1997MNRAS.286..757K
Mon. Not. R. Astron. Soc. 286, 757-764 (1997)
Global magnetic shear instability in spherical geometry
L. L. Kitchatinov1,2* and G. Riidiger1*
'Astrophysikalisches Institut Potsdam, An der Stemwarte 16, D-14482 Potsdam, Germany
2Institute for Solar- Terrestrial Physics, PO Box 4026, Irkutsk, 664033, Russia
Accepted 1996 November 22. Received 1996 October 31; in original form 1996 April 1
ABSTRACT
This paper concerns the global stability of a differentially rotating magnetized
sphere of an incompressible fluid. Rotation laws subcritical to the Rayleigh stability
criterion produce the instability in the finite interval Bulin ~B ~Bmax of the magnetic
field amplitudes. The upper, B max, and the lower, Bulin, bounds are imposed by the
finite size of the system and by finite diffusivities (magnetic resistivity and viscosity),
respectively. For high rotation rates, Bmax grows linearly with the angular velocity
gradient while Bulin approaches a constant value. The global modes with different
types of symmetry relative to the equatorial plane are identified. The modes with
symmetric magnetic field and antisymmetric flow are always dominating. Nonaxisymmetric excitations are preferred when rotation is not too slow and the field
strength is close to Bmax. The possibility of a hydromagnetic dynamo produced by the
instability in stellar radiative cores is briefly discussed.
Key words: accretion, accretion discs - instabilities - MHD - stars: magnetic
fields.
1 MOTIVATION
The magnetic shear instability has recently become a subject
of intensive study after it has been recognized as a possible
driver of turbulence in accretion discs (Balbus & Hawley
1991). The old problem of the turbulence origin is possibly
solved allowing for a (weak) external magnetic field producing the destabilizing effect. After this notion the magnetic
shear instability discovered long ago for the Couette flow
(Velikhov 1959; Chandrasekhar 1960) has got a new life.
The instability is fast, i.e. its growth rate is of order of the
rotation frequency, and it requires a rather small magnetic
field (Balbus & Hawley 1991; Hawley & Balbus 1991). It
indeed produces a turbulence in its non-linear regime
(Stone & Norman 1994; Brandenburg et al. 1995; Hawley,
Gammie & Balbus 1995; Matsumoto & Tajima 1995; Brandenburg et al. 1996; Stone et al. 1996). Its astrophysical
implications are not limited to accretion discs. The
instability can also be relevant to the problem of angular
momentum transport in stellar radiative cores (Balbus &
Hawley 1994; Balbus 1995; Urpin 1996).
In spite of the intensive studies it is not clear, however, to
what extent the instability can be modified by more detailed
physics. In particular, global treatment in disc (Papaloizou
* E-mail:
[email protected](LLK·);[email protected]
(LLK2); [email protected] (GR)
& Szuszkiewicz 1992) and cylinder (Curry, Pudritz &
Sutherland 1994; Curry & Pudritz 1995, 1996) geometries
demonstrated the influence of the boundaries as not
trivial.
In the present paper the global modes in spherical geometry are studied. There are at least two reasons to do so.
First, spherical geometry allows a completely global formulation. It has boundaries from any side, while the cylinder
geometries considered so far were unbounded in either
vertical or horizontal direction. Secondly, the magnetic
shear instability may actually work in stars, i.e. in spheres. It
may be anticipated to be relevant to stellar dynamos.
Indeed, the self-excited dynamo generated by the
instability-produced turbulence was demonstrated to exist
by Brandenburg et al. (1995, 1996), an effect that had also
been envisaged by Tout & Pringle (1992). The instability
needs, however, a decrease of the angular velocity with
distance to the rotation axis. Such a decrease may well be
present in stellar radiative cores with their extremely small
microscopic viscosities because of the continuous loss of
angular momentum through the surface by stellar winds. Of
course, the dynamo may only be found in a non-linear treatment. The solution ofthe linear global problem is, however,
a natural first step in this direction.
We do not assume axial symmetry and include finite diffusivities but neglect the effects of compressibility and stratification (cf. Stone et al. 1996). The compressibility is,
©1997 RAS
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
1997MNRAS.286..757K
758 L. L. Kitchatinov and G. Rudiger
probably, not important for the shear instability proceeding
via torsional incompressive excitations. However, if the stellar radiative cores are concerned, the stabilizing effect of
stratification on the radial displacements may, perhaps, be
significant. Then a consideration of isothermal (horizontal)
motions is likely to lead to results similar to those presented.
We shall find that the boundaries impose an upper limit,
B max, on the magnetic field amplitude. The instability is
present only for fields smaller than B max, which grows with
the rotation rate. The finite diffusivities impose the lower
bound, B min, which approaches a constant value for sufficiently rapid rotation. Close to Bmin the axisymmetric modes
are preferred while close to Bmax the growth rates of the nonaxisymmetric excitations are larger. Also different symmetries relative to the equatorial plane can be
distinguished. The modes with antisymmetric flow and symmetric magnitude field are always preferred in excitation.
2 THE MODEL
2.1 The reference state
We consider a rotating sphere of conducting fluid with
radius R. The rotation is not uniform, its angular velocity, 0,
varies with the distance, w, to the rotation axis. We do not
need to specify the physical reason for the differential rotation. It could be, e.g., some kind of the A-effect (Riidiger
1989) as in stellar convection zones or the influence of the
centrifugal force as in accretion discs.
Outside the sphere is vacuum. A uniform magnetic field,
B o, is imposed parallel to the rotation axis. Such a field
satisfies, of course, the steady induction equation inside the
sphere,
0.6
0.0 t...................~'--........._ ............J
0.0
0.2
0.4
0.6
0.8
1.0
AXIAL DISTANCE
case q = 2 is close to the critical state but is still slightly
subcritical. The majority of our results belong to q = 2. With
q = 3 there is an (outer) region where the Rayleigh criterion
is violated, ~ < O. Hence, for sufficiently low viscosity the
rotational state with q = 3 is expected to be unstable even in
the non-magnetic case.
2.2 Mathematical formulation
We start with the MHD equations for incompressible fluids
linearized around the reference state with Vo and Bo
au
at
- + (u' V) Vo + (Vo' V) u
1
= - (Bo'V)b
dw
'
ob
-=curl(u x Bo+Vo x b) +'1tlb,
at
(5)
where v is the viscosity, '1 the magnetic diffusivity both
assumed as uniform and p is the total pressure including the
magnetic term. The pressure can be excluded by curling the
momentum (first part of equation 5) hence
(3)
-=curl(Vo x w+u x wo+j x Bop) +vAw,
The parameters Wo and n will be fixed as Wo = R/2, n = 2
while three different values of q will be considered, namely
q = 1, 2 and 3. The rotation laws for these choices are shown
in Fig. 1.
The same figure also shows the standard epicyclic frequency,
w
41tp
1
- - Vp + v&t,
P
(2)
with (Donner & Brandenburg 1990)
~= z.Q d(urQ)
. AXIAL DISTANCE
Figure 1. Rotation laws (left) and the derivative of the angular
momentum density (right) for different values of the parameter q.
The rotation laws for q = 1, 2 are subcritical to the Rayleigh
instability, while with q = 3 there is a supercritical region where a
pure hydrodynamical instability may develop.
(1)
the Maxwell equations outside, and the usual continuity
condition on the boundary. So, the reference state is consistent as long as the magnetohydrodynamics applies.
Let the rotation law be prescribed by the simple parametrization,
-0.2 L.....................~..........~~-'-'-.........J
0.0
0.2
0.4
0.6
0.8. 1.0.
(4)
related to the Rayleigh criterion, ~ > 0, for hydrodynamic
stability. The rotation law for q = 1 is clearly subcritical. The
ow
at
(6)
where w andj are vorticity and current density,
1
j=-curlb.
41t
w=curlu,
Wo
(7)
is the vorticity of the reference state,
ez d[urO(w)]
W o= -
w
(8)
dw
ez is the unit vector along the rotation axis.
© 1997 RAS, MNRAS 286, 757-764
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
1997MNRAS.286..757K
Global magnetic shear instability in spherical geometry
Next, dimensionless variables marked by dashes are
introduced:
R2
t=-t',
w=Rw' ,
r=Rx,
u=ROou',
m=Oom'.
'1
b=Bob',
(9)
Then the normalized equations are
am
[wO(w)e",
~
-=Cncurl
x
at
,c]
xe
m+-~ u
20
BoR
Ha=
,
..j41tpv'1
v
Pm=-,
'1
(11)
l
v(~)l
v(~)+cur{x x v(:)
b=x x
v(~)+cur{x x
(12)
The corresponding representations for vorticity and current
density,
m=x x
j=x x
v(~)+CUrl[x x v(~)l
v(~)+cur{x x v(~)l
(16)
convenient, and identical formulations may hold also for the
other potentials. The vacuum boundary conditions for the
magnetic field are now easy to formulate in terms of the
amplitudes of the expansion (16),
OA nm n
-ax
+ -xA nm =0,
(17)
As usual the boundary conditions for the flow constitute the
requirement for the 1tr'" and 1tm cross-components of the
viscous stress, 1tij= - pv(ui,i + ui,;}, to vanish at the surface
so that
are the Cn of the dynamo theory, the Hartmann number and
the magnetic Prandtl number. The present problem is a
three-parametric one. Among the parameters the normalized angular velocity of the basic rotation, Cn, and the
external magnetic field, Ha, are the most important.
Both the vector fields in the present problem are divergence-free. They can thus be expressed in terms of the scalar
potentials defining the toroidal and poloidal parts of the
fields (Chandrasekhar 1961; Krause & Radler 1980),
u=x x
Note that the scalars B, m, andj from (12)-(14) are not the
absolute values of the vectors b, m, andj but scalar potentials defining the toroidal parts of the vectors.
The !i -operator has the spherical Legendre polynomials
as eigenfunctions. This makes the expansion in spherical
harmonics,
n,m
if the dashes are dropped. e", is the azimuthal unit vector.
The three dimensionless parameters,
D.oR2
(15)
z
(10)
Cn =--,
'1
~
10
0 1 02
.P=---sin8-+----.
sin 8 08
08 sin2 8 ocji
759
(13)
include two potential functions coincident with those in (12)
while two other functions satisfy the differential equations
~(Wnm)=o
ax r
'
2 o'l'nm
mnm+---=O.
(18)
x ox
The radial velocity must also be zero at the surface
(19)
'I'nm=O.
The stability analysis leads to the eigenvalue problem after
an exponential time dependence of the linear amplitudes is
assumed: A, B, j, W, m, 'I' ex: eat.
Substitution of (16), (12) and (13) into (10) leads to the
equations for the scalar potentials. The derivations are not
trivial. However, detailed descriptions of the procedure can
be found in the literature (e.g., Krause & Radler 1980). In
the Appendix the equation system for the global models is
given. Next we discuss briefly their symmetry properties.
In our linear theory the equations for different azimuthal
wave numbers, m, are decoupled from each other. The axisymmetric modes (m =0) and also different non-axisymmetric modes (m = 1, 2, ... ) can be considered separately.
For each value of m the equation system splits into two
independent subsystems governing the modes with different
types of symmetry relative to the equatorial plane. This
means that each subsystem includes the amplitude functions
of the expansions (16) with only odd or only even n. The
different types of symmetry are defined in Table 1.
Table 1. Two types of symmetry of the global modes.
1 ~
02'1'
m=--.P'I'-x2
or '
Sm-modes
B nm ,
n
(14)
where !i is a differential operator which in spherical
coordinates reads
= m+1,
A nm ,
n
Am-modes
Wnm, W'nm
m+3, m+5, ...
W nm '
jnm
n
= m,
Bnm,
Wnm, W'nm
m+2, m+4, ... miO
n = 2, 4, 6, ... m = 0
A nm ,
= m, m + 2, m + 4, ... m i O n = m + 1,
n = 2, 4, 6, ... m = 0
Wnm, jnm
m
+ 3,
m
© 1997 RAS, MNRAS 286, 757-764
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
+ 5,
...
1997MNRAS.286..757K
760 L. L. Kitchatinov and G. Rudiger
We shall use the notations Sm and Am for the global
modes with symmetric or antisymmetric magnetic field with
respect to the equator. The m in the notation concerns the
azimuthal wave number. It should be noted, however, that
the symmetry notation corresponds to the magnetic field.
The symmetry of the flow is opposite. The S-modes combine
symmetric magnetic field with antisymmetric flow, and for
the A-modes it is vice versa. This property of the global
modes is illustrated in Table 1 (see also Figs 4 and 5).
The eigenvalue problem is solved numerically. A finitedifference grid over radius with 101 grid points is applied.
The expansions (16) are truncated at n =m + 20. The eigenvalues and eigenmodes are found by an inverse iteration
procedure.
U
c
o
20
40
60
He
80
100 120
Figure 2. The neutral stability lines for SO-modes for the rotation
laws of the present model. Pm = 1.
3 RESULTS AND DISCUSSION
3.1 Axisymmetric modes
Figs 2 and 3 present the neutral stability lines in the Cn-Ha
plane for the SO-modes and AO-modes. Cn and Ha are the
normalized rotation rate and the external magnetic field
(11). The results are given for the angular velocity profiles
of the present model as given in Fig. 1. The curves for
symmetric and antisymmetric modes are very similar. A
closer consideration reveals, however, that the lines for Amodes are shifted upwards relative to the corresponding
lines of the S-modes. The A-modes are excited at higher
rotation rates (or lower diffusivities). The symmetric modes
are thus preferred.
Only the lines for q = 3 are crossing the Cn-axis. The
corresponding shear flow is unstable even without a magnetic field. This is, of course, no surprise as q = 3 is supercritical with respect to the Rayleigh criterion. The neutral
stability lines are representing the magnetic modification of
the hydrodynamic instability. A weak magnetic field makes
a destabilizing effect: the critical rotation rate becomes
smaller. Strong magnetic fields, however, work in the opposite direction. The system becomes stable for sufficiently
large Hartmann number (Chandrasekhar 1961).
The curves for q = 1 and q = 2 represent the magnetic
shear instability. The two lines are similar but with less shear
(q = 1) the instability requires faster rotation. For sufficiently high rotation rate a finite interval exists,
Bmin <Bo < Bmax, where the system is unstable. Hence, the
instability only exists for not too strong nor too weak magnetic fields.
A local stability analysis of ideal fluids predicts an
instability for magnetic fields of arbitrary strength. It also
predicts, however, that only excitations with wavelengths
larger than the critical value, Amin, are unstable (Balbus &
Hawley 1991):
(20)
VA is the Alfven velocity for the external magnetic field; the
proportionality constant of order unity depends on the
shear magnitude and on the fluid parameters. In our global
formulation, however, possible wavelengths (20) are limited
U
c
o
20
40
60
80
100
He
Figure 3. The neutral stability lines for AD-modes. Pm = 1.
by the size of the system hence
VA,max
1
21t--~
(21)
QaR
defines the maximum field amplitude. Equation (21) agrees
well with the estimation of Bmax by Papaloizou & Szuszkiewicz (1992) for very thick discs. This relation also explains
the linear law, Ha ~ Cn/21t approached by the right-hand
branches of the neutral stability lines of Figs 2 and 3 with
increasing Cn.
The local analysis also predicts that the wavelength corresponding to the maximum growth rate is not much larger
than ~ after (20). The spatial scale of the unstable modes
decrease with decreasing magnetic field. The decrease
should be limited by diffusion. This is a probable explanation for the existence of the minimum field amplitude, Bmin •
The consideration suggests that the left-hand branches of
the neutral stability lines in Figs 2 and 3 belong to the smallscale excitations and the right-hand branches belong to the
large-scale excitations. Figs 4 and 5 present the structure of
the neutral stability modes of these branches. A definite
difference in the spatial scales can indeed be found. It is less
© 1997 RAS, MNRAS 286, 757-764
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
1997MNRAS.286..757K
Global magnetic shear instability in spherical geometry
pronounced for the A-modes, possibly because the distance
between the left and right branches is smaller in this case.
Fig. 6 illustrates the dependence on the magnetic Prandtl
number. The reason for the change of the slope of the righthand branches in the plot is obvious. Using relation (21) and
definitions (11) we find
Co.=QoR
--
Ha
~-~21ty'Pni.
VA,max
3.2
Non-axisymmetric modes
In the following all the results are computed for Pm = 1 and
q = 2. Figs 7 and 8 present the results for the non-axisymmetric modes with m = 1 and m = 2. Modes with higher m
600
(22)
1'/
The slope is proportional to Pm°.5. The left-hand branches
of Fig. 6 are more informative. They suggest that at very fast
rotation the minimum magnetic field strength approaches
101'/
VA,min~-'
independent of the viscosity. The viscosity influences, however, the rotation rates where the estimate (23) is met.
The axisymmetric modes are steady, i.e. the imaginary
parts of the eigenvalues vanish for non-negative real parts.
Oscillatory models do also exist but not with positive growth
rates.
D
400
(23)
R
\\f.,Ol"
761
2QO
o
20
40
60
80
100
Ha
Figure 6. The neutral stability lines for various magnetic Prandtl
numbers (S-modes, q=2).
eire.
'"
".
(
~::
it'
U
rot.
Figure 4. The structure of the neutral stability SO-modes for the
left (top) and right (bottom) branches of the line with q=2 of Fig.
2 for Co = 600. Left to right: the isolines of the angular velocity
perturbations, streamlines of the meridional flow, isolines of the
toroidal magnetic field, and the poloidal magnetic field lines. Full
lines give positive values and clockwise circulation, respectively.
eire.
o
o
50
100
150
200
250
Ha
Figure 7. The neutral stability lines for Sm-modes. Non-axisymmetric modes are dominating for sufficiently large magnetic fields.
Pm=l, q=2.
t-field
U
'[I)i
I
c
ll'
,
Figure 5. The same as on Fig. 4 but for AO-modes.
o
50
100
150
Ha
Figure 8. The neutral stability lines for Am-modes. Pm=l, q=2.
© 1997 RAS, MNRAS 286, 757-764
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
1997MNRAS.286..757K
762 L. L. Kitchatinov and G. Rudiger
become unstable only at rotation rates so large that the
resolution requirements to the numerical code become
unbearable.
The slopes of the right-hand branches of the neutral
stability lines for m;60 are smaller than for the axisymmetric case. Hence, for sufficiently fast rotation and sufficiently strong magnetic fields the excitation of
non-axisymmetric modes is preferred. When the magnetic
field becomes stronger, it is more easy to interchange the
lines of force than to bend them. Only a non-axisymmetric
flow can interchange while an axisymmetric flow only
bends.
In the Figs 7 and 8 the predominance of the modes with
m = 1 can be observed. The smaller slopes of the right-hand
branches of the lines for m = 2 suggest that these modes may
eventually dominate for sufficiently large en and Ha.
We have no data for m ~ 3 owing to numerical difficulties.
We can also not admit the preference of the non-axisymmetric modes found with the present model as a general
rule.
The existence of a region in the parameter space where
non-axisymmetric excitations are preferred is, however, very
promising for a dynamo effect by the shear instability. After
Cowling's (1934) theorem self-excitation of magnetic fields
only exists with deviations from the axial symmetry. It
should be noticed in this context that with an azimuthal
background field only non-axisymmetric modes of the magnetic shear instability are excited (Ogilvie & Pringle 1996).
This statement does not cover, of course, the other instabilities (cf. Terquem & Papaloizou 1996).
In Fig. 9 one finds the growth rates for the family of the
magnetic shear instability modes. Different modes are dominating at different values of Ha. The figure also shows that
there are more members in the family than considered so
far. The above discussion only concerns the 'primary' modes
that are most easy to excite. There is, however, a large (in
some sense infinite) number of secondary modes attaining
positive growth rates at faster rotation than the primary
modes. One of the secondary SO-modes is represented by
the broken line in Fig. 9. Many modes can simultaneously be
excited even at modest values of the parameters en and Ha.
Therefore, rather complicated (turbulent) flow and mag-
120
100
(f)
1i.J
'a::4:
80
60
:r:
I-
~
40
0
a::
Cl
20
0
Sl
-20
0
50
100
150
2PO
He '
Figure 9. Magnetic field dependence of the normalized growth
rates of different modes of the magnetic shear instability for
C,,=700. The broken line shows a 'secondary' SO-mode. Pm=l,
q=2.
netic field patterns produced by the instability must be
expected.
The low diffusivities of astrophysical bodies often make
the numerical simulations problematic. The present model
also cannot go far beyond en ~ 103 • Its applicability, however, seems hardly to suffer from this limitation. At high en
our results approach very regular (linear) tendencies with a
plausible physics which should not be violated in the lowdiffusivity limit. If we apply (23) to estimate the minimum
magnetic field producing the instability in the solar radiative
core with its microscopic diffusivity of ,,~103 cm2 S-1 the
small value Brnin ~ 10- 7 G results.
This estimate must not be the final one, however, because
our model neglects stratification. The sub-adiabatic stratification of the core makes a strong stabilizing effect (Balbus
& Hawley 1994). The estimation (23) maybe expected to be
relevant only in a thin layer at the top of the core where the
stratification is still close to adiabaticity. Hence, the radiusR
in (23) should be replaced by a smaller value of the layer
thickness. The Brnin remains extremely small with any
reasonable thickness.
If we speculate further about a dynamo produced by the
instability, then the equation (21) gives the upper limit
where the dynamo should probably stop. It yields
Bm.. - 104-5 G in surprising agreement with estimates for
the convection zone-radiative core interface inferred from
the field strength in sunspots (Schiissler 1993). It should be
very tempting to attack the non-linear global problem to
check whether the magnetic shear instability can indeed
produce magnetic dynamos in spheres.
ACKNOWLEDGMENTS
The authors would like to express their thanks to Ralph E.
Pudritz who drew their attention to this problem. LLK
thanks for their support the Deutsche Forschungsgemeinschaft. This work has been also supported in part by the
Russian Foundation for Basic Research, grant No.
96-02-16019.
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763
1
Own + J
J(n)
ox
+ - [J(n + l)a J(n + 1, m) -bJ(n + 1, m)]-Own
1
J
+ - [J(n -1)a 2 (n -l,m) -b 2 (n -1, m)] - - J(n)
im
+r
ox
im 02\fn }
\fn + J(n)
(A2)
or .
The toroidal flow follows from
APPENDIX A: EQUATIONS FOR THE
GLOBAL MODES
The equation system for the global modes is given. The
subscript, m, of the azimuthal wave number in the field
amplitudes is dropped because it is the same in all terms of
the (linear) equations.
The equation for the poloidal magnetic field is
Mn
o2An
J(n)
n
-CL
J(n) I
Tt= or +7 An
+ Pn l_J(X)]
"
+ Pn,I+J (x)]
+aJ(I, m)J(I)[3Pn,I_J(x) + Yn,l-J(X)]
1
o\fn+J
J(n)
ox
+ - [bJ(n + 1,m) -J(n + l)a J(n + 1, m ) ] - 1
{bJ(I, m)J(I-l) [2rJ.nl_J(X)
+ b2 (1, m)J(1 + 1) [2rJ.n,l+ J(x)
imCn
imCn
---LrJ.n1(x)J(I)A1---wn
J(n) I
J(n)
+ J(n) [b2(n -1, m) -J(n -1) a2 (n -1, m)]
[
o\fn
ox-
- {bJ(I, m) c5n,I_J(X) +b 2 (i, m) c5n,I+J(X)
+J(I)aJ(I, m)[2rJ.n,I_J(X) + Pn,l-J(X)]
J}.
(A1)
The equation for the toroidal magnetic field is
Bn
x
J
+J(n) b2(n -1, m)--
OBn+l
+C-n L
J(n) I
[
{J(l+ 1)b2 (I,m) Pn,I+J(X)
+J(I-l)b J(I, m) Pn,l-J(X)
+J(l) a2 (1, m) [Pn,I+J (x)
+ Yn,I+J(X)]
+ [bJ(n + 1, m) -aJ(n + 1, m)J(n + 1 ) ] - ox
J}
oBn ,(A3)
+[b2(n-1,m)-a 2 (n-1,m)J(n-1)]--ox
and the vorticity equation is
+ {b 2 (i, m) rJ.n,I+J(x)[J(1 + 1) -J(n)]
+bJ(I, m) 1X",I_J(x)[J(I-l) -J(n)]
+C-n L ( [bJ(I, m){[J(I) -J(n)] Pn l_J(X)
J(n) I
'
- 2rJ.n,I_J(x)J(n) - Bn,l-J (x)}
© 1997 RAS, MNRAS 286, 757-764
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
1997MNRAS.286..757K
764 L. L. Kitchatinov and G. Rudiger
start from 1= 1 for m = 0. It is
+ b2 (1, m) ([J(l) -J(n)] f3n,l+l (x)
(n+m)(n-m)
(2n + 1)(2n -1)'
- 2an,l+l (x)J(n) - en,l+l (x)}
+ '2i2 1(1, m)J(I) [3f3n,l_l(X) + 1'n,l-l(X)]
(n +m + 1)(n -m + 1)
(2n + 1) (2n + 3)
+ [b1(1, m) ([J(l) -J(n)]an,l_l(X) + c5n,l_l(X)}
b1(n,m)= -(n+1)
+ b2(l, m) {[J(l) -J(n )]a,.,I+l(X) + c5n,l+l(X)}
(n+m)(n-m)
(2n + 1)(2n -1) ,
(n +m + 1)(n -m + 1)
(2n + 1)(2n +3)
+ 2a 1(1, m)J(I) [2an,l_l(X) + f3n,l-l(X)]
J(n) = -n(n+1).
The dependence on the angular velocity distribution comes
through the matrices
an/(x) = [ Q(ro)P,;(cos 8)P,!,(cos 8) sin 8 d8,
f
f3n/(x) =X
~ dQ(ro) -
o
1'n/(x) =r
-
- - P';(cos 8)P,!,(cos 8) sin2 8 d8,
dro
8)P,!,(cos 8) sin 8 d8,
Jo --P';(cos
dwZ
~ d2Q( ro) -
-
3
+ b2 (n -1, m)J(n ) jn-l
x
'Ojn+l
+ [b1(n + 1, m) -a1(n + 1, m ) J(n + 1)] - -
d2 Q( ro)] _
_
P';( cos 8)P,!,( cos 8) d8,
+ cos2 8 sin or - -
ax
dw
'Ojn_l}
+ [b 2(n -1, m) -a 2(n -1, m)J(n - 1 )
]-
.
ax
en/(X) =X
(A4)
The equations (14) written in terms of the amplitudes of the
expansions (16) are
'02'1'
rWn +r_n +J(n)'Pn=O,
'Or
(AS)
In these equations, the summations over I start from 1= m
for the case of the non-axisymmetric modes, m #- 0, or they
f"[
Q( ro)
d 2Q( ro)
(3 - 6 cos2 8) - - + (1- 6 cos2 8 ) r o - dro
dw
o
d3Q( ro)] _
_
- cos2 8w - - P';( cos 8)P'!'(cos 8) d8
3
dro
(A6)
where ro =X sin 8 is the axial distance, Q is the normalized
angular velocity (3), and P,; are the normalized Legendre
polynomials. The rotation law (3) is symmetric about the
equatorial plane. It implies that all elements of the matrices
(A6) with odd n + I equal zero. After this property the equation system (A1)-(AS) splits into two subsystems governing
the global modes with evenBn , Wn' 'Pn and oddAno wn,jn and
the modes with oddBn, Wn' 'Pn andevenA no wn,jn' These two
kinds of modes correspond to the two types of equatorial
symmetry specified in Section 2.
© 1997 RAS, MNRAS 286, 757-764
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System