# 80. A theoretical explanation of variation of meridional circulation with the solar cycle

Contributed by Gopal Hazra. Posted on December 19, 2017

Gopal Hazra and Arnab Rai Choudhuri
Department of Physics, Indian Institute of Science, Bangalore 560012, India

The meridional circulation is one of the most important large-scale coherent flow patterns within the solar convection zone. It has been found from helioseismic measurements that meridional circulation varies with the solar cycle1,2,3. These results are consistent with the surface measurements of Hathaway & Rightmire (Ref. 4), who found that the meridional circulation at the surface becomes weaker during the sunspot maximum by an order of 5 m s-1. One plausible explanation for this is the back-reaction of the dynamo-generated magnetic field on the meridional circulation due to the Lorentz force. The aim of our study is to develop such a model of the variation of the meridional circulation and to show that the results of such a theoretical model are in broad agreement with observational data.

To incorporate the effect of the Lorentz force on the meridional circulation, we need to consider the ϕ-component of the vorticity equation, which is

∂ωϕ/∂t+s∇.(vmωϕ / s) = s ∂Ω2 / ∂z+1/ρ2(∇ρ×∇p)ϕ+[∇×FL]ϕ + [∇×Fν(vm)]ϕ, (1)

where ωϕ is the ϕ-component of vorticity which comes from the meridional circulation vm only, s = r sinθ, FL is the Lorentz force term and Fν(vm) is the turbulent viscosity term corresponding to the velocity field vm. We now break up the the meridional velocity into two parts:

vm = v0 + v1, (2)

where v0 is the regular meridional circulation the Sun would have in the absence of magnetic fields and v1 is its modification due to the Lorentz force of the dynamo-generated magnetic field. The azimuthal vorticity ωϕ can also be broken into two parts corresponding to these two parts of the meridional circulation:

ωϕ = ω0 + ω1. (3)

We substitute Eq. (2) and (3) in Eq. (1). Then we subtract from it the version of (1) for v0 alone (FL will be absent in this case). Neglecting quadratic terms in perturbed quantities, we get

∂ω1/∂t + s∇.(v0ω1 / s) +s∇.(v1ω0 / s) = [∇×FL]ϕ + [∇×Fν(v1)]ϕ. (4)

In order to explain how the meridional circulation varies with the solar cycle, we need to solve Eq. (4) with Lorentz force calculated from the Flux Transport Dynamo model. All the parameters and solving procedures are given in Ref 5. Here we present some of the important results.

Figure 1| The time–latitude plot of perturbed velocities vθ (a) near the surface (r=0.99Rsun) and (b) at the depth r=0.75Rsun. In the northern hemisphere, positive red color shows perturbed flows towards the equator and the negative blue color shows flows towards pole. It is opposite in the southern hemisphere. Toroidal fields at the bottom of the convection zone are overplotted in both of the plots by line contours. Black solid contours indicate positive polarity and black dashed contours are for negative polarity.

Figure 1shows time-latitude plots of the perturbed velocity just below the surface and in the lower part of the convection zone from our simulation. The meridional circulation in a poloidal cut of the Sun’s Northern Hemisphere is anti-clockwise i.e., poleward at the surface and equatorward at the bottom. If the meridional circulation is to be weakened by the Lorentz forces at the time of the sunspot maximum, then we need to generate a perturbed velocity opposite to the unperturbed velocity. We find that this is exactly the case! As shown in Figure 1, during the solar maxima, the perturbed flows are towards equatorward near surface and poleward near the bottom, weakening the overall amplitude of the meridional flow. The amplitude of the perturbed meridional flow, 5 m s1 on the surface, is almost comparable with the observed value in Ref 4. This back-reaction on the meridional circulation should mainly come from the tension of the dynamo-generated toroidal magnetic field. Suppose we consider a ring of toroidal field at the bottom of the convection zone. Because of the tension, the ring will try to shrink in size and this can be achieved most easily by a poleward slip. This tendency of poleward slip would give rise to the Lorentz force opposing the equatorward meridional circulation which introduces a vorticity opposing the normal vorticity associated with the meridional circulation and, as this vorticity diffuses through the convection zone, there is a reduction in the strength of the meridional circulation everywhere including the surface as found from our simulation. Figure 2 shows the plot of peak amplitude of the total meridional circulation with time (black dashed line) overplotted with the solar cycle (red solid line). We see that the meridional circulation reaches its minimum a little after the sunspot maximum.

Figure 2| The black dashed line shows the time variation of the total meridional circulation just below the surface at 25° latitude. The red solid line shows the yearly averaged sunspot number.

In summary, we find that the back-reaction due to the Lorentz force on the meridional circulation is responsible for its variation with the solar cycle and our results are in good agreement with the observed variation of the meridional circulation.

### References

[1] Chou, D.-Y., & Dai, D.-C. 2001, ApJ, 559, L175
[2] Zhao, J., & Kosovichev, A. G. 2004, ApJ, 603, 776
[3] Komm, R., González Hernández, I., Howe, R., & Hill, F. 2015, Solar Phys., 290, 3113
[4] Hathaway, D. H., & Rightmire, L. 2010, Science, 327, 1350
[5] Hazra, G., & Choudhuri, A. R. 2017, MNRAS, 472, 2728