J. Mod. Phys., 2010, 1, 324-327
doi:10.4236/jmp.2010.15046 Published Online November 2010 (http://www.SciRP.org/journal/jmp)
Slow Plasma Dynamo Driven by Electric Current Helicity
in Non-Compact Riemann Surfaces of Negative Curvature
Luiz Carlos Garcia de Andrade
Departamento de Fisica Teorica-IF-UERJ- RJ, Maracana, Brasil
Received January 14, 2010; revised Feburary 3, 2010; accepted Feburary 1, 2010
Abstract
Boozer addressed the role of magnetic helicity in dynamos [1]. He pointed out that the magnetic helicity
conservation implies that the dynamo action is more easily attainable if the electric potential varies over the
surface of the dynamo. This provided motivated us to investigate dynamos in Riemannian curved surfaces
[2]. Thiffeault and Boozer [3] discussed the onset of dissipation in kinematic dynamos. In this paper, when
curvature is constant and negative, a simple laminar dynamo solution is obtained on the flow topology of a
Poincare disk, whose Gauss curvature is K = –1. By considering a laminar plasma dynamo [4] the electric
current helicity λ ≈ 2.34 m–1 for a Reynolds magnetic number of Rm ≈ 210 and a growth rate of magnetic
field |γ| ≈ 0.022 are obtained. Negative constant curvature non-compact H2 manifold, has also been used in
onecomponent electron 2D plasma by Fantoni and Tellez [5]. Chicone et al. (CMP (1997)) showed fast dy-
namos can be supported in compact H2. PACS: 47.65.Md.
Keywords: Plasma, Dynamos
1. Introduction
Earlier Boozer [1] has investigated magnetic helicity
driven dynamos, where the magnetic helicity constraint
is enhanced if the electric potential varies over surface of
the dynamo. He argues that in the case of the Earth, the
north-south pole variation posseses an electric potential
which varies a hundred volts. Recently helicity constraints
have also been investigated by Thiffeault and Boozer [3]
where the dissipation is taken into account. In their case,
they found that helicity generation terms are exponential
smaller than energy dissipation, so that large amounts of
energy are dissipated before any helicity can be created.
In this paper, use is made of the Riemannian geometry of
Cauchy metric in the chaotic plasma flows, where the
magnetic field is stretched in the plasma flow [6]. In their
case the high conducting fluid, with high magnetic Rey-
nolds numbers Rm of the order 108-1015, and conse-
quently very low dissipation is used.
Here one addresses the converse issue and considers
the case of a non-ideal plasma where the dynamo action
survives on the Riemannian manifold of negative con-
stant curvature in the form of a Lobachevsky plane. As it
is wellknown a compact surface can be given by a torus
and sphere, while noncompct srfaces can be given by
paraboloids or the Lobachevski plane discussed here.
This kind of negative Riemann curvature geometry in the
form of a paraboloid can be easily shown to focusing the
magnetic field orthogonal to its surface. It is important to
stress that this does not happen in the Euclidean plane, or
the spherical surface where the magnetic field lines or-
thogonal to their respective surfaces remain parallel or
diverge. This provides also another strong fountain of
motivation for investigating the magnetic flows in geo-
desic dynamos in Riemannian spaces of negative con-
stant curvature. Since as shown by Chicone et al. [7]
even fast dynamos can be supported in Riemannian
compact 2D manifolds of constant negative curvature,
the Cowling anti-fast dynamo theorem for 2D surfaces is
not violated here.
Anti-fast dynamo theorems have also been addressed
by Garcia de Andrade [2]. The slow dynamo flows ob-
tained here are shear flows, which can also be obtained
by the stretch-fold-shear dynamo mechanism investi-
gated previously by Bayly, Childress [8] and Gilbert [9].
In this paper the absence of advection terms is due to the
presence of a comoving term which makes the spatial
flow vanishes. This kind of frame is very well known in
cosmology and can be used in near future to investigate
cosmological dynamos. The dynamo flow used here is

L. C. G. D. ANDRADE 325
certainly more complex that the simple uniform stretch-
ing dynamo flow investigated by Arnold et al. [10]. An-
other motivation for the use of negative constant Rie-
mann curvature dynamo plasma surfaces, has been the
one-component two-dimensional plasma by Fantoni and
Tellez [5] in the realm of electron plasmas in 2D. Here,
as happens in general relativity the plasma undergoes a
Coriolis force which is given by the presence of the cur-
vilinear coordinates effects present in the Riemann-
Christoffel symbol in the MHD dynamo equation. In
their non-relativistic plasma limit, this geometry has
been used by Fantoni and Tellez [5] in the context of
plasma physics. They have used a Flamm’s paraboloid,
which is a noncompact manifold which represents the
spatial Schwarzschild black hole, to investigate one-
component two-dimensional plasmas.
Restoring forces and magnetic field reversals possibil-
ity are also discussed in 3D slow dynamo curved sur-
faces. Recently another sort of slow dynamos in liquid
sodium laboratory has been modelling by Shukurov at al
[11], by embedding a Moebius strip flow in the three-
dimensional space. The paper is organized as follows:
Section 2 presents the mathematical formalism necessary
to grasp the rest of the paper. In the next section the slow
dynamo solution is presented as well as the non-geodesic
equations is computed. In this Section 3, the sign of
magnetic helicity in the exponential growth of the slow
dynamo is shown to be important to the slow dynamo
action. Both helicities are computed on the hyperbolic
Poincare disk. Discussions and conclusions are presented in
Section 4.
2. Slow Dynamo Plasmas in Curved Surfaces
In the Euclidean three-dimensional space R3 described
by Lobachevsky plane geometry can be presented here
for the benefit of non-mathematically inclined reader.
The Lobachevsky metric is given by
2222
dsy dx dy



(1)
where H2 = (w = x + iy; y > 0) is the hyperbolic plane in
its half-upper part. Here 1i is the imaginary unit
of the complex plane C. The Ricci ten Lobachevsky met-
ric
11 2
1
Ry  (2)
22 2
1
Ry  (3)
2R (4)
12
21 22
1
y
 (5)
2
11
1
y

 (6)
Riemann curvature tensor is given by
1212 4
1
Ry  (7)
The Kretschmann scalar invariant, so much used in
GR to determine whether a singularity is not a true sin-
gularity or a horizon, just in Schwarzschild black hole
geometry, is given by
1212
1212 1RRR

 (8)
which shows that the line y = 0 represents a fake singu-
larity or an event horizon of the 2D section of the uni-
verse. The process by which the particles are stretched in
the plasma flow to give rise to dynamo, is the geodesic
equation
2
2() 0
dJ KsJ
ds

 (9)
whose solution for the negative curvature hyperbolic
space is


00
()sinsin ()
J
sJh KsJhs (10)
Note that the force-free dynamo equation yields
2
()BcurlcurlB B

 (11)
where curlB B


(12)
is the force-free Beltrami equation. From the assumption
that the comoving frame is used here, one obtains


0cur lVB (13)
The expression for the self-induction equation is

BB VB 

 

(14)
where 2

 is the Laplacian in general curvilinear
coordinates. Therefore the calculation of this term shall
be fundamental in our case. Let us expand this term in
terms of Cartesian coordinates Laplacian
22
F
lat xy

 (15)
and the Riemann-Christo_el connection
,, ,
1
2
iil
j
kljklkjjkl
gg gg 



 (16)
where (i; j = 1,2,3). Now let us consider the above MHD
dynamo equation in curvilinear coordinates. Since the
advection term is in principle not present, our first worry
should be to compute the first

1ij
ij
BggB
g
 (17)

L. C. G. D. ANDRADE 326
Here, the rate of the ampli_cation of the magnetic field
from the ansatz
0()rt
BBxe (18)
The covariant expression for the Laplacian operator
then becomes
2
ij j
ij
F
g

  

(19)
here
F
 is the at gradient in Cartesian (x,y) coordinates.
Here

:
iij i
j
jk
gTr  (20)
is the trace of the above Riemann-Christoffel symbol. To
derive the expression (19) one used the Riemannian ge-
ometry identity for the trace of Riemann-Christoffel sys-
tem
1
:
ii
g
g


 

 (21)
By taking the solenoidal constraint on the magnetic
field divB = 0 one obtains the form of the field as
2
0
iirt
BBey (22)
Note that this expression shows that, unless the y co-
ordinate is bounded the magnetic field grows spatially
without bounds. Since the only constraint on y is that it is
positive, this certainly may be the case. If one uses the
Riemann-Christo_el connections of the above Lo-
bachevsky-Poincar? hyperbolic disk, one may find the
first two terms in the general Laplace-Beltrami operator.
By using the force-free condition above one obtains the
dynamo equation as
2ii
BB   (23)
The Maxwell magnetic two-form F is
:ij
ijxy z
F
F dxdxB dydzBdzdxB dxdy
(24)
where  symbol means the wedge skewsymmetric
product. Note that the Bz is the component of the mag-
netic field orthogonal to the Lobachevsky plane. Focus-
ing of the negative curvature surface magnetic field can
be done by the orthogonal magnetic fields to its surface.
A simple drawing of the paraboloid can show that these
magnetic lines converge to some focusing point outside
the paraboloid. It is easy to check that the expression (21)
yields a solution for the dynamo Equation (22) as long as
2

  (25)
which shows that by the slow dynamo condition
0
limRe( )0


 (26)
Of course in fast dynamos expression (26) would be
positive. Here Re represents the real part of the growth
rate scalar. Note that from this expression the constraint
0 implies that either the dynamo slowliness is enhanced
or the dynamo is marginal (2

= 0). This result is ob-
tained since the slow dynamo criteria predominates over
magnetic field decay.
3. Electric and Magnetic Helicities and
Force-Free Slow Dynamos
Now let us compute the electric current helicity

which in the force-free dynamo case is given by
2
.jB
B
  (27)
here by Maxwell equations the electric current j is given
by
jB

 (28)
From the closed two form dB = 0 of the magnetic field
yields
:
x
yz zy
BAA

 (29)
:
y
xz zx
BAA

  (30)
:
z
xy yx
BAA

 (31)
From this definition one is able to determine the elec-
tric helicity and the magnetic helicity H [6]
1[.]
H
AB
g
 (32)
Since
x
yx z
jAB

 (33)
and Bx the electric helicity vanishes while the magnetic
helicity can be expressed in terms of the magnetic vector
potential as
y
xz
A
B

 (34)
yields
22
0
() t
x
A
yBeydy
 
 (35)
which since the electric helicity

vanishes reduces to
22
23
00
1
() 3
tt
x
A
yBeydyBey
  

 (36)
By considering that only Az and By vanish and that the
gauge vector magnetic potential is given by
1
A


  (37)
where

(y,t) is the electric potential, the magnetic
helicity may be computed as

L. C. G. D. ANDRADE 327
2
28
0
t
HBye

 
 (38)
whose electric potential is given by
2
4
5t ye




 (39)
This shows that the electric potential on non-compact
Riemannian surfaces of negative curvature, which can be
bound in the boundary of the Poincare discs, and decays
in time. One also notes that in the ideal plasma case
where the resistivity

vanishes the gauge condition
does not lead to the Weyl condition
0A (40)
unless at the center of the Poincare disc. The magnetic
helicity also vanishes very fast as one approaches y = 0.
However this is forbidden in the Lobachevsky-Poincare
plane, since there y > 0. Thus not only electric potential
but also magnetic helicity never vanish spatially at the
Poincare disc, unless as t → ∞.
4. Conclusions
In general, fast dynamo are investigated in compact
Riemannian manifolds, as has been shown by Arnold et al.
[10] and by Chicone and Latushkin [7]. In this paper,
slow dynamos have been investigated in non-compact
Riemannian manifolds. Here a toy model for a spatial
hyperbolic section of a possible astrophysical dynamos
in Lobachevsky plane is considered in 3D. This can
serve as a disc dynamo in astrophysics or hyperbolic
section of a cosmological model or even to investigate
disc plasmas in laboratory as done by Fantoni et al. In
the cosmological model the magnetic helicity can be
investigated along with current helicity in the case of
dynamos. These quantities are also useful in laboratory
dynamos [11]. Slow cosmic dynamos in plasmas can be
obtained in laboratory as has been shown by Colgate et
al. [12]. The investigation of restoring and viscous forces
in the model may also serve as models for the geodyna-
mos. Note that here, despite of the fact that both mag-
netic and electric helicities vanish, the slow dynamo ac-
tion in non-compact Riemannian manifolds of constant
negative curvature. From the geodynamo and convection
point of view in an interesting paper H Busse [13]
showed that the presence of curvilinear coordinates in-
troduce new features on the rotating spherical shells that
could be considered as Riemannian surface of positive
Gaussian curvature. By considering that the plasma dy-
namo ow topology of the Poincare disc has a treshold in
the growth rate of

0:022, performed in the laminar
plasma dynamo experiment by Wang et al. [4], from a
Rm ≈ 210. From the expression 2

  one obtains
that the electric current helicity can be determined as

≈ 2.34 m-1. In this computation, the inverse relation be-
tween the diffusion constant

and the magnetic Rey-
nolds number Rm was used. All these physical applica-
tions make the model presented here useful in physical
realistic situations and deserve further study.
5. Acknowledgements
I am very much indebt to J-Luc Thiffeault, Dmitry
Sokoloff, Yu Latushkin and Rafael Ruggiero for reading
for helpful discussions on the subject of this work. I ap-
preciate financial supports from UERJ and CNPq.
6. References
[1] A. H. Boozer, Physics of Fluids B, Vol. 7, 1993, p. 2271.
[2] L. C. Garcia de Andrade, Physics of Plasmas, Vol. 15,
2008.
[3] J. L. Thiffeault and A. D. Boozer, Physics of Plasmas,
Vol. 10, No. 1, 2003, p. 259.
[4] A. Wang, V. Pariev, C. Barnes and V. Barnes, Physics of
Plasmas, Vol. 9, 2002, p. 1491.
[5] R. Fantoni and G. Tellez, Journal of Statistical Physics,
Vol. 133, 2008, p. 121.
[6] J. L. Thiffeault, Journal of Physics A, Vol. 34, No. 29,
2001, p. 5575.
[7] C. Chicone and Yu. Latushkin, “Evolution Semigroups in
Dynamical Systems and Differential Equations,” Ameri-
can Mathematical Society, 1999.
[8] B. Bayly, Physical Review Letters, Vol. 57, 1986, p.
2800.
[9] A. Gilbert, Proceedings of the Royal Society London A,
Vol. 433, 1993, p. 585.
[10] V. Arnold, Ya. B. Zeldovich, A. Ruzmaikin and D. D.
Sokoloff, JETP, Vol. 81, No. 6, 1981, p. 2052.
[11] A. Shukurov, R. Stepanov and D. D. Sokoloff, “Dynamo
Action in Moebius Flow,” Physical Review E, Vol. 78,
2008, p. 025301.
[12] V. Pariev, S. Colgate and J. M. Finn, Astro-Physical
Journal, Vol. 658, No. 128, 2007.
[13] H. Busse, Physics of Fluids B, Vol. 14, 2003, p. 1301.