A&A 421, 305-322 (2004)
DOI: 10.1051/0004-6361:20040121
N. S. Dzhalilov^{1,2,3} - J. Staude^{2}
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
3 - Shamakhi Astrophysical Observatory of the Azerbaidjan Academy of
Sciences, Baku 370096, Azerbaidjan Rep.
Received 15 May 2002 / Accepted 13 October 2003
Abstract
The general partial differential equation governing linear adiabatic nonradial
oscillations in a spherical, differentially and slowly rotating non-magnetic
star is derived. This equation describes mainly low-frequency and high-degree
g-modes, convective g-modes, rotational Rossby-like vorticity modes, and
their mutual interaction for arbitrarily given radial and latitudinal
gradients of the rotation rate. Applying to this equation the "traditional
approximation'' of geophysics results in a separation into radial- and
angular-dependent parts of the physical variables, each of which is described
by an ordinary differential equation.
The angular parts of the eigenfunctions are described by the Laplace tidal
equation generalized here to take into account differential rotation.
It is shown that there appears a critical latitude in the sphere where the
frequencies of eigenmodes coincide with the frequencies of inertial modes.
The resonant transformation of the modes into the inertial waves acts as a
resonant damping mechanism of the modes. Physically this mechanism is akin
to the Alfvén resonance damping mechanism for MHD waves. It applies
even if the rotation is rigid.
The exact solutions of the Laplace equation for low frequencies and rigid
rotation are obtained. The eigenfunctions are expressed by Jacobi polynomials
which are polynomials of higher order than the Legendre polynomials for
spherical harmonics. In this ideal case there exists only a retrograde wave
spectrum. The modes are subdivided into two branches: fast and slow modes.
The long fast waves carry energy opposite to the rotation direction, while
the shorter slow-mode group velocity is in the azimuthal plane along the
direction of rotation. It is shown that the slow modes are concentrated
around the equator, while the fast modes are concentrated around the poles.
The band of latitude where the mode energy is concentrated is narrow, and
the spatial location of these band depends on the wave numbers (l, m).
Key words: hydrodynamics - Sun: activity - Sun: interior - Sun: oscillations - Sun: rotation - stars: oscillations
In a recent paper Dzhalilov et al. (2002; Paper I) investigated which lowest-frequency eigenoscillations can occur in the real Sun and what role they might play in redistributing angular momentum and causing solar activity. We found that such waves could only be Rossby-like vorticity modes induced by differential rotation. However, the general nonradial pulsation theory adopted from stellar rotation has some difficulties for such low-frequency oscillations. For slow rotation, when the sphericity of the star is not violated seriously, the high-frequency spherical p- and g-modes are no longer degenerate with respect to the azimuthal number m (Unno et al. 1989). Independent of the spherical modes, non-rotating toroidal flows (called "trivial'' modes with a zero frequency) become quasi-toroidal with rotation (called r-modes with a nonzero frequency; Ledoux 1951; Papaloizou & Pringle 1978; Provost et al. 1981; Smeyers et al. 1981; Wolff 1998). Although rotation removes the degeneracy of the modes, it also couples the modes with the same azimuthal order, and this makes the problem more difficult. For the high-frequency modes (Rossby number 1, where and are the angular frequencies of oscillations and of stellar rotation, respectively) this difficulty is resolved more or less successfully. For this case the small perturbation rotation theory is applied: the eigenfunctions are represented by power series, the angular parts of which are expressed by spherical harmonic functions Y^{m}_{l} (Unno et al. 1989). These power series are well truncated, unless , when the role of the Coriolis force increases.
Most pulsating stars point out the low-frequency instability (Cox 1980; Unno et al. 1989). Rotation causes strong coupling of the high-order and the convective g-modes, and the r-modes with and with the same m, but different l (Lee & Saio 1986). Generally the matrix of the coupling coefficients to be determined is singular (e.g., Townsend 1997). In all papers on the eigenvalue problem of nonradially pulsating stars, there exists a "truncation problem'' for the serial eigenfunctions, the angular parts of which are represented by spherical harmonics (e.g., Lee & Saio 1997; Clement 1998).
The governing partial differential equations (PDEs) of the eigenoscillations of rotating stars are complicated from the point of view of mathematical treatment, even if the motions are adiabatic. This difficulty arises because in a spherical geometry an eigenvalue problem with a singular boundary condition has to be solved. The singularity of the governing equations on the rotation axis is not a result of the applied physical approximations (small amplitude linearization, adiabaticity of the processes, incompressibility of motion, rigid rotation approximation, etc.), but is a property of the spherical geometry, permitting special classes of motion which may become the global eigenmotion of the star. Due to these difficulties it is very important to obtain some analytical solution of the eigenvalue problem with rotation, even if rather special approximations are used.
For the Coriolis force is the dominant term in the equation of motion. In this case this force causes the motion to be preferentially horizontal, . With decreasing wave frequency, , the flows become practically horizontal, . Here v_{r} is the radial component and are the surface components of the velocity. This is why in many investigations of global linear or nonlinear motions in differently rotating astrophysical objects the radial velocity component and the radial changes are ignored, i.e., v_{r}=0, . That is, 2D flows are considered, e.g. Gilman & Fox (1997), Levin & Ushomirsky (2001), Cally (2001), Charbonneu et al. (1999). Nevertheless, even such a 2D approximation describes the basic physics of the global r-modes of rotating stars, e.g., Levin & Ushomirsky (2001).
However, real situations (many stars show nonradial pulsations with rather low frequencies, e.g. Unno et al. 1989) require the inclusion of . It is important to study, e.g., the g-mode rotation interaction, the transport of rotation angular momentum, the mixing problems of the stellar interior, etc. The consideration of a small nonzero radial component of velocity is the next important step for improving the 2D approximation. Essentially, in geophysics the traditional approximation takes into account in the governing equations. This approximation is also often applied to global motions in astrophysical objects, for example, Bildstein et al. (1996), Lou (2000), Berthomieu et al. (1978), Lee & Saio (1997), Dziembowski & Kosovichev (1987). To avoid misunderstanding, we will give here a short excursion into this approximation; see, for example, Unno et al. (1989), Shore (1992). To show the meaning of the traditional approximation clearly let us use a local analysis procedure.
In the case of a homogeneous rotating fluid,
is
included into the wave equation only in the term
In the present work for the non-magnetic and non-convective cases we obtain one PDE in a spherical geometry for the adiabatic pressure oscillations in the differentially rotating star ( ) with arbitrary spatial gradients of rotation (Sect. 2). This general equation is split into - and r-component ODEs, if the traditional approximation is applied (Sect. 3). The -component equation is Laplace's tidal equation generalized for the differentially rotating case. In Sect. 4 we analyse this equation more qualitatively. We find the general condition for the shear instability due to differential rotation in latitude. We find that the smallest rotation gradient is responsible for the prograde (seen in the rotating frame) vorticity wave instability, while a stronger gradient causes the retrograde wave instability. For solar data (small rotation gradients) the m=1 prograde mode instability is possible (Sect. 4.4). The possible existence of such a global horizontal shear instability on the Sun has been investigated by Watson (1981) and Gilman & Fox (1997), that of shear and other dynamic instabilities and of thermal-type instabilities in stars also by Knobloch & Spruit (1982) and others. Laplace's tidal equation for low frequencies in the rigid-rotation case is investigated in detail in Sect. 5. It is shown that the eigenfunctions are defined by Jacobi polynomials which are of higher order than the Legendre polynomials.
The motion of the fluid in a self-gravitating star, neglecting magnetic field
and viscosity, may be described in an inertial frame by the hydrodynamic
equations. These equations in conventional definitions are written as
We suppose that the equilibrium state (zero indices of the variables) of the
star is stationary and that its differential rotation is axially symmetric:
(5) |
Small amplitude deviations from the basic state of the star may be
investigated by linearizing Eqs. (1)-(4). For Eulerian
perturbations (variables with a prime) the equation of motion becomes
(Unno et al. 1989)
Excluding
from Eq. (7) by using Eq. (12) and
taking its r and
components we get
(17) |
(18) |
Now adding to these equations the
component of Eqs. (7)
and (9) we get our set of equations:
For the general case the system of Eqs. (21)-(24) has been
reduced in Appendix A to one PDE for the pressure perturbation:
(28) |
Let
(29) | |||
b_{4} = 1-2b_{1} + b_{2} , | |||
We assume here an extra condition for the rotation rate: . Such a representation of the variable part of the rotation rate is no strong restriction. For the Sun, for example, except for the solar tachocline. Depending on the problem to be solved we can simplify the calculations ignoring either or .
In that way we may separate the variables
Equation (32) is the generalized Laplace equation if differential rotation is
present,
.
For rigid rotation,
,
Eq. (32) becomes the standard Laplace equation:
In astrophysics the r-component Eq. (33) appears, and the two equations must be solved together to find both spectral parameters. In the rigid-rotation case Lee & Saio (1997) looked for numerically in an approach similar to that in geophysics, fixing in Eq. (34). Here we offer another approach, where we will find from the r-equation for a given .
It is convenient to introduce into Eq. (32) the new variable
:
From Eqs. (A.2), (A.3) and (A.5) we can derive in the traditional
approximation the following formulae for the components of the fluid velocity:
Now we will impose a restriction on
const, the logarithmic latitudinal
gradient of the rotation frequency. We might take the linear dependence
,
but for such a profile the structure of the solutions is
not changed qualitatively.
Really
const. corresponds to a rotation profile
Let us introduce the new dependent variable
In our case the physical equations are reduced to Heun's Eq. (47) for which the Riemannian relation (48) is satisfied. This means that the regular eigenfunctions in the hemisphere including the singular pole and the equator exist and we can find them. A direct numerical solution of this equation is practically impossible and we intend to use the P-branches to find the analytical eigenfunctions. However, the main difficulty of Heun's Eq. (47) is connected with the "accessory'' parameter h which is not included in the Riemannian scheme. The arbitrariness of h does not make it possible to represent the solutions by a single P-function. Therefore, hypergeometric and confluent hypergeometric equations, the differential equations of Lamé, Mathieu, Legendre, Bessel, and Weber, those of the polynomials of Jacobi, Chebyshev, Laguerre and Hermite as well as those of Bateman's k-functions are special or limiting cases of Heun's equation. Due to the parameter h the solutions of Heun's equation are usually searched for as series of P-functions. For example, Erdélyi (1942) has represented in such a series the P-branches by the hypergeometric functions. In physical tasks, e.g. the damping of MHD waves in resonant layers, the same equation arises (Dzhalilov & Zhugzhda 1990). The exact solutions of Eq. (47) will be considered in the next paper of this series. Here we will restrict ourselves to a qualitative analysis and to a limiting case.
Generally for low frequencies ( ) three singularities of Heun's equation may be realized in a hemisphere: 0 < a < 1. Note that the singularity in the Riemann scheme does not occur in our problem. At the equator (x=1), Eq. (47) has two solutions with the exponents 0 and . If we put the P-solutions with these exponents into Eqs. (39)-(41), we can see that the first solution is divergent, while the second solution provides the limited and at the equator. In the same manner, at the pole (x=0) one of the solutions with the exponents "0'' and () may provide the limited and only for some selected values of (see the next subsection). In the construction of the global solution in the hemisphere, the boundary conditions at the equator and at the pole should remove the divergent solutions.
To provide regularity of the global solution at the middle singularity (x=a) the two P-solutions with the exponents 0 and ( ) must be regular. This is because every retained regular solution at the equator or at the pole will, after its analytical continuation, be expressed as the sum of these two independent solutions at x=a. The second exponent (at x=a) if or if . If then . This implies, that the second independent solution of Eq. (47) with the exponent is regular at the singular point x=a for all variables, as follows from Eqs. (39)-(41). For the first solution, however, with the exponent of "0" at x=a the physical variables and are limited only if a is complex, i.e. is complex. In this way we conclude that the solutions of the singular boundary value problem will be complex and that the process of wave-rotation interaction produces a mechanical instability.
In our initial adiabatic motion approximation for which the final
Eq. (47) is derived, the complex
should be related to one of
the following physical mechanisms:
1) Shear instability:
The latitudinal differential rotation with a gradient
can
favour the development of mechanical shear instability. This instability
is akin to the classical Rayleigh instability mechanism or to the
Kelvin-Helmholtz instability with gravity. The latitudinal instability cannot
be prevented by gravity in the horizontal direction. The development of
this instability will strongly depend on the values and the sign of
as well as on the wave frequency.
2) Resonant damping:
The existence of a critical latitude
may be the reason for
the damping of the eigenmodes. For the solar rotation the location of this
latitude in the hemisphere depends weakly on
,
see
Eq. (37). The critical latitudes exist even if the rotation is rigid at
the inertial frequency
.
The
appearance of the critical latitudes on the rotating spherical surface and
their role in the formation of a wave guide around the equator is a well
known phenomenon in geophysics (e.g., Longuet-Higgins 1965). A concentration
of the wave amplitude at the singular latitude is very similar to the
damping of adiabatic waves in MHD resonant layers (recall that the
nature of the Coriolis force is very similar to that of
the ponderomotive force). Physically the damping of MHD waves, for example,
in the resonant Alfvén layer where
,
means the
transformation of waves into Alfvén waves which are concentrated along
the magnetic field in the resonant layer; here
is the radial
dependence of the Alfvén velocity, k_{x} is the wave number along the
magnetic field. This mechanism is an important damping mechanism of MHD waves; it has been used tentatively as a corona heating mechanism (e.g., Ionson
1978, 1984; Hollweg 1987).
In our case the Alfvén waves are changed to inertial oscillations.
The transformation of rotation-gravity waves to inertial waves in the narrow
range of critical latitudes acts as a resonant damping mechanism. The resonance
of the local inertial oscillations (frequency
)
with
the eigenmodes (frequency )
occurs in those places where their
frequencies are very close to each other. For low frequency modes,
,
the resonant layer is located close to the equator, and
for higher frequency modes,
,
near the pole. It can be shown
that generally wave motion with a
frequency of
must be produced to ensure conservation of
potential vorticity in the rotating system.
The width of the critical layer will depend on the values of the imaginary
parts of the eigenfrequencies. Formally, if there exists a continuous
spectrum, then the critical layers may be very wide. In
reality, however, we may expect the formation of some "active''
latitude belts if only some selected discrete modes are excited. In these
belts the greatest part of the mode energy will be concentrated in the
inertial modes. The further interaction of these waves with rotation can
change the rotation velocity. The evolution of the produced resonant
inertial waves in their initial state should be investigated separately on
the base of our general PD Eq. (27). For the further devolopment another
dissipation effect as well as nonlinearity should be taken into
consideration. We are not aware of investigations of the resonant damping of
rotation waves through the critical latitudes in the physics of stellar
pulsations.
3) Tunneling of waves:
It is well known that the tunnel leakage of wave energy from a cavity leads
to a decrease of wave amplitudes in time, i.e. the tunnel effect can
work if the wave frequency is complex. This is very effective for
the p-modes in the solar atmosphere (Dzhalilov et al. 2000). Equation (35)
or Eq. (47) can easily be transformed into a two-terms form:
Z'' +
U(x) Z'=0. Due to the singular points the behavior of the wave potential U(x) is complicated, especially if
and
(we will omit here a detailed analysis of U(x)). However, a series of
changes of the sign of U(x) shows the existence of cavity (oscillation) and
tunnel (non-oscillation) zones in the area 0<x<1. Close to the equator at
for all cases U>0, and for most cases near the pole ()
U<0. This means that the eigenmodes are formed in the cavity which is
located around the equator. Near the pole we must choose only a decreasing
solution. For example, in the rigid rotation case we have
.
However, for any
(differential
rotation) and complex
we will have a complex wave vector in the
-direction. This means that we will have runnig waves toward the
pole. As the amplitude of these waves is zero at the pole we have no
reflected waves from the pole. Thus we may conclude that the tunnel effect
can work throughout each hemisphere.
These questions are important, but a detailed discussion may be given only after Heun's equation has been solved.
Here we discuss two conditions: the regularity of the solutions at the pole and the approximate instability condition for the modes. If we put the P-solutions with the exponents 0 and into Eq. (43) we get . Then means that for the regularity of the solutions at the pole the condition must be obeyed.
On the other hand, an instability is possible when the eigenfrequencies are complex, implying a complex , as the wavenumber is complex (for tunneling) if is complex. Otherwise, the solution of the eigenvalue problem gives a dispersion relation for which becomes complex if the parameter is complex (of course, this is one possibility). This is seen, for instance, directly from the dispersion relation (68, see below) for the lower frequency limit. Here |m| should be changed to . For the instability it follows from Eq. (44), that the necessary condition is S_{2}>0. It is clear that the axially-symmetric mode with m=0 is excluded in this case.
For lower values of the rotation gradient
the necessary condition S_{2}>0 demands
for the prograde waves (
)
the condition
,
which
is more realistic for stellar situations (equatorward spinning up at the
surface with radius r). Rayleigh's necessary condition for
instability (Rayleigh 1880; Watson 1981) says that the function
(gradient of
vorticity) must change its sign in the flow. Rewriting this function in our
definitions we get that
(51) |
The sufficient condition for instability is obtained from Eq. (44) and reads .
The regularity condition at the pole
can be rewritten
as
Figure 1: The domains of validity of the solution regularity and of the instability conditions in the phase space for given values of the rotation gradient . a) Shows the values , Eq. (52). In the area the solutions are limited at the pole. b) Shows the behavior of (dashed) and (solid). Between the solid and dashed lines with the same labels (values of ) mode instability is possible. | |
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For (strong gradients) the condition Eq. (52) is met only for retrograde waves in the range (region II). For we get for any that . This is the line between regions II and III. All three regions are shown in Fig. 1a. It is seen that the regularity condition is working for and if . For very small only modes with large mare possible. The smallest modes may appear in the limit . These conclusions are correct only if the instability occurs.
Now let us consider the second condition, the complex frequency condition
.
This inequality may be rewritten as
The total condition for the existence of spatially stable but temporally
unstable waves reads as follows:
(56) |
Figure 2: The wave instability areas (hatched) in the phase space from an overlap of Figs. 1a and b and from the condition for the integer m (solid line in a) and dashed in b)). The labels in the areas are the values. Both for prograde and retrograde waves is limited: for prograde waves the instability is possible if , for retrograde waves the instability is possible if . | |
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Figures 2a,b show the validity ranges of this condition for some typical values of . The hatched areas are places where mode instability is possible. These figures are obtained by overlapping Figs. 1a and b. The hatched areas of prograde waves are strongly restricted also by the condition because the integer . For prograde waves on both sides these hatched areas become very narrow: with decreasing the extent of the hatched area decreases and tends to the point . We will see that this is the solar case. The hatched instability areas disappear with decreasing . This means we have a lower limit .
For retrograde waves it is sufficient to write the condition as . In Fig. 2b is the dash-dot curve and is the solid curve. Instability is possible if for .
In Fig. 2 only the hatched areas for given are ranges of possible solutions, if wave instability occurs. Outside these hatched areas regular solutions are impossible. The case without instability (neutral oscillations) must be investigated separately.
Let us consider at which places we might expect mode instability in the Sun.
Unfortunately, it is not clear how the core rotates. Nevertheless some
rotation gradients close to the Sun's centre might exist, and we could expect
mode instability there. It is known from helioseismology that the radiative interior
has a very small
,
but the exact value is unknown. We have
better information on the rotation profile of the solar envelope, including
the tachocline. Helioseismology data may be described by different
approximate formulae. One of these is (Charbonneau et al. 1998)
Using this formula we show in Fig. 3 the dependence for different . We see that in the Sun , and the maximum is in the photosphere. Figure 2 implies that unstable retrograde waves are not present in the solar case. Instability of prograde waves in the Sun occurs in the upper right corner of Fig. 2a which is enlarged in Fig. 4. Here we see that the instability area disappears when 10^{-4}. This boundary is located at the bold horizontal line in Fig. 3. Thus the prograde waves become unstable in the Sun in those places where 3 10^{-4} 0.15. This means that instability is possible in the area which includes the greater part of the tachocline, the convective zone, and the photosphere. With increasing r the instability zone expands from middle to high latitudes. Figures 2a and 4 show that the instability occurs at high frequencies ( ) and on global scales ( ). Considering that is an integer we get m=1.
However, our instability analysis is based on the general Riemann scheme of Heun's equation, which is valid only if the middle singular point x=a is far from the other edges at x=0 and x=1. Thus, the limiting cases and (the latter is more important for the solar case) should be considered separately. In these limiting cases the regularity condition Eq. (48) may be changed, and the curve in Fig. 4 limiting the instability areas from below may be shifted. In this case instability with higher m-modes should be expected.
Figure 3: The local estimate of the logarithmic gradient of the solar rotation frequency for real solar data from helioseismology depending on the co-latitude and the radial distance (the labels are values of ). The bold horizontal line is 10^{-4}, above which the prograde waves become unstable (see next picture). | |
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Figure 4: Enlarged part of Fig. 2a for the smallest gradients of rotation. The labels 1, 2,..., 6 correspond to 10^{-4}, 5 10^{-4}, 1 10^{-3}, 2 10^{-3}, 3 10^{-3}, 4 10^{-3}. Areas of instability exist only if 10^{-4}. | |
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After the qualitative analysis of Heun's Eq. (47) we can start a quantitive analysis. Note that the qualitative conclusions drawn above are valid for the more general Eq. (35) with the term. Heun's equation with four singularities in the general case is solved by a series of hypergeometric Gauss functions. A similar task has been considered for the damping of MHD waves at resonance levels by Dzhalilov & Zhugzhda (1990). We will start to study Eq. (47) for some simple limiting cases. At high frequencies ( , where shear instability is acting in the Sun) and at low frequencies ( , where the waves are stable against shear instability in the Sun) Heun's equation is much simpler. In these cases the singular level x=a is shifted either to the pole or to the equator. For both cases the solutions are expressed by one hypergeometric function.
In the present work we consider particularly the second case. Let
.
Then we have
and
.
Equation (47) is now the hypergeometric equation:
(59) |
(61) |
Using the conditions
and
the
parameters in the solution Eq. (60) are greatly simplified. Because
only a regular solution at the pole (x=0) will be left.
In the standard definitions of hypergeometric functions (Abramowitz & Stegun
1984) we have
The new dispersion relation Eq. (69) completely differs from the
dispersion relation of the almost toroidal r-modes. Their dispersion relation
can be derived from Eq. (69) if we formally set
.
Then
Figure 5: The spectrum of low-frequency retrograde modes. The numbers on the curves are the l values. For the calculation of the spectrum, Eq. (69), nHz is used. | |
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Figure 6: The normalized group velocity as a function of the azimuthal numbers for given degrees l. | |
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Figure 7: The possible domain of the existence of eigenmodes for given Rossby numbers (the numbers on the curves). For example, the case (close to the 22-year modes) is emphasized. All possible values of the azimuthal numbers (m=m_{1} and m=m_{2}) are on this curve for given discrete l in the range (see text). If , the eigenmodes disappear. | |
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From this we get an upper limit for l if is given: . As we get an approximate condition for the existence of the modes: . For such degrees of l the azimuthal numbers are also limited: . For we have , , and . However, considering the regularity of the solutions Eq. (66), we must take . m_{1}=1 if . For we have m_{1}=m_{2}. This situation is shown in Fig. 7 for different values of . A decrease of the frequency decreases the domain of existence of the modes.
Putting Eq. (73) into Eqs. (71), (72) gives
Figure 8: The phase velocities of fast (solid) and slow (dashed curves) modes versus l. The numbers at the curves are values, similar to Fig. 7. Here the velocities are normalized to ( ). Every curve is restricted to the range . | |
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Here is the phase velocity of the fast modes and that of the slow modes. For we have and . In Eq. (73) and in Fig. 7 the m_{1} branch corresponds to the fast mode, but m_{2} to the slow modes, since . In Fig. 8 the normalized phase velocities (with inverse sign) for the selected values of in Fig. 7 are shown versus l. Both branches are retrograde modes ( ). Using km s^{-1} for the Sun, we get from Fig. 8 very slow phase velocities. The fast wave velocity (solid lines) depends more strongly on l. With increasing both branches are accelerated.
In Fig. 9 the group velocities are presented in the same way. For fast waves the group velocity is always parallel to the phase velocity ( ), while for the slow waves we have the opposite behavior . Slow mode packets carry off energy in the rotation direction. is valid everywhere. With deceasing the range shifts to the right, and it is seen in Fig. 9 that for such low is almost zero.
Note that m=l modes are always fast modes.
Figure 9: Absolute values of the group velocities of fast (solid) and slow (dashed) modes normalized to versus l for selected . For fast waves , for slow modes . For we have . | |
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Taking into account the quantization condition Eq. (68) in the
solutions Eqs. (63), (67), (39), and (41) we obtain
the eigenfunctions. Turning from complex velocities into the real
displacements,
(remember that
is the
velocity seen in the rotating frame), we get
(88) |
Figure 10: The amplitudes of eigenfunctions versus co-latitude for given pairs (l,|m|), Eqs. (83)-(85). The first row shows the pressure function normalized to its own maximum. From left to right all curves are related to the wave numbers first given above. The middle and last rows are similar to the first row, but now show the latitudinal ( ) and azimuthal ( ) amplitude functions, normalized to the maximum of ( ). | |
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Table 1: Results for the 22-year period: frequency deviations for permitted quantum numbers (l, |m|).
Let us consider some particular cases.
(89) | |||
(90) | |||
(91) |
In Fig. 10 for some typical selected pairs of (l,m) the amplitude functions, Eqs. (83)-(85), are shown as functions of . The first row is the pressure function function normalized to its maximum. The first l=1 figure represents for different |m|(|m| increases from left to right). As the eigenfunctions are multiplied by a factor, the amplitudes are strongly suppressed around the pole. Increasing |m| for a given l shifts the maxima toward the equator. l is the surface node number of the function. On the contrary, increasing l for a given |m| (the second figure of the first row with m=1 in Fig. 10) suppresses the amplitudes around the equator, and the maximum is shifted toward the pole. l=|m| is the equilibrium case. In the third figure of the first row the equilibrium latitude with maximum amplitude is defined by for all l=|m| modes.
The second and third rows of Fig. 10 are the latitudinal ( ) and azimuthal ( ) eigenfunction amplitudes, respectively, normalized to the maximum of , see Eqs. (81) and (82). The latitudinal amplitude behaviour is similar to that of the pressure. The azimuthal amplitudes are smaller than the latitudinal amplitudes, but with a change of l a redistribution of the amplitudes will not take place. has practically the same amplitude at all latitudes and for all l.
Figure 10 implies that we can expect an interesting behaviour of the eigenfunction amplitude, when both l and |m| are large. Suppression from two sides may give rise to a concentration of wave energy in narrow latitude bands. For example, this is the case for the 22-year solar mode.
For the 22-year modes we take (1.441 nHz), for which . Then we derive from the equations after Eq. (73) the limiting values of the integer l, . For all lin this range we find from Eq. (73), rounding off, integer azimuthal numbers m_{1} and m_{2} for the fast and slow modes, respectively. Putting these integer numbers into Eq. (69) we find the deviation from the central frequency due to the integer azimuthal numbers. Additional slow modes are also possible in the interval l=0-10. The results are given in Table 1 in Appendix A. It is seen that the fast modes with low l have larger deviations. This table includes all possible (l,|m|) pairs which correspond to the 22-year period. For some example pairs of (l,|m|) we plot in Fig. 11 the latitude dependence of the quantity , averaged over the wave period, which characterizes the energy density of the modes. The hemisphere is divided into two equal parts: slow modes are located around the equator (solid lines), fast modes are concentrated around the pole. Each (l,|m|) pair is located in a narrow latitude band. Note that the slow modes (the group velocity of which is in the rotation direction) with sunspot-like spatial scales are at latitudes of .
The eigenfunctions Eqs. (76)-(79) allow us to discuss the
flow character produced by the waves, even if the solution of the radial
equation Q(r) is unknown. Excluding from these equations the time-dependent
phase we can obtain the trajectory equations of the fluid elements. In the
meridional
plane we have
Figure 11: The normalized energy density of the 22-year fast (dashed) and slow (solid) modes versus co-latitude. The numbers on the curves are (l,|m|) pairs taken from Table 1. | |
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At the surface of the cone over
the trajectory of each
fluid element is an ellipse around the equilibrium point. The cone
displacement equation is
(94) |
On the surface of any sphere with a radius r the motion of the fluid
elements is on trajectories of the ellipse
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In the present work we have derived the general PDE governing non-radial adiabatic long-period (with respect to the rotation period) linear oscillations of a slowly and differentially rotating star. This general equation includes all high-order g-modes and all possible hybrids of rotation modes as well as their mutual interaction. The main objective of the present paper was to obtain an analytic solution of this equation for the extreme low-frequency limit and to show that the solutions cannot be expressed by the associated Legendre functions. Thus, in the low-frequency limit the pulsation theory of stars meets serious difficulities. For this purpose we used some simplifying approximations. The geophysical "traditional approximation'' considerably simplifies this general equation, and we get two ODEs for the r- and -components instead of one with arbitrary gradients of rotation . We have obtained a more stringent condition for the applicability of this approximation to the pulsation of stars. Only for very low frequencies is this restriction the same as that for the standard case. We also imposed some restrictions on the rotation profile used. All of these restrictions have been compared with empirical solar rotation profiles and, thus, we justified their use in our modelling.
The -equation is Laplace's equation generalized to a latitudinal differential rotation. Without solving this equation we found qualitatively the condition for the appearance of a global instability. This instability is driven by the latitudinal shear of rotation. It is not influenced by buoyancy, unlike the Kelvin-Helmholtz instability, and therefore it may easily be realized in stars, even if the horizontal gradient of rotation is small.
We have shown the possible appearance of critical latitudes at which there occurs a resonant interaction of eigenmodes with the inertial modes for a frequency . The transformation of modes with a frequency in narrow latitude belts acts as a resonant damping mechanism. With decreasing mode frequency these belts are shifted toward the equator. This mechanism may play an important role in the redistribution of rotation angular momentum.
The appearance of mode instability strongly depends on the Rossby number, on the azimuthal wave numbers, and on the latitudinal rotation gradients. Very large gradients produce retrograde waves (seen in the rotating frame), while a slower rotation gradient is responsible for the prograde mode instability. The rotation gradient has a lower boundary below which an instability of the modes is impossible for any Rossby number or azimuthal number m.
We have applied the instability condition to helioseismological data. Here a global instability is possible for the m=1 mode at practically all latitudes. Radially the instability zone extends from the greater part of the tachocline up to the photosphere. The shear instability for the Sun was first obtained by Watson (1981). According to his results the instability is possible only in the photospheric layers. Later Gilman & Fox (1997) showed that such an instability is possible in the tachocline too, if strong toroidal magnetic fields are included. Our results show that the instability of the m=1 modes and other modes (m>1) are possible without magnetic fields, in contradiction to Gilman & Fox (1997; see also Charbonneau et al. 1999). This difference is probably connected with the incompleteness of the equations used by Watson (1981) and by Gilman & Fox (1997); their equations are two-dimensional only.
The exact solutions of Laplace's tidal equation for lower frequencies are expressed by Jacobi's polynomials. Just for lower frequencies the numerical calculations of stellar pulsation analysis meet great problems, when one is looking for the eigenfunctions as infinite series of Legendre functions. The eigenfunctions, defined by higher-order Jacobi polynomials, cannot be expressed by convergent series of associated Legendre functions. Every Legendre function is a particular case of a Jacobi polynomial.
It has been shown here that the retrograde (slow and fast) modes with high surface wave numbers (l,m) are energetically concentrated in narrow latitude bands. This analysis was done for the 22-year modes as an example. Such a concentration of mode energy in a narrow spatial area makes such modes vulnerable to different instability mechanisms such as the -mechanism considered in Paper 1.
All of our results have been obtained in the traditional approximation. This approximation does not work for waves propagating strictly within the equatorial plane. In our work this case is excluded. Motions which were considered in our paper can never cross the equator, and our boundary condition is when . In the paper we derived all the solutions which obey the traditional condition: high (l,m) - short waves and .
For the independent variable
and the definitions
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Acknowledgements
The authors gratefully acknowledge the critical and very different, but constructive comments by two anonymous referees who helped to improve the paper. Moreover, we would like to thank George Isaak and Jet Katgert for careful reviews of the paper and suggestions for improving the language. The present work has been financially supported by the German Science Foundation (DFG) under grants Nos. 436 RUS 113/560/4-1 and 436 RUS 113/689/2-1 and by the Russian Foundation of Basic Research under RFBR No. 04-02-16386.