OBSERVATIONS AND MODELING OF NORTH–SOUTH ASYMMETRIES USING A FLUX TRANSPORT DYNAMO

Based on the observations of sunspots on the surface of the Sun and the relatively well organized poloidal field, which changes polarity approximately every 11 yr, it has been universally accepted that the solar magnetic cycle is a dynamo process involving the transformation of the polar field into the toroidal field and subsequent conversion of the toroidal field into the poloidal field of opposite polarity over the course of approximately 11 yr (see, e.g., Babcock 1961 and its citations). The generation and propagation of large-scale magnetic fields via the dynamo mechanism are considered to be a two-step process. The first step involves shearing of the poloidal component of the magnetic field by differential rotation, which gives rise to the azimuthally directed toroidal magnetic field. This toroidal field then gives rise to the formation of the sunspots and the active regions (ARs). The second step is the formation of the poloidal component from the toroidal component, which occurs from the magnetic flux being liberated by the growth and the decay of the sunspots, with the leading polarity flux moving toward the equator and the following polarity toward the pole. In some models, movement of the following polarity fields toward poles is due to the meridional circulation, as illustrated with both the kinematic dynamo and flux-transport dynamo models (see, e.g., Wang et al. 1991; Choudhuri et al. 1995 and references therein).

The study of the solar cycle on long timescales indicates that the solar cycle is virtually symmetric between the northern and southern hemispheres, in the sense that the average amplitudes, shapes, and durations of cycles are very similar (see, e.g., Goel & Choudhuri 2009). However, there are individual cycles that are known to be stronger in one hemisphere than the other. For example, just after the Maunder Minimum, almost all the sunspots were observed in the southern hemisphere (see, e.g., Ribes & Nesme-Ribes 1993). Asymmetries between the two hemispheres have also been observed in various solar activity phenomena such as sunspot area, sunspot numbers, faculae, coronal structure, posteruption arcades, coronal ionization temperatures, polar field reversals, and solar oscillations (see, e.g., Chowdhury et al. 2013; Sýkora & Rybák 2010; Gao et al. 2009; Li et al. 2009; Temmer et al. 2006; Knaack et al. 2004, 2005; Tripathi et al. 2004; Ataç & Özgüç 1996; Oliver & Ballester 1994; Zolotova et al. 2010; Svalgaard & Kamide 2013 and references therein), in addition to long-term hemispheric asymmetries in solar activity in previous solar cycles (see, e.g., Vizoso & Ballester 1989; Carbonell et al. 1993; Norton & Gallagher 2010).

Further, the rise and fall of solar cycle 23 has been discussed by many authors; it has been found that the behavior of solar cycle 23 is very peculiar for an odd-numbered cycle (Chowdhury et al. 2013). Cycle 23 showed a slow rise compared to other odd-numbered cycles and was found to be weak compared to other odd-numbered cycles (Li et al. 2009; Chowdhury et al. 2013). Additionally, it shows an unusual second peak during the declining phase (Li et al. 2009; Mishra & Mishra 2012). Moreover, the temporal characteristics of cycle 23, such as sunspot number and sunspot area, are similar to those of the Gleissberg global minimum cycles 11, 13, and 14, which occurred between 1880 and 1930, as well as solar cycle 20 (Krainev 2012). Analysis of polar field patterns indicates that polar field reversal was slower than in the previous two cycles, as discussed in (Dikpati et al. 2004), which could have delayed the rise of solar cycle 24. The first two years of cycle 24, with low solar activity concentrated in the south, are similar to the cycle that immediately followed the Maunder Minimum (Krainev 2012).

Our motivation for this research is to investigate solar cycle 23 and the rise of solar cycle 24 by computing the AR fluxes that form the toroidal fluxes. We then concentrate on the solar cycle minimum and investigate the asymmetry in the hemispheres with the aim of addressing the issue related to the deep minimum observed in cycle 23. To support our observations, we have carried out dynamo simulations as mentioned in Belucz & Dikpati (2013). We also investigated the role of meridional circulation combined with flux asymmetry for solar cycle 23 by discussing different cases related to the asymmetries found. The rest of the paper is organized as follows. In Section 2 we present the observations and data selection, followed by magnetic flux analysis and results in Section 3. In Section 4 we discuss dynamo simulations and relate them to our observations. In Section 5 we summarize the results and science.

We calculate the line-of-sight (LOS) component of the magnetic field (hereafter B_⊥) using the magnetograms recorded by the Michelson Doppler Imager (MDI; Scherrer et al. 1995) on board the Solar Heliospheric Observatory (SOHO). MDI is an instrument used to observe signs and strength of the LOS component of the photospheric magnetic field. MDI images the Sun using a 1024 × 1024 CCD camera and acquires one full-disk LOS magnetic field each 96 minutes (5 minutes averaged magneto grams), among other observing sequences, which is free from atmospheric noise. A full-disk magnetogram of the Sun has a resolution of ∼4'' (2'' × 2'' pixel⁻¹) and a field of view of 34' × 34'. Preflight per-pixel error in the flux was estimated at 20 G (20 Mx cm⁻²) (Scherrer et al. 1995), which was found to be 14 G in flight, as was reported by Hagenaar (2001).

In the present work, we have used MDI magnetograms to compute the daily magnetic flux of ARs observed by solarmonitor.org on the solar disk from 1996 May 6 to 2010 April 12 (approximately 5100 days), which covers the final stages of solar cycle 22, the complete cycle 23, and the rising phase of cycle 24. During this period, we have manually monitored evolution of 1948 ARs, which include 286 AR nests and 6 AR evolutions, where the dispersion stage of ARs was observed to be persistent over multiple revolutions. We preferred using a manual approach over an automated one, as discussed in Zhang et al. (2010) and Stenflo & Kosovichev (2012), so that we could eliminate multiple counting of the same AR due to solar rotation.

Figure 1 shows the location of ARs on the solar disk during the declining phase of cycle 22 (top left), the rising phase of solar cycle 23 (top right), the declining phase of solar cycle 23 (bottom left), and the rising phase of solar cycle 24 (bottom right). The two hemispheres are separated by the black line representing the solar equator. We further represent our observations in a butterfly diagram in Figure 2 that shows the location of the AR observed and the time. Here, in the case of AR nests, we have noted the position and date of the AR that was first detected in solarmonitor.org. We then concentrate our analysis on the 238 ARs observed on the solar disk from 2005 July 22 to 2010 April 12, which covers the declining phase of cycle 23 and the rising phase of cycle 24.

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To refine the data set, we ignore those ARs that dispersed in less than two days, since they generally lacked sunspots. We also exclude ephemeral ARs; their fluxes are discussed in Hagenaar (2001) and Hagenaar et al. (2003). The MDI magnetogram shows noise of 002 on the limbs, with additional noise on the right limb due to wavelength changes in the Michelson filters Θ (Wenzler et al. 2004). Hence, we ignore all ARs born on the right limb at longitudes ≳60°. For ARs born on the right limb with an angle ≳60° the angle between LOS magnetic field and perpendicular magnetic field (⊥) tends to zero (Hagenaar 2001; Hagenaar et al. 2003). Hence, we have omitted all readings beyond ≳60°.

After choosing the ARs, we select a box containing isolated region(s) and calculate the highest value of B_⊥, viz., B_⊥max in each box. Thereafter we separate the negative and positive polarities of ARs, using contours with levels at B_⊥ = 0, B_⊥ < −14 G and B_⊥ > 14 G. Then we define a contour at 99% of the calculated B_⊥max values to eliminate the magnetic fields outside the ARs. Finally, we use Equation (1) to calculate the magnetic fluxes (Φ) of each region separately,

$$\begin{equation} \Phi = \int B \cdot ds. \end{equation} \tag{ 1 }$$

Since the AR magnetic field is bipolar, the net unbalanced magnetic flux in an AR should be nearly zero. To check whether our observations were accurate, we noted the net unbalanced flux; if it has a value other than zero, then we return to the first step again and reselect the boxes. This parameter also meant that we consider the AR nest as a single large AR that occurred mostly during maximum of solar cycle 23. During the minima, we found two AR nests and were able to resolve them into individual ARs. In order to remove the error due to LOS effects, we calculate the angle Θ of each pixel from the disk center following the method described in Hagenaar (2001) using

$$\begin{equation} \Theta = \sin ^{-1}\, \left[ \frac{\sqrt{ \left(\begin{array}{c}x_{\bot }^{2} \end{array} \right) + \left(\begin{array}{c}y_{\bot }^{2}\end{array} \right)}}{R} \right] , \end{equation} \tag{ 2 }$$

where x_⊥, y_⊥ are radial coordinates of each pixel within the AR on the solar disk and R is the radius of the Sun in pixels.

To calculate the LOS magnetic flux of ARs, we have used the relation that the LOS magnetic flux is related to the perpendicular component of magnetic flux using the formula

$$\begin{equation} \Phi _{{\rm LOS}}= \Phi _{\bot } \cdot \cos \Theta. \end{equation} \tag{ 3 }$$

In order to understand magnetic flux evolution, we produce magnetic flux versus time graphs as shown in Figures 3 and 4. We summarize the calculated fluxes from the complete database in Figure 3. We represent fluxes from NOAA 07961 as 0 (1996 May 6) on the x-axis as the beginning of the data set and proceed with computing daily fluxes. Since we intended to study both hemispheres independently, we have represented AR fluxes in the northern hemisphere by black solid lines and those in the southern hemisphere by red solid lines. It can be easily noticed from Figure 3 that solar cycle 23 was significantly asymmetric in terms of dominating hemispheres.

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Stenflo & Kosovichev (2012) and Zhang et al. (2010) have carried out detailed analyses of ARs observed by MDI with their respective techniques. Stenflo & Kosovichev (2012) performed a critical analysis of a data set using automated techniques and studied various properties of these bipolar regions, including their orientation and the tilt angle variation. Zhang et al. (2010), on the other hand, studied the basic physical parameters, including the magnetic flux of individual ARs and their distribution with respect to size as well as magnetic flux. They found the solar cycle between 1996 and 2008 to be very asymmetric. In terms of number of ARs, they found 938 ARs in the south and 792 in the north. We found similar results: we found 1071 ARs in the south and 872 in the north, which makes the asymmetry approximately 12%. Zhang et al. (2010) also studied the asymmetry in the magnetic flux and found that the southern hemisphere was stronger than the northern hemisphere during the declining phase of solar cycle 23; we found a similar asymmetry, where we found 10 times more flux emergence in the southern hemisphere. These results are also in agreement with those obtained by (Li et al. 2009) and (Chowdhury et al. 2013), based on sunspot observations.

Figure 3 suggests that the behavior of deep minima may be related to AR fluxes related to the later part of the declining phase of cycle 23. Thus, we concentrate our analysis on the latter part of cycle 23. In order to study the flux behavior therein, we selected the final 1694 days from Figure 3 (i.e., data between the 3468th day and the 5162th day) and represented in the graphs in Figure 4. Here we represent our observations with NOAA AR 10791 (northern hemisphere, observed on 2005 July 22 and represented by 0 on the x-axis in the top panel of Figure 4). In the south we began with NOAA AR 10794 (southern hemisphere, observed on 2005 August 1), both occurring roughly midway during the declining phase of cycle 23. We continue until the dispersion of NOAA AR 11060 (in the northern hemisphere on 2010 April 12, represented by 1694 on the x-axis in Figure 4) in the north. In the plots, the blue line indicates the magnetic flux behavior during the declining phase of cycle 23, and the black line indicates the magnetic flux behavior during the rising phase of cycle 24.

Figure 4 clearly indicates that photospheric magnetic flux during the final four years of solar cycle 23 was dominant in the southern hemisphere, producing a profound north–south asymmetry in terms of AR numbers and magnetic flux. During this period, we found 121 ARs in the southern hemisphere as compared to 60 ARs in the northern hemisphere. This asymmetry becomes even more pronounced as the cycle progresses. The AR fluxes in the southern hemisphere were approximately 10 times stronger than those in the northern hemisphere. But this behavior changed completely during the rising phase of cycle 24, for which the strength of fluxes in the north is four times that in the south. This was observed with 36 ARs emerging in the north compared to 23 in the south. Proceeding toward cycle 24, we find that (see Figure 4) the new cycle began in the north on the 875th day, which is 2007 December 13, when the emergence of an opposite polarity sunspot was observed. In the south the opposite polarity sunspot was observed on the 1013th day, which is 2008 May 3. This is 143 days (about five months) after the reversal of polarities in the northern hemisphere. Moreover, Figure 4 also clearly shows that the change of polarity in the northern hemisphere occurred smoothly and quickly, with a mixture of ARs from both cycles for a period of 200 days, whereas in the southern hemisphere, the new emerging flux showed a delay.

In order to understand how the solar dynamo works and to investigate the reason behind our observational results concerning the asymmetry between hemispheres, as well as to understand the effect of this asymmetry on the deep minima, we have carried out sets of dynamo simulations. Belucz & Dikpati (2013) have shown that differences in the form and amplitude of meridional circulation between northern and southern hemispheres can cause significant differences in the poloidal, polar, and toroidal fields produced there. The longer the meridional circulation differences persisted, the larger the differences became. Belucz & Dikpati (2013) focused on global changes in meridional circulation, including amplitude changes of the whole circulation and differences between one and two cells in either latitude or depth. The meridional motions of sunspots, pores (Ribes & Bonnefond 1990, and their citations), and other magnetic features (Komm 1994; Meunier 1999) are related to the generation of inflows. Observations by Gizon & Birch (2005) show that there are also significant meridional circulation patterns in the Sun that are not global, including one that is associated with ARs themselves. In particular, there are inflows into ARs from lower and higher latitudes that can be as large as 50 m s⁻¹. When averaged in longitude, these inflows can create a meridional circulation signal of 5 m s⁻¹ or more. With more ARs in one hemisphere compared to the other, the average inflow should also be larger in the more active hemisphere. In addition, since ARs are the source of surface poloidal flux that migrates toward the poles and causes polar field reversals, the effect of the meridional circulation from the inflows may in fact be larger than represented by the full longitude average. It has also been shown that the inflows may play an important role in the generation of the poloidal field during the final stages of the solar cycle (Cameron & Schüssler 2012; Jiang et al. 2010). Therefore, we need to assess the role of AR inflows and their differences between northern and southern hemispheres to see how much difference in solar cycles they can produce.

We have carried out flux-transport dynamo simulations using the same model as used in Belucz & Dikpati (2013), in order to see the role of AR inflows. For the sake of completeness, we briefly repeat the setup of the simulation runs in the following subsection (Section 4.1), which describes the dynamo equations, mathematical forms of the dynamo ingredients, and boundary and initial conditions. Section 4.2 presents the detailed formulation of the treatment of inflow cells, and Section 4.3 the consequences of the inflow cells.

4.1. Dynamo Simulation Setup

Our starting point is the setup of Belucz & Dikpati (2013). We write the dynamo equations as

$$\begin{eqnarray} \frac{\partial A}{\partial t} &+& \frac{1}{r\sin \theta }({\bf u}.\nabla) (r\sin \theta A) = \eta \left({\nabla }^2 - \frac{1}{r^2 \sin ^2 \theta }\right) A\nonumber\\ && +\left[S_{\rm BL}(r,\theta) + S_{\rm tac} (r,\theta)\right]\nonumber\\ &&\times {\left[1+{\left(\frac{B_{\phi }|_{ov}(\theta ,t)}{{B_0}}\right)}^2 \right]}^{-1} \thinspace B_{\phi }|_{ov}(\theta ,t), \end{eqnarray} \tag{ 4a }$$

$$\begin{eqnarray} \frac{\partial B_{\phi }}{\partial t} &+& \frac{1}{r} \left[\frac{\partial }{\partial r}(r u_r B_{\phi }) + \frac{\partial }{\partial \theta }(u_{\theta } B_{\phi }) \right]\nonumber\\ &&= r\sin \theta ({\bf B}_p.\nabla)\Omega -{{\bf \hat{e}}}_{\phi }\thinspace. \thinspace \left[ \nabla \eta \times \nabla \times B_{\phi }{{\bf \hat{e}}}_{\phi }\right]\nonumber\\ &&\quad +\eta \left({\nabla }^2 -\frac{1}{r^2 \sin ^2 \theta }\right)B_{\phi }, \end{eqnarray} \tag{ 4b }$$

in which A(r, θ, t) denotes the vector potential for the poloidal field, B_ϕ(r, θ, t) the toroidal field, u_r(r, θ), u_θ(r, θ) the meridional flow components, Ω(r, θ) the differential rotation, η(r) the depth-dependent magnetic diffusivity, S_BL(r, θ) the Babcock–Leighton-type surface poloidal source, S_tac(r, θ) the tachocline α-effect, and B₀ the quenching field strength, which we set to 10 kG in this calculation.

We use the following expressions respectively for the Babcock–Leighton surface source and tachocline α-effect:

$$\begin{eqnarray} S_{\rm BL}(r,\theta) &=&\frac{s_1}{4}\left[1+{\rm erf}\left(\frac{r-r_1}{d_1}\right)\right]\nonumber\\ &&\times \left[1-{\rm erf}\left(\frac{r-r_2}{d_2}\right) \right]\, {2\, \sin \theta \cos \theta }. \end{eqnarray} \tag{ 5 }$$

For 0 ⩽ θ ⩽ π/2,

$$\begin{eqnarray} S_{\rm tachocline}(r,\theta) &=& \frac{s_2}{4} \left[1+{\rm erf}\left(\frac{r-r_3}{d_3}\right)\right] \left[1-{\rm erf}\left(\frac{r-r_4}{d_4}\right) \right]\nonumber\\ &&\cdot \frac{\sin \left(6(\theta -\frac{\pi }{2})\right) e^{-\gamma _2 \thinspace {(\theta -\frac{\pi }{4})}^2}}{[e^{\gamma _3(\theta - \pi /3)} +1]}, \end{eqnarray} \tag{ 6a }$$

and for π/2 ⩽ θ ⩽ π,

$$\begin{eqnarray} S_{\rm tachocline}(r,\theta) &=& \frac{s_2}{4} \left[1+{\rm erf}\left(\frac{r-r_3}{d_3}\right)\right] \left[1-{\rm erf}\left(\frac{r-r_4}{d_4}\right) \right]\nonumber\\ &&\cdot \frac{\sin \left(6(\theta -\frac{\pi }{2})\right) e^{-\gamma _2 \thinspace {(\theta -\frac{3\pi }{4})}^2}}{[e^{\gamma _3 (2\pi /3-\theta)} +1]}. \end{eqnarray} \tag{ 6b }$$

The parameter values used in Equations (5), (6a), and (6b) are s₁ = 2.0 m s⁻¹, s₂ = 0.5 m s⁻¹, r₁ = 0.95 R_☉, r₂ = 0.987 R_☉, r₃ = 0.705 R_☉, r₄ = 0.725 R_☉, d₁ = d₂ = d₃ = d₄ = 0.0125 R_☉, γ₂ = 70.0, and γ₃ = 40.0. Note that the values of s₁ and s₂ determine the amplitude of the Babcock–Leighton poloidal source term and the tachocline α-effect, respectively, but the maximum amplitudes of S_BL and S_tachocline are not exactly 2 m s⁻¹ and 50 cm s⁻¹, but instead ∼1.93 m s⁻¹ and ∼37 cm s⁻¹, respectively, for the parameter choices given above. This happens because of the modulation of error functions used in Equations (5), (6a), and (6b).

The diffusivity profile is given by (for more details, see Dikpati et al. (2002)

$$\begin{equation} \eta (r)\!=\!\eta _{\rm core}+\frac{\eta _{\rm T}}{2}\left[\!1+{\rm erf}\left(\!\frac{r-r_5}{d_5}\!\right)\!\right]+\frac{\eta _{\rm super}}{2}\left[\!1+{\rm erf} \left(\!\frac{r-r_6}{d_6}\!\right)\!\right], \end{equation} \tag{ 7 }$$

in which η_core = 10⁹ cm² s⁻¹, η_T = 7 × 10¹⁰ cm² s⁻¹, η_super = 3 × 10¹² cm² s⁻¹, r₅ = 0.7 R_☉, r₆ = 0.96 R_☉, d₅ = 0.00625 R_☉, d₆ = 0.025 R_☉. These choices make this profile possess a supergranular-type diffusivity value (η_super) in a thin layer at the surface, which drops to a turbulent diffusivity value (η_T) in the bulk of the convection zone, and at the base of the convection zone the diffusivity drops quite sharply to a much lower value (η_core) to mimic the molecular diffusivity.

The stream function ψ for the steady part of the meridional circulation is given by

$$\begin{eqnarray} {\psi }r\sin \theta &=&{\psi }_0 \frac{(\theta -{\theta }_0)}{(\theta +\theta _0)} \sin \left[k \frac{\pi (r-R_b)}{(R_{\odot }-R_b)}\right]\nonumber\\ &&\times \left(1-e^{-{\beta }_1 r{\theta }^{\epsilon }}\right) \left(1-e^{{\beta }_2 r(\theta -\pi /2)}\right) e^{-{((r-r_0)/\Gamma)}^2}.\quad \end{eqnarray} \tag{ 8 }$$

The streamline flow can be obtained in the northern hemisphere by plotting the contours of ψ r sin θ. The streamlines in the southern hemisphere can be obtained by implementing mirror symmetry about the equator. The parameter values for this stream function are k = 1, R_b = 0.69R, β₁ = 0.1/(1.09 × 10¹⁰) cm⁻¹, β₂ = 0.3/(1.09 × 10¹⁰) cm⁻¹, = 2.00000001, r₀ = (R_☉ − R_b)/5, Γ = 3 × 1.09 × 10¹⁰ cm, and θ₀ = 0. This choice of the set of parameter values produces a flow pattern that peaks at 24° latitude.

In order to perform simulations in nondimensional units, we use 1.09 × 10¹⁰ cm as the dimensionless length and 1.1 × 10⁸ s as the dimensionless time. These choices respectively come from setting the dynamo wavenumber, k_D = 9.2 × 10⁻¹¹ cm⁻¹, as the dimensionless length, and the dynamo frequency, ν = 9.1 × 10⁻⁹ s⁻¹, as the dimensionless time, which means that the dynamo wavelength (2π × 1.09 × 10¹⁰ cm) is 2π and the mean dynamo cycle period (22 yr) is 2π in our dimensionless units. Thus, in nondimensional units, the parameters that define the meridional circulation given in Equation (5) are R_b = 4.41, β₁ = 0.1, β₂ = 0.3, = 2.00000001, r₀ = (R_☉ − R_b)/5, Γ = 3, and θ₀ = 0. The latitude of the peak flow can be varied by changing β₁ and β₂; for example, changing β₁ from 0.1 to 0.8 and β₂ from 0.3 to 0.1, a flow pattern can be constructed that peaks at 50°, but for the present study, we fix the latitude of the peak flow at 24°.

Considering an adiabatically stratified solar convection zone, we take the density profile as

$$\begin{equation} \rho (r) = \rho _b {\left[ (R_{\odot }/r)-1 \right]}^m, \end{equation} \tag{ 9 }$$

in which m = 1.5. However, in order to avoid density vanishing at r = R_☉, which would cause an unphysical infinite flow at the surface, we use ρ(r) = ρ_b[(R_☉/r) − 0.97]^m in our simulations. Using the constraint of mass conservation, the velocity components (v_r, v_θ) can be computed from

$$\begin{equation} u_r=\frac{1}{\rho r^2 \sin \theta }\frac{\partial }{\partial \theta } (\psi r \sin \theta), \end{equation} \tag{ 10a }$$

$$\begin{equation} u_{\theta }=-\frac{1}{\rho r \sin \theta }\frac{\partial }{\partial r} (\psi r \sin \theta). \end{equation} \tag{ 10b }$$

The peak flow speed is determined by a suitable choice of ψ₀/ρ_b. We use a peak flow speed of 14 m s⁻¹ in all simulations.

4.2. Formulation of Inflow Cells

In order to include inflow cells into the steady meridional circulation pattern, described in Section 4.1, we incorporate a time-dependent stream function (Ψ_inflow), which is prescribed as follows:

$$\begin{equation} \Psi _{\rm inflow} = {\psi }_{0_{\rm inflow}}\sin \left[\frac{\pi (r-r_{\rm inflow})}{(R_{\odot }-r_{\rm inflow})}\right] \sin \left[\frac{2\pi (\theta - \theta _{\rm high})}{(\theta _{\rm low})}\right]. \end{equation} \tag{ 11 }$$

In Equation (11), ${\psi }_{0_{\rm inflow}}$ determines the velocity amplitude of the inflow cells, r_inflow determines how deep down the inflow cells extend from the surface, θ_high and θ_low determine their extent in θ (colatitude) coordinates, θ_high denoting the cell boundary at the poleward side and θ_low at the equatorward side. Since the inflow cells are normally associated with ARs, their θ locations have to be a function of time. We implement the time dependence in the θ coordinate of the inflow cells in accordance with the migration of the latitude zone of sunspots. Thus, we prescribe θ_high and θ_low as follows:

$$\begin{equation} \theta _{\rm high}=\theta _{\rm center} - \pi /18, \end{equation} \tag{ 12a }$$

$$\begin{equation} \theta _{\rm low}=\theta _{\rm center} + \pi /18, \end{equation} \tag{ 12b }$$

$$\begin{equation} \theta _{\rm center}=\theta _{\rm center_{\rm initial}} + \delta t \frac{\pi /6}{\tau }. \end{equation} \tag{ 12c }$$

Here θ_center is the center of the inflow cells, $\theta _{\rm center_{\rm initial}}$ is the starting location of the center of the inflow cells, and (π/6/τ) is the migration speed of the center of the inflow cells. Note that the θ-extent of each of the pair of inflow cells is 10°. So in order to make the inflow cells migrate from ∼50° latitude to the equator, we have to make their center migrate from ∼40° latitude to ∼10° latitude. Here τ is approximately one sunspot cycle period (i.e., half of a magnetic cycle period), and δt is the time step for dynamo field evolution. For simplicity, we assume in this calculation that their extent in depth remains the same. We take $r_{\rm inflow}=0.9 \_ R_{\odot }$.

Figure 5 shows the prescribed form of the inflow circulation cells (see Equations (11), (12a)–(12c)) we have included in the dynamo model. In the left panel we have superimposed the inflow circulation streamlines on the single-celled global meridional circulation. As mentioned earlier, in our calculations we have allowed the inflow circulation to reach to a depth of 0.9 R_☉ and to latitudes of 10° poleward and equatorward of the AR latitude. The right panel in Figure 5 shows the total streamlines for a case for which the peak global circulation is 14 m s⁻¹ and the peak inflow is 15 m s⁻¹.

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4.3. Effect of North–South Asymmetry in Inflow Cells

In the simulations, the inflow pattern is introduced into both the northern and southern hemispheres, but with a much stronger peak in the south (15 m s⁻¹ in the south versus 1.5 m s⁻¹ in the north). The choice of this difference is motivated by the fact that there were many more ARs in the south compared to the north in the declining phase of cycle 23 (see Figure 1), and the flux in the southern hemisphere was observed to be about a factor of 10 higher than in the northern hemisphere. In both hemispheres, the inflow circulation pattern is propagated toward the equator at a rate consistent with the equatorward migration of the latitudes of AR appearance. Figure 6 shows the patterns of meridional circulation (panels (a)–(d)), toroidal field contours (panels (e)–(h)), and poloidal field lines (panels (i)–(l)) for a sequence of time intervals separated by 2.7 yr within a single sunspot cycle. The simulation was begun a few cycles earlier with the same weak inflow in both hemispheres; the stronger inflow in the south was introduced a few months before the first frames shown.

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We see in panel (i) that there is an immediate effect on the surface poloidal field in the south. By counting the number of contours of poloidal field lines, we can see this effect in the form of more concentrated flux in the neighborhood of the inflow pattern. Because the extra inflow near the surface is slowing down the migration of poloidal flux toward the pole, by panel (j) 2.7 years later, the polar field in the south is weaker and is reversing sign later than in the north. Again the number of contours reveals that this results in less poloidal flux being transported to the bottom in high latitudes to cancel out the previous poloidal fields. This, in turn, allows the toroidal field near the bottom in the south to become significantly stronger than in the north (see panels (f), (g), and (h)). Therefore, we see that the stronger inflows associated with one cycle in one hemisphere can lead to stronger toroidal fields in that hemisphere in the next cycle. This suggests that in the nearly independent northern and southern hemispheres the strength of one hemisphere compared to the other may persist for more than one cycle. There is observational evidence for this persistence, which is discussed in Dikpati et al. (2007, and reference therein), as well as differences in time of sunspot maximum.

In this particular simulation, the difference in inflow speed was introduced for a duration of about 12 yr, after which the inflow in the south returned to the same lower value as in the north. Figure 7 shows a butterfly diagram for several cycles that includes the time with different inflows. On this diagram the extra inflow in the south occurred for years 5–17. Shading is for the poloidal field amplitudes, contours for the toroidal field amplitude. If we focus on the toroidal field contours, we can see that the stronger toroidal field in the south persists for more than one cycle after the extra inflow has been switched off. This effect can clearly be seen in Figure 8, in which the tachocline toroidal fields, taken from 45° latitude, have been plotted in the north (dashed black) and south (solid red) in the top panel. In the bottom panel the polar field patterns in the north (dashed black) and south (solid red) are presented. The effect of even a temporary increase in inflow affects the dynamo well beyond the duration of the extra inflow; even though the maximum effect on the polar field, namely, a continuous increase in the south polar field, can be seen during the drifting inflow cells until the sunspot minimum, the effect on the tachocline toroidal fields is more enhanced in the succeeding cycle, because of the increased polar fields being advected there by the time of the start of the next sunspot cycle, thus providing a stronger seed magnetic field. Cameron & Schüssler (2012) found a similar effect, namely, an increase in polar field at the end of a sunspot cycle and an increase in the sunspot cycle strength in the succeeding cycle, due to the presence of inflow cells in their surface transport model. In reality the extra inflow would persist as long as more ARs are produced in the south, so the effect of this extra inflow would be even more pronounced and persistent, and inherently nonlinear.

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The peculiar behavior of solar cycle 23 and its prolonged minima has attracted much attention from researchers over the past few years. There have been various studies taking many different parameters into account. In the present paper we have discussed the contribution of AR fluxes and their asymmetries in the northern and southern hemispheres, during solar cycle 23 and the rising phase of solar cycle 24, with the aim of addressing the issue of the deep minimum observed in solar cycle 23. The observations showed that cycle 23 was highly asymmetric. During the rise phase of cycle 23, the northern hemisphere was dominant over the southern hemisphere, which reversed during the decline phase of the cycle.

Further, we concentrated our analysis on the declining phase of cycle 23 and the rising phase of cycle 24. The analysis shows that the magnetic flux in the southern hemisphere is about 10 times stronger than that in the northern hemisphere during the declining phase of solar cycle 23. The trend, however, reversed during the rising phase of solar cycle 24, and magnetic flux becomes stronger (about a factor of four) in the northern hemisphere. Moreover, it was found that there was significant delay (about five months) in changing the polarity in the southern hemisphere in comparison with the northern hemisphere. These results may provide us with hints about how the toroidal fluxes would have contributed to the solar dynamo during the prolonged minima in solar cycle 23 and in the rise phase of solar cycle 24.

It has been shown previously by Belucz & Dikpati (2013) that the degree of asymmetry in amplitude between the northern and southern hemispheres can be changed significantly by differences in meridional circulation amplitude and/or profile between north and south. Here we have demonstrated that the difference between hemispheres in axisymmetric inflow into AR belts can lead to differences in peak amplitude that can last for more than one sunspot cycle. In the example shown here, we find that an increase in inflow in the south, which would accompany more solar activity there, leads to stronger toroidal fields in the south for substantially more than one cycle even after the extra inflow has been shut off. Therefore, this mechanism can lead to persistence of one hemisphere dominating over the other for multiple cycles, as is often observed. In effect, once a larger inflow is established in one hemisphere, its existence provides reinforcement for stronger cycles in that hemisphere to follow. An interesting question is then how the Sun eventually breaks out of this asymmetric pattern to a new one in which the other hemisphere dominates. Among other possibilities, this could occur when some other feature of meridional circulation, such as its amplitude or profile, changes in one hemisphere relative to the other.

We thank the referee for important input that has made the manuscript more comprehensive. The authors would also like to thank Robert Cameron for valuable input on the manuscript. Juie Shetye thanks IUCAA for the excellent hospitality during her visit. This project was started at IUCAA. We gratefully acknowledge Professor John Gerard Doyle at Armagh Observatory for the important correction and suggestions. Research at Armagh Observatory is grant-aided by the Leverhulme Trust, Grant Ref. RPG-2013-014, and the work at High Altitude Observatory, National Center for Atmospheric Research, in Boulder, Colorado, is partially supported by NASA's LWS grant with award number NNX08AQ34G. The National Center for Atmospheric Research is sponsored by the National Science Foundation. SOHO is a project of international cooperation between ESA and NASA.