Coupling between radial and torsional oscillations in a magnetized
plasma and solar cycle radius variation
T.V. Zaqarashvili1 and G. Belvedere2
1. Abastumani Astrophysical Observatory, Georgia
2. Astrophysics Section, Department of Physics and Astronomy, Universitá di Catania, Italy
Abstract
The weakly non-linear interaction of slow magnetosonic and torsional Alfvén waves propagating along an
unperturbed poloidal magnetic field in a stellar interior is studied in the high plasma β limit. It is shown
that slow magnetosonic waves parametrically drive torsional modes with the half frequency and wavenumber.
Therefore global slow magnetosonic waves which have a radial velocity polarization, may amplify the torsional
oscillations. Possible applications of this mechanism to the Sun is briefly discussed.
Introduction
Although Walén’s original suggestion (Walén, 1949) that torsional azimuthal Alfvén oscillations may exist
in stellar interiors, the energy source sustaining these oscillations is not yet known (Rosner & Weiss, 1992).
Without an energy source, the oscillations are believed to be damped by phase mixing or transport of magnetic
flux towards the surface. Several different mechanisms have been suggested from time to time to support
torsional oscillations, however none of them may account for the energy source. The main problem is that
torsional Alfvén waves are pure electromagnetic oscillations in magnetohydrodynamics and one needs to have
an azimuthal force which may support them against damping. The absence of such a force suppressed the
interest in these oscillations as a possible mechanism for stellar activity cycles. However no one paid attention
to the fact that it is not necessary to have a direct driving force for the amplification of the oscillations. The
oscillation can also be amplified through the periodical variation of the system parameters and the process
is called parametric resonance. The well-known example of such system is the mathematical pendulum with
periodically varying length. When the frequency of the parameter variation is twice the frequency of the system
oscillation, then the oscillations will exponentially grow in time. The phase speed of torsional Alfvén waves
depends on the magnetic field and medium density. So if some external force (or oscillation) causes a periodical
variation of either quantities (or both together) then the Alfvén waves will be amplified. The pioneering work of
1
Zaqarashvili (2001) suggested that the periodical density variation caused by sound waves in the high plasma
β limit may amplify the shear Alfvén waves. Then, Zaqarashvili & Roberts (2002a) showed the resonant
energy transfer from fast magnetosonic waves propagating across the unperturbed magnetic field to Alfvén
waves propagating along the field. The energy transfer is relative to the velocity component of resonant Alfvén
waves normal to both magnetic field and direction of propagation of the magnetosonic waves. This appeared
to be a very interesting phenomenon. In both papers the temporal behaviour of Alfvén waves was found to
be governed by Mathieu equation. Thus the frequency of resonant Alfvén waves was the half frequency of
the driving compressible oscillations. The natural astrophysical suggestion was that stellar pulsation in the
fundamental mode may amplify torsional Alfvén waves in a seed magnetic field (Zaqarashvili & Roberts, 2002b;
Zaqarashvili et al. 2002). The process can be of importance in stellar chromospheres and coronas as the Alfvén
waves propagate upwards carrying energy and momentum and so may contribute to chromospheric and coronal
heating. On another hand, the fundamental frequency of stellar pulsation is too high for resonant torsional
Alfvén waves to give rise to large-scale torsional oscillations in a stellar interior. Therefore one needs a longer
period pulsation in order to suggest them as an energy source for torsional oscillations.
Here we suggest that the slow magnetosonic waves propagating along the magnetic field in the high plasma
β limit (sometimes they are called as pseudo-Alfvén waves) are effectively coupled to the torsional Alfvén waves
propagating in the same direction. The velocity polarization of slow magnetosonic waves propagating along
a straight uniform magnetic field is along the radial direction in the case of a cylindrical coordinate system.
Therefore fixing the boundaries along the axis of the cylindrical system originates a standing pattern of waves
leading to the pulsation of the magnetic cylinder with a spatial scale which corresponds to the first or second
eigenmode. In the case of stars with a poloidal magnetic field in the interior, the fundamental mode of the
slow magnetosonic waves will have a very low frequency due to the small phase speed (note, that the phase
speed of slow magnetosonic waves is the Alfvén speed in the high plasma β limit). So it will lead to long
period stellar pulsation. Since the velocity of the slow magnetosonic waves has the radial direction, any force
(gravity, radiation pressure etc) may support this oscillation. If the oscillation transfers energy to torsional
Alfvén waves, then they will lead to the set up of torsional oscillations, as their wavelength will comparable
to the stellar radius. So the torsional oscillations will gain energy from the radial pulsation and thus can be
sustained against damping. Here we consider a simple case of wave coupling in a non-rotating star with a
poloidal magnetic field.
Equations and mathematical developments
We consider a non-turbulent medium with zero viscosity, infinite conductivity and negligible displacement
current, which can be described by the magnetohydrodynamic (MHD) equations:
∂B
= ∇×(V×B),
∂t
ρ
∂V
1
+ ρ(V·∇)V = −∇P + (∇×B)×B − ρ∇Ψ,
∂t
µ
∂ρ
+ ∇(ρV) = 0,
∂t
∇2 Ψ = 4πGρ,
(1)
(2)
(3)
(4)
where ρ is the medium density, P is the pressure, V is the velocity, B is the magnetic field, G is the gravitational
constant, Ψ is the gravitational potential and µ is the magnetic permeability. To this system we add the equation
of state
P = P (ρ, T ),
(5)
where T is the temperature, which connects the thermodynamic variables (pressure, density and temperature or
entropy). We adopt a cylindrical coordinate system (R, φ, Z) and for simplicity consider only the axisymmetric
problem so everywhere ∂/∂φ = 0 is assumed. We suppose, that an uniform poloidal magnetic field, B0 , exists
inside the star directed along the Z axis (see Fig.1).
The star is supposed to be in hydrostatic equilibrium:
∇P + ρ∇Ψ = 0.
2
(6)
Figure 1: The cylindrical coordinate system and poloidal magnetic field in stellar interior.
In this paper always we consider the magnetic energy smaller than the gravitational one, so that the Lorentz
force (if B0 is not uniform) does not change the equilibrium significantly.
For the study of the coupling between different oscillations we use the perturbation method i.e. all variables
are perturbed around their equilibrium values and their consequent evolution is studied.
We first briefly describe the linear evolution of the perturbations in form of linear waves and then study
the weakly non-linear interaction between the waves, which leads to the coupling of our interest.
The linear approximation
All physical variables are presented as a sum of unperturbed and perturbed parts ξ = ξ0 + ξ1 . In the linear
approximation only the first order terms are retained. This allows us to consider the evolution of different
waves separately.
Torsional Alfvén waves have azimuthal velocity and magnetic field components and represent the purely
electromagnetic part of oscillations. They are described by equations
∂Vφ
∂bφ
= B0
,
∂t
∂Z
(7)
B0 ∂bφ
∂Vφ
=
,
(8)
∂t
µ ∂Z
where ρ0 is unperturbed density, while Vφ , bφ are the velocity and magnetic field perturbations. The unperturbed density is considered to be homogeneous. Of course, it has a much more complicated spatial structure
in a stellar interior, but in order to understand the physics of the wave coupling we choose the simplest
model. Once the physical grounds of the coupling are clear, future models may include a more realistic spatial
distribution of the physical variables (including the unperturbed magnetic field).
The restoring force for the torsional waves is the tension of magnetic field lines only. The waves propagate
along the magnetic field lines with the Alfvén speed
ρ0
B0
.
cA = √
µρ0
3
(9)
Consequently the dispersion relation of the waves is
ω
= cA ,
k
(10)
where ω is the frequency and k is the wave number.
The waves are strictly transversal and they do not cause a density perturbation in the linear regime.
The remaining linearized equations are
∂VR
∂bR
= B0
,
(11)
∂t
∂Z
B0 ∂(RVR )
∂bZ
=−
,
(12)
∂t
R ∂R
∂Ψ0
∂p1
B0 ∂bR
∂bZ
∂VR
− ρ1
=−
+
−
,
(13)
ρ0
∂t
∂R
µ
∂Z
∂R
∂R
∂p1
∂Ψ0
∂VZ
=−
− ρ1
,
∂t
∂Z
∂Z
1 ∂(RVR ) ∂VZ
∂ρ1
,
= −ρ0
+
∂t
R ∂R
∂Z
ρ0
(14)
(15)
where VR , VZ , bR , bZ , ρ1 , and p1 are the perturbations of velocity, magnetic field, density and pressure
respectively. Here we neglect the perturbation of the gravitational potential Φ1 , however, if the spatial scale of
oscillations is comparable to the stellar radius, then the term ρ0 ∇Φ1 has the same order of magnitude as the
term ρ1 ∇Φ0 , so can not be neglected. We will return to this point later.
Equations (11)-(15) include magneto-acoustic-gravity waves. When the hydrodynamical pressure is much
stronger than the magnetic one (i.e. β = 8πP/B 2 ≫ 1), which is the case in a stellar interior, then these
waves can be split into high-frequency and low-frequency branches. The pressure gradient and gravity lead to
the high-frequency oscillations, while the Lorentz force leads to the low frequency ones. Therefore the highfrequency branch includes fast MHD and acoustic-gravity waves, while the low-frequency branch includes slow
MHD waves. In this paper we are not interested in the high-frequency branch, so in remaining part of the
paper it will be ignored.
Thus we consider only the slow MHD waves, which differ from the torsional modes in the velocity polarization: the velocity of slow waves is polarized along R, while the velocity of the torsional waves is polarized
along φ.
The equations for linear slow waves can be written as
∂bR
∂VR
= B0
,
∂t
∂Z
B0 ∂(RVR )
∂bZ
=−
.
∂t
R ∂R
B0 ∂bR
∂bZ
∂VR
=
−
.
ρ0
∂t
µ
∂Z
∂R
(16)
(17)
(18)
The velocity component along the Z axis belongs to the high-frequency branch and thus will be neglected
hereinafter.
Differentiating equation (18) in time and using equations (16)-(17) we obtain the second order differential
equation
2
2
∂
1 ∂(RVR )
2 ∂
2 ∂
−
c
V
=
c
.
(19)
R
A
A
∂t2
∂Z 2
∂R R ∂R
Now, taking VR = u(R) exp (iωt + ikZ), we obtain
(k 2 c2A − ω 2 )u(R) = c2A
1 ∂(Ru(R))
∂
.
∂R R
∂R
(20)
When the wave propagation is not strictly along the magnetic field, this equation leads to Bessel equation.
The phenomenon is well studied in magnetic tube theory. The external medium affects the wave propagation
4
inside the tube through the boundary conditions imposed at its surface and thus the waves do not propagate
with the Alfvén speed. The boundary conditions imply that the normal component of the velocity and the
total (hydrodynamical and magnetic) pressure must be continuous through the surface. Notice that usually the
surface is assumed to be straight i.e. the curvature due to the wave propagation is neglected. This is a quite
good approximation when the wavelength is much longer than the tube radius. However, when the wavelength
is comparable to the tube radius (or less), the tension force becomes stronger and should be considered in the
boundary conditions. Due to the strength of the tension force, the waves may propagate with the Alfvén speed
along the tube and the external medium does not affect the propagation. An analog of this process can be the
”ripple” waves at the water surface due to the surface tension. But the study of this phenomenon is not the
scope of the present paper, however it is worth to look at it in future. In our case the problem is simpler as we
can impose free boundary conditions at the stellar surface.
Let us consider the slow waves propagating along the magnetic field. Then the restoring force of the waves
is the Lorentz tension force and the waves propagate with the Alfvén speed. Thus the equation
1 ∂(Ru(R))
∂
=0
(21)
∂R R
∂R
is satisfied. The solution of this equation is
u(R) =
C00
C01
R+
,
2
R
(22)
where C00 and C01 are integration constants. Imposing that the velocity is finite on the axis (R = 0) we obtain
C01 = 0. Therefore we get
C00
R.
(23)
u(R) =
2
This solution diverges for large R, which means that it can be valid only in a bounded medium like magnetic
tubes or stellar interiors. In our case the velocity has, at the stellar surface, the value u(R0 ) = C200 R0 , where
R0 is the stellar radius. It is seen from equations (16)-(18) that, for u(R) expressed by (23), bZ does not depend
on R, while also bR is a linear function of R. So we can summarize the properties of slow magnetosonic waves
as
R
(24)
VR = uR (t, Z) ,
R0
R
bR = hR (t, Z) ,
(25)
R0
bZ = hZ (t, Z).
(26)
The wave dispersion relation is the same as for the torsional waves (equation (10)).
An interesting point is connected with the density. Due to (23), the continuity equation implies that
ρ1 = ρ1 (t, Z).
(27)
So, contrary to the torsional modes, the propagation of transversal slow waves leads to density variations.
However they propagate with the Alfvén speed not with the adiabatic sound speed. In order to have a
zero velocity along the magnetic field in equation (14), these density variations should be balanced by either
temperature or gravitational potential variations.
In principle, it is possible to distinguish Alfvén waves and slow waves propagating along the magnetic field
in a cylindrical geometry: always they have azimuthal and radial velocity polarization respectively. But one
can not distinguish these waves in a rectangular geometry if they are linearly polarized.
Now let us consider the weakly non-linear interaction between slow and torsional Alfvén waves.
The non-linear coupling
Taking into account the non-linear terms, equations (16)-(18) and (7)-(8) become
B0 ∂
1 ∂
∂bz
=−
(RVR ) −
(RVR bZ ),
∂t
R ∂R
R ∂R
5
(28)
!
Vφ2
∂VR
∂VR
=
+ VR
−
(ρ0 + ρ1 )
∂t
∂R
R
B0 + bZ ∂bR
bφ ∂(Rbφ )
∂bZ
=
−
−
,
µ
∂Z
∂R
Rµ ∂R
∂Vφ
∂
∂
∂bφ
= B0
+
(Vφ bZ ) −
(VR bφ − Vφ bR ),
∂t
∂Z
∂Z
∂R
∂Vφ
VR ∂(RVφ )
(ρ0 + ρ1 )
+
=
∂t
R ∂R
=
bR ∂(Rbφ )
(B0 + bZ ) ∂bφ
+
,
µ
∂Z
Rµ ∂R
(29)
(30)
(31)
ρ0 ∂
1 ∂
∂ρ1
=−
(RVR ) −
(RVR ρ1 ).
(32)
∂t
R ∂R
R ∂R
In equations (7)-(8) the solutions for the torsional modes have an arbitrary R-dependence. So, if we take
Vφ = uφ (t, Z)
R
,
R0
(33)
bφ = hφ (t, Z)
R
,
R0
(34)
and take into account equations (24)-(27), then the R-dependence is cancelled in equations (28)-(32) and we
obtain the system
hZ u R
∂hZ
= −2B0 1 +
,
(35)
∂t
B0 R0
#
"
u2R − u2φ
hZ
B0 R0
hZ ∂ 2 hZ
∂uR
1+
+
−
=−
1+
B0
∂t
R0
2µρ0
B0 ∂Z 2
2h2φ
,
µρ0 R0
u R hφ
hZ ∂uφ
∂hφ
= B0 1 +
−2
,
∂t
B0 ∂Z
R0
∂uφ
2uR uφ
B0
hZ ∂hφ
hZ
+
=
1+
−
1+
B0
∂t
R0
µρ0
B0 ∂Z
−
−
Notice that
hφ ∂hZ
,
µρ0 ∂Z
hZ
ρ1
=
ρ0
B0
and
2
hR
∂hZ
=−
.
R0
∂Z
(36)
(37)
(38)
(39)
(40)
Equations (35)-(38) describe the non-linear behaviour of Alfvén and slow magnetosonic waves propagating
along the unperturbed magnetic field in the high β plasma limit. The cross non-linear terms in equations
(36)-(38) describe the coupling of the waves. The inspection of these equations reveals that the action of these
waves has a different nature. The torsional waves act directly on the slow waves, because the terms due to
the torsional modes in equation (36) do not include the components of the slow modes (at least at the second
order). On another hand, the influence of slow waves on torsional waves has a parametric origin, because the
non-linear terms in equations (37)-(38) include at least one component of the torsional waves. It means that
slow magnetosonic waves affect the Alfvén waves parametrically through the variation of the Alfvén speed
6
during the propagation. Recently Zaqarashvili & Roberts (2002c) suggested that sound and linearly polarized
Alfvén waves are coupled when they propagate with the same speed along the magnetic field (i.e. β ∼ 1) and
the frequency (as well as the wave number) of the sound waves is twice the frequency of Alfvén waves. In that
case the action of the sound waves had a similar parametric origin. So a similar phenomenon can occur here,
as slow and torsional Alfvén waves propagate with the same speed and the slow waves cause the periodical
variation of the Alfvén speed. Therefore, in order to solve equations (35)-(38) we use the same formalism of
weakly non-linear interaction. So we consider the perturbations much smaller than the unperturbed values.
Then the variation of the wave amplitudes due to the non-linear interaction will be a slow process. Therefore
the perturbed physical quantities can be represented as the product of a slowly varying amplitude Cj (t) and a
rapidly oscillating term,
hZ = C1 (t)eiψ1 (t,Z) + C1∗ (t)e−iψ1 (t,Z) ,
(41)
uR = C2 (t)eiψ1 (t,Z) + C2∗ (t)e−iψ1 (t,Z) ,
iψ2 (t,Z)
(42)
C3∗ (t)e−iψ2 (t,Z) ,
(43)
uφ = C4 (t)eiψ2 (t,Z) + C4∗ (t)e−iψ2 (t,Z) .
(44)
ψ1 = ω1 t + k1 Z,
(45)
ψ2 = ω2 t + k2 Z,
(46)
hφ = C3 (t)e
+
Here
where ω1 , k1 , ω2 , k2 are the frequencies and wave numbers of slow and Alfvén waves respectively. The
frequencies and wave numbers satisfy the dispersion relation (10) which is the same for both waves.
Substitution of expressions (41)-(44) into equations (35)-(38), and averaging over rapid oscillations in phase,
leads to the disappearance of all exponential terms, so that only the first and third order (with Cj ) terms
remain. In the first approximation (neglecting all second and third order terms), the slow and Alfvén waves
are decoupled and the amplitudes Cj are constant. Third order terms with Cj (which are due to the advective
terms in the momentum equation) are significant only in the case of very large amplitudes and so can be
ignored. However, when
ψ1 = ±2ψ2 ,
(47)
the second order terms (with Cj ) also remain after averaging and consequently the wave amplitudes become
time-dependent. So if the frequencies and wave numbers satisfy the conditions
ω1 = 2ω2 , k1 = 2k2 ,
(48)
ω1 = −2ω2 , k1 = −2k2 ,
(49)
the slow and Alfvén waves are resonantly coupled. In other words, they may exchange energy during the
propagation.
According to condition (48), averaging equations (35)-(38) leads to the equations which govern the temporal
behaviour of the complex amplitudes:
2B0
C2 = 0,
R0
(50)
C2
2C32
B0 R0 2
k1 C1 − 4 +
= 0,
2µρ0
R0
µρ0 R0
(51)
Ċ1 + iω1 C1 +
Ċ2 + iω1 C2 −
Ċ3 + iω2 C3 − ik2 B0 C4 + ik2 C1 C4∗ +
Ċ4 + iω2 C4 −
2
C2 C3∗ = 0,
R0
(52)
C1 ∗
2
C1
iB0 k2
C3 +
Ċ4 − iω2 C4∗ +
C2 C4∗ +
µρ0
B0
B0
R0
+
3ik2
C1 C3∗ = 0.
µρ0
(53)
These equations describe the process of energy exchange between slow and Alfvén waves. The influence
of the torsional waves is expressed by the last two terms in equation (51) and the parametric influence of the
7
slow waves is expressed through the last two terms in equation (52) and the last four terms in equation (53).
Substituting C2 from equation (50) into equations (51)-(53), we obtain
C̈1 + 4iω2 Ċ1 +
2B0 2
4B0 2
C4 −
C = 0,
2
R0
µρ0 R02 3
Ċ3 + iω2 C3 − ik2 B0 C4 −
2iω2
1
Ċ1 C3∗ −
C1 C3∗ +
B0
B0
+ik2 C1 C4∗ = 0,
Ċ4 + iω2 C4 −
(54)
(55)
C1 ∗ Ċ1 ∗
C1
iB0 k2
C3 +
Ċ −
C − 3iω2 C4∗ +
µρ0
B0 4 B0 4
B0
+
3ik2
C1 C3∗ = 0.
µρ0
(56)
Here we are interested in the influence of slow waves on torsional ones. So, in order to study this process,
we assume that the slow wave amplitude is larger than the amplitude of torsional waves. Therefore we neglect
the terms which describe the back-reaction of torsional waves i.e. ∼ C32 and ∼ C42 . So we take C1 (and
consequently C2 ) as constant. The physical meaning is that the energy of slow waves is considered to be much
larger than the energy of initial torsional waves. This is a quite good approximation at the initial stage, when
the back-reaction can be neglected.
So we have
2iω2
C1 C3∗ + ik2 C1 C4∗ = 0,
(57)
Ċ3 + iω2 C3 − ik2 B0 C4 −
B0
Ċ4 + iω2 C4 −
iB0 k2
C1 ∗
C1
C3 +
Ċ − 3iω2 C4∗ +
µρ0
B0 4
B0
+
3ik2
C1 C3∗ = 0,
µρ0
(58)
where C1 = C10 + iC11 = const.
Now let us search for solutions of (57)-(58) in the exponential form
C3 = (C30 + iC31 )eδt ,
(59)
C4 = (C40 + iC41 )eδt ,
(60)
where C30 , C31 , C40 and C41 are constants. In expressions (59)-(60) δ is real, otherwise equations (57)-(58) are
not satisfied. A positive δ implies amplification of the initial perturbation, while a negative δ implies damping.
We substitute expressions (59)-(60) into equations (57)-(58) and, without loss of generality, we take C11 = 0.
Finally, we get equations
2C10
C10
δC30 − ω2 1 +
C31 + k2 B0 1 +
C41 = 0,
(61)
B0
B0
C10
2C10
C30 − k2 B0 1 −
C40 = 0,
(62)
δC31 + ω2 1 −
B0
B0
C10
3C10
δ 1+
C40 − ω2 1 +
C41 +
B0
B0
k2 B0
3C10
+
1+
C31 = 0,
(63)
µρ0
B0
3C10
C10
C41 + ω2 1 −
C40 −
δ 1−
B0
B0
k2 B0
3C10
−
1−
C30 = 0.
(64)
µρ0
B0
8
0.05
0.05
R
0.1
0
−0.05
−0.1
0
u
bZ
0.1
−0.05
0
10
20
tω
30
−0.1
40
0.01
0.005
0.005
0
10
20
tω
30
40
0
10
20
tω
30
40
u
b
φ
0.01
φ
0.015
0
0
−0.005
−0.005
−0.01
−0.01
0
10
20
tω
30
−0.015
40
Figure 2: Amplification of torsional Alfvén waves (below) with the period twice than the period of slow mode
(up).
This system has a non-trivial solution if the system determinant is zero, which gives a fourth order equation
for δ. After dropping higher order terms, we get
δ≈
ω2 C10
.
2 B0
(65)
Thus we get that torsional Alfvén waves grow exponentially in time and their growth rate depends on the
relative amplitude of slow waves i.e. C10 /B0 , which is typical of weakly non-linear processes.
In order to check the analytical solution, we performed the numerical simulation of spatially averaged
equations (35)-(38) with k1 = 2k2 , so only time dependence is retained. The back-reaction of the torsional
waves is again neglected. So the numerical calculation is straightforward and we find that torsional waves have
an exponentially growing solution when their period is twice the period of the slow waves i.e. ω1 = 2ω2 (Fig.2).
The upper plots are the magnetic field and velocity components of the slow waves (bZ and uR ) and the plots
below are the components of the torsional waves (hφ and uφ ).
Discussion
We have seen that slow magnetosonic (or pseudo Alfvén) and torsional Alfvén waves are resonantly coupled
when they propagate with the same Alfvén speed along an unperturbed magnetic field in a β ≫ 1 medium. The
slow magnetosonic wave parametrically triggers the torsional wave with the half frequency and wave number.
The growth rate of the torsional mode depends on the amplitude of the slow wave: the larger is the amplitude
of the slow wave, the faster is the energy transfer to torsional waves, which is a common scenario for non-linear
processes.
The coupling between these waves may have very interesting applications in astrophysics. The existence of
torsional oscillations in stellar interiors is a long time dilemma. Hydromagnetic torsional oscillations of a seed
magnetic field have been suggested to take place in stellar interiors, which may lead to the cyclic change in sign
of the solar toroidal magnetic field (Walén, 1949; Cowling, 1953; Piddington, 1971; Gough, 1988). However
the oscillations probably are damped because of the transport of the magnetic flux to the surface and/or the
9
phase mixing. Therefore the absence of a plausible energy source which can support torsional oscillations has
been an argument against their existence. However, several different sources have previously been suggested
to support torsional oscillations, but they are not convincing.
Recently stellar radial pulsation in the fundamental mode, which can be considered as a standing fast
magnetosonic wave in the presence of magnetic field, has been suggested as a source for torsional Alfvén waves
(Zaqarashvili & Roberts, 2002b; Zaqarashvili et al. 2002). However, amplified torsional waves can not give rise
to torsional oscillations due to their short wavelength as compared to the stellar radius (because the Alfvén
speed is much smaller than the sound speed). Therefore, rather than give rise to large-scale torsional oscillations,
they will propagate upwards to the stellar surface, and can be of remarkable importance for chromospheric and
coronal activity as they carry energy and momentum.
As is well known, it is a natural physical process that in closed systems (the Sun or a simple tuning fork)
almost all energy is stored in the fundamental modes. For example, take an ordinary tuning fork: any force
(whatever its origin is) leads to an acoustic oscillation in the fundamental frequency. Obviously, there are
other overtones, but with much smaller amplitudes. However, if one puts the tuning fork in a magnetic field
embedding a plasma, then the acoustic frequency will split into two frequencies corresponding to fast and
slow magnetosonic waves. Similarly, the large-scale magnetic field may cause the splitting of the fundamental
acoustic mode into high and low frequency branches, corresponding to fast and slow magnetosonic waves.
Therefore a small deviation of the star from equilibrium must lead to the oscillation in both, fast and slow
fundamental modes. Due to the small phase velocity, the period of standing slow waves will be much longer
than the period of the fast fundamental mode. In the case of the adopted simple configuration of poloidal
magnetic field in a stellar interior (see Fig.1), the standing slow magnetosonic waves can be pictured as radial
transversal oscillations with velocity nodes along the Z axis (notice that there are no velocity nodes along the
radial direction; see equations (24)-(27)). The number of nodes depends on which harmonic is excited. As the
phase speed is Alfvénic, the first harmonic will have a period of order 2R0 /cA , the second of order R0 /cA etc.
Due to the radial polarization of the velocity, the oscillations may easily gain energy from various sources like
gravity, pressure gradient, rotation etc.
Because of the parametric coupling, these oscillations will transfer energy to the torsional Alfvén waves with
twice the period and wavelength (see equation (48) and Fig.2). The amplified Alfvén waves may easily lead
to the set up of torsional oscillations as their wavelength now is comparable to the stellar radius. Torsional
oscillations now can be sustained against damping, as they may continuously gain energy from the radial
pulsation.
The energy that supports the slow fundamental mode may be huge. With reference to subsection 5.1.,
the gravitational energy of the Sun (i.e. the work required to dissipate solar matter to infinity), with radius
R0 ≈ 7·1010 cm and mass M ≈ 2·1033 g, is 6.6·1048 erg, even larger than the total internal radiant energy
≈ 2.8·1047 erg. The global density variation due to the slow fundamental mode is accompanied by a gravitational
potential variation (see Poisson equation: equation (4) in our paper). If one considers short wavelength
oscillations (compared to the stellar radius), one can neglect the gravitational potential variation (this is the
so called Cowling approximation in helioseismology), but for fundamental modes it must be retained (see
the classical paper by Ledoux & Pekeris, 1941). Suppose that the radius variation ∆R due to the slow
fundamental mode is as small as ∆R/R0 ∼ 10−5 (for the Sun it will be ∼ 0.01′′ , however observations show
the radius variation with ∼ 0.02′′ (Emilio et al. 2000) and even more with ∼ 0.08′′ (Laclare et al., 1996).
The gravitational energy change for a homogeneous sphere (Chandrasekhar, 1939) will be of the order of
6.6·1048 ∆R/R0 ∼ 6.6·1043 erg (this is the energy stored in the slow fundamental mode). Suppose that only
0.1% of this energy goes into the energy of torsional oscillations. Then the corresponding magnetic energy
variation will be of the order of ∼ 6.6·1040 erg. Even if the magnetic field is redistributed throughout the
whole stellar volume, we get a toroidal magnetic field variation with an amplitude of > 104 G. So the energy is
more than enough for supporting the torsional oscillation against ohmic diffusion or phase mixing. Moreover,
the amplification of the toroidal magnetic field component will be accompanied by the magnetic buoyancy
instability, which leads to the eruption of magnetic flux towards the surface. Thus torsional oscillations can not
cause a significant back-reaction on the pumping radial pulsation as their further amplification will be balanced
by the energy transport towards the surface in form of magnetic flux tubes, which leads to chromospheric and
coronal activity.
However, it must be mentioned that the many simplifications of the model (e.g. uniform density, cylindrical
geometry) will cause some limitations to the application of the mechanism to real situations in stars. For
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instance, notice that the density (and the magnetic field) can be considered as uniform only in a narrow
cylindrical shell, not throughout the star, but the physics of the mechanism still remains substantially the
same. Considering spherical symmetry instead of cylindrical one will be a further improvement, however it will
cause additional mathematical complications. The aim of this paper is to show the existence of such an energy
transformation mechanism, which can be developed and applied to more realistic astrophysical situations in
future.
0.1
Solar radius variation and torsional oscillations
The variation of the solar radius during the 11-year period activity cycle is a long time problem. The characteristics of the variation may reveal some basic mechanisms of activity, therefore many efforts have been
made in order to find observational evidence of a correlation between radius variation and activity parameters.
The observational results are controversial and somewhat inconsistent. Some observations indicate a negative
correlation (Laclare et al., 1996), some results show no systematic variation in time (Brown & ChristensenDalsgaard, 1998). Most observations reveal a positive correlation (Ulrich & Bertello, 1995; Basu, 1998; Emilio
et al., 2000; Noel, 2004). The observed amplitude of the variation is also controversial. Based on Mount
Wilson data, Ulrich & Bertello (1995) found ∼ 0.2′′ ; Noel (2004) suggested a similar amplitude ∼ 0.2′′ , while
in Laclare et al. (1996) the amplitude is ∼ 0.08′′ . Probably, the most realistic value of the amplitude ∼ 0.02′′
was obtained by Emilio et al. (2000) by analyzing SOHO-MDI data. The variation of the solar radius during
the activity cycle naturally leads to the idea of its connection to the magnetic field pressure variation in the
interior due to the large-scale turbulent dynamo. However, even a very small variation of the radius requires a
huge magnetic energy to compensate the corresponding gravitational energy change (see the above estimate).
Therefore, the reverse process is energetically much more convenient i.e. the process in which the slow mode
radial pulsation drives the activity cycle.
Then the proposed wave coupling may successfully work in the Sun. As we mentioned above, the fundamental period of slow pulsations will be of order R0 /cA . So, in order to obtain a 11-year period pulsation (which
in our framework corresponds to the 11-year period of solar radius variation), the mean Alfvén speed in the
interior needs to be of order 200 cm s−1 . This implies the mean magnetic field strength in the solar interior to
be ∼ 103 G. The radial pulsation will amplify the torsional oscillation of the unperturbed magnetic field in the
radiative interior with twice the period i.e. with a period of 22-years, which is the period of the solar magnetic
field oscillation. So, if this is the case, the energetical problem for the hydromagnetic oscillator model of the
solar cycle no longer exists, as the pulsation may gain energy from various sources, as discussed above.
Conclusion
In this paper we considered the weakly non-linear coupling between slow magnetosonic and torsional Alfvén
waves in the high plasma β limit. It has been shown that slow waves propagating along the unperturbed
magnetic field may resonantly amplify torsional Alfvén waves with the half frequency and wave-number of slow
waves. The growth rate of torsional Alfvén waves depends on the amplitude of slow waves. Therefore the
global slow magnetosonic waves, excited due to the action of various forces (gravity, gas and radiation pressure
gradients, rotation etc.) in a stellar interior, may give rise to large-scale torsional oscillations. The process may
be of remarkable importance in solar and stellar physics.
Acknowledgments
The work of T.Z. was supported by the NATO Reintegration Grant FEL.RIG 980755. T.Z also thanks to
Organizing Committee for financial support.
References
Basu, D. 1998, Sol. Phys., 183, 291
Brown, T.M. & Christensen-Dalsgaard, J. 1998, ApJ, 500, L195
11
Chandrasekhar, S. 1939, An introduction to the study of stellar structure, (University of Chicago Press, Chicago)
Cowling, T.G. 1953, in The Sun, ed. G.P. Kuiper (University of Chicago Press, Chicago), 575
Emilio, M., Kuhn, J.R., Bush, R.I. & Scherrer, P. 2000, ApJ, 543, 1007
Gough, D.O., 1988, Nature, 336, 720
Laclare, F., Delmas, C., Coin, J.P. & Irbah, A. 1996, Sol. Phys., 166, 211
Ledoux, P. & Pekeris, C.L. 1941, ApJ, 94, 124
Noel, F. 2004, A&A, 413, 725
Piddington, J.H., 1971, Proc. Astr. Soc. Australia, 2, 54
Rosner, R. & Weiss, N.O. 1992, The solar cycle, ASP Conferense Series (ASP: San Francisco), 27, 511
Ulrich, R.K. & Bertello, L. 1995, Nature, 377, 214
Walén, C. 1949, Ark. Mat. Astr. Fys., 33A, 18
Zaqarashvili, T.V. 2001, ApJ, 552, L81
Zaqarashvili, T.V. & Roberts, B. 2002a, Phys. Rev. E 66, 026401
Zaqarashvili, T.V. & Roberts, B. 2002b, Radial and Nonradial Pulsations as Probes of Stellar Physics, eds. C.
Aerts, T.R. Bedding and J. Christensen-Dalsgaard, IAU Colloquium 185, 484
Zaqarashvili, T.V., Javakhishvili, G.S. & Belvedere, G. 2002, ApJ, 579, 810
Zaqarashvili, T.V. & Roberts, B. 2002c, Proc. 10th. European Solar Physics Meeting, ”Solar Variability: From
Core to Outer Frontiers”, ESA SP-506, 79
12