4 Results

In this section we present the result of the numerical experiment. We will concentrate on the flows with the geometries s=8, t=8 (sectoral cells or banana cells) and s=8, t=0 (zonal cells or meridional rolls) both with characteristic parameters $u_{\rm char}=1,\dots, 100$ m s^-1 (strength of the flow field, defined in Eq. (19)). We also did calculations with flow fields with geometries $s=6, t= 0,\dots,6; s=8, t= 0,\dots, 8; s=10; t=0,\dots,10$ and characteristic parameters $u_{\rm char}=1,\dots, 500$ m s^-1, but all the inversions show similar results and are best discussed in terms of the chosen cases.

Figure 8 displays the inversion results for flow fields with degree s=8 and orders t=8 (sectoral cells or banana cells) and t=0 (zonal cells or meridional rolls). The characteristic parameter $u_{\rm char}$ increases from top to bottom from 1  m s^-1 to 100  m s^-1.

We note that the maximal velocity of the flow in the radial direction is in general not identical with $u_{\rm char}$, because the maximum amplitude of the spherical harmonics given in Eq. (2) for our calculated cases is close to but not necessarily equal to one. The true maximal amplitude of u_s^t(r) depends on $u_{\rm char}$, s and t. It is $0.51 u_{\rm char}$ for the example s=t=8 and $1.16 u_{\rm char}$ for $s=8,\ t=0$. For these examples the locations in the solar model where the maximal amplitude is reached can be identified in the left and right plots of Fig. 7 in the middle of the convection zone, at longitudes that are multiples of $\pi/8$ for t=8 and at the poles for t=0. The corresponding normalization factors for the other geometries analyzed are in the same range. Therefore, the detectability limits derived in this section are valid for them as well.

4.1 Sectoral cellular flows

The influence of the additional poloidal field that is placed in the convection zone becomes noticeable at amplitudes higher than $u_{\rm char}= 10$  m s^-1 for the sectoral cells (Fig. 8, left two columns). As the flow field in direction of rotation is given by the $\phi$-component of the vector $\vec{P}_s^t$

$${P_s^t}_\phi\propto v_s^t Y_s^t(\theta,\phi)$$ (27)

the low latitudes are supposed to be affected more than the high latitudes. This is exactly what the inversion shows, the response decreases with increasing latitude. The plot with $u_{\rm char}=30$ m s^-1 shows that the low latitudes are already strongly distorted, while the high latitudes show the usual inversion results.

With further increasing amplitude the inversion shows that the higher latitudes also become affected. Moreover, artifacts below the convection zone become apparent, where a rigid body rotation was the input flow field. These artifacts are e.g., a deeper lying tachocline, and regions of alternating retarded and accelerated differential rotation. We will give an explanation for these artifacts later in Sect. 4.3.

4.2 Zonal meridional flows

For the zonal meridional flow field the inversions themselves do not show any influence of the poloidal convection cells (Fig. 8, right two columns). This is what we expect, because the zonal field does not have flow components in direction of the rotation. Thus, it gives confidence in the numerical simulations and the analysis techniques. But focusing on other quantities that are helpful like the expansion coefficients a_s or the $(m,\nu)$ diagrams of the multiplets, the additional velocity field can be noticed. Concerning the a_s the meridional flows are only reflected in the even coefficients; for the latter an increasing amplitude leads to an increasingly strong distortion of the multiplet. With large flows the structure of a multiplet can be disrupted to a degree that is not observed, cf. Fig. 9, where we used a rather unrealistic amplitude of 500 m s^-1 to emphasize the distortion of the multiplet.

Figure 9: Disrupted multiplet l=34, n=15 for differential rotation plus a poloidal flow s=8, t=4, $u_{\rm char}=500$ m s-1. Dashed line: undisturbed multiplet from Fig. 2.

Each point in a $(l,~\nu)$ diagram represents a multiplet; if a multiplet is strongly distorted it cannot be fitted by only a few orthogonal polynomials (Eq. (20)). If that happens the whole multiplet is not taken into account for the inversion. This leads to pronounced gaps in the $(l,~\nu)$ diagram for $u_{\rm char} > 100$ m s^-1. As Fig. 10 (right column) shows, the number of distorted multiplets increases with increasing amplitude and these multiplets are located along straight lines in the $(l,~\nu)$ diagram. The modes that drop out are those with the same penetration depth, i.e., they all have the same ratio $\nu/L$, indicating the same lower turning point. These modes therefore are the ones that will have the largest influence on the inferred rotation rate in this region. These modes probe the zones where the flow field is strongest.

For the rotation inversion the odd a_s coefficients are used. The meridional flows and the rotation are orthogonal to each other; this is also mirrored in the splittings. As we have seen in Sect. 2, the expansion of the rotational splitting is given by the odd and the expansion of the meridional-flow-splitting is given by the even a_s coefficients. We note that the calculated even a_scoefficients are zero in the absence of meridional flows. As the orthogonality of the Wigner 3j symbols (Edmonds 1960) is given by

$$\sum_{l_3m_3}(2l_3 + 1) \left(\begin{array}{ccc} {l_1}&{l_2}&... ...'}&{m_3} \end{array}\right) = \delta_{m_1m_1'}\delta_{m_2m_2'}$$ (28)

and

 

$$\sum_{m_1m_2} \left(\begin{array}{ccc} {l_1}&{l_2}&{l_3}\\ {m_1}... ...right) =\frac{\Delta(l_1,l_2,l_3)}{2 l_3 + 1} \delta_{l_3l_3'}\delta_{m_3m_3'},$$ (29)

where $\Delta(l_1,l_2,l_3)$ means that this factor is non-zero only if the conditions in Eq. (14) are satisfied. The Wigner 3j symbols (s l l/0 m -m) and (s l l'/0 m -m'), that correspond to rotational or meridional flows, respectively, are orthogonal with regard to s. That means that the meridional and the rotational flows can be disentangled. Focusing on the even a_s-coefficients (Fig. 10 left column) the meridional flow becomes noticeable at an amplitude $u_{\rm char}= 10$ m s^-1.

Figure 10: Left: coefficients a2 divided by the standard deviation of a2. With increasing $u_{\rm char}$ the influence on the coefficient becomes more visible as increasing deviation from zero. Right: $(l,~\nu)$ diagrams. Bold symbols indicate multiplets where a2 is more than three standard deviations away from zero. The number of distorted multiplets increases with increasing amplitude $u_{\rm char}$. Both indicators show that the modes probing the region of the strongest velocity field (in this examples s=8, t=0) become affected.

Hence, even if the inversion of the odd a_s-coefficients itself does not show any changes in the rotation rate, because the rotation rate is not changed as there are no additional flows in the direction of the rotation, we can nevertheless detect the meridional flows, if their amplitude is larger than 10 m s^-1. The new diagnostics are the distortion of the multiplets and the even a_s-coefficients.

4.3 Model artifacts below the convection zone

In this section we explain the emergence of artifacts below the convection zone for the poloidal flow components with $t\not=0$. The main reason for this behavior is that we are inverting for differential rotation. However, the poloidal flow field components bring in additional flows in the direction of the rotation. Hence the odd a_s-coefficients and therefore the inferred rotation rate are changed due to crosstalk. This change in the coefficients does not vanish by the usage of averaging kernels in Eq. (26), because these kernels consists of eigenfunctions that do not necessarily couple by the poloidal flow.

As the Wigner 3j symbols (s l l/0 m -m) and (s l l'/t m -m') with $t\not=0$ are not orthogonal, and as the basis of the functions used (s l l/0 m -m) is not complete, the two flows, differential rotation and poloidal field, cannot be disentangled. As a consequence, the total splittings in the multiplets are misinterpreted in terms of zonal toroidal flows. The inversion will then lead to a distorted differential rotation profile that is compatible with the distorted multiplets calculated from the combined effect of regular rotation and additional poloidal flow. For low amplitudes $u_{\rm char}$ this is attained by extending the differential rotation deeper into the sun and changes in the rotation rate at low colatitudes, because the modes concentrated to the equator and penetrating deeply are affected most. For high amplitudes $u_{\rm char}$ all modes are affected; the inversion results in a changed near surface rotation rate and in a strong latitudinal variation of the differential rotation, that is extended deep into the sun.

We conclude from our numerical experiment that pure meridional flows and pure sectoral cellular flows in certain convective cells can be detected with global helioseismic data. The detectability limit that we can derive on the basis of our models (Eqs. (2)-(9), (18), and (19)) of such large-scale flows is of the order of $\vert\vec{v}\vert\approx10$ m s^-1.