A&A 384, 282298 (2002)
DOI: 10.1051/00046361:20011836
N. S. Dzhalilov^{1,2}  J. Staude^{2}  V. N. Oraevsky^{1}
1  Institute of Terrestrial Magnetism,
Ionosphere and Radio Wave Propagation of the Russian Academy of Sciences,
Troitsk City, Moscow Region 142190, Russia
2  Astrophysikalisches Institut Potsdam, Sonnenobservatorium Einsteinturm,
14473 Potsdam, Germany
Received 27 June 2001 / Accepted 20 December 2001
Abstract
Retrograde waves with frequencies much lower than the rotation frequency
become trapped in the solar radiative interior. The eigenfunctions of the
compressible, nonadiabatic (mechanism and radiative losses taken
into account) Rossbylike modes are obtained by an asymptotic method assuming
a very small latitudinal gradient of the rotation rate. An integral
dispersion relation for the complex eigenfrequencies is derived as a
solution of the boundary value problem. The discovered resonant cavity modes
(called Rmodes) are fundamentally different from the known rmodes:
their frequencies are functions of the solar interior structure, and the
reason for their existence is not related to geometrical effects. The most
unstable Rmodes are those with periods of 13yr, 1830yr,
and 150020000yr; these three separate period ranges are known from
solar and geophysical data. The growing times of those modes which are
unstable with respect to the mechanism are
10^{2}, 10^{3},and 10^{5} years, respectively. The amplitudes of the Rmodes are growing
towards the center of the Sun. We discuss some prospects to develop the
theory of Rmodes as a driver of the dynamics in the convective zone which
could explain, e.g., observed shortterm fluctuations of rotation, a control
of the solar magnetic cycle, and abrupt changes of terrestrial climate in the
past.
Key words: hydrodynamics  Sun: activity  Sun: interior  Sun: oscillations  Sun: rotation
The 22year magnetic cycle of solar activity is the most prominent phenomenon of several largescale dynamic events that occur in the Sun. (Really, the magnetic half cycles or sunspot number cycles vary in length between 913 years, and 11 yr is an average of the 20 halfcycles available.) An explanation of the basic mechanism underlying this fameous phenomenon is the fundamental challenge of solar physics. The achievements of the theory of the  dynamo turned out to be a great success. However, neither all observations of magnetic and flow fields nor the radiation fluxes which are related to this phenomenon and which are measured at the surface of the Sun or indirectly, by helioseismology, in its interior, can be explained unambigously in this way. Although our present work is not directly related to the dynamo theory, we will outline here those difficulties which have common points with our results.
As a consequence of our imperfect knowledge of basic characteristics of turbulent convection as well as meridional circulation and details of the rotation of the Sun's interior, the solutions of the dynamo equations become functions of many free, unknown parameters (e.g. Stix 1976). For instance, by clever combinations of these parameters it is possible to get from kinematic theory an oscillatory magnetic field with a 22year period and a growing amplitude. However, another choice of these parameters leads to waves of growing amplitude for other periods. So one could draw a butterfly diagram not only with an 11year periodicity. It remains still an open question which of the clever combinations resulting in a solarlike 22year activity cycle is realized in the Sun. We could not find a work on the dynamo wave problem, showing that just the 22year period is preferred among others with a maximum growth rate and with the spatial scales required for solar activity. Instead, many authors pointed out that the cycle period of 22 years is hard to explain (Stix 1991; Gilman 1992; Levy 1992; Schmitt 1993; Brandenburg 1994; Weiss 1994; Rüdiger & Arlt 2000).
From the solution of the inverse problem of helioseismology (e.g. Tomczyk et al. 1995) it is known that the convective envelope of the Sun is rotating with a latitude dependence of the angular velocity similar to that of the surface but almost rigid in radial direction. A stronger radial gradient which is required for the  dynamo mechanism is located in a shallow layer (thickness (Kosovichev 1996), where is the solar radius) immediately below the convective zone  the tachocline (Spiegel & Zahn 1992). Below the tachocline up to a depth of at least the radiative interior is rotating with an angular velocity law similar to that of a solidbody. The question arises: what compels the Sun to rotate in such a strange manner, which is different from the generally accepted, theoretically predicted stable rotation law? How to handle a dynamo theory for which the "" area is separated from the "" area over a large part of the extent of the convective zone? In order to solve this problem Parker (1993) has put forward the idea of an interface dynamo, the basic features of which existed already in earlier dynamo models (Steenbeck et al. 1966). To close the cycle of such a stretched dynamo it is necessary to have some mechanism delivering toroidal magnetic flux, arising by the shear of differential axisymmetric rotation (Cowling 1953) in the tachocline, to the "" dynamo area (e.g. Moffatt 1978; Krause & Rädler 1980). To get a solarlike magnetic activity it is necessary to suppose the existence of a huge (10^{5}G) toroidal magnetic field to create enough magnetic buouancy for the leakage of magnetic flux and to solve the tilt problem of lifting loops (e.g. Caligari et al. 1998). Moreover, a high magnetic diffusivity contrast between the convective envelope and the underlying radiative core should be assumed to solve the quenching problem of the effect (see, e.g., Fan et al. 1993; Cattaneo & Hughes 1996). However, it is a major challenge for any dynamo model to produce such strong fields.
The idea of the interface dynamo was further developed, e.g. by Charbonneau & MacGregor (1997). Later, a fit to the real solar rotation profile with its latitudinal and radial dependencies has been included by Markiel & Thomas (1999), but so far no satisfactory solarlike oscillatory solutions for the interface dynamo have been found. Growing wave solutions are suppressed by the latitudinal shear.
Mechanisms for braking the solar internal rotation are also under discussion. The character of the core rotation is not clear because here the accuracy of helioseismic inversions gets worse (Chaplin et al. 1999) and the results seem to be in contradiction with the oblateness measurements (Paterno et al. 1996). There are some suggestions that a deceleration of the radiative interior depends on the transport of angular momentum between this region and the convective zone. For instance, Mestel & Weiss (1987) supposed that even a weak largescale magnetic field would be sufficient to couple very efficiently the interior and the convective zone, leading essentially to solid body rotation. In this way the magnetic torques can also extract angular momentum from the radiative interior (e.g. Charbonneau & MacGregor 1993).
The wave mechanism for the solution of this problem is more popular. Schatzman (1993), Zahn et al. (1997), and Kumar & Quataert (1997) have concluded that the solid rotation of the radiative interior is a direct consequence of the effect of internal gravity waves. Gravity waves generated near the interface between the convective and radiative regions transport retrograde angular momentum into the interior, thereby spinning it down. Here the main idea is that the isotropically generated gravity waves become anisotropic due to Doppler shifts of frequencies in the differentially rotating Sun. In that way for anisotropic retrograde and prograde waves the radiative damping is different, and the residual negative angular wave momentum may compel the solar radiative interior to corotate with the convective zone. This idea has been further developed by Kumar et al. (1999) including a toroidal magnetic field to explain the existence of the unstable shear layer "tachocline". However, Ringot (1998) has shown that a quasisolid rotation of the radiative interior cannot be a direct consequence of the action of internal gravity waves produced in the convective zone. Gough (1997) questioned this idea emphasizing that the mechanism can work only if the waves are generated with strong amplitudes to transport the required angular momentum. This means, resonance waves are required, but such waves may penetrate only to distances less than beneath the convective zone due to the strong radiative damping. These waves must deposit their angular momentum before returning to the convective zone, but not before penetrating far into the radiative interior.
For the wave mechanism the question of an anisotropic propagation relative to the azimuthal rotation is a key moment. Fritts et al. (1998) have shown that convection, penetrating into the stratified and strongly sheared tachocline, can produce preferentially propagating gravity waves.
There have also been speculations that the rotation of the core may be variable, perhaps with a time scale of the solar cycle (e.g. Gough 1985). The present paper is along these lines.
From our short discussion we conclude that the convective envelope and the radiative interior are coupled to each other through a certain global agent, resulting in almost corotation. To advance the solution of the problem the dynamo theory should take into account the presence of this global agent. We suppose that really this agent is provided by waves with the following properties:
Waves should represent largescale global eigenoscillations of the Sun. Their origin must be related to rotation, they must be strongly anisotropic with respect to the azimuthal angle. Looking at the characteristics of the solar cycle we immediately see the high coherency of these global motions (the constant periods, phase shifts, amplitudes, the latitude appearence, etc.). Activity grows in the first phase with a timescale which is considerably shorter than the decay time in the second phase; this fact and the quick eruptive release of energy by the reconnection mechanism indicate that the waves must be unstable.
It is noteworthy that the inner gravity waves do not fulfill these requirements. The quesion is whether the rmodes do?
In a nonrotating sphere (, where is the angular frequency of solar rotation) the wave motion is subdivided into two noncoupling components: spheroidal p, f and gmodes (for which the main restoring forces are pressure gradient and buouancy) and toroidal modes (e.g. Unno et al. 1989). Toroidal modes are degenerated horizontal eddy motions confined to a spherical surface with a radius r for which , , and . Here Y^{m}_{l} is the spherical harmonic with a degree l and order m, is the colatitude, is the azimuthal angle in the spherical polar coordinates, is the fluid velocity field, is the angular frequency of the fluid motion, and Q^{m}_{l}(r) is an arbitrary amplitude function. Toroidal modes have zero radial velocity but have nonzero radial vorticity, (for the spheroidal modes it is vice versa). These modes do not alter the equilibrium configuration.
When a slow rotation ( ) is included the spheroidal modes are slightly modified but they keep their main properties. Degeneracy of toroidal modes is removed only partially by the rotation, and quasitoroidal waves  known as rmodes  appear with a nonzero frequency of in the rotating frame (Papaloizou & Pringle 1978; Brayn 1889). Usually the governing equations of the rmodes are obtained by expanding the initial physical variables of the equations in the rotating system into power series with respect to the small parameter (10^{4.7} for the Sun, e.g. Papaloizou & Pringle 1978; Provost et al. 1981; Smeyers et al. 1981; Saio 1982). These power series practically describe the deviation of the surface of the star from its initial spherical state, resulting from rotation through Coriolis and centrifugal forces. As a result of the deformation of the spherical surface with a radius r the radial vorticity of the toroidal modes cause a surface pressure perturbation through the Coriolis force. However, the rmodes practically keep the main properties of toroidal flows: , . The degeneracy of the rmodes is that their frequencies hardly depend on Q^{m}_{l}(r), i.e. they are independent from the inner structure of the star. For the l=1 modes the frequency in the inertial system is again close to zero, (Papaloizou & Pringle 1978). The rmode equations define the amplitudes Q^{m}_{l}(r), and taking into account the next terms with small in the series practically does not change the frequencies.
Due to the fact that the rmodes are practically surface deformation waves, some similarity of these waves to the surface gravity waves or to the fmodes is apparent. For high l the fmodes are an analogy to surface gravity waves in a planeparallel fluid with . In the Cowling approximation fmodes with l=1 have zero frequency too, (Unno et al. 1989). This corresponds to a parallel displacement of the whole star. For high l the fmode frequencies are also independent of the inner structure, with (Gough 1980). So, rmodes are also fundamental rotating modes with an inertial frequency, .
For the Sun the properties of rmodes have been investigated in great detail by Wolff et al. (1986) and Wolff (1998, 2000; and Refs. therein).
Some properties of the rmodes are also similar to those of the Rossby waves in geophysics (Pedlosky 1982). Similar to the Rossby waves and unlike the gmodes the rmodes are strongly anisotropic. They propagate only in azimuthal direction, opposite to rotation (i.e. they are retrograde waves in the corotating frame). Because we are interested in length scales corresponding to those of large sunspots, we have to consider rmodes with . To get oscillations with periods of years ( ) we must choose , just such rmodes are physically more interesting (Lockitch & Friedman 1999). However, in the case of high l the amplitudes of the rmodes will be concentrated near the surface of the Sun (Provost et al. 1981; Wolff 2000), and so they can actively interact with convective motions (Wolff 1997, 2000). Because for these modes and , their chance to take part in the redistribution of angular rotation momentum in the radiative interior is low. Note that the slow solar differential rotation does not change the behavior of such rmodes with (Wolff 1998).
Looking for further analogies between waves connected with gravity and with rotation, we remember that beside the surface gravity waves there exist internal gravity waves with , the frequencies of which depend on the inner structure (N is the BruntVäisälä frequency). Similar to these waves there exist "true" Rossby (not deformation) waves, the frequency of which depends also on the internal structure.
We include here a short review on the main features of Rossby waves; they have been investigated in great detail in geophysics (e.g. Pedlosky 1982; Gill 1982). In the simplest case, that is in a planeparallel, homogeneous, rotating layer, the dispersion relation for the Rossby waves is . Here k_{x} is the wave number perpendicular to the rotation axis, k_{z} is expressed by the internal deformation radius of Rossby which depends on the BruntVäisälä frequency, is the transverse gradient (in y direction) of the Coriolis parameter: a vertical component of the "planetary" vorticity in the given local point. Unlike the rmodes the Rossby wave frequencies are functions of the internal structure and have maximum dependence on the gradient : if and if .
Any disturbance of the local flow in a rotating frame may generate waves of the Rossby type. These waves exist only if there is a gradient of the potential vorticity . Here an absolute vorticity is the sum of the relative and the planetary vorticities, , is any conserved scalar quantity, (for instance, for adiabatic motion that could be the entropy or the density in the case of incompressible plasma). The Rossby wave motion is a solution of the nonlinear equation for transport of . The potential vorticity is conserved if the medium is barotropic ( ) and if there are no torques. The rotation of the frame is added to any vorticity in the velocity field. Any motion within a rotating fluid serves as a potential source for vorticity.
The relative vorticity may be evoked by the geometrical surface as well as by internal gradients. It depends on the choice of the function and on . For example, an unevenness of the ocean bottom causes the topographic Rossby waves, or a dependence of the Coriolis parameter on the earth latitude ( , where is the geographic latitude) is the main cause of atmospheric Rossby waves.
In the solar dynamo context the ability of Rossby waves to induce solarlike magnetic fields has been considered by Gilman (1969). Here the mechanism for sustaining the Rossby waves is a latitudinal temperature gradient in a thin, rotating, incompressible convective zone. To interpret the dynamical features of largescale magnetic fields the Rossby vorticies excited within a thin layer beneath the convective zone are considered by Tikhomolov & Mordvinov (1996) as the result of a deformation of the lower boundary of the convective zone.
From the discussion in Sect. 1.4 we conclude that just Rossbylike waves could be suitable for our requirements. As the main driving mechanism we choose a latitudinal (or horizontal) differential rotation, . Baker & Kippenhahn (1959) have pointed out that the uniform rotation of a star is not a typical case. Low frequencies (periods of years) could easily be obtained searching for the eigenoscillations of the Sun's radiative interior, where the gradient of the rotation speed is close to zero (in accordance with the helioseismology results). Large scales such as those associated with sunspots ( ) decrease the frequencies too. Similar to the rmodes the Rossby waves are strongly anisotropic (retrograde waves), but unlike the rmodes these waves are concentrated close to the solar center. These results have already been obtained by Oraevsky & Dzhalilov (1997), who investigated the trapping of adiabatic, incompressible Rossbylike waves in the solar interior. In the present work we take into account compressibility for the nonadiabatic waves. We look for unstable waves. It is clear that the necessary condition for the KelvinHelmholtz shear instability, (Ando 1985), is not fulfilled in the radiative interior. Then we decided to include the thermal mechanism of instability which is favoured at low frequencies (Unno et al. 1989). To balance the mechanism the radiative losses in the diffusion regime are included. To exclude all geometrical effects we ignore the influence of the spherical surface at the given radius. The dispersion relation in the limit of adiabatic incompressiblity and at very low frequencies is the same as that for Rossby waves in geophysics. In order to distinguish these rotational body waves from the rmodes we call them Rmodes (Rossby rotation).
The governing fourth order equation is obtained from the basic equations in Sect.2. Some qualitative analysis of the wave cavity trapping is done for the simpler adiabatic case in Sect.3. Using the asymptotic solutions obtained in Sect.4 the complex boundary value problem is solved in Sect.5. The calculation of the eigenfrequencies and the instability analysis are done in Sect.6. The obtained unstable modes are shortly discussed in Sect.7.
Let us investigate global motions with large timescales such that the Rossby
number is small,
.
Before the appearence of any
disturbances the basic stationary state of the rotating star is defined
mainly by the balance of pressure gradient, gravity force, and forces
exerted by the noninertiality of the motion (Coriolis and centrifugal forces).
In the case of an incompressible fluid with a homogeneous rotation rate
usually this state is called "geostrophic balance". A star disturbed by an
external force tends to return to this basic state. Our aim is to study for
the Sun the dynamics of small deviations from the steady geostrophic balance.
For this purpose it is natural to write the dynamic equations in a frame
rotating together with the Sun. The magnetic field will be ignored. For
arbitrary
the equation of momentum in conventional
definitions is given by
The next simplification of Eq. (2) is connected with the quasirigid
rotation of the inner part of the Sun below the convective zone, which
is known from the solution of the inverse problem of helioseismology. In this
case we can omit the third term of the l.h.s of Eq. (2). Such a
restriction of the gradients of
requires to obey the
conditions:
The next equations are the mass and energy conservation equations in the
standard form:
In the limit of incompressible fluid, or , it follows from Eq. (6) that the condition is not needed to satisfy . Hence, in a dissipative ( ), incompressible fluid sound cannot propagate instantaneously. It means,we cannot use the condition of to get the incompressible limit for nonadiabatic waves.
Now we will try to simplify the energy loss function
assuming
reasonable approximations for the Sun's interior. We use the formula for the
heat conductivity of a fully ionized gas to show that in the Sun radiative
transport of energy is more important than that by particle heat
conductivity:
(7) 
(9) 
Let us take an arbitrary point at the surface of the rotating sphere. The
position of this point is defined by its radius r, its colatitude
,
and its azimuth angle. We assign to this point a local,
lefthanded Cartesian system of coordinates ,
where the axis is directed along the radius (local vertical), the direction of the
axis is meridional (towards the pole), and that of the axis
azimuthal. In this frame of reference
.
Strictly speaking, the
axis coincides with the rotation axis only at the pole
(
). In the case of a homogeneous fluid
is included into
the wave equation only in the term
In order to construct the "plane" limit we expand
around
a fixed
(
):
.
Here
,
,
and
Note that such a ""limit is applicable also around the equator plane, where the traditional approximation does not fit. The advantage of this limit is the possibility to include as a parameter in the Cartesian system. In this way we use here an inertial Cartesian system of coordinates (x,y,z) in a frame rotating with an angular frequency . All nonperturbed model variables are functions of z only, and for the gravity acceleration we have . For the observer from the nonrotating frame the elements of fluid are moving due to rotation with a velocity , where in the frame of our approximation . V_{0x}<0 means that the xaxis is directed opposite to the rotation.
For linearization each physical variable f=f_{0}+f' is decomposed into a mean
term f_{0} and a small fluctuating term f'. Neglecting terms of higher
order than the first one we get our oscillation equations
We define
=  
=  
= 
(14) 
(22)  
(23)  
(24) 
(28) 
Our next step is to separate the z and y dependence of the variables in
the governing equations to have finally one ordinary differential equation.
In the general case such a separation is not possible, and we consider here
the very simple case when the function
is
independent of y. Only in this case it is possible to separate the
equations with respect to the variables y and z and we can write
.
Here k_{y} should be determined
from the boundary conditions, and a complex k_{y} is not excluded. In this
way the system of partial differential equations in the planeparallel
approximation is reduced to ordinary differential equations. Now we assume
the following formula for the rotation profile
(32) 
Our next step is to derive one differential equation for the temperature
perturbations. The variable v_{z} is easily excluded from
Eqs. (25)(27). Then we get for the pressure perturbations
(34) 
Introducing a new dependent variable
(35) 
(36')  
A_{1}=a_{1}a_{3}+a_{2}+a_{3}',  
(37')  
Because we are interested in very low frequency oscillations with periods of
120 years, we take
s^{1},
,
and we have a small parameter for our task
(38) 
For idealized adiabatic waves (
)
we have a second order equation,
(39) 
As we consider adiabatic waves, the transition to the incompressible
case can be done by
.
In this case
and instead of Eq. (41) we
obtain
For waves running in opposite direction to rotation in the azimuth ( ), the cavities (trapped wave area) can form between the bottom of the convective zone and almost the center of the Sun as well as in the outer part of the Sun where N^{2}>0.
Waves propagating parallel to rotation may be trapped only in the convective zone (N^{2}<0). To the outer and inner sides from the convective zone the amplitude of these waves decrease exponentially.
For the incompressible case it is easy to solve the eigenvalue problem of the cavity oscillations, because Eq. (42) has no singularity. Such a task has been solved by Oraevsky & Dzhalilov (1997). However, in the nonadiabatic case we cannot apply the limit . Therefore we have to investigate the more complicated compressible case.
To investigate the function I(z) given by Eq. (41) in a compressible plasma we need the orders of the quantities entering the function I(z). To estimate these values let us consider a linear profile of temperature, , where the gradient , is the effective temperature. Then we have a limit for the parameter from the center (z=0) up to the surface (z=1): . The other parameters have the same order, . Then we get also: . Now we can estimate the sign behavior of I(z). We shall consider more characteristic places of the Sun. In the following the condition will be supposed.
At the center, where
kms^{1}, if
we have
The area around the convective zone is more complicated.
The function I(z) has singularities at the points where d=0 and a=0;
d is connected with the BruntVäisälä frequency and
a is defined mainly by the rotation gradient, Eqs. (37').
The two points of d=0 correspond to the bottom and the upper boundary
of the convective zone. The function a(z) is defined mainly by two terms
Figure 1: Scheme of the dependence of the wave potential I(z) defined by Eq. (41) on the distance normalized to the solar radius. Zeros of this function I(z)=0 (turning points) divide the wave zone into transparent (cavity) and opaque (tunnel) parts. In the upper part of the Sun there are singular points of the function I(z) which are located between the turning points: the circles correspond to a(z)=0 and the boxes correspond to N^{2}(z)=0 (boundaries of the convective zone). The narrow area at the bottom of the convective zone between circles and boxes is the tachocline. The main internal cavity comprises the whole radiative interior as the convective zone becomes opaque for the waves.  
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Thus if we approach the convective zone from below there exists a sequence of special points: four times the function I(z) crosses zero (turning points), and between these zeros the singular points are placed (see Fig. 1). The turning points are determined practically by the condition of and the singularities by the conditions a=0 (circles in Fig. 1) and N^{2}=0 (boxes in Fig. 1). Due to the very low frequency and the sharp decrease of N^{2}(z) the turning point of the main inner cavity is very close to the first singular point where a=0.
In this way for waves with k_{x}>0 a main large internal cavity is placed practically between the center and the bottom of the tachocline. The solar atmosphere is a wavepropagating zone. Between the inner cavity and the solar surface a dark convective tunnel is placed; a very narrow wavetrapping zone around the bottom of convective zone is also possible. It is clear that a tunneling of waves across the magnetized and turbulent convective area to the surface is probably possible. For waves with k_{x}<0 the convective zone becomes a cavity. In this case the waves cannot be propagating at the solar surface.
Not all singularities of the wave potential I(z) are singular levels of the physical variables. Around the point where we can write . Then we have the equation x^{2}Y'' +2 Y=0, where . The solutions of this equation are . As we get from Eqs. (33) + (40) that if . It means, that the boundaries of the convective zone are not singular levels for the initial physical variables.
Another situation exists at the point where a=0. If we denote now x as , our Eq. (40) around the point is x^{2}Y''Y3/4=0, the solutions of which are . Hence, for these solutions we have const. Then we get from Eq. (33) that the second solution in p'/p_{0} diverges at a=0.
There exist methods to construct asymptotic solutions of differential equations of second order with a singular turning point. However, we are now returning to our fourth order Eq. (37) for two reasons: our intent is to consider the instability problem of the eigenmodes, and consequently in the complex plane the singularity at is removed from the real zaxis.
The existence of the small parameter
in Eq. (37) allows us to
apply asymptotic methods to solve this equation. Here we shall construct
the inner cavity solutions only. As it has been discussed above the
coefficients of Eq. (37) vary over a wide range. Very crudely, we have
the following estimates from the center of the Sun to the bottom of the
convective zone:


and
.
Assuming that
10) we can separate the
variable part of
as
and rewrite Eq. (37) in a convenient form:
The solutions corresponding to the dissipative modes are searched for in the
form:
S^{2}  =  (47)  
B_{1}  =  (48)  
F_{0}  =  (49)  
F_{1}  =  S(2SB_{0}''+3S'B_{0}'+S''B_{0})+3[S(SB_{0})']'  
(50) 
Thus we have a radiation boundary condition for the dissipative modes: from the
point
(turning point of adiabatic waves) the dissipative waves
are radiated. In all directions the amplitudes of these modes
must decrease, and while setting
these modes must disappear.
The following solution obeys these conditions:
Now we will search for "slow" quasiadiabatic wave solutions in the form
=  (61) 
=  (62) 
=  (63) 
(65) 
Now we have to determine from the boundary conditions one of the unknown
constants C_{1,2}. Here we assume the next simplification. In the real solar
situation we should include the tunneling of waves through the opaque
convective zone to the transparent solar surface. However, in this work we do
not complicate the situation by including this important effect. Here we are
interested in the eigenoscillation spectrum of the main interior cavity. So our
solutions must be finite in the whole domain of integration:
.
In the limit of
we have the asymptotics
(67) 
In this section we impose physically reasonable boundary conditions to the general solution of (69) to determine the spectrum of the eigenoscillations of the inner cavity.
At the solar center (z=0) where
,
and
we apply a rigid boundary condition: v_{z}(0)=0. As the function
is
finite at the center Eq. (25) reads as
.
Using Eqs. (27) + (21) with dimensionless parameters we have
Let us consider the location of the upper turning point (
): there the
equation
is fulfilled, from which we may define the
parameter d,
.
Here we can apply a free boundary condition, because at the bottom of the
convective zone the BruntVäisälä frequency and the rotation rate
change very sharp. (A more realistic approach would be to
include the tunneling of the waves.) At the free surface the Lagrangian
pressure must be constant,
,
where p=p_{0} +p' is the total
pressure. From here we have
(76) 
From Eqs. (72) + (78) we get the general dispersion
relation
Now we can determine the complex eigenfrequencies from the integral dispersion relation Eq. (86) for the given s^{1}. Recall that is the latitudinal gradient of the rotation rate, is the temperature exponent of the nuclear reaction power, , and determines the power exponent of the temperature of the nuclear reactions, (Eq. (5)). For the pp reactions we will use . The dependence of this equation on the free parameter k_{y} is a drawback of this equation. In real situations the waves cannot be progressive across the shear in ydirection. In the general case the complex is a solution of the twodimensional eigenvalue problem. To simplify our task for a very small latitudinal gradient of rotation we have introduced the free parameter k_{y}. To simulate the decay of the wave amplitudes towards the pole, in y direction, we shall consider only the case with real k_{y}^{2}<0 . A negative spectral parameter k_{y}^{2} could also be confirmed by the tidal equation of Laplace (e.g. Lee & Saio 1997) for very low frequencies. This equation includes the influence of sphericity on the angular dependence of the eigenfunctions, if simple rigid rotation is considered in the frame of the traditional geophysical approximation (which fails near the equator).
Ando (1985) has derived a local dispersion relation for waves around the stellar equator. Solving his Eq. (15) with respect to k (that is our k_{y}) for low frequencies we get two solutions: k^{2} = m^{2} and , where m is the azimuthal wave number. As , the second solution is strongly damped in ydirection. Our case corresponds to the first solution, k^{2} = m^{2}.
From geophysical applications (Pedlosky 1982; Gill 1982) we infer that
is obeyed for the ageostrophic wave propagation
across the shear flow. Really, it is seen from Fig. 2, that the mode
separation with respect to n is essential for the solar situation if the
condition
is fulfilled. In Fig. 2 the Re
dependence is shown for the case
and
for
example. Here, a very small imaginary part of the frequency is supposed,
,
where
(87) 
Figure 2: Mode separation region versus . On the vertical axis the absolute values of the frequencies normalized to are shown. The maxima from top to down in the curves correspond to n=1, 3, 5, ..., 19. Calculations are done for the case . This figure does not yet show the eigenfrequencies, but the domain of k_{y}^{2} where the ndependence of the frequencies is obvious. This domain is .  
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Figure 3: The harmonic numbers ( ) limited by the nonadiabatic effects versus k_{x} and . The quantities on the curves correspond to the gradients of the rotation rate . Such a limit for high n does not exist for the adiabatic waves.  
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Figure 4: Real ( ) and imaginary () parts of the eigenfrequencies ( a) and b), respectively) normalized to versus the radial harmonic number n. The numbers on the curves 1, 2, ..., 10 correspond to different values of the latitudinal gradient of the rotation rate , . Positive corresponds to the growth rate of the waves due to the mechanism. For negative the waves become damped in time due to radiative losses.  
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Figure 5: Growth rate of the instabilities of the eigenmodes of the differentially rotating Sun. Three modes with periods of 13yr, 1830yr (these ranges depend on ) with a small additional peak at 100 yr, and 150020000yr (independent of ) become maximum unstable () for high orders n. The solid, dashed, and dotted lines correspond to the small latitudinal gradients of the rotation rate of , and 10^{5} at .  
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It follows from Eq. (86) that for a given the frequencies decrease with increasing harmonic number n( ) in the adiabatic case ( ). At very high we have almost steady motions with a frequency . Howewer, the situation is changed if dissipation is taken into account, . In this case smallscale motions are quickly damped, very low frequency oscillations with high n could not become trapped and cannot manifest themselves as eigenmodes. Really, as the nonadiabatic parameter , at very low frequencies the solutions of the initial equations must have a dissipative character. An investigation of the initial equations for a steady flow with is a separate task. Here we restrict ourselves only to our dispersion relation Eq. (86) with increasing n. We can rewrite this equation as . With decreasing the righthand part decreases, and then it becomes independent from . It means, that n is limited: . Hence, it is a results of the dissipative effects, that only limited modes become trapped. The value of depends on k_{x} and . In Fig. 3 the dependence is presented. It follows from this figure that for a given k_{x} the number increases strongly by increasing the rotation gradient . So, if the solar interior is rotating similar to a solidbody, very longperiod oscillations (almost steady flows) should be suppressed. As the values of are sufficiently high, the accuracy of the asymptotic solutions is high.
Now we consider which spectrum of trapped waves with is possible. To calculate the wave spectrum from Eq. (86) we use the standard model of Stix (private communication; Stix & Slaley 1990) for the solar interior. For the special case (i.e. km) the ndependence of the real and imaginary parts of the eigenfrequencies are shown in Figs. 4a,b. The calculations were done for different small values of the rotation rate gradient , covering a wide range: 10^{4}. This is done because we know from helioseismology only that the parameter is small, but the exact value is not yet known. In Figs. 4a,b and are normalized to the cycle frequency of the 22year oscillations: s nHz). As expected the frequencies decrease with n and increase with or k_{x}. The imaginary parts oscillate around the zero value: if , the waves are unstable and their amplitudes increase with time; in the opposite case, if , we have stable/damped waves. In Fig. 4b we have two positive maxima: the first corresponds to shortperiod oscillations of 13 yr ("quasibiennial modes") and the second one to mediumperiod oscillations of 1830yr ("22yr modes"). The position of the quasibiennial modes versus n is stable and is 15. For the 22yr modes n_{22} is slightly increased with an increase of . It is seen from Fig. 4a that n must increase to keep the same frequency with increasing . For smaller frequencies this shift is stronger. Figure 4b shows that the instability gets stronger if increases: unstable waves become more unstable and damped waves are stronger damped. Waves with high n>200 at very low frequencies also show instability which cannot be shown in Fig. 4b due to the scaling.
Of course, will change with the radius in the real solar radiative interior. Hence, those places, where becomes relatively large, may become sources of unstable waves. Our calculations can easily be generalized for any k_{x} as Eq. (86) is a function of only.
To characterize the mode instability, the behavior of the parameter is more important. is the growth rate (increment) of the instability of the modes if and the damping rate (decrement) if . In Fig. 5 we present the dependence for the whole range of frequencies ( ) for which our asymptotic theory is valid. We have three distinguished global maxima for the growth rate which correspond to period ranges of 13yr, 1830yr with a small additional peak at 100 yr, and 150020000yr ("4000yr modes") of the eigenmodes. These modes have radial node numbers near n=850, 60, and 15, respectively. It was already mentioned, that for the unstable modes the value of n is slightly changed with a change of . So for the unstable modes n is high and the asymptotic results are reliable. The growth rates of the 22yr oscillations are always greater than those of the quasibiennial modes. The characteristic growing time for the 22year modes is 1000yr as .
Figure 6: The existence area of the 22yr (solid lines) and 2yr (dashed lines) eigenmodes versus the parameters , and . Figure 6a shows that the wavenumber and the rotation gradients ( ) are limited as . The two narrow maxima of the growth rate in Fig. 6b indicate that for the strongly fixed values of the 22yr and 2yr modes become unstable (see text).  
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Now we consider in which range of n and
the unstable modes
are located. For this aim we fix the frequency as
s^{1}, and for a given
we find
the complex root of the dispersion relation Eq. (86). The parameters
and
correspond to this root. The same calculations
were done for the 2yr oscillations. The results are shown in Fig. 6. Here the
solid lines correspond to the 22yr and the dashed lines to the
quasibiennial modes. A sharp increase of n to
with
in Fig. 6a indicates some upper limit for the parameter
.
For the 22yr and 2yr modes these limits are approximately
and
,
respectively. Figure 6b (where the
increment/decrement is presented) shows for which values of
and n the 22yr and 2yr modes become unstable. Two maxima for each mode
appear in .
These are for the 22year modes:
and
;
for the 2yr modes:
and
.
These estimates could be used to define a possible value of
inside of the Sun. This will be possible if we can identify these modes
from observations. For instance, at
(sunspot scale)
a gradient of the rotation rate with
or
is needed to excite the 22yr modes. Here the
possibility of an excitation of both the n=15 and the n=55 modes is not
excluded if
is changed with the radius.
A better way to define
would be to identify both the 22yr and the
quasibiennial modes, with different k_{x} at the surface of the Sun.
In the present paper we have shown that toroidal eddy flows which are degenerated in a nonrotating fluid can become a reservoir of various branches of oscillatory modes when the degeneracy is removed by rotation. The mechanism depends on the condition for the existence and alteration of the relative vorticity as well as on the stellar rotation rate and its gradient. Apparently at least for slowly rotating stars ( ) the rotation waves could be divided into two types: rmodes with high frequencies ( ) which are independent from the inner structure and mainly caused by geometrical effects, and the Rmodes with low frequencies ( ) which depend on the inner structure and are considered in the present paper. This classification is similar to that of f and g modes or to that of surface and body tube modes of magnetic cylinders. Note that the properties of Coriolis forces and ponderomotive forces in MHD are very similar to each other. Both rotation modes are prototypes of the geophysical Rossby waves.
We investigated the instability problem of the Rmodes sustained by a very small latitudinal gradient of the rotation rate in the solar radiative interior. The problem has been solved for a realistic solar model without an arbitrary choice of free parameters except , the product of and the horizontal wave number. Among the eigenoscillations three modes with periods of 13yr, 1830yr, and 150020000yr turn out to be maximum unstable to the mechanism. Here the smoothing effect is the radiative damping. All of these instabilities are in the range of high radial node numbers n which indicates that the applicability of the asymptotic solution is satisfied.
The 22yr modes with a growing time of 1000 yr are of particular
interest with respect to the solar activity cycle problem. In the simpler
case when adiabatic Rmodes are considered in an incompressible fluid,
in Eq. (86) is independent of the wave number and of the
frequency for very low frequencies. Then in the azimuthal direction the phase
and group velocities are
(88) 
A nice property of the Rossby waves is that every monochromatic mode is a solution of the full nonlinear hydrodynamic equations. It means, that we should expect the development of nonlinear Rmodes with large amplitudes. We could also expect that just in this nonlinear regime the toroidal magnetic flux will be lifted from the upper boundary of the cavity (the tachocline) to the surface. The energy release of the nonlinear waves could be accomplished by magnetic reconnection. Here it is possible that toroidal currents are generated via a twist of toroidal magnetic field lines by the cyclonic flows of regular Rmodes with fixed characteristics. Parker (1955) as well as Steenbeck et al. (1966; see also Krause & Rädler 1980) have suggested for the dynamo process that such a mechanism, the effect, is working by turbulent motions under the influence of Coriolis forces.
Our present model points out the possibility of forced oscillations instead of a selfexcited dynamo to solve the solar cycle problem, and this with the correct period of 22 yr. Similar ideas are due to Tikhomolov (2001) who has recently suggested a hydrodynamic driving of the 11yr sunspot cycle. There is still a smaller peak of the growth rate (Fig. 5) at 100 years; such a period is observed as a modulation of the 11/22 yr cycles.
We expect that in a nonlinear stage of our model a huge toroidal magnetic field of 10^{5}G will no longer be required to cause a buoyant rise of magnetic flux tubes, because the external nonzero upflow produced by the regular vortical Rmodes could trigger the eruption of stable magnetic flux tubes stored in the overshoot region. However, we did not yet calculate the details of the rise of the flux tubes, thus the problem of their appearance at the surface with small tilt angles and at low latitudes remains still unresolved.
There is observational evidence for the shortperiod oscillations as well: from helioseismic sounding Howe et al. (2000, 2001) have recently discovered variations of solar rotation with a period of 1.3 yr in the lower convective zone. Quasitwo year modes are very likely seen regularly in various solar data (e.g. Waldmeier 1973; Akioka et al. 1987; Rivin & Obridko 1992). The existence of two magnetic cycles (the main 22yr and the quasibiennial period) on the Sun has been reported by Benevolenskaya (1996, 1998). So far the origin of these modes was not yet clear. Terrestrial quasibiennial oscillations have been clearly seen in tropical meteorological radiosonde data, and a possible solar origin by related phenomena in the solar interior, Rossby waves in particular, has been discussed as well (McIntire 1994).
The longperiod oscillations in the broad range  yr, with a maximum growth rate around 4500 yr, could be the cause of abrupt changes of the global terrestrial climate in the past: DansgaardOeschger events, these are abrupt onsets of warm periods during the last ice age, had mean distances of 4500 yr, but they were distributed over a larger period range, similar to that in our model, with shortest distances often around 1500 years (see, e.g., Ganopolski & Rahmstorf 2001). These events were caused by changes of the thermohaline circulation of the ocean, which in its turn were probably triggered by changes in the solar energy output.
Acknowledgements
Michael Stix kindly provided detailed tables from his internal solar model calculations. The critical comments and suggestions by Kris Murawski, KarlHeinz Rädler, and the referee J. Andrew Markiel helped to improve earlier versions of the paper. The authors gratefully acknowledge financial support of the present work by the German Science Foundation (DFG) under grant No. 436 RUS 113/560/11 and 3, by the German Federal Ministry of Education and Research through the German Space Research Center (DLR) under grant No. 50QL96019, and by the Russian Foundation for Basic Research (RFBR) under grant No. 000216271.