A&A 471, 1011-1022 (2007)
DOI: 10.1051/0004-6361:20077418
A. F. Lanza
INAF - Osservatorio Astrofisico di Catania, via S. Sofia 78, 95123 Catania, Italy
Received 6 March 2007 / Accepted 7 June 2007
Abstract
Context. The solar torsional oscillations, i.e., the perturbations of the angular velocity of rotation associated with the eleven-year activity cycle, are a manifestation of the interaction among the interior magnetic fields, amplified and modulated by the solar dynamo, and rotation, meridional flow and turbulent thermal transport. Therefore, they can be used, at least in principle, to put constraints on this interaction. Similar phenomena are expected to be observed in solar-like stars and can be modelled to shed light on analogous interactions in different environments.
Aims. The source of torsional oscillations is investigated by means of a model for the angular momentum transport within the convection zone.
Methods. A description of the torsional oscillations is introduced, based on an analytical solution of the angular momentum equation in the mean-field approach. It provides information on the intensity and location of the torques producing the redistribution of the angular momentum within the convection zone of the Sun along the activity cycle. The method can be extended to solar-like stars for which some information on the time-dependence of the differential rotation is becoming available.
Results. Illustrative applications to the Sun and solar-like stars are presented. Under the hypothesis that the solar torsional oscillations are due to the mean-field Lorentz force, an amplitude of the Maxwell stresses
10^{3} G^{2} at a depth of 0.85
at low latitude is estimated. Moreover, the phase relationship between
and
can be estimated, suggesting that
below 0.85
and
above.
Conclusions. Such preliminary results show the capability of the proposed approach to constrain the amplitude, phase and location of the perturbations leading to the observed torsional oscillations.
Key words: Sun: rotation - Sun: activity - Sun: magnetic fields - Sun: interior - stars: rotation - stars: activity
Doppler measurements of the surface rotation of the Sun show bands of faster and slower zonal flows that appear at midlatitudes and migrate toward the equator with the period of the eleven-year cycle, accompanying the bands of sunspot activity. The amplitude of such velocity perturbations, called torsional oscillations, is 5 m s^{-1} and faster rotation is observed on the side equatorward of the sunspot belt (Howard & LaBonte 1980). Helioseismology has revealed that the torsional oscillations are not at all a superficial phenomenon but involve much of the convection zone, as shown, for example, by Howe et al. (2000), Vorontsov et al. (2002), Basu & Antia (2003) and more recently by Howe et al. (2006,2005). The amplitude of the angular velocity variation is nHz at least down to 10%-15% of the solar radius, although the precise depth of penetration of the oscillations is difficult to establish given the present uncertainties of the inversion methods in the lower half of the solar convection zone (e.g., Howe et al. 2006). In addition to such a low-latitude branch of the torsional oscillations, helioseismic studies have detected the presence of a high-latitude branch (above latitude) that propagates poleward and the amplitude of which is about nHz (e.g., Toomre et al. 2000; Basu & Antia 2001). Such a branch seems to propagate almost all the way down to the base of the convection zone.
A general description of the perturbation of the angular velocity of the torsional oscillations, given the present accuracy of the observations, is provided by the simple formula (e.g., Howe et al. 2005; Vorontsov et al. 2002):
More recent studies by Covas et al. (2004,2005) present models based on the simultaneous solution of non-linear mean-field dynamo equations and the azimuthal component of the Navier-Stokes equation with a uniform turbulent viscosity. They reproduce the gross features of the torsional oscillations and of the solar activity cycle with an appropriate tuning of the free parameters. Rempel (2006,2007) considers the role of the meridional component of the Navier-Stokes equation in mean-field models and finds that the perturbation of the meridional flow cannot be neglected in the interpretation of the torsional oscillations. His models suggest that the low-latitude branch of the torsional oscillations cannot be explained solely by the effect of the mean-field Lorentz force, but that thermal perturbations in the active region belt and in the bulk of the convection zone do play an active role, as proposed by Spruit (2003).
In the present study, the angular momentum conservation is considered and the relevant equation in the mean-field approximation is solved analytically for the case of a turbulent viscosity that depends on the radial co-ordinate. A general solution is derived independently of any specific dynamo model, allowing us to put constraints on the localization of the torques producing the torsional oscillations. An illustrative application of the proposed methods is presented using the available data.
The observations of young solar-like stars by means of tomographic techniques based on high-resolution spectroscopy have provided recent evidence for time variation of their surface differential rotation (e.g., Donati et al. 2003; Jeffers et al. 2007). Lanza (2006a) has recently shown how such variations can be related to the intensity of the magnetic torque produced by a non-linear dynamo in their convective envelopes, in the case of rapidly rotating stars for which the Taylor-Proudman theorem applies. In the near future, the possibility of measuring the time variation of the rotational splittings of p-mode oscillations in solar-like stars may provide us with information on the changes of their internal rotation, although with limited spatial resolution. In the present study, we extend the considerations of Lanza (2006a) to the case of a generic internal rotation profile, not necessarily verifying the Taylor-Proudman theorem, to obtain hints on the amplitude of the torque leading to the rotation change.
We consider an inertial reference frame with the origin in the barycentre of the Sun and the z-axis in the direction of the rotation axis. A spherical polar co-ordinate system is adopted, where r is the distance from the origin, the co-latitude measured from the North pole and the azimuthal angle. We assume that all variables are independent of and that the solar density stratification is spherically symmetric.
The equation for the angular momentum conservation in the mean-field approach reads
(e.g., Rüdiger & Hollerbach 2004; Rüdiger 1989):
(4) |
(5) |
The equation for the conservation of the angular momentum can be recast in the form:
The solar angular velocity can be split into a time-independent component
and a time-dependent component ,
i.e., the torsional oscillations:
(11) |
The general solution of Eq. (12) with the boundary conditions (15) can be obtained by the method of separation of the variables and expressed as a series of the form (e.g., Lanza 2006b, and references therein):
To find the solution appropriate to the solar torsional oscillations as specified by Eqs. (1) and (2), it is useful to derive an alternative expression for the functions
as follows. Substituting Eq. (13)
into Eq. (25) and taking into account that the element of volume is
,
the right hand side of Eq. (25) can be recast in the form of a volume integral extended to the solar convection zone:
(28) |
Equations (33) can be used to compute the angular velocity perturbation when
is known.
Note that in the case in which the Lorentz force due to the mean field and
the meridional flow are the only sources of angular momentum redistribution, Eq. (14) gives:
The results derived above allow us to introduce methods to localize the torques producing the torsional oscillations in the convection zone. Suppose that the observations provide us with the functions
and
appearing in Eq. (1). The functions
can be written as:
(36) |
The divergence of
can be obtained from
Eqs. (13) and (24) as:
(42) |
(45) |
The statistical errors in the measurements of the angular velocity variations can be easily propagated through the linear Eqs. (37), (39), (40) and (44) to find the errors on the estimates of
or the average of
.
For instance, if we consider the standard deviations
of the data d_{i}, i.e., the rotational
splittings or the splitting coefficients from which the internal rotation is derived, the standard deviation
of the integral in Eq. (44) is:
(47) |
(48) |
Note that a constant relative error in the measurements of A^{(c,s)} leads to the same relative error in Eq. (40) and in Eqs. (44) and (46), given the linear equations that relate the corresponding quantities. Indeed, there is also a systematic error in our inversion method related to the poor knowledge of the turbulent viscosity that determines the form of the radial eigenfunctions .
The variation of the kinetic energy of rotation associated with the torsional oscillations, averaged over the eleven-year cycle, can be computed according to Lanza (2006b) and is:
(50) |
Most models of the torsional oscillations assume that they are due to the Lorentz force
produced by the mean field as derived from dynamo models. Therefore, let us consider the case in which only the mean-field Maxwell stresses contribute to the perturbation, i.e.:
(52) |
(53) |
Sequences of Doppler images can be used to measure the surface differential rotation of solar-like stars and its time variability, as done by, e.g.,
Donati et al. (2003) and Jeffers et al. (2007) in the cases of AB Dor and LQ Hya. Lanza (2006a) discussed the
implications of the observed changes of the surface differential rotation on the internal
dynamics of their convection zones, assuming that the angular velocity is
constant over cylindrical surfaces co-axial with the rotation axis. The present model
allows us to relax the Taylor-Proudman constraint on the internal angular velocity, but
some different assumptions must be introduced to obtain the internal
torques in the active stars. Here we assume that the variation of the surface
differential rotation is entirely due to the Maxwell stresses of the internal magnetic
fields, localized in the overshoot layer below the convection zone, as in the interface dynamo model by, e.g., Parker (1993). The interior model of the Sun can also be applied to AB Dor and LQ Hya because they have a similar relative depth of the convection zone. Therefore, the basic quantities can be scaled according to the stellar parameters, as explained in Lanza (2006a).
Following Donati et al. (2003), we consider a surface differential rotation of the form:
(55) |
(57) |
This result made use of the limited information we can get from surface differential rotation. However, in the near future, the observations of the rotational splittings of stellar oscillations promise to give information on the internal rotation and its possible time variations. Since only the modes of low degrees ( ) are detectable in disk-integrated measurements, the spatial resolution of the derived internal angular velocity profile is very low. Lochard et al. (2005) considered the case in which only the mean radial profile of is measurable.
In view of such an information accessible through asteroseismology,
let us consider a more general case
in which some average of the internal angular velocity of the star, say
,
can be
measured as a function of the time:
(61) |
(62) |
(65) |
(66) |
A model of the solar interior can be used to specify the functions
and
that appear in our equations. While the density stratification can be determined with an accuracy better than 0.5%, the turbulent dynamical viscosity is uncertain by at least one order of magnitude and it is estimated
from the mixing-length theory according to the formula:
Figure 1: The ratio of the density to the density at the base of the convection zone ( - solid line) and the ratio of the turbulent dynamical viscosity to the turbulent viscosity at the base of the convection zone ( - dotted) versus the fractionary radius in our solar interior model. | |
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The basic equations of our model (i.e., 12, 13 and 14) can be made nondimensional by adopting as the unit of length the solar radius , as the unit of density , i.e., the density at the base of the solar convection zone, and as a unit of time , where is the turbulent viscosity at the base of the convection zone. Indeed, the value of the diffusion coefficients estimated from the mixing-length theory leads to a too short period for the solar cycle in mean-field dynamo models. Covas et al. (2004) adopted a turbulent magnetic diffusivity 10^{11} cm^{2} s^{-1} to get a sunspot cycle of 11 yr. This implies 10^{10} g cm^{-1} s^{-1} in our model.
The radial eigenfunctions and the Jacobian polynomials P_{n}^{(1,1)} have been computed from the respective Sturm-Liouville problem equations by means of the Fortran 77 subroutine sleign2.f^{} (Bailey et al. 2001). For the radial eigenfunctions , the Sturm-Lioville problem has been solved with Neumann boundary conditions at both ends, set at and to avoid divergence at the surface. For the Jacobian polynomials, limit point boundary conditions have been adopted at .
Note that, to be rigorous, it would be better to use the helioseismic estimate of at r=0.99 where we fixed our outer boundary for the computation of the radial eigenfunctions instead of the stress-free boundary condition that is valid only at the surface. However, the differences are confined to the outermost layer of the solar convection zone, the moment of inertia of which is so small that there are no pratical consequences.
The eigenvalues and the eigenfunctions computed by sleign2.f have been compared with those computed by means of the code introduced by Lanza (2006b). The relative differences in the eigenvalues and in the eigenfunctions are lower than 1.5% for , . However, some problems of convergence of the numerical algorithm used by sleign2.f have been found for , particularly for , so we decided to limit its application up to k = 19.
The Jacobian polynomials and the eigenvalues computed by sleign2.f are very good up to n =30, as it has been found by comparison with their analytic expressions up to n=10 and their asymptotic expressions for . We conclude that for and it is better to use the asymptotic formulae (22) and (18) instead of the numerically computed and P_{n}^{(1,1)}.
The eigenvalue gives the inverse of the characteristic timescale of angular momentum transfer of the mode corresponding to under the action of the turbulent viscosity. The longest timescale corresponds to the lowest eigenvalue, i.e., in nondimensional units. It corresponds to a timescale of 0.86 yr with the turbulent viscosity given by the mixing-length theory and to 39.5 yr with 10^{10} g cm^{-1} s^{-1}.
The available data on the torsional oscillations are displayed with a typical radial resolution of 0.05 and a latitudinal resolution of in Howe et al. (2006,2005). The rotational inversion kernels of Schou et al. (1998) show a higher radial resolution close to the surface, but, given the small amplitude of the torsional oscillations, the choice of a uniform resolution of 0.05 seems to be better.
We have found that the best results of the localization of the source term with the method
outlined in Sect. 2.4 are obtained with localization functions that depend on
r or
only and that have a smooth derivative. As a typical function to probe the
radial localization, we adopt:
Figure 2: The modulus of the radial localization kernel versus the relative radius for the function (69), as obtained by truncating the series of the radial eigenfunctions at , for three different intervals centered at r=0.725 ( upper panel), 0.85 ( middle panel) and 0.965 ( lower panel), all with an amplitude of r_{2} - r_{1} = 0.05 . The modulus of the derivative has been normalized to its maximum value. | |
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The performance of the localization method introduced in Sect. 2.4 has been tested with simulated data in the absence of noise. The results of two such tests are plotted in Fig. 3 where we consider the case of a purely radial perturbation localized within an interval of amplitude 0.05 . Its time dependence is assumed to be purely cosinusoidal and the corresponding coefficients are computed by means of Eq. (30) up to the orders n=38 and k=19. From the , the coefficients are computed by means of Eq. (26) and the simulated angular velocity perturbation follows from Eq. (16).
When the interval in which is localized coincides with one of the inversion intervals [r_{1}, r_{2}], the lower limit for turns out to be 80%-90% of the value assumed in the simulation (Fig. 3, upper panel). The agreement increases up to 90%-95% if we consider a case in which has a constant sign and put the value of the derivative in the denominator of Eq. (46) instead of its modulus. This happens because the modulus of the derivative increases the effects of the sidelobes by increasing the value of the denominator in Eq. (46). When the interval in which the assumed is localized does not coincide with an interval of the inversion grid, the inversion method still performs well distributing the contributions among neighbour intervals nearly in the correct proportion (Fig. 3, lower panel). The case of simulations including a noise component is straightforward to treat thanks to the linear character of our inversion method. The inverted value turns out to be the sum of the inverted noiseless value and of the contribution coming from the inversion of the noise. Their amplitude ratio is equal to the ratio of the amplitudes of the input noise to the input signal (for constant relative signal errors), as discussed in Sect. 2.4.
Figure 3: Upper panel: test case of the application of the inversion method from Sect. 2.4. An input profile with in nondimensional units between 0.80 and 0.85 (solid line) is used to simulate a noiseless profile of angular velocity perturbation. The reconstructed lower-limit profile according to Eq. (46) is plotted as a dotted line. Lower panel: same as in the upper panel, but with an input profile localized between 0.775 and 0.825 . | |
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The localization in latitude can be sampled by means of a localization function of the kind:
Figure 4: The modulus of the latitudinal localization kernel versus for the function (70), obtained by truncating the series of the Jacobian polynomials at , for three different intervals of in the Northern hemisphere, i.e., [0, 0.26] ( upper panel), [0.5, 0.7] ( middle panel) and [0.87, 0.99] ( lower panel). Note the symmetry of the modulus of the kernel with respect to the equator. The value of is normalized at its maximum at . | |
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We conclude that our choice of and is perfectly adequate to invert the available data on the solar torsional oscillations to derive information on the location of the perturbation term within the convection zone.
The data plotted in Fig. 4 of Howe et al. (2006) can be used for an illustrative application of the inversion methods introduced in Sect. 2.4. They are given with a sampling of between the equator and of latitude, i.e., in the range in which the rotational inversion techniques perform better (Schou et al. 1998). The features distinguishable on the plots indicate an actual radial resolution of 0.05 , in agreement with the sampling adopted in Fig. 3 of Howe et al. (2005). An average statistical error of about 30% can be assumed for the amplitude, whereas the phase errors become very large below 0.80 , especially at low latitudes, because of the uncertainty in the reconstruction of the signal in the deep layers. Note that the error intervals reported in Fig. 4 of Howe et al. (2006) indicate only how the particular method solution (here an OLA inversion of SoHO/MDI data) would vary with a different realization of the input data affected by a randon Gaussian noise. Unfortunately, they do not give the statistical ranges in which the true values of the amplitude and phase are likely to lie. This is not a major limitation in the context of the present study because we aim at illustrating the capabilities of the proposed approach rather than derive definitive conclusions.
To perform our inversion, we interpolate linearly the values of the amplitude and phase over the grid used to compute the radial eigenfunctions and the Jacobian polynomials. We assume that the amplitude is zero at the poles and increases linearly toward of latitude whereas the phase is constant poleward of of latitude.
The divergence of the angular momentum flux perturbation
can be obtained from Eq. (40). We define its amplitude and phase as:
Figure 5: The isocontours of the function as defined by Eq. (71) in the case of 10^{12} g cm^{-1} s^{-1}. The scale on the left indicates the ranges corresponding to the different colors and is in units of 10^{5} g cm^{-1} s^{-2}. The relative statistical uncertainty of is 30%, as it follows from the relative uncertainty of the data. | |
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Figure 6: The isocontours of as defined by Eq. (72) in the case of 10^{12} g cm^{-1} s^{-1}. The phase ranges from to . The scale on the left indicates the phase ranges corresponding to the different colors and is in radians. | |
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Figure 7: As for Fig. 5 in the case of 10^{10} g cm^{-1} s^{-1}. The scale on the left is in units of 10^{3} g cm^{-1} s^{-2}. The relative statistical uncertainty of is about 30%. | |
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Figure 8: As for Fig. 6 in the case of 10^{10} g cm^{-1} s^{-1}. | |
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It is interesting to note that the dependence of the amplitude and phase of the torsional oscillations on the turbulent viscosity can lead to its estimate in the framework of mean-field models (e.g., Rüdiger et al. 1986; Rüdiger 1989). Specifically, Rempel (2007) finds that a mean turbulent kinematic viscosity about one order of magnitude smaller than the mixing-length estimate is needed to reproduce the polar branch of the torsional oscillations.
Lower limits for the modulus of the angular momentum flux vector can be derived from Eq. (46). From the lower limits on and , a lower limit on can be derived and it is plotted in Figs. 9 and 10 for the radial and latitudinal localizations described in Sect. 3.2, respectively. Given the uncertainty in the penetration depth of the torsional oscillations, in addition to the results for a penetration down to the base of the convection zone, we also plot those for penetration depths of 0.80 and 0.90 , respectively. They are obtained simply by assuming that the oscillation amplitude has the value in Fig. 4 of Howe et al. (2006) above the penetration depth and drops to zero immediately below it.
When the oscillations penetrate down to the base of the convection zone, the maximum of the perturbation term is reached between 0.80 and 0.85 . It is mainly localized into two latitude zones, i.e., within from the equator and between and . Note that our data refer mainly to the low-latitude branch of the oscillations, so the possible source at latitude > cannot be detected. When the turbulent dynamical viscosity is reduced with respect to the mixing-length value, the amplitude of the perturbation term drops, but its radial and latitudinal localizations are not greatly affected, except for a shift of the nearly equatorial band towards higher latitudes.
According to the hypothesis that the Lorentz force is the only source of the torsional oscillations, the amplitude of the Maxwell stress can be estimated from the lower limit of and is 2.7 10^{5} G^{2} in the case of 10^{12} g cm^{-1} s^{-1}. Considering that the poloidal field is of the order of 1-10 G, this leads to very high toroidal fields in the bulk of the convection zone, which would be highly unstable because of magnetic buoyancy. On the other hand, with 10^{10} g cm^{-1} s^{-1}, the Maxwell stress is 10^{3} G^{2} which leads to a toroidal field intensity of the order of 10^{3} G. It may be stably stored for timescales comparable to the solar cycle thanks to the effects of the downward turbulent pumping in the convection zone (Brandenburg 2005, and references therein).
Note that the maximum radial stress is reduced by a factor of 30 when is decreased from 2.56 10^{12} to 5.62 10^{10} g cm^{-1} s^{-1}, whereas the maximum of is reduced only by a factor of 4 (Figs. 9 and 10). This mainly reflects the predominance of the radial gradient over the latitudinal gradient of the angular velocity perturbation in the deeper layers of the solar convection zone.
The average phase lags and between and and the azimuthal field , as derived by the method in Sect. 2.6, are plotted in Figs. 11 and 12, respectively. It is interesting to note that below 0.85 when 10^{10} g cm^{-1} s^{-1}, in agreement with the finding of most dynamo models in which the azimuthal field is produced by the stretching of the radial field in the low-latitude region, where for r < 0.95 (e.g., Schlichenmaier & Stix 1995; Rüdiger & Hollerbach 2004). Above 0.85 , the phase relationship between the radial and the azimuthal fields leads to with a phase lag of , in agreement with the early finding that the photospheric zone equatorward of the activity belt is rotating faster than that at higher latitudes (Rüdiger 1989). Note that becomes negative for r > 0.95 at low latitudes, leading to a reversal of the phase relationship between the two field components in the outer layers of the solar convection zone. Our limited spatial resolution and the uncertainty of the measurements of the torsional oscillations inside the Sun might explain why we find the transition from a mostly positive to a negative at 0.85 .
Figure 9: Upper panel: the lower limit of the amplitude of the perturbation term averaged over spherical shells of thickness 0.05 versus the relative radius; different linestyles and colors refer to the depth at which the torsional oscillations are assumed to vanish: black solid line - oscillations extending down to the base of the convection zone; green dotted line - oscillations extending down to 0.80 ; red dashed line - oscillations extending down to 0.90 . Results are obtained assuming a turbulent dynamical viscosity at the base of the convection zone 10^{12} g cm^{-1} s^{-1}. Lower panel: as in the upper panel, but for 10^{10} g cm^{-1} s^{-1}. The relative statistical uncertainties are in all cases about 30%, as follows from the uncertainties of the data. | |
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Figure 10: Upper panel: the lower limit of the amplitude of the perturbation term averaged over different latitude zones versus their average value of . Different linestyles and colors are used to indicate the results for different penetration depths of the torsional oscillations, as in Fig. 9. Results are obtained assuming a turbulent dynamical viscosity at the base of the convection zone 10^{12} g cm^{-1} s^{-1}. Lower panel: as in the upper panel, but for 10^{10} g cm^{-1} s^{-1}. The relative statistical uncertainties are in all cases about 30%, as follows from the uncertainties of the data. | |
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The phase lag between and depends remarkably on the colatitude for 10^{12} g cm^{-1} s^{-1}, whereas it is almost constant for 10^{10} g cm^{-1} s^{-1}, leading in the latter case mostly to . This, together with , suggests that the toroidal field is mainly produced by the stretching of the radial field.
On the other hand, if the angular momentum transport leading to the torsional oscillations is produced only by a perturbation of the meridional flow, as in the thermal wind model, then a very small perturbation follows from the lower limit of . For 10^{12} g cm^{-1} s^{-1}, we find a minimum amplitude of the meridional flow component oscillating with the eleven-year cycle of 3 cm s^{-1} at a depth of 0.85 and a latitude of . Note, however, that if we estimate from its divergence and the typical lengthscale of its variations in Fig. 5, we find a value about one order of magnitude larger, that is in agreement with the estimate by Rempel (2007). A similar argument applies also to the estimate of the Maxwell stresses made above.
Figure 11: Upper panel: the phase lag between and as derived by Eqs. (54) averaged over spherical shells of thickness 0.05 versus the relative radius. Different linestyles are used to indicate the results for different penetration depths of the torsional oscillations, as in Fig. 9. Results are obtained assuming a turbulent dynamical viscosity at the base of the convection zone 10^{12} g cm^{-1} s^{-1}. Lower panel: as in the upper panel, but for 10^{10} g cm^{-1} s^{-1}. | |
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Figure 12: Upper panel: the phase lag between and , as derived by equations analogous to (54), averaged over different latitude zones versus their average value of . Different linestyles are used to indicate the results for different penetration depths of the torsional oscillations, as in Fig. 9. Results are obtained assuming a turbulent dynamical viscosity at the base of the convection zone 10^{12} g cm^{-1} s^{-1}. Lower panel: as in the upper panel, but for 10^{10} g cm^{-1} s^{-1}. | |
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The average kinetic energy variation associated with the torsional oscillations, as computed from Eq. (49), is 10^{28} erg, whereas the average dissipated power is 6.7 10^{26} erg s^{-1}, when 10^{12} g cm^{-1} s^{-1}, and only 1.5 10^{25} erg s^{-1}, when 10^{10} g cm^{-1} s^{-1}.
When the torsional oscillations are assumed to be confined to shallower and shallower layers, the location of the perturbation term is shifted closer and closer to the surface and it becomes more and more uniformly distributed in latitude. Its amplitude shows a remarkable decrease because of the smaller moment of inertia of the surface layers.
The results of Sect. 2.7 can be applied to the variation of the surface differential rotation observed in, e.g., LQ Hya between 1996.99 and 2000.96 (see Table 2 of Donati et al. 2003). Assuming an internal structure analogous to the solar one and that the differential rotation variations observed at the surface are representative of those at a depth of 0.99 R, we can estimate a lower limit for in the overshoot layer from Eq. (59). This is used to estimate the Maxwell stresses by assuming that: , where R is the radius of the overshoot layer and R its thickness. As such, the minimum magnetic field strength is: G. However, if we assume a poloidal field strength of 100 G, as indicated by the Zeeman Doppler imaging, we find an azimuthal field of 10^{4} G, which is in the range of the values estimated by Lanza (2006a) in the framework of the Taylor-Proudman hypothesis. Note that the result is independent of in the limit . Although this application is purely illustrative, it suggests that our method can be applied to derive estimates of the internal magnetic torques (as well as other sources of angular momentum transport) when asteroseismic results are available, in combination with surface rotation measurements, to further constrain rotation variations in solar-like stars.
We introduced a general solution of the angular momentum transport equation that takes into account the density stratification and the radial dependence of the turbulent viscosity . Its main limitation is due to the uncertainty of the turbulent viscosity in stellar convection zones. It is interesting to note that the method of the separation of variables to solve Eq. (3) can be applied also when is the product of a function of the radius by one of the latitude. However, when depends on the latitude, the angular eigenfunctions are no longer Jacobian polynomials.
Our formalism can be applied to compute the response of a turbulent convection zone to prescribed time-dependent Lorentz force and meridional circulation. From the mathematical point of view, it is a generalization of that of Rüdiger et al. (1986) and can be easily compared with it by considering that: , where P_{n} is the Legendre polynomial of degree n and . Moreover, our method can be used to estimate the torques leading to the angular momentum redistribution within the solar (or stellar) convection zone, thus generalizing the approach suggested by Komm et al. (2003). The main limitation, in addition to the uncertainty of the turbulent viscosity, comes from the low resolution and the limited accuracy of the present data on solar torsional oscillations, particularly in the deeper layers of the convection zone. In fact, these layers are the most important because torsional oscillations with an amplitude of 0.5%-1% of the solar angular velocity extending down to the base of the convection zone lead to Maxwell stresses with an intensity of at least 8 10^{3} G^{2} around 0.85 or a perturbation of the order of several percents of the meridional flow speed at the same depth (e.g., Rempel 2007). If the torsional oscillations are due solely to the Maxwell stresses associated with the mean field of the solar dynamo, we can also estimate the phase relationships between , and . Our preliminary results indicate that in the layers below 0.85 and in the outer layers which, together with helioseismic measurements of the internal angular velocity, suggest that the toroidal field is mainly produced by the stretching of the poloidal field by the radial shear.
Future helioseismic measurements may improve our knowledge of the torsional oscillations, essentially by extending the time series of the data or by means of space-borne instruments, such as those foreseen for the Solar Dynamic Observatory (e.g., Howe et al. 2006). On the other hand, asteroseismic measurements may open the possibility of investigating similar phenomena in solar-like stars, particularly in those young, rapidly rotating objects showing variations of the angular velocity one or two orders of magnitude larger than the Sun.
Acknowledgements
The author wishes to thank an anonymous Referee for valuable comments and Professor G. Rüdiger for interesting discussion. Solar physics and active star research at INAF-Catania Astrophysical Observatory and the Department of Physics and Astronomy of Catania University is funded by MIUR (Ministero dell'Università e della Ricerca), and by Regione Siciliana, whose financial support is gratefully acknowledged.
This research has made use of the ADS-CDS databases, operated at the CDS, Strasbourg, France.
The convergence of the Green function series in Eqs. (34) can be studied by considering
the asymptotic formulae for
and the
Jacobian polynomials, i.e., for
and .
In the asymptotic limit, those series can be written as:
This can be proven by considering the inequalities in (21) and the formula:
(A.4) |
Equation (A.3) follows from the equality (1.217.1) of Gradshteyn & Ryzhik (1994). In this way, we find: