Synchronized Helicity Oscillations: A Link Between Planetary Tides and the Solar Cycle?

Abstract

Recent years have seen an increased interest in the question of whether the gravitational action of planets could have an influence on the solar dynamo. Without discussing the observational validity of the claimed correlations, we examine which possible physical mechanism might link the weak planetary forces with solar dynamo action. We focus on the helicity oscillations that were recently found in simulations of the current-driven, kink-type Tayler instability, which is characterized by an \(m=1\) azimuthal dependence. We show how these helicity oscillations may be resonantly excited by some \(m=2\) perturbations that reflect a tidal oscillation. Specifically, we speculate that the tidal oscillation of 11.07 years induced by the Venus–Earth–Jupiter system may lead to a 1:1 resonant excitation of the oscillation of the \(\alpha\)-effect. Finally, we recover a 22.14-year cycle of the solar dynamo in the framework of a reduced zero-dimensional \(\alpha\)\(\Omega\) dynamo model.

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Acknowledgements

This work was supported by the Deutsche Forschungsgemeinschaft in the frame of the SPP 1488 (PlanetMag), as well as by the Helmholtz-Gemeinschaft Deutscher Forschungszentren (HGF) in the frame of the Helmholtz alliance LIMTECH. Wilcox Solar Observatory data used in this study were obtained via the web site wso.stanford.edu (courtesy of J.T. Hoeksema). The sunspot data are SILSO data from the Royal Observatory of Belgium, Brussels, obtained via www.sidc.be/silso/infosnytot . F. Stefani thanks R. Arlt, A. Bonnano, A. Brandenburg, A. Choudhuri, D. Hughes, M. Gellert, G. Rüdiger, and D. Sokoloff for fruitful discussion on the solar-dynamo mechanism.

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Appendix: The Numerical Model

Appendix: The Numerical Model

In this appendix we sketch the integro-differential equation scheme that was used in Section 2 to calculate the oscillations of the helicity and \(\alpha\). More details can be found in Weber et al. (2013, 2015). For an alternative numerical method to treat the TI, see Herreman et al. (2015).

In our code we circumvent the usual \(\mathit{Pm}\) limitations of pure differential-equation codes by replacing the solution of the induction equation for the magnetic field by invoking the so-called quasi-static approximation (Davidson, 2001). We replace the explicit time stepping of the magnetic field by computing the electrostatic potential by a Poisson solver and by deriving the electric-current density. In contrast to many other inductionless approximations in which this procedure is sufficient, in our case we cannot avoid computing the induced magnetic field as well. The reason for this is the presence of an externally applied electrical current in the fluid. Computing the Lorentz-force term it turns out that the product of the applied current with the induced field is on the same order as the product of the magnetic field (due to the applied current) with the induced current. The induced magnetic field is computed as follows: in the interior of the domain, we apply the quasi-stationary approximation and solve the vectorial Poisson equation for the magnetic field that results when the temporal derivative in the induction equation is set to zero. At the boundary of the domain, however, the induced magnetic field is computed from the induced current density by means of Biot–Savart’s law. In this way, we arrive at an integro-differential equation approach, similar to the method used by Meir et al. (2004).

In detail, the numerical model as developed by Weber et al. (2013) works as follows: it uses the OpenFOAM library to solve the Navier–Stokes equations (NSE) for incompressible fluids

$$\begin{aligned} \dot{\boldsymbol {u}} + ({\boldsymbol {u}}\cdot \nabla){\boldsymbol {u}} = - \nabla p + \nu\Delta{\boldsymbol {u}} + \frac{\boldsymbol {f}_{\mathrm {L}} }{\rho}\quad \textrm{and}\quad\nabla\cdot\boldsymbol {u} = 0, \end{aligned}$$
(5)

with \(\boldsymbol {u}\) denoting the velocity, \(p\) the (modified) pressure, \(\boldsymbol {f}_{\mathrm {L}} = \boldsymbol {J} \times\boldsymbol {B} \) the electromagnetic Lorentz force density, \(\boldsymbol {J}\) the total current density, and \(\boldsymbol {B}\) the total magnetic field. The NSE is solved using the PISO algorithm and applying no-slip boundary conditions at the walls.

Ohm’s law in moving conductors

$$\begin{aligned} {\boldsymbol {j}} = \sigma(-\nabla\varphi+ {\boldsymbol {u}}\times {\boldsymbol {B}} ) \end{aligned}$$
(6)

allows us to compute the induced current [\(\boldsymbol {j}\)] by previously solving a Poisson equation for the perturbed electric potential [\(\varphi= \phi-J_{0}z/\sigma\)]:

$$\begin{aligned} \Delta\varphi= \nabla\cdot({\boldsymbol {u}} \times{\boldsymbol {B}} ). \end{aligned}$$
(7)

We concentrate now on cylindrical geometries with an axially applied current. After subtracting the (constant) potential part [\(J_{0}z/\sigma\)], with \(z\) as the coordinate along the cylinder axis, we use the simple boundary condition \(\varphi= 0\) at the top and bottom and \(\boldsymbol {n}\cdot\nabla \varphi=0\) at the mantle of the cylinder, with \(\boldsymbol{n}\) as the surface normal vector.

The induced magnetic field at the boundary of the domain can then be calculated by Biot–Savart’s law

$$\begin{aligned} {\boldsymbol {b}}({\boldsymbol {r}}) = \frac{\mu _{0}}{4\pi} \int \mathrm{d}V' \, \frac{{\boldsymbol {j}}({\boldsymbol {r}}') \times ({\boldsymbol {r}}-{\boldsymbol {r}}')}{\vert {\boldsymbol {r}}-{\boldsymbol {r}}'\vert ^{3}}. \end{aligned}$$
(8)

In the bulk of the domain, the magnetic field is computed by solving the vectorial Poisson equation

$$\begin{aligned} \Delta{\boldsymbol {b}}=\mu_{0} \sigma\nabla \times( { \boldsymbol{u}} \times{\boldsymbol{B}} ), \end{aligned}$$
(9)

which results from the full time-dependent induction equation in the quasi-stationary approximation.

Knowing \(\boldsymbol {b}\) and \(\boldsymbol {j}\), we compute the Lorentz force \({\boldsymbol {f}}_{\mathrm {L}}\) for the next iteration. For more details about the numerical scheme, see Sections 2 and 3 of Weber et al. (2013).

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Stefani, F., Giesecke, A., Weber, N. et al. Synchronized Helicity Oscillations: A Link Between Planetary Tides and the Solar Cycle?. Sol Phys 291, 2197–2212 (2016). https://doi.org/10.1007/s11207-016-0968-0

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Keywords

  • Solar cycle
  • Models helicity
  • Theory