrical models
of magnetic ropes in the solarterrestrial environment,
twodimensional axisymmetric model
of atmospheric ball lightning in spherical coordinates
(Tsui 2006; Tsui et al. 2006),
active galactic nucleus jets in astrophysical plasmas
(Tsui & Serbeto 2007),
traditionally treated as a steady state accretionejection
MHD transport phenomenon from the accretion disk to the polar axis
(Blandford & Payne 1982; Pelletier & Pudritz 1992).
As for threedimensional selfsimilar MHD solutions,
there is one earlier case of interplanetary solar magnetic ropes
(Gibson & Low 1998),
which is not appropriate for astrophysical applications.
The basic MHD equations in a Eulerian fluid description are given by

(1) 

(2) 

(3) 

(4) 

(5) 

(6) 
Here,
is the mass density,
is the bulk velocity,
is the current density,
is the magnetic field,
p is the plasma pressure,
is the free space permeability,
is the polytropic index, and M is the central mass
that provides the gravitational field.
This set of MHD equations describes the selfconsistent
interactions of a conducting fluid with the magnetic field.
The fluid is governed by the NaviaStoke equations
coupled to the magnetic field
through the
magnetic force.
The magnetic field is governed
by the slowly time varying Maxwell equations
coupled to the fluid
through the
infinite conductivity condition.
We have modelled the planetary nebula gas
as a fully ionized plasma,
although there is neutral material in the gas.
If the gas is partially ionized,
the neutral components could be collisionally
coupled to the ionized component as a load.
Consequently, the neutral transport properties,
such as density and pressure profiles,
are related to the ionized component.
We consider a radially expanding plasma and seek selfsimilar
solutions in time where the time evolution is described by the
dimensionless evolution function y(t).
For this purpose, it is most convenient to think of
a Lagrangian fluid description,
and consider the position vector of a given
laminar flow fluid element .
Under selfsimilarity, the radial profile is time invariant
in terms of the radial label
,
which has the dimension of r.
Physically,
is the Lagrangian radial position
of a fixed fluid element.
With a finite plasma, the domain of
is bounded by mass
conservation

(7) 
As for the plasma velocity, we consider selfsimilar structures
derived from a spherically symmetric radial velocity which
can be written as

(8) 
Our selfsimilar parameter ,
defined through
the Lagrangian fluid label, explicitly represents
the fluid velocity by the time evolution function y(t).
We note that y(0)=1 is the initial condition,
so that
corresponds to the initial radial position.
This initial plasma sphere,
,
corresponds to the distribution in space of the ejected
fast wind from the hydrogen layer,
together with the entrained magnetic field lines,
throughout the entire strong pulsation period of the AGB star.
This initial sphere then expands selfsimilarly
in accordance with the evolution function y(t) such that
.
The evolution function will be solved selfconsistently
with respect to the spatial structures of the plasma.
Since we are considering an isotropic radial plasma flow,
we might wonder why structures other than unidimensional ones could arise.
Indeed, for a purely hydrodynamic system without instabilities,
only unidimensional configurations can emerge,
with radially dependent mass density
and plasma pressure p.
Nevertheless, for MHD systems, complex structures
with inhomogeneous mass density and plasma pressure will emerge,
because of the interactions between the conducting plasma
and the inherently two or threedimensional magnetic field.
The independent variables are now transformed from
to
.
We then determine the explicit
dependence of y on each one of the physical variables with
this radial velocity using functional analysis.
First, making use of Eqs. (8), (1) becomes
To reach the second equality, we note that the first bracket
in the first equality corresponds to the total time derivative
of an Eulerian fluid element which amounts to the time derivative
of a Lagrangian fluid element. As for the second bracket, it can
be reduced by using
and
.
Solving this equation for
the y scaling by separating the time part gives

(9b) 
Here, we have used the barred variables to denote the spatial part,
where the temporal part is explicitly solved
in terms of the evolution function.
As for Eq. (6), with
where
is a constant that carries the physical dimension so that F is
a dimensionless function, it follows

(10a) 

(10b) 
As for Eq. (3), with the aid of Eq. (5), the magnetic fields are







(11a) 

(11b) 







(12a) 

(12b) 

(13a) 

(13b) 
Making use of Eq. (9b), we derive the plasma pressure from Eq. (10b)
Since Eqs. (12a) and (13a) are of the same form, we conclude that,
under selfsimilarity,
is a linear function of
with

(15) 
Making use of Eq. (4) to eliminate the current density in Eq. (2),
we obtain the momentum equation which has three components.
The ,
,
and r components are respectively







(16) 







(17) 







(18) 
We have reduced the general set of timedependent ideal
MHD equations, Eqs. (1)(6), to a set of selfsimilar equations with
appropriate time scalings, Eqs. (7)(13).
The general ideal MHD set has nonlinear terms of convective
type
.
By using the fluid label description, the
convective terms are absorbed in the Lagrangian time derivative
representation.
The structure of the nonlinear terms,
absorbed in the Lagrangian fluid label formulation,
will appear in the profile of the system.
After this selfsimilar formulation,
we have to solve Eqs. (16)(18) for the selfsimilar configurations.
For this, we first separate the radial variable
from the other two variables by writing

(19) 
where we have used tilded variables for the angular part,
and likewise for
and
.
Furthermore, we take

(20) 
Here, ,
,
and
are dimensionless functions,
and A_{0} and p_{0} carry the dimensions of magnetic
field and pressure respectively.
Since magnetic pressure is a quadratic quantity
of the magnetic field,
we have used this to write
the plasma pressure as .
We take A_{0}=1 for unit amplitude magnetic fields,
such that p_{0} is relative to this amplitude.
Specifically, we take

(21) 
to represent a power law field decaying with distance,
where a is a normalizing parameter of .
Considering
Eqs. (16), (17) are respectively







(23) 
We now separate the azimuthal dependence by writing

(24a) 

(24b) 

(24c) 
where
.
These functional
dependences give Eq. (22) as



(25a) 
Considering the separation constant im, such that

(26b) 

(25) 
we then have



(26) 
which identifies .
Following the same procedures,
Eq. (23) reads



(27) 
The left sides of these two equations are the same,
which allows the right sides to be equated to give

(28a) 

(28b) 
To solve for
,
we integrate Eq. (28a) to get
where we have multiplied and divided the right side by
to implement the integration, and
.
This solution of
is singular at x=+1 for positive m, and x=1 for negative m.
Such a solution results in jet features
on the magnetic field lines and plasma density.
We note that
is a composite variable
that contains
of the radial magnetic field
and
of the plasma pressure.
This composite variable
,
therefore,
reflects the partition of the magnetic energy
and the plasma thermal energy through p_{0}.
To get ,
instead of solving either Eqs. (26) or (27),
we make use of Eq. (5) to get



(30a) 

(30b) 
Subsituting Eqs. (30b) into (26) and (27) respectively gives

(31a) 

(31b) 
It can be shown readily that these two equations
are equivalent to Eq. (28a).
We note that Eq. (30b) couples
with
.
To decouple these two functions,
we consider the special case of
Coincidentally, although this gives an inverse squared
dependence on distance of
by Eq. (21),
this is not the motivation for choosing n=2.
The basic motivation is to decouple Eq. (30b) from
.
The case of
is discussed in Sect. 7.
Multiplying by
and defining
,
Eq. (30b) can be integrated to give

(33a) 

(33b) 

(33c) 

(33d) 
where Eq. (33c) is obtained from Eq. (31a), and
can be recovered from
of Eq. (28b).
From these solutions, we see that B_{r} and p are singular
at x=+1 through
and
,
whereas
and
are singular
at x=1 through .
A singular magnetic field by itself,
such as a dipole field at the center, is acceptable.
To avoid unphysical results,
we require the magnetic flux be finite
even though the magnetic field is infinite.
In other words, we require the singularities
be integrable in x,
which demands the power of the singularities be less than unity.
By inspection of the terms, we conclude that m<1/3.
Let us take, as an example,

(34) 
to show
,
,
and
in Figs. 13 respectively.
In Fig. 1, the amplitude of the physical quantity
is plotted against the polar angle
so that the projections of the plot
on the x and y axes give
and
respectively.
In this polar plot, the magnitude of the physical variable
is given by the radial length from the origin.
The y and x axes in the figure stand for
the polar z axis and the projected radius
on the equatorial plane of the planetary nebula.

Figure 1:
The function
with
against the polar angle
with projections on the vertical and horizontal axes
and
respectively,
indicating an integrable singularity of finite flux
at x=+1 for a jet structure. 
Open with DEXTER 

Figure 2:
The function
with
in a polar plot indicating an integrable singularity
at x=+1 for a jet structure. 
Open with DEXTER 
Figures 1 and 2 show an integrable singularity at x=+1,
or .
Furthermore, Fig. 3 shows a weaker singularity at x=+1 than the one of Fig. 2.
However, Fig. 3 shows another singularity at x=1,
while Fig. 2 is regular at that location.
As for
,
the second term of Eq. (33d)
exceeds the first term when
or as x gets very close to unity.
Writing
,
we get
,
or
.
With Eq. (34), we have
.
Consequently,
gets smaller as x approaches unity, and it vanishes at
.
Beyond this point, it becomes negative,
giving a jet structure as in Fig. 4.
This jet structure is along the radial magnetic field
given by
.
This negative sign can be absorbed
in the
factor of Eq. (24c)
as a phase shift by writing

Figure 3:
The function
with
indicating integrable singularities
at x=+1 and at x=1 for knot structures. 
Open with DEXTER 
As for the radial component, with
,
Eq. (18) reads
With
as the separation constant, we have
We write the mass density as

(37) 
where
carries the dimension of mass density,
and
and
are dimensionless functions.
We can identify immediately from Eq. (35a) that

(38a) 
The second equation reads



(39) 
This mass density has a singularity at x=+1.
With m=1/4, we note that the numerator of the last term
in Eq. (38),
(1+x)^{(2m1)/2}, has a negative power.
This gives a singularity at x=1. Since the power of
this singularity is 1/4, it is also integrable.
The mass density distribution is shown in Fig. 5
with a jet structure in the x=1 direction,
because of the radial magnetic field.
If we consider m<0,
Figs. 15 would be inverted.
Consequently, the jet structures would be on both sides
of the polar axis.

Figure 4:
The function
with
indicating a plasma pressure jet
in a very narrow cone about x=+1. 
Open with DEXTER 

Figure 5:
The function
with
indicating a mass density jet
in a very narrow cone about x=+1. 
Open with DEXTER 
For the time evolution function of Eq. (36b),
we multiply over by d
to get

(40a) 
where H is an integration constant
which stands for the energy density of the system.
This equation is subject to the initial condition
of y(0)=1 by definition of the Lagrangian fluid label.
The solution of the evolution function
depends on the nature of the physical system,
represented by
and H.
Considering that the stellar wind is launched
above the escape velocity
together with the magnetic fields,
H would be positive.
This implies that planetary nebulae
are open systems that expand indefinitely.
Let us consider
positive,
such that
is a monotonically decreasing function.
Since the right side of Eq. (40a) has to be positive,
we conclude that
.
This gives the plasma velocity a slow start
with
that finishes with a fast terminal velocity

(40b) 
which is qualitatively compatible with observations
of high momentum and energy (Bujarrabal et al. 2001).
We have modelled the stellar wind by an isotropic radial flow
which could be driven by the stellar magnetic field
with unspecified topology.
Driven by selforganization,
the interactions between the conducting plasma
and the magnetic fields
steer the MHD system to a highly inhomogeneous
spatial distribution
with plasma jets in the polar directions,
although the plasma velocity is radially symmetric.
This redistribution in space among the magnetic energy,
plasma kinetic energy, and plasma thermal energy
is the basic mechanism that drives the jet momentum,
expressed through the time evolution function.
For this reason, we are dealing with a spatially
focusing effect,
where energies are rearranged in space,
instead of a plasma accelerating mechanism.
With the momentum equation solved, the magnetic fields
are given by

(41a) 

(41b) 

(41c) 
The magnetic field lines are given by

(42a) 
which can be written as
With
as an integration constant, the first equality gives

(43a) 
which can be integrated numerically, as is shown in Fig. 6.
Since the singularity is integrable,
is finite at x=+1.
This result in the radial
can then be converted to
radial position r(t) through the selfconsistently
determined evolution function y(t) by
.
The second equality corresponds to
and can be integrated to give

(43b) 
where K is an integration constant.
With the interval of x between (1,+1),
or
between ,
the left side of Eq. (43b),
(1+x)^{m}/(1x)^{m} labelled on the left axis,
covers an interval
with positive m,
and is plotted in Fig. 7 against x between (1,+1) labelled on the bottom axis.
The right side,
labelled on the right axis,
is also plotted against
on the top axis,
with a scale between 0 and ,
in the same figure.
In order to view the mapping
between the right side and the left side,
we assign a large constant K.
With m=1/4 and K=3, as x departs from 1,
or
from ,
departs from 0,
and it maps a root of .
As x approaches +1, or
approaches 0,
reaches ,
which takes
over .
On the return path of the field lines,
x decreases from +1 back to 1,
bringing
from
to
along the descending branch of
,
which takes
over .
This descending branch, which is the continuation of Fig. 7,
is not shown.
To summarize, the field lines starting at x=1 and
go through the
x=(1,+1,1) cycle once,
with
,
while completing the
cycle once,
covering
,
before closing on themselves again.
This generates helical field lines, as shown in Fig. 8,
on the surface of revolution of Fig. 6.
Since the singularities at
are integrable,
and also because of the circulating nature of
the fields
and
,
the magnetic field lines converge to
at x=1 axis and to
at x=+1 axis, as shown in Fig. 6.
These locations correspond to pointsymmetric magnetic
knots, where the field strength is infinite.
The filaments correspond to the helical
magnetic field lines in space, as shown in Fig. 8.
These same field lines have a different shape
when they are viewed at different orientations,
such as in Figs. 9 and 10.
These field lines in terms of
expand in space
with radial position r(t) according to
.
Further field lines can be generated with, for example,
m=1/3.5=2/7=4/14.
In this case, the field lines starting at x=1 and
go through the
x=(1,+1,1) cycle once,
with
,
while completing the
cycle once,
covering
.
Since the lowest
multiple of the cycle is four,
the field lines have to complete four cycles of x and
of
to make
cover
,
such that the field lines can close on themselves again.

Figure 8:
The threedimensional magnetic field lines,
wound on the surface of revolution of Fig. 6,
viewed parallel to the xy plane at about 45 degrees. 
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Figure 9:
The threedimensional magnetic field lines,
wound on the surface of revolution of Fig. 6,
viewed down the z axis. 
Open with DEXTER 
A range of
in space corresponds to a range of a.
With a range of
of Eq. (43a),
we have a distribution of ,
and we can fill the space with shells of field lines
of some m<1/3 up to
of Eq. (7),
and generate a sequence of magnetic knots
on the downward polar axis,
.
Despite the radial component at x=+1 giving the plasma
jet structures, the field lines are closed
at a line segment
on the upward polar axis, ,
because of the circulating nature of the meridian and
azimuthal components.
With m<0, the mirror images of the knots and
field lines can be superimposed on those with m>0,
generating more lines of knots and concentric shells
of magnetic field lines.
We now solve Eq. (30b) for an arbitrary n.
Combining Eqs. (30b) and (31a), we get



(44a) 
where
and
is given by Eq. (29). If the boundary condition is known,
this equation can be integrated numerically to get



(44b) 
To obtain the boundary condition at x=1, we note that
the right side of Eq. (44a) is equal to
2m(n2)(1+x)^{m+1}/(1x)^{m1}.
Since m<1 but positive, this term vanishes at x=1 and
at x=+1 as well.
As a result, in the neighborhood of ,
P^{2}(x) is described by the homogeneous version of Eq. (44a),
where the right side is null. The homogeneous solution
can be solved readily as

(45) 
which provides the needed boundary condition.
The selfsimilar functions are, therefore, given by

(46a) 

(46b) 

(46c) 
The magnetic field lines are described by

(47a) 

(47b) 
The second equation remains the same, while the first equation
can be integrated numerically to get the field lines.
With n=3, the corresponding selfsimilar functions
of Figs. 26 are evaluated anew, and are presented in Figs. 1115.
From Figs. 12 and 15, we can see that the
source term in Eq. (44a) makes
and the
mapping more symmetric.
The corresponding magnetic field lines of Figs. 810
are also shown in Figs. 1618, with knots,
and ,
symmetrically placed at equal distances from the center.

Figure 10:
The threedimensional magnetic field lines,
wounded on the surface of revolution of Fig. 6,
viewed along the x axis. 
Open with DEXTER 

Figure 11:
The function
,
with n=3,
indicating an integrable singularity
at x=+1 for a jet structure. 
Open with DEXTER 

Figure 12:
The function ,
with n=3,
indicating a more symmetric structure. 
Open with DEXTER 

Figure 13:
The function
,
with n=3,
indicating a plasma pressure jet
in a very narrow cone about x=+1. 
Open with DEXTER 

Figure 14:
The function
with n=3 indicating a mass density jet
in a very narrow cone about x=+1. 
Open with DEXTER 

Figure 15:
The magnetic field line
dependence,
with n=3,
showing a symmetric structure. 
Open with DEXTER 

Figure 16:
The threedimensional magnetic field lines,
with n=3,
viewed parallel to the xy plane at about 45 degrees. 
Open with DEXTER 

Figure 17:
The threedimensional magnetic field lines,
with n=3,
viewed down the z axis. 
Open with DEXTER 

Figure 18:
The threedimensional magnetic field lines,
with n=3, viewed along the x axis. 
Open with DEXTER 
We have presented an analysis that describes planetary nebula
morphologies as threedimensional selforganized structures.
Analytically, because of selforganization,
the morphologies cannot be linked
to the central AGB star in a deterministic way,
although numerically we can always simulate turbulence
from the beginning to reach the end states,
given enough computing power and time,
as in meteorological models of hurricane formation.
Because of our mostly Newtonian scientific tradition
with a direct and explicit link
between cause and consequence,
we are not accustomed to see phenomena
from a selforganizational perspective.
The situation is somewhat analogous to
dealing with uncertainty principle and duality of light
in quantum mechanics.
In order to get the end configurations without specifying
the initial conditions,
we have used selfsimilar MHD solutions with selfconsistent
evolution functions to represent the selforganized states
emerging from turbulence.
In our model, the key role of plasma pressure
is represented in the momentum equation, Eq. (2),
which is solved together with the magnetic field.
Although the plasma velocity is isotropically radial,
due to selforganization,
complex threedimensional structures can arise
because of the interactions
between magnetic fields and the conducting plasma.
These structures correspond to the spatial redistribution
of plasma and magnetic fields,
which amounts to focusing of magnetic energy,
plasma thermal energy, and plasma momentum in space.
With this selforganization model
through selfsimilar configurations,
we have revealed that magnetic fields and plasma
are structured in a narrow cone along the polar axis.
The magnetic field strength, plasma pressure, and mass density
are singular there but integrable.
The plasma pressure and mass density along the jet
decrease monotonically with distance according to a power law.
The resulting plasma jets are under the local pressure
of the magnetic field, and, as a result,
they can carry large amounts of momentum and energy.
Some examples of these energetic jets
appear in the M29 Twin Jet Nebula,
CRL 2688 Egg Nebula, NGC 3242, NGC 6826, NGC 7009.
The magnetic fields have their lines of force
circulating in space forming filaments
that converge at given locations on the ploar axis,
giving pointsymmetric knots.
The radial field, which is also singular on the axis,
preserves the divergencefree nature of the magnetic knots.
Examples of knots can be found in the M29 Twin Jet Nebula,
NGC 5307, and filaments in MyCn 18 Hourglass Nebula,
NGC 6543 Cat's Eye Nebula, NGC 2392 Eskimo Nebula,
M29 Twin Jet Nebula, NGC 6543.
In a recent publication (Tsui & Serbeto 2007)
of an axisymmetric twodimensional model,
extragalactic polar jets are described
from the same selforganizational perspective
through an eigenvalue equation with regular eigenfunctions.
Jets are driven by plasma pressure to
progressively collimate along the polar axis.
In particular, most of the ejected mass is distributed
incompact plasma tori along the polar jets.
The fact that jets, knots, and other sturctures
are ubiquitous and reproducible phenomena
on stellar and galactic scales has led us
to believe that there is an underlying order
by selforganization.
Without the astrophysical observations,
these configurations would be just exercises
of selfsimilar mathematics.
Under the matching morphologies,
we may see planetary nebulae
as the result of selforganization,
rather than being direct consequences
of the AGB central engine.
Selfsimilarity with selfconsistent evolution functions
is sufficient to reveal the structures.
Acknowledgements
The author is deeply grateful to Dr. B. C. Low for the inspiring
thoughts and physical insights of selfsimilar solutions,
and to Prof. Akira Hasegawa for the very essential concept
of selforganization in fluids and plasmas.
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