Estimates of Current Helicity and Tilt of Solar Active Regions and Joy’s Law

Abstract

The tilt angle, current helicity and twist of solar magnetic fields can be observed in solar active regions. We carried out estimates of these parameters by two ways. Firstly, we consider the model of turbulent convective cells (super-granules) which have a loop floating structure towards the surface of the Sun. Their helical properties are attained during the rising process in the rotating stratified convective zone. The other estimate is obtained from a simple mean-field dynamo model that accounts magnetic helicity conservation. The both values are shown to be capable to give important contributions to the observable tilt, helicity and twist.

APPENDIX

1.1 A. MOMENTUM EQUATION IN RELAXATION APPROXIMATION

We use the momentum equation applying the anelastic approximation at the boundary between the solar convective zone and the photosphere. The terms appeared in equation (1) are ${{p}_{{{\text{tot}}}}} = p + \frac{{\rho {{{\mathbf{u}}}^{2}}}}{2} + \frac{{{{{\mathbf{B}}}^{2}}}}{{8\pi }}$ for the total pressure, where $p$ is the hydrodynamic pressure, $\rho$ density, ${\mathbf{B}}$ is the magnetic field, ${\mathbf{g}}$ the acceleration due to gravity, $S$ the entropy, so that $- {\mathbf{g}}S$ is the buoyancy force; ${{\rho }_{0}}{{{\mathbf{F}}}_{{{\text{mag}}}}} = \frac{{({\mathbf{B}} \cdot \nabla ){\mathbf{B}}}}{{4\pi }} - \left( {\frac{{\nabla {{\rho }_{0}}}}{{{{\rho }_{0}}}}} \right)\frac{{{{{\mathbf{B}}}^{2}}}}{{8\pi }}$ the non-gradient part of the magnetic force in density stratified fluid, where the first turm stands for magnetic stress while the second term for magnetic buoyancy; ${{{\mathbf{F}}}_{{{\text{hd}}}}} = {\mathbf{u}} \times {\mathbf{w}}$ local Coriolis force from nonlinear local fluid motion, and ${{\rho }_{0}}{{{\mathbf{F}}}_{{{\text{visc}}}}} = \nu {{\rho }_{0}}\left[ {{{\nabla }^{2}}{\mathbf{u}} - \frac{2}{3}\nabla ({\text{div}}{\mathbf{u}})} \right]$ the viscous force, where $\nu$ is the molecular viscosity, and ${{\rho }_{0}}{{{\mathbf{F}}}_{{{\text{cor}}}}} = 2{{\rho }_{0}}{\mathbf{u}} \times {{\Omega }_{ \odot }}$ the Coriolis force from solar global rotation, ${\mathbf{w}} = {\text{curl}}\,{\mathbf{u}}$ for the vorticity.

In order to eliminate the terms containing gradients of pressure we calculate curl of that equation (1), to obtain the equation for vorticity $w$, and we are interested in the radial component ${{w}_{r}}$ only. We assume for the rough estimate that the contribution from the ${{({\text{curl}}{{{\mathbf{F}}}_{{{\text{mag}}}}})}_{r}}$, ${{({\text{curl}}{{{\mathbf{F}}}_{{{\text{hd}}}}})}_{r}}$, ${{({\text{curl}}{{{\mathbf{F}}}_{{{\text{visc}}}}})}_{r}}$, ${{({\text{curl}}{{{\mathbf{F}}}_{{{\text{cor}}}}})}_{r}}$, can be replaced by a relaxation term as ${{ - {{w}_{r}}} \mathord{\left/ {\vphantom {{ - {{w}_{r}}} {{{\tau }_{D}}}}} \right. \kern-0em} {{{\tau }_{D}}}}$, where ${{\tau }_{D}}$ has a meaning of the sunspot twisting time.

Under these assumptions the radial component of the equation for the vorticity in the spherical coordinates reads

$$\begin{gathered} \frac{{{{w}_{r}}}}{{{{\tau }_{D}}}} = 2{{({\text{curl}}[u \times \Omega ])}_{r}} = 2\left[ {(\Omega \cdot \nabla ){{u}_{r}} - {{\Omega }_{r}}{\text{div}}u} \right] \\ = 2\Omega \left( {\cos\theta \frac{{d{{u}_{r}}}}{{dr}} - \sin\theta \frac{1}{r}\frac{{d{{u}_{r}}}}{{d\theta }} - \cos\theta {\text{div}}{\mathbf{u}}} \right) \\ = - 2\Omega \left[ {\cos\theta \left( {\frac{{{{u}_{r}}}}{{{{H}_{\rho }}}} - \frac{{d{{u}_{r}}}}{{dr}}} \right) + \sin\theta \frac{1}{r}\frac{{d{{u}_{r}}}}{{d\theta }}} \right], \\ \end{gathered} $$

(14)

where $\Omega$ is the solar angular rotation approximately corresponding to Carrington rotation with the siderial rotation period of approximately 25 days.

The radial derivative of the vertical convective velocity can be estimated as $\frac{{\partial {{u}_{r}}}}{{\partial r}} \approx - \frac{{{{u}_{r}}}}{{{{H}_{\rho }}}}$. Here the negative sign reflects the effect of slow-down the velocity in the rising flux tubes.

1.2 B. ESTIMATION OF THE RATIO OF THE TOROIDAL AND POLOIDAL FIELDS IN DYNAMO MODELS

In order to estimate the ratio of the toroidal and poloidal fields we refer to the formalism of the paper by Kleeorin et al. (1995), especially their equation (3) and below. Their equations are the non-dimensional $\alpha \Omega$-dynamo system for non-linear evolution of the poloidal (azimuthal component of the vector potential) $A$ and toroidal (azimuthal component of the magnetic field vector) $B$ fields.

The parameter $D = {{R}_{\alpha }}{{R}_{\Omega }}$ is the dimensionless dynamo number, characterising the intensity of dynamo action that is defined using the typical values of functions $\alpha$, $\Omega$, and ${{\eta }_{T}}$, where ${{R}_{\alpha }}$ is the dimensionless number characterising the efficiency of the $\alpha$—effect, and ${{R}_{\Omega }}$ is the dimensionless number characterising the differential rotation with respect to turbulent diffusivity ${{\eta }_{T}}$.

In the linear problem if the $\alpha$-coefficient is of the order of unity, ratio ${A \mathord{\left/ {\vphantom {A B}} \right. \kern-0em} B}$ is of order of ${1 \mathord{\left/ {\vphantom {1 {\sqrt {\left| D \right|} }}} \right. \kern-0em} {\sqrt {\left| D \right|} }}$. Typical values of the dynamo number in developed nonlinear regime $D$ usually exceed the critical value ${{D}_{{{\text{cr}}}}}$ by factor $3{\kern 1pt} - {\kern 1pt} 10$, and the range of ${{R}_{\alpha }}$ is typically of order of $1 - 3$. In the non-linear evolution, the effective $\alpha$—coefficient is reduced by the order of $\xi = {{{{D}_{{{\text{crit}}}}}} \mathord{\left/ {\vphantom {{{{D}_{{{\text{crit}}}}}} D}} \right. \kern-0em} D}$, where ${{D}_{{{\text{crit}}}}}$ is the threshold value of the dynamo number for generation of a marginally unstable mode. Thus, in the nonlinear regime the ratio ${A \mathord{\left/ {\vphantom {A B}} \right. \kern-0em} B}$ becomes of order of $\sqrt {{\xi \mathord{\left/ {\vphantom {\xi {\left| D \right|}}} \right. \kern-0em} {\left| D \right|}}} = {{\sqrt {\left| {{{D}_{{{\text{crit}}}}}} \right|} } \mathord{\left/ {\vphantom {{\sqrt {\left| {{{D}_{{{\text{crit}}}}}} \right|} } {\left| D \right|}}} \right. \kern-0em} {\left| D \right|}}$. We can estimate the values of the dynamo number, e.g., using some simple 1D dynamo models reproducing basic regularities and irregularities of the solar cycle, see Kleeorin et al. (2016). They result in ${{D}_{{{\text{crit}}}}} \approx - 2 \times {{10}^{3}}$ and $D \approx - 8 \times {{10}^{3}}$, therefore, the ratio is ${A \mathord{\left/ {\vphantom {A B}} \right. \kern-0em} B} \approx 6 \times {{10}^{{ - 3}}}$.