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A steady state of the disc dynamo
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DOI: 10.1080/03091929208225248
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A steady state of the disc dynamo
Ladislav Kvasz
a
, Dmitry Sokoloff
b
& Anvar Shukurov
ca
Mathematics Department , Comenius University , Bratislava,Czechoslovakia , 84215
b
Physics Department , Moscow State University , Moscow, USSR ,119899
c
IZMIRAN, Academy of Sciences , Troitsk, Moscow Region, USSR ,142092Published online: 19 Aug 1992.
To cite this article:
Ladislav Kvasz , Dmitry Sokoloff & Anvar Shukurov (1992) A steady state of thedisc dynamo, Geophysical & Astrophysical Fluid Dynamics, 65:14, 231244
To link to this article:
http://dx.doi.org/10.1080/03091929208225248
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Geophys.
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Fluid
Dynamics,
Vol.
65,
pp. 231244 Reprints available directly from the publisher Photocopying permitted
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A STEADY STATE
OF
THE DISC DYNAMO
LADISLAV KVASZ
Mathematics Department, Comenius University, Bratislava 84215, Czechoslovakia
DMITRY SOKOLOFF
Physics Department, Moscow State University, Moscow 119899,
USSR
ANVAR SHUKUROV
IZMIRAN,
Academy
of
Sciences, Troitsk, Moscow Region 142092, USSR
(Received
2
June
1990;
in final
form
25
July
1991)
We discuss the steady states of the amdynamo in a thin disc which arise due to aquenching. Two asymptotic reDmes are considered,
one
for the dynamo number
D
near the generation threshold
Do,
and the other for
ID1
>>
1.
Asymptotic solutions for
~DDo~<<~D0~
ave rather universal character provided only that the bifurcation is supercritical. For
ID1
>>
1
the asymptotic solution crucially depends on whether
or
not the mean helicity
a,
as a function
of
B,
has a positive root (here
B
is the mean magnetic field). When such a root exists, the field value in the major portion of the disc is
0(1),
while near the disc surface thin boundary layers appear where the field rapidly decreases to zero (if the disc is surrounded by vacuum). Otherwise, when
a=O(\Bi )
for
IBl+cu,
we demonstrate that
IBI
=O(lDI'/')
and the solution
is
free
of
boundary layers. The results obtained here admit direct comparison with observations of magnetic fields in spiral galaxies,
so
that an appropriate model
of
nonlinear galactic dynamos hopefully could be specified.
KEY WORDS:
Nonlinear dynamos, asymptotic analysis, galactic dynamos.
1. INTRODUCTION Disc models of the turbulent meanfield dynamo are sufficiently simple to admit relatively easy numerical or analytical solution. The interest in these models is due to the fact that they describe the physics of well explored dynamosystems in spiral galaxies. In this paper we propose asymptotic solutions for the simplest model of nonlinear meanfield dynamoc when the mean helicity coefficient
u
is supposed to be an explicit function of the mean field intensity
B,
u
=
u(r,
B).
The asymptotic parameter is the dynamo number
D.
We consider separately two asymptotic regimes, one for
ID
Do[
<<
[Do[
with
Do
the critical dynamo number, and another for
ID[
>>
1. We consider a onedimensional (thin disc) amdynamo which
is
a zerothorder approximation to galactic dynamo models. The thindisc amdynamo has a remarkable property that over a wide range of dynamo numbers,
102D2
300,
the only excited dynamo mode is a
steady
quadrupole (Ruzmaikin
et
al.,
1980).
(for positive
D
the lowest mode grows only for
D
500.
Note that typically
D

10
in spiral galaxies.) This fact makes the nonlinear behaviour of the thindisc amdynamo
23
I
D o w n l o a d e d b y [ P r o f e s s o r A n v a r S h u k u r o v ] a t 0 9 : 2 8 1 8 S e p t e m b e r 2 0 1 4
232
L.
KVASZ,
D.
SOKOLOFF
AND
A. SHUKUROV
quite simple: for
ID1
c
00
we do not expect to encounter such nonlinear phenomena as limit cycles, chaotic behaviour, etc. which are often observed in spherical dynamos or in (oscillatory) onedimensional a2dynamos (e.g. Brandenburg
et
al.,
1989;
Meinel and Brandenburg,
1990).
Note that the numerical results which Meinel and Brandenburg
(1990)
obtained for steady nonlinear a'dynamos, agree reasonably with the results obtained below. Thus, the second type of asymptotic solutions considered below are actually intermediate asymptotics valid for
1
<<
ID1
<<
O(300).
For
IDDoI<<IDoI
we have
IBI<<1
and any nonlinear function
a(r,B)
can be approximated as
a
=
olo(z)[l g(z)lB[']
with certain functions
a, z)
and
g(z)
and constant
s
(here
z
is the coordinate measured across the disc,
121
<
1
).
Choosing
g(z)
>
0
guarantees these the bifurcation at
D
=
Do
is supercritical. It is almost obvious that in this asymptotic regime
IBI
=
O(l1
D/DOl1 )
while the field distribution is close to the eigenfunction obtained for
D
=Do.
Asymptotic solutions for
ID1
>>
1
are more versatile. It is crucial for them whether or not
a
as a function of
IBI
has a positive root. When such root exists [e.g.,
a=a,(l gB2)],
the steadystate field strength is of the order of
g1/2 =0(1)].
Vanishing of
a
implies that the appropriate boundary value problem is singularly perturbed and boundary layers arise at
IzI
=
1
(the disc surface). Otherwise, when
a
does not vanish identically for any
B
[e.g.,
cc=a,(l +gIBI')'],
the steadystate field strength is
of
the order of
101
while the field distribution over
z
is governed by the ratio
ao(z)/g(z).
The solution
is
free of boundary layers. The asymptotic analysis proposed here should be extended in two important directions. First, one should prove the existence and uniqueness of the exact solution approximated by the derived asymptotic forms. Basic experience with existence theorems indicates that such a proof can be much more complicated than the derivation of asymptotic solutions themselves. The proof of the existence (and uniqueness) theorems for similar problems is given by Belyanin
et
al.
(1991);
we have every reason to expect that their results can be extended to the problems discussed below. Second, the obtained solutions should be shown to be stable. It can be shown, that the equation for a linear perturbation reduces to the basic kinematic dynamo equation but with
D
replaced by
20
for
cc=a,(l gB2)
and by
(1
s)DIB,j
for
a
=
ole
1
+glBI )
with
B,
the unperturbed field. Recalling that growing solutions for positive
D
exist only for very large
D
E
500
(and for
s
>
l),
we see that the solutions obtained here are stable under typical galactive conditions,
ID1
100.
2.
BASIC EQUATIONS Let
us
consider the generation of the mean magnetic field in a thin turbulent disc. It is widely believed that the effect of the mean magnetic field
B
on the fluid motions, results primarily in reduction of the mean helicity of the turbulence. Therefore the helicity coefficient
a
becomes a function of
IBI.
We consider two simple specific forms
D o w n l o a d e d b y [ P r o f e s s o r A n v a r S h u k u r o v ] a t 0 9 : 2 8 1 8 S e p t e m b e r 2 0 1 4
NONLINEAR
DISC DYNAMOS
233
of nonlinearity, and These forms are very similar to those commonly used in the meanfield dynamo models. We slightly generalize the form of nonlinearity by taking into account that
g(z)'I2
is a characteristic steadystate strength of the mean field, which can be a function of position because, e.g. the ambient density varies with
z.
It is clear that for
s=2
and
g(z)lBJ2<<
the forms (1) and
(2)
are equivalent. From the viewpoint
of
nonlinear asymptotic analysis discussed below, the only principal difference between the forms (1) and
(2)
is that the former has a zero (at
(B(
g(z) I2)
while the latter is definite positive (see Figure
1).
(We restrict ourselves to supercritical bifurcations at
D
=Do.
The situation is more complicated for subcritical bifurcations cf. Krause and Meinel, 1988). Since any more or less simple explicit form of nonlinearity belongs to one of the two classes, we believe that our results are more general than a simple analysis of two specific forms
(1)
and
(2).
We introduce the nonlinearity into the simplest, but nevertheless realistic, dynamo model: a dynamo in an infinitely thin differentially rotating disc surrounded by vacuum
:
=
Br RR,(aB,)+ d2B, at
aZ
az
(3)
known as the crodynamo equations (for the considered solutions,
B,
is negligible in comparison to
B,
and
B,J
Here where
j?
is the turbulent magnetic diffusivity, and the asterisk refers to the characteristic value of the corresponding quantity (the angular rotation velocity
0
he halfthickness of the disc
h
and the mean helicity coefficient
a).
A
detailed discussion of the linear solutions of equations
(3)(4)
can be found in, e.g., Moffatt (1978), Parker (1979) and Ruzmaikin
et
al.
(1988b). Equations
(3)(4)
must be supplemented by boundary conditions. For the disc surrounded by vacuum and for axisymmetric solutions we have
B,=B,=O
at
z=
f
1,
5)
G.A.F.D.J
D o w n l o a d e d b y [ P r o f e s s o r A n v a r S h u k u r o v ] a t 0 9 : 2 8 1 8 S e p t e m b e r 2 0 1 4