THE QUASI-BIENNIAL PERIODICITY AS A WINDOW ON THE SOLAR MAGNETIC DYNAMO CONFIGURATION

The GONG has provided nearly continuous and stable velocity images of the Sun since 1995 May. It consists of six instruments deployed worldwide, based on a Michelson interferometer using the Ni line at 676.8 nm. Mode parameters (frequency, full width, and amplitude) for each (n, ℓ, m) are estimated up to ℓ = 150 by applying the standard GONG analysis (Anderson et al. 1990). The peak-fitting algorithm has two criteria to judge the quality of the fit to a mode (Hill et al. 1998) and based on these quality flags, we removed few outliers from the analysis. The individual p-mode parameters are afterward made publicly available (http://gong2.nso.edu/archive/). In this work, we analyze the temporal variations of p-mode frequencies covering almost 17 years of observations starting from 1995 May up to 2012 January.

The spatial structure of the acoustic modes in the Sun can be described by associated Legendre functions P^m_ℓ(x), where ℓ is the degree, m is the azimuthal order running from −ℓ to ℓ, and x is the cosine of the colatitude θ. The degeneracy among the modes of the same ℓ and m is broken by rotation and asphericity. The labeling of a mode is completed by the radial order n (number of radial intersections or harmonics). The frequencies of a mode of a specific n, ℓ multiplet are given using the following polynomial expansion:

$$\begin{equation} \nu _{n, \ell , m}=\nu _{n,\ell }+\sum a_{j}(n,\ell)P_{j}^{\ell }(m), \end{equation} \tag{ 1 }$$

where the basic functions are polynomials related to the Clebsch–Gordon coefficients C^{ℓ, m}_j0ℓm (Ritzwoller & Lavely 1991) by

$$\begin{equation} P_{j}^{\ell }(m)=\frac{\ell \sqrt{(}2\ell -1)!(2\ell +j+1)!}{(2\ell)!\sqrt{(}2\ell +1)}C^{\ell , m}_{j,0, \ell , m}. \end{equation} \tag{ 2 }$$

The frequency ν_nℓ is the so-called central frequency of the multiplet, and we will also use this parameter as it provides a measure of the global activity of the Sun.

In this work, we look for temporal variations in p-mode frequencies caused by changes in magnetic activity levels. Within this context the frequency shift can be defined as the difference between the frequencies of the corresponding modes observed at different epochs and the reference values taken as average over the minimum phase of solar activity (δν ≡ ν_i − ν(B₀)) or as the difference between the mode frequency at certain date and its temporal mean ($\delta \nu _{i}\equiv \nu _{i}(B)-\bar{\nu _{i}}$) (Howe et al. 2002). We choose the first approach as it provides the advantage of directly comparing the shift with other publications, because it does not depend on the inclusion of new data sets. Since the frequency shifts have a well-known dependency on frequency and mode inertia (Jain et al. 2000), we consider only those modes that are present in all data sets, and the shifts are scaled by the mode inertia (Christensen-Dalsgaard & Berthomieu 1991). The mean frequency shift is calculated from the following relation:

$$\begin{equation} \delta \nu (t)=\frac{\sum _{n,\ell ,m}{\frac{Q_{n,\ell }}{\sigma ^{2}_{n,\ell ,m}}}\delta \nu (t)}{\sum _{n,\ell ,m}\frac{Q_{n,\ell }}{\sigma ^{2}_{n,\ell ,m}}}. \end{equation} \tag{ 3 }$$

The weighted averages of these frequency shifts were then calculated in two different frequency bands:

• 1.  
  low-frequency band 1600 μHz ⩽ν ⩽ 2500 μHz;
• 2.  
  high-frequency band 2500 μHz ⩽ν ⩽ 3500 μHz.

This frequency dependence analysis will tell us to which depths the 2-year signal is the most sensitive. In fact, the mode frequency determines the position of the upper turning point (UTP) of the waves since the mode frequency increases as the UTP approaches the solar surface. Seismic observations over the 11-year cycle have already shown that the increase in the magnitude of the shift is predominantly a subsurface phenomenon as it strongly increases in the upper few hundred kilometers (Chaplin et al. 2001).

We investigate the solar cycle changes in p-mode central frequency averaged over spherical degree ℓ = 0–120. In order to extract the quasi-biennial signal from the total shifts, we subtracted the 11-year envelope by using a boxcar average of 2.5-year width (Fletcher et al. 2010). Figure 1 compares the temporal changes in p-mode frequency in the two frequency bands over solar cycle 23 (left panel) and more specifically for the 2-year period (right panel). The error in shifts is of the order of 10⁻³ μHz for both frequency bands. The presence of a quasi-biennial modulation is clearly visible in both frequency bands and over the whole period of observation. There are several interesting features to underline in the magnitude of the shift over the 2-year cycle and to which one can compare to previous studies on low-degree modes:

• 1.  
  it is rather weak;
• 2.  
  it increases by a factor of ≈3 from a low- to high-frequency band. This enhancement is smoother compared to the stronger increase over the 11-year cycle;
• 3.  
  it becomes faint over the descending phase of solar cycle 23 and the low- and high-frequency bands are identical in this descending phase.

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The other interesting feature is that over the ascending phase of solar cycle 23, the signal is stronger in the high-frequency band, while this does not occur over the ascending phase of solar cycle 24. This might imply that the ongoing cycle 24 will be weaker compared to solar cycle 23.

We investigate the significance of the QBP signal in p-mode central frequency averaged over spherical degree ℓ = 0–120 and in the two frequency bands by applying the wavelet analysis developed by Torrence & Compo (1998). The upper panels of Figure 2 show the temporal evolution of p-mode frequency shifts over the period 1996–2012, while the lower panels identify periods in which the signal reaches 90% confidence level for both frequency bands. This occurs during periods around the solar maximum (1998–2004) with several significant periodicities in the range of 1.99 years ⩽ T ⩽ 3.9 years in both frequency bands. This shorter cyclic component characterizes different phases of solar magnetic activity, in contrast with activity proxies where the QBP signal is mainly prominent over periods coinciding with solar maximum. This peculiar feature seems to further confirm the persistent nature of the QBP signal, in agreement with previous findings based on the analysis of low-degree modes (Simoniello et al. 2012a). This finding is important to advance in dynamo theories that already take into account both periodic components of solar magnetic activity. Understanding the 2-year signal might give further constraints to these models.

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The p-mode spatial configuration can be described by spherical harmonics with each mode characterized by its spherical harmonic degree ℓ and azimuthal order m. The spherical degree defines the penetration depth through the following relation (Christensen-Dalsgaard & Berthomieu 1991):

$$\begin{equation} r_{t}=\frac{c(r_{t})\sqrt{\ell (\ell +1)}}{2\pi \nu }, \end{equation} \tag{ 4 }$$

where c is the sound speed and r_t is the LTP radius. A higher value of ${\nu }/{\sqrt{\ell (\ell +1)}}$ denotes a smaller value of r_t and hence a greater depth. We carried out a depth dependence analysis in four different regions of the Sun's interior (core, radiative zone, tachocline, and convection zone) to verify that the shifts we discuss are independently confined. Then we used the ratio ${\nu }/{\sqrt{\ell (\ell +1)}}$ to select only the modes having their LTP between 0.90 ⩽ (r_t/R) ⩽ 1 solar radius. This would verify the frequency dependence of the shift in the subsurface layers and exclude different behaviors over the two cycles due, for example, to the presence of a second dynamo mechanism. Simultaneously, we used the ratio m/ℓ to select the acoustic modes more sensitive to lower or to higher latitudes. We decided to select three different latitudinal bands corresponding to equatorial latitudes (0° ⩽ θ ⩽ 30°), mid-latitudes (30° ⩽ θ ⩽ 60°), and high latitudes (60° ⩽ θ ⩽ 90°). Some authors have speculated that the QBP signal might be induced by a second dynamo mechanism located at 0.95 solar radius (Benevolenskaja 1998a, 1998b). It is known that the strong shear present in the subsurface layers increases toward latitudes higher than 60° (Schou et al. 1998). If this layer is acting as a second dynamo mechanism and it is causing the QBP, we might find an increase of the 2-year signal at higher latitudes. Figure 3 shows the changes during solar cycle 23 as a function of latitude for the three ranges.

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The figures present several interesting features:

• 1.  
  the magnitude of the shift decreases with increasing latitudes over both periodic components characterizing solar magnetic activity. This might indicate that the larger magnitude of the signal at equatorial latitudes could be due to the presence of strong toroidal fields located between −45° ⩽ θ ⩽ 45° (de Toma et al. 2000) during the course of the 11-year cycle;
• 2.  
  the size of the shift is larger at equatorial latitudes and over periods coinciding with solar maximum at all latitudes and over both periodic components, in agreement with previous findings averaged over all ℓ values (Simoniello et al. 2012b);
• 3.  
  the 11-year envelope over the 2-year cycle is clearly visible at all latitudes.

All these features clearly show that the QBP signal is highly coupled to the main dynamo driving the 11-year cycle.

The characterization of the shift with frequency, depth, and latitude over the 11- and 2-year cycles has shown differences in the enhancement rate of the magnitude of the shift. We now attempt to provide an explanation for it. Figure 4 shows the frequency dependence of the frequency shifts taken over 2001 August 8–2001 October 19 (top panels). This period corresponds to the maximum amplitude of the shift for both components of solar activity cycles. It shows that the enhanced amplitude follows an exponential behavior over solar cycle 23, in agreement with previous findings, and a slower enhancement rate over the 2-year cycle. The bottom panels in Figure 4 show the magnitude of the shift as a function of the position of the upper turning point. Those values have been calculated from model S of Christensen Dalsgaard (Chaplin et al. 2001). A sudden increase in the size of the shift is clearly visible in the last few hundred kilometers beneath the solar surface for both periodic components, although it is reduced over the 2-year period. The acoustic time of the modes is larger in these layers as the sound speed is smaller so any change of magnetic field strength in these layers is amplified.

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The solar magnetic signature along solar cycle 23 has been extracted from the even-order splitting coefficients of the high-degree acoustic modes observed with MDI at 0.996 and 0.999 R_☉. This study shows that poloidal and toroidal magnetic field strengths decrease toward the solar surface (Baldner et al. 2009). Figure 5 shows the strength of these two magnetic field components along solar cycle 23 and also deduces the 2-year cycle components at (r_t/R) = 0.999 and (r_t/R) = 0.996. Signatures of the 2-year signal reach 90% confidence level around the solar maximum (1998–2004) with a period of 2.3 years for both components. This periodicity agrees reasonably well with our findings. We now attempt to interpret our findings by taking into account the magnetic field strength behavior in the subsurface layers and the increase of the amplitude of the modes near the solar surface. We have been misleading if one has deduced that the magnetic field strength should also increase. This is not the appropriate interpretation. In the Sun's interior, β = (P_gas/P_mag) ≫ 1, as the gas pressure is extremely large compared to the magnetic pressure. Only very close to the surface does this value become smaller (Kosovichev 2008). The gas pressure undergoes a strong decay near the surface, while the magnetic pressure, over the same range of depths, undergoes a slower decay (B²). As a consequence, β reduces dramatically at the vicinity of the surface, so the magnetic pressure plays a direct role on the mode frequency in very close layers near the surface, inciting, consequently, the observed shift. The rapid enhancement in the size of the shift is, therefore, a combination of the β ratio and longer acoustic time of the modes in these layers. The reduced enhancement size over the 2-year cycle may be interpreted in terms of magnetic field strength or topology (see Section 6.3).

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The order of magnitude of the magnetic field deduced from splitting values of the subsurface layers (Baldner et al. 2009) gives a first insight into what phenomenon can influence the shifts that is discussed in this paper. In parallel to this classical approach, dedicated to secular 1D models of the Sun in agreement with deeper helioseismic observations, 3D hydro and magnetohydrodynamic simulations have been developed by Stein & Nordlund (1998) through the code STAGGER. They have shown that the order of magnitude of the turbulent pressure represents about 10% of the gas pressure in the simulations of the subsurface layers of the Sun (Nordlund & Stein 2000). Even more interesting, they have shown evidence that this turbulent term concerns a larger region than previously thought in 1D model, typically at least 1000 km in depth (Rosenthal et al. 1999). Magnetohydrodynamical simulations 48 Mm large and 20 Mm deep have been performed recently, using a horizontal magnetic field chosen at the bottom of this box compatible with values extracted from Baldner et al. (2009) and shown in Figure 5. Two complementary effects are visible in the simulations: a slow reduction of the turbulence by the deeper magnetic field in the lower part of the box and a direct effect of the magnetic pressure in the last 500 km just below the surface (see also Piau et al. 2011). These simulations agree reasonably well with what we observe in the present study, in the sense that the turbulent pressure plays a role in the absolute values of all the acoustic mode frequencies and the magnetic effect is directly connected to the small temporal variations that we study here. Of course, this last effect is clearly connected to the near surface region for all the acoustic mode frequencies.

In agreement with earlier findings, we note that the size of the shift, over the 11-year cycle, undergoes a sudden and strong enhancement ≈200 km below the photosphere, while over the same range of depths we found a reduced gain over the 2-year cycle. Because of this, some authors invoked the action of a second dynamo mechanism (Fletcher et al. 2010; Broomhall et al. 2012) due to the subsurface rotational shear extending 5% below the solar surface (Schou et al. 1998). It is worth noting that dynamos with separate generation regions could have either a common period or two distinct periods, as shown in Moss & Sokoloff (2007). Therefore, a shallower dynamo does not imply a different periodicity from the 11-year one. However, we do not want to address this issue, as we aim to look for observational evidences pointing to a separate and/or different mechanism from the main dynamo behind the 2-year periodicity. In Section 3.4, we have shown that the origin and the behavior of the shift for both periodic components of solar magnetic activity in the subsurface layers is due to the sudden and stronger decrease of β. Therefore, we do not visualize a need to invoke a further dynamo mechanism to explain the origin of the shift induced by the 2-year cycle. Moreover, the latitudinal dependence of the size of the shift does not differ over the two cycles and the 11-year envelope clearly modulates the amplitude of the 2-year signal at all latitudes, suggesting that the two periodicities are intimately coupled. Thus, our seismic analysis does not provide observational evidences in favor of a mechanism different and separate from the main dynamo behind the origin of the 2-year cycle. We also want to stress that our analysis does not exclude the possibility that the strong rotational shear layer might have a role in the generation of toroidal fields (Pipin & Kosovichev 2011), since our analysis indicates that a dynamo independent of its location is behind both components of solar magnetic activity.

The instability of magnetic Rossby waves in the tachocline might enhance the 2-year periodicity when the magnetic field strength is greater than 10⁵ G (Zaqarashvili et al. 2010). It is difficult for such strong magnetic fields to exist in the tachocline. In fact using the splitting of modes with their lower turning points around the base of the convective zone gave an upper limit of 0.3 MG (Basu 1997), in good agreement with similar upper values of 0.3–0.4 MG found by other investigators (Antia & Basu 2000). Furthermore, it is still an ongoing investigation how the magnetic Rossby waves vary with latitude in slow rotator stars. In order to fit with our results, it should be proven that the Rossby waves are smoothly redistributed along the latitudes with a maximum at low latitude and the poloidal wave number should be such that no nodes will rise from the equator toward the polar regions. However, recently it has been shown that specific magnetic field configurations can deplete the Rossby waves near the equator in favor of Alfvén waves (Tobias et al. 2011).

Helioseismic observations have shown contradictory results in the temporal variations of the rotation rates at the tachocline. In fact, while Howe et al. (2000) found periodic variations of almost 1.3 years, others cast some doubts on these findings (Antia & Basu 2000). Therefore, it is still an open debate if temporal variations of the rotation rates indeed exist and if they differ with depths. To verify whether or not the spatiotemporal fragmentation indeed occur in the Sun's interior, the investigation should be carried out by using the even- and odd-order Clebsch Gordon coefficients, as they provide information on the rotation rates and magnetic fields at different depths. The odd-order splitting coefficients are caused by the rotation of the Sun, while the even-order coefficients are caused by second-order effects of rotation and by the effects of magnetic fields or any other departure from spherical symmetry in the solar structure. Therefore, it would be extremely important to look for periodicities at different depths in the temporal evolution of both orders of splitting coefficients. If signatures of the 2-year periodicity occur only in the even-order coefficients and it is present at all depths, this finding might provide further evidences in favor of the link between the 2-year periodicity and the dynamo magnetic field configuration (see next sections).

We noticed that the largest active regions tend to always appear at similar longitudes, which are called active longitudes. These are persistent during many cycles, but it can suddenly shift by 180° to the other side of the star (Berdyugina & Usoskin 2003). This phenomenon is known as the flip-flop cycle (Jetsu et al. 1991) and it seems to be rather common in stars (Berdyugina 2006; Usoskin 2007). The origin of the flip-flop cycle is not yet fully understood, but it could be explained to be the result of the excitation of a global non-axisymmetric (quadrupole-like) dynamo mode (Moss et al. 1995; Tuominen et al. 2002; Moss 2004). Such a field configuration plausibly exists in the Sun along with the dipole-like one as inferred by solar dynamo models (Moss & Brooke 2000). The relative strengths of the two dynamo modes should define the amplitudes of the observed cycles. Within this formalism it is also predicted that the amplitude of the secondary cycle, in general, is expected to have a lower amplitude, as part of the energy is transferred from the primary magnetic configuration (dipolar) to the secondary one (quadrupolar). This feature agrees well with our findings, as we have shown that the secondary cycle in the Sun is an additive contribution to the main cycle, whose signal strength is rather weak. The period of the oscillations of the axisymmetric mode should, instead, define the lengths of the observed cycles. Therefore, the full flip-flop cycle is expected to have the same length as the axisymmetric mode (Berdyugina et al. 2002). In the Sun the major spot activity switches between the active longitudes in about 1.8–1.9 years and on average it has been observed to make six switches of the active longitude during the 11-year sunspot cycle. These values have been obtained by analyzing solar cycles 18 up to 22 whose lengths were shorter than 11 years. Our findings seems to fit well with predictions and observations of flip-flop cycles. We found a significant periodicity at ≈2 years, a bit higher than the usual flip-flop findings, but solar cycle 23 lasted longer than usual (about 12.6 years). We, then, might expect that all periodicities between 1.5 years ⩽T ⩽ 4 years (corresponding to individual periods of one longitude's dominance) might be seen as the visible manifestation of the same physical mechanism. Recently, some other authors also concluded that the non-constant period length is the manifestation of a unique quasi-biennial cycle (Vecchio & Carbone 2008). Furthermore, the magnitude of the shift increases toward equatorial latitudes over both periodic components of solar magnetic activity, and this feature comes up naturally within this formalism.