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$\begin{figure} \par\includegraphics[height=14.2cm,width=8.3cm,clip]{3696-f1.ps}\end{figure}$ Figure 1: The domains of validity of the solution regularity and of the instability conditions in the phase space $\{\varepsilon_{{\mbox{\tiny R}}}/m, \varepsilon^2_{{\mbox{\tiny R}}}\}$ for given values of the rotation gradient $\beta _{\mu }$ . a) Shows the values $\varepsilon_{{\mbox{\tiny R}}}/m = \chi_3$ , Eq. (52). In the area $\varepsilon_{{\mbox{\tiny R}}}/m \le \chi_3$ the solutions are limited at the pole. b) Shows the behavior of $\chi _1$ (dashed) and $\chi _2$ (solid). Between the solid and dashed lines with the same labels (values of $\beta _{\mu }$ ) mode instability is possible.
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In the text

$\begin{figure} \par\includegraphics[height=14cm,width=8.3cm,clip]{3696-f2.ps}\end{figure}$ Figure 2: The wave instability areas (hatched) in the phase space from an overlap of Figs. 1a and b and from the condition $\vert\epsilon _R/m\vert < \sqrt {\epsilon _R^2}$ for the integer m (solid line in a) and dashed in b)). The labels in the areas are the $\beta _{\mu }$ values. Both for prograde and retrograde waves $\beta _{\mu }$ is limited: for prograde waves the instability is possible if $\beta _{\mu }<0.5$ , for retrograde waves the instability is possible if $\beta _{\mu }>3.5$ .
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In the text

$\begin{figure} \par\includegraphics[height=7.1cm,width=8.4cm,clip]{3696-f3.ps}\end{figure}$ Figure 3: The local estimate of the logarithmic gradient of the solar rotation frequency $\beta _{\mu }$ for real solar data from helioseismology depending on the co-latitude and the radial distance (the labels are values of $r/R_{\odot }$ ). The bold horizontal line is $\beta _{\mu }= 3$ $\times$ 10^-4, above which the prograde waves become unstable (see next picture).
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In the text

$\begin{figure} \par\includegraphics[height=7cm,width=8.4cm,clip]{3696-f4.ps}\end{figure}$ Figure 4: Enlarged part of Fig. 2a for the smallest gradients of rotation. The labels 1, 2,..., 6 correspond to $\beta _{\mu }= 3$ $\times$ 10^-4, 5 $\times$ 10^-4, 1 $\times$ 10^-3, 2 $\times$ 10^-3, 3 $\times$ 10^-3, 4 $\times$ 10^-3. Areas of instability exist only if $\beta _{\mu } \ge 3$ $\times$ 10^-4.
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In the text

$\begin{figure} \par\includegraphics[height=7cm,width=8.1cm,clip]{3696-f5.ps}\end{figure}$ Figure 5: The spectrum $\omega (l,m)$ of low-frequency retrograde modes. The numbers on the curves are the l values. For the calculation of the spectrum, Eq. (69), $\Omega =\Omega _{\rm eq}=460.7$ nHz is used.
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In the text

$\begin{figure} \par\includegraphics[height=7cm,width=8.35cm,clip]{3696-f6.ps}\end{figure}$ Figure 6: The normalized group velocity $v_{\rm g}/2\Omega r_0\sin\theta_0$ as a function of the azimuthal numbers for given degrees l.
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In the text

$\begin{figure} \par\includegraphics[height=7.1cm,width=8.3cm,clip]{3696-f7.ps}\end{figure}$ Figure 7: The possible domain of the existence of eigenmodes for given Rossby numbers $\varepsilon_{{\mbox{\tiny R}}}$ (the numbers on the curves). For example, the case $\varepsilon_{{\mbox{\tiny R}}}=0.002$ (close to the 22-year modes) is emphasized. All possible values of the azimuthal numbers (m=m₁ and m=m₂) are on this curve for given discrete l in the range $l \le l_{\rm max}$ (see text). If $\varepsilon_{{\mbox{\tiny R}}}>0.1$ , the eigenmodes disappear.
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In the text

$\begin{figure} \par\includegraphics[height=7.1cm,width=8.35cm,clip]{3696-f8.ps}\end{figure}$ Figure 8: The phase velocities of fast (solid) and slow (dashed curves) modes versus l. The numbers at the curves are $\varepsilon_{{\mbox{\tiny R}}}$ values, similar to Fig. 7. Here the velocities are normalized to ( $-2\Omega~ r_0\sin \theta_0$ ). Every curve is restricted to the range $l \le l_{\rm max}$ .
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In the text

$\begin{figure} \par\includegraphics[height=7.1cm,width=8.35cm,clip]{3696-f9.ps}\end{figure}$ Figure 9: Absolute values of the group velocities of fast (solid) and slow (dashed) modes normalized to $2\Omega~ r_0\sin\theta_0$ versus l for selected $\varepsilon_{{\mbox{\tiny R}}}$ . For fast waves $v_{\rm gr}^+<0$ , for slow modes $v_{\rm gr}^->0$ . For $l=l_{\rm max}$ we have $v_{\rm gr}^+=v_{\rm gr}^-=0$ .
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In the text

$\begin{figure} \par\includegraphics[height=14.9cm,width=17.1cm,clip]{3696-f10.ps}\end{figure}$ Figure 10: The amplitudes of eigenfunctions versus co-latitude for given pairs (l,|m|), Eqs. (83)-(85). The first row shows the $\Theta (\theta )$ 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 ( $\xi _{\theta A}$ ) and azimuthal ( $\xi _{\phi A}$ ) amplitude functions, normalized to the maximum of ( $\xi _{\theta A}$ ).
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In the text

$\begin{figure} \par\includegraphics[height=7.4cm,width=8.2cm,clip]{3696-f11.ps}\end{figure}$ 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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In the text

$\begin{figure} \par\includegraphics[height=7.1cm,width=8.4cm,clip]{3696-f3.ps}\end{figure}$	Figure 3: The local estimate of the logarithmic gradient of the solar rotation frequency $\beta _{\mu }$ for real solar data from helioseismology depending on the co-latitude and the radial distance (the labels are values of $r/R_{\odot }$ ). The bold horizontal line is $\beta _{\mu }= 3$ $\times$ 10^-4, above which the prograde waves become unstable (see next picture).
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$\begin{figure} \par\includegraphics[height=7cm,width=8.4cm,clip]{3696-f4.ps}\end{figure}$	Figure 4: Enlarged part of Fig. 2a for the smallest gradients of rotation. The labels 1, 2,..., 6 correspond to $\beta _{\mu }= 3$ $\times$ 10^-4, 5 $\times$ 10^-4, 1 $\times$ 10^-3, 2 $\times$ 10^-3, 3 $\times$ 10^-3, 4 $\times$ 10^-3. Areas of instability exist only if $\beta _{\mu } \ge 3$ $\times$ 10^-4.
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$\begin{figure} \par\includegraphics[height=7cm,width=8.1cm,clip]{3696-f5.ps}\end{figure}$	Figure 5: The spectrum $\omega (l,m)$ of low-frequency retrograde modes. The numbers on the curves are the l values. For the calculation of the spectrum, Eq. (69), $\Omega =\Omega _{\rm eq}=460.7$ nHz is used.
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$\begin{figure} \par\includegraphics[height=7cm,width=8.35cm,clip]{3696-f6.ps}\end{figure}$	Figure 6: The normalized group velocity $v_{\rm g}/2\Omega r_0\sin\theta_0$ as a function of the azimuthal numbers for given degrees l.
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$\begin{figure} \par\includegraphics[height=7.1cm,width=8.35cm,clip]{3696-f8.ps}\end{figure}$	Figure 8: The phase velocities of fast (solid) and slow (dashed curves) modes versus l. The numbers at the curves are $\varepsilon_{{\mbox{\tiny R}}}$ values, similar to Fig. 7. Here the velocities are normalized to ( $-2\Omega~ r_0\sin \theta_0$ ). Every curve is restricted to the range $l \le l_{\rm max}$ .
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$\begin{figure} \par\includegraphics[height=7.1cm,width=8.35cm,clip]{3696-f9.ps}\end{figure}$	Figure 9: Absolute values of the group velocities of fast (solid) and slow (dashed) modes normalized to $2\Omega~ r_0\sin\theta_0$ versus l for selected $\varepsilon_{{\mbox{\tiny R}}}$ . For fast waves $v_{\rm gr}^+<0$ , for slow modes $v_{\rm gr}^->0$ . For $l=l_{\rm max}$ we have $v_{\rm gr}^+=v_{\rm gr}^-=0$ .
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$\begin{figure} \par\includegraphics[height=7.4cm,width=8.2cm,clip]{3696-f11.ps}\end{figure}$	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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