2
nd
Chaotic Modeling and Simulation International Conference, 1-5 June 2009, Chania Crete Greece
1
A TRANSMISSION LINE MODEL FOR THE SPHERICAL BELTRAMI
PROBLEM
C. D. Papageorgiou
1
, T. E. Raptis
2
1
Dept. of Electrical & Electronic Engineering,
National Technical University of Athens, Greece
E-mail:
chrpapa@central.ntua.gr
2
Division of Applied Technologies ,
National Centre for Science and Research “Demokritos”,
Patriarchou Grigoriou & Neapoleos, Athens, Greece
E-mail:
rtheo@dat.demokritos.gr
Abstract
We extend a previously introduced model for finding eigenvalues and eigenfunctions of PDEs with
a certain natural symmetry set based on an analysis of an equivalent transmission line circuit. This
was previously applied with success in the case of optical fibers [8], [9] as well as in the case of a
linear Schroedinger equation [10], [11] and recently in the case of spherical symmetry (Ball
Lightning) [12]. We explore the interpretation of eigenvalues as resonances of the corresponding
transmission line model. We use the generic Beltrami problem of non-constant eigen-vorticity in
spherical coordinates as a test bed and we locate the bound states and the eigen-vorticity functions.
1. Introduction
The notion of a Beltrami field appears first in hydrodynamics in the studies of Eugenio Beltrami
[1] at the end of the 19
th
century where it has found widespread application in vortical fluid states
with a characteristic helical geometry. Its rediscovery in modern electromagnetism took place at the
middle of the 20
th
century when Lundquist [2] proposed the same model for galactic magnetic fields
where the field appears to be parallel to its own rotation (neglecting displacement current). Later,
Lust and Schluter [3] and Chandrasekhar and Woltjier [4] proposed a theoretical justification of
these states in terms of magnetic plasma equilibrium. Several types of solutions are reviewed in [6]
and [7].
Nowadays, the Beltrami equation serves as a model of
a) “Force-Free Fields” for which the magnetic part of the Lorentz force disappears (J // B) in
equilibrated plasma and
b) for parallel E // B components when applied to the vector potential directly (in which case we get
A
r
B
A
E
),(, t
λ
ω
=
−
=
).
The Beltrami problem is defined by at least two and at most three PDEs given by
A
r
A
),( t
λ
=
×
∇
(1a)
0
=
•
∇
=
•
∇
AA
λ
(1b)
The second condition is usefull in applications in electromagnetism when A stands either for the
vector potential or the magnetic field. Applying twice the rotation operator in the first of (1) gives
the most general representation of the non-linear inhomogeneous Helmholtz operator for Beltrami
flows as
2
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Chaotic Modeling and Simulation International Conference, 1-5 June 2009, Chania Crete Greece
2
AArA
×∇=−∇
λλ
),(
22
t
(2)
In the case of constant
λ
the inhomogeneous part is of course zero. Solutions of this class are usually
termed linear “
Trkalian
” fields as they were first studied extensively by Viktor Trkal [5].
In the next paragraph we turn to equation (1) and attempt to locate eigen-modes in terms of the
orthogonal basis of vector spherical harmonics. In section 2, we perform the expansion and derive
the associated ODEs and in section 3, we introduce the transmission line model for this system. In
section 4, we solve the model for the associated angular momentum eigenvalue and we approximate
the specific eigen-mode for a “bound state”.
2.
Spherical expansion of the Beltrami equation
In what follows we restrict attention to purely radial functions for the eigen-vorticity
λ
(r).
We
introduce the usual basis of Vector Spherical Harmonics as },{
, lmlmlm
r
Ψ∇×=ΦΨ∇=ΨΨ=Υ
rr
r
r
r
and we seek an appropriate decomposition of the field A and its eigen-vorticity
λ
. Then there must
be a unique reference frame for which both equations (1a-b) can be satisfied by taking
[
]
∑∑
−=
∞
=
Φ+Ψ+Υ=
l
lm
lmlmlmlmlmlm
l
rcrbra
r
r
r
)()()(
0
A
(5)
Then condition 0
=
∇
A is satisfied if
0
)1(
2
0)(
)1(
2
0
=
+
−+
=Ψ
+
−
+=∇
∑ ∑
∞
= −=
lm
lm
lm
l
l
lm
lmlm
lm
lm
b
r
ll
r
a
a
rb
r
ll
r
a
a
&
&
A
(6a)
while the choice of the λ function is restricted by 0
=
•
∇
A
λ
. For a purely radial λ function it
suffices to take the additional condition that the radial part of
A is set to zero which is a very
stringent condition on the coefficients of (5). We note however that in the more general case we may
allow for violations of condition (1b). In electromagnetic applications we may instead assume the
presence of static charges satisfying the Lorenz gauge
AA •∇=∇−=∂
−
λλφ
1
cc
t
.
From now on we will deal solely with the first condition (1a) for the rotation of
A
. We now
turn back to the first of (1) for the rotation operator which obtains
∑ ∑
∞
= −=
Φ
−++Ψ
+−Υ
+
−=×∇
0
)1(
l
l
lm
lm
lmlm
lmlm
lm
lmlmlm
r
a
r
b
b
r
c
cc
r
ll
r
&
r
&
r
A
(7)
For the rotation to be parallel to
λ
A we should have
0
0
)1(
=−−+
=++
=
+
−
lm
lmlm
lm
lm
lm
lm
lmlm
c
r
a
r
b
b
b
r
c
c
ac
r
ll
λ
λ
λ
&
&
(9)
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Chaotic Modeling and Simulation International Conference, 1-5 June 2009, Chania Crete Greece
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Using the first of (9) results in the ODE system
)()(
)(
)1(
)(
2
rbrc
rc
r
ll
rb
λ
λ
λ
−=
′
+
−=
′
(10)
3.
The Transmission Line model
At this section we introduce an isomorphism between the system (10) and a lossless transmission
line via the mapping
V
dr
dI
rbI
I
dr
dV
rcV
λ
γ
λ
j
j
j
2
−=→=
−=→=
(14)
where we have introduced the propagation constant
)1(,
2
2
2
2
+=−= llL
r
L
λγ
. Equivalently we may
define the transmittances to be
λ
γ
λ
2
1
1
j
j
=Υ
−
=
Ζ
(15)
Then the complex impedance is defined through
2
11
22
2
1
1
1
γ
γ
γ
λ
=ΥΖ
−==
Υ
Ζ
=Ζ
r
L
jj
(16)
From standard transmission line theory we have that propagation is defined for
)1(
||
0
2
+
><
ll
ror
λ
γ
(18a)
while damping (evanescent modes) for
)1(
||
0
2
+
<>
ll
ror
λ
γ
(18b)
An element of such a transmission line corresponds to a lumped-element model of the initial domain
[0, ∞] of the radius r into a set of mesh points
n
rrr ,...,,
10
with each element interpolating into the
arbitrary intervals ],[
1
+
nn
rr .We then take each element to correspond to a segment of infinitesimal
width
nn
rrr
−
=
∆
+1
for which the γ factor can be taken approximately constant. For such an element
we may take the equivalent relations
2
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Chaotic Modeling and Simulation International Conference, 1-5 June 2009, Chania Crete Greece
4
=
+
+
)(
)(
)cosh()sinh(
)sinh()cosh(
)(
)(
1
1
n
n
nn
nn
n
n
rI
rV
rI
rV
ξξ
ξξ
(19)
where we have put rn
n
∆= )(
γξ
. The value of the impedance at each point of the line will be given
by the recursive formula
nnn
nnn
n
n
n
n
ZrZ
ZrZ
Z
rI
rV
rZ
+
+
==
+
+
)tanh()(
)tanh()(
)(
)(
)(
1
1
ξ
ξ
(21)
and the boundary conditions are defined through the respective limits )(),0(
∞
→
→
rZrZ based on
the assumption that the asymptotic limit of a lossless transmission line corresponds to the fixed point
of (19) given by
n
Z according to (16).
4.
Trapped modes and bound states
The form of the propagation constant leads to the possibility of “trapped” waves and their
respective eigenmodes. A necessary condition for trapping is to have at least one region
21
rrr ≤≤
where
(
)
0
2
<r
γ
allowing propagation and two disparate regions
21
, rrrr ><
with
(
)
0
2
>r
γ
where
only evanescent modes exist.
This situation is reminiscent of the bound states of quantum mechanical systems with the
function )(
2
r
λ
playing a role analogous to that of the potential in the radial Schroedinger equation.
Evanescent modes are then analogous to the scattered wave functions above a threshold. On the
other hand, a constant value of λ allows only a simple root
03
/)1(
λ
+= llr
beyond which
propagation is allowed up to the infinity. We will next evaluate one such example of an eigen-
vorticity function supporting a trapped mode.
According to a standard technique introduced in [11] and [12], a method for locating the eigen-
values and compute the associated eigen-mode is based on a simple resonance theory for
transmission lines. In this viewpoint, we may take the bound state to correspond to an area where
ordinary resonance takes place through the condition
CL
ZZ = between the “capacitive” and
“inductive” elements in the imaginary part of the respective complex impedance. This can
effectively be restated as
{
}
0)()(Im
=+
−+
cc
rZrZ
(22)
Evolution of the above quantity in the region
],[
21
rr
prescribes also the associated eigen-mode. In
order to locate a certain mode for a given value of angular momentum we iterate the mapping (21)
forward and backward (
nn
rr −→
) in the interval
],[
21
rr
and expand the interval until we find a root
of (22).
A simple choice of a
γ
function with a trapping region is given by
rrrr
rrrrL
21
4
21
2
2
))((
+
−−
=
γ
(23)
2
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Chaotic Modeling and Simulation International Conference, 1-5 June 2009, Chania Crete Greece
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1212211
2
2
1
2
2
/),/(1
,
)(
RrrRrrR
RrRr
L
=+=
+
=
λ
(24)
In Fig. 1 we show the exact form of the propagation factor while for the eigen-vorticity we show
that
2
λ
is positive definite everywhere. In Fig. 2 and 3 we show the evolution of the difference of
the imaginary parts for 1
=
l
and 2
=
l
and boundary conditions
9
10)(,0)0(
j≅∞=
ZZ
.
5.
Beltrami solutions through the Lorentz group
From the previous analysis a fundamental observation occurs based on the hyperbolic relation
connecting the triplet
{
}
rL /,,
λγ
which can be written as
0)()(
2
2
22
=−+
r
L
rr
λγ
(25)
Going to the transformed variables
(26)
leads to the Lorentzian manifold M(2,1) with the usual Minkowski metric of signature (1,1,-1) that
has an invariant “null” length (virtual photon path)
0)(
2222
=−+=
LTYXs
(27)
and a “speed-of-light” c =
L
.
This means that given a set of “core” functions },{
00
λγ
and a angular momentum value
L
we can
produce an infinite family of deformed solutions which are connected by both proper and improper
Lorentz transforms. Thus, for the choice represented by (23) we may find a continuous trajectory
passing through different families of functions },{
nn
λγ
by a direct application of the generic linear
transformation ),(),(:),(),(
11
)()(
++
ΛΛ
→
′′
→
νννν
λγλγ
LL
TXTX
where the matrix Λ(
L
) is
parametrized according to the four conjugacy classes of the full Lorentz group (elliptic, hyperbolic,
parabolic and Loxodromic) thus leading to different families of solutions.
6.
Conclusions
We showed that a general class of solutions of the spherical Beltrami problem can be found in terms
of vector spherical harmonics of which the resulting system is equivalent with a generic lossless
transmission line. When applied to a specific choice of eigen-vorticity function it allows finding the
set of parameters necessary for each mode to exist. We have given explicitly an example of such a
function for which trapped modes can exist in a region. We believe that this phenomenon deserves
further examination especially with recourse to electromagnetic applications.
References
rT
rY
rX
/
)(
)(
2
2
j=
=
=
λ
γ