SOLAR FLOWS AND THEIR EFFECT ON FREQUENCIES OF ACOUSTIC MODES

The difference between QDPT and DPT lies in the choice of the set of NRSSM eigenfunctions (s_k), whose linear combinations are used to express the perturbed eigenfunctions (s'_k), e.g.,

$$\begin{equation} {\bf s_k^{\prime }} = \sum _{k^{\prime }\in K} a_{k^{\prime }} {\bf s_{k^{\prime }}}. \end{equation} \tag{ 5 }$$

In DPT, only the eigenfunctions from the standard solar model which are exactly degenerate constitute the eigenspace K. In contrast to this in QDPT, the set consists of all eigenfunctions of the standard solar model with nearly degenerate frequencies. It is well known that simple harmonic oscillators couple strongly only if the natural uncoupled frequencies of the oscillators are nearly degenerate. Hence, it is important to determine which of the NRSSM eigenfunctions should be included in the analysis. Let the eigenspace of the suitable eigenfunctions be called K. Qualitatively, the quasi-degenerate condition implies that we include an eigenfunction ${\bf s}_{nlm}(=\xi _{nl}(r)Y_{l}^m{\bf {\hat{r}}}+\eta _{nl}(r){\bf {\nabla }}_{h}Y_l^m) \in K$ iff |ω_nlm − ω_ref| < , where is the radius of the neighborhood about the central frequency. We have generally used = 100 μHz in this work. This is comparable to asymptotic spacing between two consecutive modes in radial order n. Moreover, no significant change in calculated frequency shifts is observed for >100 μHz. Here, ω_ref is the frequency of the central eigenmode for which we require the frequency shift. The size of the eigenspace K will be determined by the (1) desired level of accuracy which is set by and (2) the selection rules of the perturbation operator.

In the NRSSM, the frequencies are independent of the azimuthal-order m, and this degeneracy is lifted by departures from spherical symmetry due to rotation or magnetic field. Helioseismic data traditionally give the splitting coefficients for all modes that are detected. These coefficients are defined by (e.g., Ritzwoller & Lavely 1991)

$$\begin{equation} \omega _{nlm} = \omega _{nl} + \sum _{q}a_q^{(nl)}\mathcal {P}^{l}_q(m), \end{equation} \tag{ 6 }$$

where ω_nl is the mean frequency of the multiplet, $\mathcal {P}_q^{l}(m)$ are the orthogonal polynomials of degree q, and a^(nl)_q's are the so-called splitting coefficients.

The equations of motion for a mode k with eigenfrequency ω_k for a NRSSM and a model perturbed by addition of differential rotation and/or large-scale flow can be respectively represented by

$$\begin{eqnarray} {\mathcal {L}_0}{\bf s_k}&=&-\rho _0\omega _k^2 {\bf s_k}, \end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray} {\mathcal {L}_0}{\bf s^{\prime }_k}+{\mathcal {L}_1}{\bf s^{\prime }_k}&=&-\rho _0{\omega ^{\prime }_k}^2 {\bf s^{\prime }_k}, \end{eqnarray} \tag{ 8 }$$

where ω'_k is the perturbed frequency and s'_k given by Equation (5) is the perturbed eigenfunction. Taking scaler product with s_j in Equation (8) and using the notation ${\mathcal H}_{jk^{\prime }} = - \int {\bf s_{j}}^{\dagger } {\mathcal L}_1 {\bf s_{k^{\prime }}} dV$ and the definition $\mathcal {L}_1 {\bf s_k} = -2i\omega _{{\rm \scriptsize ref}}\rho _0({\bf v.\nabla)s_k}$, we obtain the matrix eigenvalue equation,

$$\begin{equation} \sum _{k^{\prime } \in K} \big\lbrace {\mathcal H}_{jk^{\prime }} + \delta _{k^{\prime }j} \big(\omega _{k^{\prime }}^2 - \omega _{{\rm ref}}^2\big)\big\rbrace a_{k^{\prime }} = \big({\omega ^{\prime }_k}^2-\omega _{{\rm ref}}^2\big)a_{j}, \end{equation} \tag{ 9 }$$

with eigenvalue λ = (ω'²_k − ω²_ref) and eigenvector X_j = {a_j}. Here, ω_ref is a reference frequency which approximates ω'_k. In this work, we use ω_ref = ω_k, the frequency of the mode being perturbed. It will be clear from Section 3.2 that the perturbation matrix $[\mathcal {H}_{n^{\prime }n,l^{\prime }l}^{m^{\prime }m}]$ is Hermitian for both differential rotation and poloidal flows. For example, if we consider only two modes with frequencies ω₁ and ω₂, then QDPT gives the following coupling matrix for the mode ω₂:

$$\begin{equation} \left[\begin{array}{@{}cc@{}} {\mathcal H}_{11} -\Delta & {\mathcal H}_{12}\\ {\mathcal H}_{21} & {\mathcal H}_{22}\\ \end{array} \right], \end{equation} \tag{ 10 }$$

with Δ = ω²₂ − ω²₁. We are interested in the eigenvalues λ of $[\mathcal {H}_{ij}]$ corresponding to perturbation in ω₂ which may be easily shown to be

$$\begin{equation} \lambda \sim {\mathcal H}_{22} -\frac{{|\mathcal H}_{12}|^2}{{\mathcal H}_{11} -{\mathcal H}_{22} -\Delta }\;. \end{equation} \tag{ 11 }$$

Let ω'₂ be the modified frequency of the mode which initially has a frequency ω₂. Then from Equation (11) we have

$$\begin{equation} \omega ^{\prime }_2 \sim \left[\omega _2^2+{\mathcal H}_{22} -\frac{|{\mathcal H}_{12}|^2}{{\mathcal H}_{11} -{\mathcal H}_{22} -\Delta }\right]^{{\textstyle {1\over 2}}}\;. \end{equation} \tag{ 12 }$$

If DPT were used then the last term involving the off-diagonal term $\mathcal {H}_{12}$ would not be present, and hence this term gives the correction arising from using the QDPT. It is clear that this correction can be large if the two modes are nearly degenerate.

If |H₁₂| ≈ |Δ|, then the frequency shift due to use of QDPT would be of order of Δ/ω₂, which is comparable to the difference ω₂ − ω₁. On the other hand, if |H₁₂| ≪ |Δ|, then the frequency shift would be much less. For a typical p-mode, the spacing ω₂ − ω₁ is of order of a few μHz. For a rotation velocity of 2000 m s⁻¹, the frequency shift is of the order of 450 m nHz, which would define the magnitude of diagonal terms. Thus, for meridional flow with an averaged velocity of 20 m s⁻¹, we may expect the matrix elements H₁₂/ω₂ ∼ 4.5l nHz. However, in this case the diagonal elements are zero, and we need the off-diagonal element which involves cross product of two different eigenfunctions in the integration (cf., Equation (15)), and (for the same velocity) this integral would be more than order of magnitude smaller than that in the diagonal term for rotation. Thus, the magnitude of H₁₂ would be an order of magnitude less, giving a frequency shift of order of 0.2l²/(ω₂ − ω₁) nHz, taking the maximum value of m. For l ≈ 100, this would give a shift of order of a few nHz. This is much smaller than that obtained by Roth & Stix (2008).

In the following section, we shall give the expression for the perturbation matrix elements $[\mathcal {H}^{mm^{\prime }}_{nn^{\prime },ll^{\prime }}]$ due to rotation as well as poloidal flows to be used for solving the eigenvalue Equation (9).

The Wigner–Eckart theorem (Equation (5.4.1) of Edmonds 1960) states that the general matrix element of any tensor perturbation operator can be expanded in terms of Wigner 3j symbols whose coefficients of expansion are independent of azimuthal-order m and m':

$$\begin{equation} \mathcal {H}^{m^{\prime }m}_{n^{\prime }n, l^{\prime }l} = (-1)^{m^{\prime }} \left(\begin{array}{@{}ccc@{}} l^{\prime } & s & l \\ -m^{\prime } & t & m \end{array}\right) (n^{\prime }l^{\prime }||\mathcal {L}_s^{t}||nl). \end{equation} \tag{ 13 }$$

The coefficient of the Wigner 3j symbol in Equation (13) due to coupling by the differential rotation is given by

$$\begin{eqnarray} (n^{\prime }l^{\prime }||\mathcal {L}_s^{t}||nl)& =& 8 \pi \omega _{{\rm ref}}(1-(-1)^{l+l^{\prime }+s})\nonumber \\ &&\times \gamma _{l}\gamma _{l^{\prime }}\int _{0}^{R_{\odot }} \gamma _s w_s^0 \rho r^2 dr T_s(r), \end{eqnarray} \tag{ 14 }$$

where

$$\begin{eqnarray} \nonumber T_s(r) &=& -\frac{1}{r}\bigg\lbrace \xi ^{\prime }\eta +\eta ^{\prime }\xi -\xi ^{\prime }\xi \\ \nonumber && -\frac{1}{2}\eta ^{\prime }\eta [l(l+1)+l^{\prime }(l^{\prime }+1)-s(s+1)]\bigg\rbrace \\ &&\times \Gamma _{l}\Gamma _{l^{\prime }}\left(\begin{array}{@{}ccc@{}} l^{\prime }& s & l \\ -1& 0 & 1 \end{array}\right), \end{eqnarray} \tag{ 15 }$$

$\Gamma _q = \sqrt{q(q+1)/2}$ and $\gamma _q = \sqrt{(2q+1)/4\pi }$. ξ and η are radial and horizontal components of the eigenfunction defined in Section 3.1.

Similarly, we have derived the coefficient of the Wigner 3j symbol in Equation (13) due to coupling by the large-scale poloidal flow

$$\begin{eqnarray} &&\nonumber \big(n^{\prime }l^{\prime }||\mathcal {L}_s^{t}||nl\big) = 8 \pi i \omega _{{\rm ref}} (1+(-1)^{l+l^{\prime }+s}) \\ &&\times \ \gamma _{l}\gamma _{l^{\prime }}\int _{0}^{R_{\odot }} \gamma _s u_s \rho r^2 dr \left[ R_s - \frac{\partial {H_s}}{\partial {r}}\right], \end{eqnarray} \tag{ 16 }$$

where

$$\begin{eqnarray} \nonumber R_s(r) &=& { {1\over 4}}\left(\xi ^{\prime }\frac{\partial {\xi }}{\partial {r}}-\frac{\partial {\xi ^{\prime }}}{\partial {r}} \xi \right) \left(\begin{array}{ccc} l^{\prime } & s & l\\ 0 & 0 & 0 \end{array}\right) \\ &&+ { {1\over 2}}\Gamma _{l}\Gamma _{l^{\prime }}\left(\eta ^{\prime }\frac{\partial {\eta }}{\partial {r}}-\frac{\partial {\eta ^{\prime }}}{\partial {r}} \eta \right)\left(\begin{array}{@{}ccc@{}} l^{\prime } & s & l\\ -1 & 0& 1\end{array}\right), \end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray} \nonumber H_s(r)&=&{\dsty {1\over 2}}[l(l+1)-l^{\prime }(l^{\prime }+1)] \\ &&\times \left[{\dsty {1\over 2}}\xi \xi ^{\prime }\left(\begin{array}{@{}ccc@{}} l^{\prime } & s& l\nonumber \\ 0 & 0 & 0\end{array}\right) - \eta \eta ^{\prime } \Gamma _{l}\Gamma _{l^{\prime }}\left(\begin{array}{@{}ccc@{}} l^{\prime } & s & l \\ -1 & 0 & 1\end{array}\right)\right]\nonumber \\ && -\eta ^{\prime }\xi \Gamma _{l^{\prime }}\Gamma _{s}\left(\begin{array}{@{}ccc@{}} l^{\prime } & s & l \\ -1 & 1 & 0\end{array}\right)+\xi ^{\prime }\eta \Gamma _{l}\Gamma _{s}\left(\begin{array}{@{}ccc@{}} l^{\prime } & s&l \\ 0 & 1 &-1\end{array}\right).\nonumber \\ \end{eqnarray} \tag{ 18 }$$

The selection rules for the Wigner 3j symbol to be non-zero are:

• 1.  
  p/2 ⩾ max(l, l', s), where p = l + l' + s,
• 2.  
  m − m' + t = 0.The selection rule for non-trivial reduced matrix elements due to differential rotation is given by
• 3.  
  l + l' + s must be odd.This means that the ∂_θY⁰₃ component of differential rotation couples modes with l' = l, l ± 2. The set of modes that constitutes the eigenspace K must follow selection rules (1), (2), and (3) and the quasi-degenerate condition |ω_nlm − ω_ref| < 100 μHz. The selection rule for reduced matrix elements due to large-scale poloidal flow to be non-zero is
• 4.  
  l + l' + s must be even.

In this case, the set of modes in the eigenspace K must follow selection rules (1), (2), and (4) and the quasi-degenerate condition as before. Note that the matrix elements for rotation are real and symmetric, whereas those for poloidal flow are imaginary and antisymmetric. The perturbation matrix $[\mathcal {H}_{n^{\prime }n,l^{\prime }l}^{m^{\prime }m}]$ is Hermitian for both cases and will have real eigenvalues.

We show in this section that QD treatment would give non-zero even coefficients a_2q even in the absence of the centrifugal term, and their effect on a_{2q + 1} is indeed small as argued by Lavely & Ritzwoller (1992). The determination of even splitting coefficients due to differential rotation is important as this may be used to remove the effect of rotation from the observed splitting coefficients to isolate the effect of the magnetic field and other large-scale flows in the observed splitting coefficients. However, if DPT is used, only the odd coefficients a_{2q + 1} are non-zero if we neglect the centrifugal force which is of second order in Ω. Further, in this approximation only, the north–south symmetric component of rotation gives non-zero frequency shifts. Lavely & Ritzwoller (1992) applied the QDPT to differential rotation to find that for intermediate l modes (l ∼ 50) the quasi-degenerate coupling will have little effect on the modal frequencies and can be ignored for all practical purposes.

It is easy to verify from Equation (13) that for pure rotation $({\bf v}=\Omega (r, \theta)r\sin \theta \hat{\phi })$ the symmetric matrix elements are odd functions of m. Hence, from Equations (6) and (11), one can infer that the perturbed frequency will have both odd and even splitting coefficients. It can be easily shown that there will be no additional effect in QDPT due to component w⁰₁ of rotation velocity as the off-diagonal elements in the resulting matrix would vanish. Thus, the leading effect of rotation arises from w⁰₃ term. We calculate this contribution by using

$$\begin{equation} w_3^0(r)=\left\lbrace \begin{array}{@{}ll@{}}34.9 \hbox{ m s$^{-1}$} & {\rm if}\ r \ge 0.7\,R_{\odot }\\ 0, & {\rm otherwise}.\end{array}\right. \end{equation} \tag{ 19 }$$

This value is chosen such that the DPT gives a value of a₃ close to the observed value for most modes trapped in the convection zone. The QDP treatment modifies the odd splitting coefficient la₁ by up to 0.4 nHz (see Figure 2(a)) and la₃ by up to 0.2 nHz, which are a couple of orders of magnitude smaller than the typical errors in these coefficients. In Figure 2(b), we plot the splitting coefficient la₂ for all the modes with frequency less than 4.5 mHz and l ⩾ 10, against r^nl_t, the lower turning point of the mode.

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The calculation has been done using an equation similar to Equation (11) but for all the modes in the neighborhood of radius 100 μHz about the central mode. The mathematical details of the calculation has been discussed in Section 3.2 (see Equations (14) and (15)). The maximum value of la₂ ∼ 30 nHz which may be significant depending upon the magnitude of the large-scale flow or magnetic field perturbations. Note that the effect of centrifugal force on la₂ also happens to be of the same order (see Figure 5 of Antia et al. 2000). We have emphasized earlier that in order to spot the signatures of the large-scale flows and magnetic fields, we need to remove the effect of rotation on a₂.

It has been found from Doppler measurements that there exists a hemispheric asymmetry in surface rotation having an angular dependence ∂_θY⁰₂ (e.g., Hathaway et al. 1996). This component has an amplitude of w⁰₂ = −7.8 ± 0.3 m s⁻¹, which indicates that the southern hemisphere was rotating slightly faster than the northern hemisphere during the period of the data. We have also calculated the splitting coefficient la₂ due to north–south asymmetry assuming that it is constant throughout the convection zone, and they happen to be small with la₂ ≲ 0.1 nHz.

In this section, we investigate theoretical frequency shifts due to meridional circulation. We start with s = 2, t = 0, which is the dominant component in the observed meridional flow near the surface. In our analysis, we have included all f- and p-modes with frequency less than 4.5 mHz and l ⩾ 10. We consider a meridional circulation with one cell in each hemisphere and maximum horizontal velocity at the surface equal to 30 m s⁻¹ which is consistent with the Doppler measurements and ring diagram analysis. This gives u₀ = 9 m s⁻¹ in Equation (4). From Equations (17) and (18), we see that the coupling matrix for poloidal flows is Hermitian with zero diagonal elements. This means that degenerate perturbation treatment cannot be applied to calculate the frequency shifts, and it becomes essential to use the QDPT, unlike for rotation where QDPT just provides a second-order correction. In the presence of zero diagonal elements of the matrix $[\mathcal {H}_{ij}]$, Equation (11) for the frequency shift for coupling between two modes simplifies to

$$\begin{equation} \delta \nu = \frac{\omega ^{\prime }_2 -\omega _2}{2\pi } \sim \frac{{\mathcal {H}}_{12}^2}{4\pi \omega _2\Delta }\;. \end{equation} \tag{ 20 }$$

It may be noted that the sign of the frequency shift for the central multiplet with ω_ref depends on the sign of Δ = ω²_ref − ω²₁, where ω₁ is the nearest mode to the central frequency. If there are many modes with frequency close to frequency of the central mode, then we can expect these contributions to be added. In this case, if the sign of frequency differences is not the same, the terms will partially cancel each other. If for some pair of modes which satisfy the selection rules, the frequency difference is very small, then their frequencies would be shifted significantly.

In Figure 3, we show an example of the frequency shift δν(m) of the multiplet (n, l) = (1, 292), which was also considered by Roth & Stix (2008). The Wigner 3j symbol with t = 0 (implying m' = m) in Equation (13) is an odd function of m and so $\delta \nu (m) \propto \mathcal {H}_{12}^2$ is symmetric about m = 0 for all the multiplets. Another important point to note from Equation (20) is that δν ∝ u²₀. This has also been verified by numerical calculations. The maximum shift we get for this multiplet from our calculations is ∼8.5 nHz in contrast to Figure 2 of Roth & Stix (2008) where they obtain a maximum shift of 0.1 μHz. On comparison with their Equation (15), we find a difference of a factor of 2 in the first term in the expression of H_s(r). Even after using their version of Equation (15) in our calculations, we were not able to reproduce the large shifts reported by them. The reason for this discrepancy is not clear. One possibility is the normalization of velocity amplitude. The velocity decreases rapidly with height, and if the normalization is applied at a higher level, then the entire velocity profile will be scaled up. In these calculations, we have normalized u₀ by comparing it with the surface velocity. We find that the peak value of v^t_s(r) in Figure 1 occurs a little below the surface. For instance, for a flow with s = 2 and a velocity of 30 m s⁻¹ near r = R_☉, this peak in v^t_s(r) happens to be 72 m s⁻¹ at r = 0.996 R_☉. However, in reality such a rapid increase in v_s has not been found from local helioseismology which can probe up to a depth of 0.96 R_☉. Hence, in our opinion the theoretical frequency shift calculated using such a radial profile of v^t_s(r) is an upper limit to the actual shift. Ideally, we should normalize u₀ to get the maximum value of the horizontal velocity to be 30 m s⁻¹. That will bring down the splitting coefficients by a factor of 4 or more. On the other hand, if we normalize the velocity further up then for the same value of velocity the calculated splitting coefficients would go up. However, that is not realistic as in that case the maximum velocity would be much larger than the observed value. In Figures 4(a) and (b), we plot the frequency shifts averaged over m for each multiplet, whereas Figure 4(c) shows the splitting coefficient la₂ calculated for all the multiplets.

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We consider only even values of s so that the meridional flow does not have cross-equatorial components. As we increase s, selection rules allow more and more multiplets in a radius of 100 μHz to couple, and there are many more instances of "near degeneracy," so that the δν ∝ 1/Δ in Equation (20) for some multiplets becomes quite large. For example, the multiplets (17, 56) and (16, 64) which have a frequency difference of −0.06 μHz can be coupled by the meridional flow with s = 8 to give la₂ ∼ 480 nHz. In these calculations, we have used u₀ = 100 m s⁻¹. In Figure 3, we have plotted the frequency shifts for the pair of multiplets (15, 34) and (14, 40) also considered by Roth et al. (2002), coupled by the same flow. The result for all multiplets with frequency less than 4.5 mHz and l ⩾ 10 is shown in Figures 5(a) and (b). It is to be noted from Figure 5(b) that only some multiplets have a₂ larger than the errors in a₂ for those modes in observational data. It is only these multiplets which we may hope to detect in the observations. In this case, we have taken the errors from the GONG data set centered at 2002 November 19. We present comparisons with GONG as well as MDI data in Section 4.4. Apart from the nearly degenerate modes the splitting coefficients of other modes are larger than that found for s = 2 with a similar value of horizontal velocity near the surface. It may be noted that for large values of s, the maximum value of horizontal velocity would be quite different from v^t_s(r) because of the additional factor arising from the gradient of spherical harmonics (cf. Equation (1)). In the case, where we normalize u₀ such that v⁰_s(R_☉) = 30 m s⁻¹, we also obtain an increase with s in the splitting coefficients as reported by Roth & Stix (2008). This increase is not seen if the maximum value of the horizontal velocity near the surface is normalized to 30 m s⁻¹. Of course, observations near the solar surface do not show such large magnitudes for these higher order components of meridional flow. If realistic amplitudes are used then the effect will be negligible.

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In this section, we calculate the splittings due to poloidal flows with angular dependences that depend on longitude, ϕ. We consider the flow given by spherical harmonics Y⁴₈(θ, ϕ) and Y⁸₈(θ, ϕ) (also called sectoral rolls or banana rolls). This calculation is little more involved than that in Section 4.2 as a non-zero t allows coupling between different m and m' in Equation (13). Hence, it becomes important to take into account the effect of rotational splitting on Δ before calculating the effect of these kind of poloidal flows. It is easy to see from the properties of the Wigner 3j symbols that the s = 8, t = 8 flow couples the mode (n, l, m) with (n', l', m ± 8). Hence, the difference of the square of frequencies Δ is no longer independent of m unlike in Section 4.2.

Let us consider the perturbation in the frequency of the multiplet (15, 34). The nearest multiplet to this happens to be (14, 40) with a frequency difference of 4.12 μHz, and the next one is (16, 28) with the frequency difference 19.01 μHz. This is the same mode shown by Figure 3 of Roth et al. (2002). We use the rotational splittings calculated from temporally averaged rotation rate as inferred from the GONG data to calculate Δ. The result of our calculation including only the nearest multiplet (14, 40) has been shown in Figure 6. The asymmetric frequency shift shown by the solid line in Figure 6(a) denotes the change in the frequency of the mode (15, 34, m) due to coupling with the mode (14, 40, m − 8), whereas the dashed line denotes the change due to coupling with (14, 40, m + 8). As shown in Equation (20), a crucial component of the shift comes due to the difference of the squared frequencies $\Delta _{m,m^{\prime }}$ of the modes coupling according to the QDPT condition and the selection rules. The corresponding Δ_{m,m − 8} and Δ_{m,m + 8} are plotted in Figure 6(b). The total shift is given by the sum of the solid and dashed curves. The couplings within a multiplet, i.e., (n, l, m) ⇄ (n, l, m ± 8) happen to be zero because of the antisymmetry of the matrix elements. The values of splitting coefficients la₁ and la₂ for the mode (15, 34) are calculated to be −12.2 nHz and −5.7 nHz and that for (14, 40) are 11.1 nHz and 5.3 nHz, respectively.

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We repeat this calculation for all modes with frequency less than 4.5 mHz and find that some of the multiplets have quite high values of a₂ (see Figure 9(b)). For example, the multiplet (18, 61) which couples with the nearest multiplet (17, 69) with a frequency difference of −3.4 μHz. In Figure 7(a), we have plotted δν(m) versus m for the coupling (18, 61, m) ⇄ (17, 69, m − 8) which seems to have a discontinuity at m = −22. This is due to the fact that Δ_{m,m − 8} for this coupling also changes sign at m = −22. Note that two modes (18, 61, m) and (17, 69, m − 8) having a mean frequency difference of 4.12 μHz become "nearly" degenerate at m = −22. This jump presumably gives rise to high values of splitting coefficients. The other coupling (18, 61, m) ⇄ (17, 69, m + 8) is smooth and so is the Δ_{m,m + 8} as shown in Figure 7(b) by the dashed line. We also calculate the total shift experienced by the multiplet (18, 61) while coupling with three nearest modes (17, 69), (19, 55), (19, 53) and find no significant difference with Figure 7(a). We have also shown a calculation for the asymmetric shifts for the multiplet (14, 34) and (14, 40) coupled due to a flow varying as Y⁴₈(θ, ϕ) in Figure 8.

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The asymmetric shift due to these couplings gives rise to non-zero odd splitting coefficients in addition to the even coefficients. Since rotation inversions neglect this contribution, the result may not be correct if the giant cells have significant amplitudes. The effect of giant cells on rotation inversions can be estimated by performing rotation inversion using odd splitting coefficients from these calculations. The resulting rotation rate should be subtracted from the actual rotation inversions. It is likely that if these multiplets are used for helioseismic inversion using the rotation kernel, they will give rise to distortion in the inverted rotation profile. The effect on the odd splitting coefficients, a₁ and a₃ is shown in Figures 9(a) and (c) in terms of observational errors in those multiplets. We use the theoretically calculated a₁ and a₃ to perform a 1.5d helioseismic rotation inversion using the formulation due to Ritzwoller & Lavely (1991). We use 1.5d inversion as we have calculated only two splitting coefficients, a₁ and a₃. For this purpose, we use only those modes which are present in a GONG data set and use the errors in those modes for inversions using a regularized least squares method using the same smoothing as used for inverting real data (e.g., Antia et al. 1998). The results are shown in Figure 10, which shows the resulting w⁰₁(r) and w⁰₃(r) in terms of the estimated errors. Because of the presence of modes with large a₁, a₃ with r_t ≈ 0.7 R_☉, there are some oscillations in this region, but their amplitudes are less than the estimated errors. Thus, the effect on rotation inversion may not be significant even when the giant cells have velocity of 100 m s⁻¹. With increasing velocity, the effect would be larger, and it may be possible to detect this effect as it will give an oscillatory signal in rotation inversion at these depths. This has been pointed out by Roth et al. (2002). They find a much larger effect as their estimate of frequency shift is generally larger than what we find. In real data, the effect may not be as dramatic as shown by Figure 8 of Roth et al. (2002) since many of the modes with large shifts in a_{2q + 1} would yield large residuals in inversions, and any reasonable inversion would eliminate such modes. In our calculation, we found residuals going to about 4σ level and did not eliminate any modes. Thus, it is difficult to put any limits on magnitude of flows using inversion results.

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The even coefficient a₂/σ₂ is also shown in Figure 9(b). If we find some multiplets with 0.6 < r_t/R_☉ < 0.8 having magnitudes larger than the errors in observations as in Figure 9(b), we should be able to throw light on the flow velocities in giant cells. We shall look for such features in Section 4.4.

We use seven different data sets from the GONG observations, covering the late descending part of cycle 22 to end of cycle 23, i.e., from 1995 June to 2008 November. We choose only those modes for which a₂ is available in the data sets. Since the different data sets can have different set of modes we repeat the calculation for all theseven data sets. Most of the contribution to observed splitting coefficients appear to come from near surface effects, while we are interested in modes with turning point close to the base of the convection zone. Thus, we first separate the surface effects in the data by fitting a cubic spline in terms of frequency to lI_nla_2k/I_nl0Q_lk versus frequency. Here, I_nl is the mode inertia of the mode (e.g., Christensen-Dalsgaard 2002), while I_nl0 is the mode inertia for the l = 0 mode at the same frequency, while Q_lk is a geometric factor as defined by Antia et al. (2001). After subtraction of the surface effect from the observed splitting coefficients, all the multiplets are sorted in the order of increasing r_t and then an error weighted 100 point average is applied. In Figures 11(a) and (b), we show the result of the averaging for the GONG 108-day averaged data sets centered on 2002 November 19 and 2007 September 18, respectively. To get an idea of the magnitude of the giant cell flows, one can consider comparing the size of the valley and the hump in the region 0.6 < r_t/R_☉ < 0.8 in the theoretical curve with that in the observation. It is difficult to make out if the expected signature is present in the observed data sets. In most cases, no significant hump is seen in this region, though in some cases like in Figure 11(a) there is some hint of hump which is comparable to error bars. The main problem with this approach is that the running mean tends to average modes with positive and negative values of a₂, thus reducing the significance of the results. Thus, it may be better to separate out these modes to look for the signal.

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We have said earlier that only way to put an upper limit on the flow velocity is to look for signatures of nearly degenerate modes in the data. In order to do that we divide the multiplets from the theoretical calculations in two groups: one for which la₂ > 10 nHz and the other with la₂ < −10 nHz. Amongst 3700 theoretically evaluated multiplets, we find only about 250 multiplets satisfying either criteria. Then, we further search for these multiplets in the GONG data sets and perform an averaging over the available multiplets to get an estimate of a₂ from theory and error σ₂ in a₂ from the observations. Only about 10–12 of the 250 multiplets are usually found to be present in the GONG data sets. We find that for all the data sets, |la₂| ∼ 22 nHz for theoretically calculated coefficients, whereas the error in la₂ (in observed set) is ∼14 nHz for the Y⁴₈(θ, ϕ) flow with u₀ = 100 m s⁻¹. The corresponding values for the Y⁸₈(θ, ϕ) flow with a u₀ = 50 m s⁻¹ are 28 nHz and 13 nHz, respectively. Since the observed data do not show these values (in both >10 nHz and <−10 nHz sets) we can only put an upper limit on velocities in such flows. For this purpose we compare the a₂ values from theory with observational errors to calculate the confidence level of the upper limit. So we can say that up to a 1.5σ level we can rule out the Y⁴₈(θ, ϕ) dependent flow with u₀ > 100 m s⁻¹, and the existence of the Y⁸₈(θ, ϕ) flow with u₀ > 50 m s⁻¹ can be eliminated with a confidence level of 2σ. We have also performed some calculations with Y³₈(θ, ϕ) angular dependence and can say with a 1.5σ confidence level that such flows also cannot have u₀ > 70 m s⁻¹. Our results for multiplets in all the seven GONG data sets, divided into two groups according to the theoretical value of la₂, are represented in Table 1. MDI data also give similar results. By looking at detailed peak profiles of these abnormal modes in observed data, it may be possible to put more stringent limits on velocity of such flows.