Magnetohydrodynamic shallow water equations with the alpha effect: Rossby-dynamo waves in solar--stellar tachoclines

TL;DR

This work addresses how overshoot convection in the solar-like tachocline can modulate large-scale magnetic dynamics. By extending the MHD shallow-water equations to include a vertical gradient of the dynamo coefficient ($\alpha_z$) via the term $\nabla \times (\alpha \mathbf{B})$, the authors derive a dispersion relation that reveals a pure-$\alpha$ branch and a coupled Rossby-dynamo spectrum. The resulting Rossby-dynamo waves exhibit energy exchange between convective and rotational components and yield timescales ranging from $\sim$150 days (Rieger-type) to several decades (Schwabe cycle) for plausible parameter values, linking internal tachocline physics to surface magnetic activity. This framework offers magneto-seismological potential for probing stellar interiors and suggests a unified mechanism for solar-like cyclic magnetism driven by Rossby-dynamo coupling.

Abstract

The activity of Sun-like stars is governed by the magnetic field, which is believed to be generated in a thin layer between convective and radiative envelopes. The dynamo layer, also called the tachocline, permits the existence of Rossby waves (r-modes) described by magnetohydrodynamic shallow water models, which may lead to short-term cycles in stellar activity. Convective cells penetrate into the layer creating an overshoot upper part, where they transport an additional energy for vigorous activity. The aim of this paper is to study the influence of overshooting convection on the dynamics of Rossby waves in the tachoclines of Sun-like stars. Here we write the magnetohydrodynamic shallow water equations with the effect of the penetrative convection and study the dynamics of wave modes in the layer. The formalism leads to the excitation of new oscillation modes connected with the dynamo coefficient, alpha, causing periodic modulations of all parameters in the tachocline. The modes are coupled with the Rossby waves resulting mutual exchange of convective and rotation energies. The timescales of Rossby-dynamo waves, for certain parameters, correspond to Schwabe (11 years) and Rieger (150-170 days) cycles as observed in solar activity. The waves provide a new paradigm for internal magnetism and may drive the dynamos of Sun-like stars. Theoretical properties of the waves and observations can be used for magneto-seismological sounding of stellar interiors.

Figures (5)

• Figure 1: Schematic representation of convective penetration into the tachocline, the thin interface layer between convective (upper part) and radiative (lower part) envelopes in Sun-like stars. The temperature gradient is sub-adiabatic in the tachocline and radiative envelope, but approaches super adiabaticity immediately above the tachocline where the convection starts. The green arrows represent the convective cells, which create the overshoot area in the upper half of the tachocline through penetration, while the red arrows represent the radiative transport of energy by photons. The influence of small-scale turbulent motions on the large-scale dynamics of magnetic field expressed by the $\alpha$ effect operates in the overshoot layer and causes the coupling of Rossby and dynamo waves. The image is not to scale.
• Figure 2: Phase ($\omega/k_x$, upper panel) and group ($\partial \omega/{\partial k_x}$, lower panel) speeds of Rossby-dynamo waves vs toroidal wavenumber for the toroidal magnetic field strength of 10 kG at the latitude 30$^{0}$ of the solar tachocline according to Eq. (16). The negative (positive) speeds correspond to the retrograde (prograde) propagation. The speeds are normalized by the tachocline rotation speed $\Omega R$, where $\Omega=2.9 \times$10$^{-6}$ rad s$^{-1}$ is the sidereal angular velocity and $R \sim 5 \times 10^{10}$ cm is the distance from the solar center. The toroidal wavenumber $k_x$ is normalized by $R$. Blue and red lines indicate the phase and group speeds for the dynamo coefficient, $\alpha_z$ (normalized by $\Omega$), as 0.003 and 0.3, respectively. The dimensionless value of the surface gravity speed, $c/\Omega R=\sqrt{gH}/\Omega R$, where $g$ is the reduced gravity and $H=10^7$ m is the thickness of the overshoot layer, here is equal to 0.33. The reduced gravity is related with the fractional deference between actual and adiabatic temperature gradients, $\delta$, as $gH/\Omega^2 R^2\sim 10^{3} \delta \sim 10^{3} |\nabla-\nabla_{ad}|$, where $\nabla=d{\ln}T/d{\ln}P$ and $\nabla_{ad}=\left ({\partial \ln T}/{\partial \ln P} \right )_{ad}$. The high frequency inertia gravity waves are not shown in these plots. We note that the group speeds of Rossby and dynamo waves reverse the propagation for higher value of dynamo coefficient (red lines): one of the dynamo waves changes from prograde to retrograde propagation at $k_x \sim 1.5$ and the Rossby wave changes from the retrograde to the prograde propagation at $k_x \sim 2$. Here $k_y=0$ is taken during computation.
• Figure 3: Wave frequency (normalized by rotation frequency) vs dimensionless reduced gravity, $gH/\Omega^2 R^2$ computed from Eq. (16) at the latitude 45$^{0}$ of the solar tachocline for the magnetic field strength of 10 kG and the wavelength of $k_x R=1$. The dashed lines correspond to the solutions with $\alpha_z/\Omega=$ 0, while the solid lines show the solutions for the dynamo coefficient of $\alpha_z/\Omega=$ 0.3. The blue and green lines show the fast and slow magneto-Rossby waves, respectively. The red lines correspond to the modified dynamo waves. The Rossby waves have very low frequency for the lower value of reduced gravity. The frequency of slow magneto-Rossby waves remains low, but that of fast magneto-Rossby waves increases for the larger value of reduced gravity. When approaching the dynamo wave solution (for the reduced gravity of $\sim$ 0.5), the two waves do not cross, but rather switch properties (called the avoided crossing). High frequency inertia gravity waves are not shown in the plot.
• Figure 4: Period of Rossby-dynamo waves vs magnetic field strength for the dynamo coefficient, $\alpha_z/\Omega$, of 0.12 and $k_x R=1$ computed from Eq. (16) at the latitude 30$^{0}$ of the solar tachocline. The blue and red lines correspond to the reduced gravity of $gH/\Omega^2 R^2=0.003$ and $gH/\Omega^2 R^2=0.005$, respectively. Upper panel: Modified dynamo waves in the magnetic field interval of 20--70 kG, which have the Rieger cycle timescales of 130--200 days. Lower panel: Modified fast magneto-Rossby waves in the same interval of magnetic field strength, which have solar cycle timescales of 10--40 years.
• Figure 5: Period of Rossby-dynamo waves vs magnetic field strength for the dynamo coefficient, $\alpha_z/\Omega$, of 0.0015 and $k_x R=1$ computed from Eq. (16) at the latitude 30$^{0}$ of the solar tachocline. The blue and red lines correspond to the reduced gravity of $gH/\Omega^2 R^2=0.085$ and $gH/\Omega^2 R^2=0.095$, respectively. Lower panel: Modified dynamo waves in the magnetic field interval of 0-20 kG, on a solar cycle timescale of 15-45 years. The two solutions of modified dynamo waves correspond to a solar cycle timescale of 15-45 years (lower solution) and to Gleisberg cycle timescales of 45-160 years. Upper panel: Modified fast magneto-Rossby waves in the same interval of magnetic field strength on Rieger cycle timescales of 165-185 days.