Effects of radiative interior on solar inertial modes

TL;DR

The paper investigates how solar inertial modes interact with the Sun's radiative interior by extending a differential-rotation CZ model with a subadiabatic RZ down to $0.5\,R_\odot$ and solving for low-frequency linear eigenmodes using the Dedalus code, with free-surface boundaries. Most CZ inertial modes experience only minor frequency shifts ($|\Delta\omega|$ typically $\lesssim 10\,\mathrm{nHz}$) and surface eigenfunctions that remain nearly unchanged, while dissipation in the overshoot layer increases damping; the RZ hosts Rossby modes with dispersion $\omega^{\rm R}_{\ell,m}$ that can couple to CZ modes to form CZ-RZ mixed modes under near-resonant conditions. These CZ-RZ mixed modes have significant energy in both CZ and RZ but possess large mode masses in the RZ, making stochastic excitation by CZ convection unlikely and complicating their observational detectability. The work emphasizes the role of horizontal motions in the overshoot region for tachocline dynamics and suggests that while the CZ-RZ coupling can occur, purely surface-based observations may struggle to distinguish mixed modes from pure CZ inertial modes. Overall, the extended modeling clarifies when the radiative interior matters for solar inertial modes and highlights the balance between coupling strength, dissipation, and observational accessibility.

Abstract

Solar inertial modes are expected to play important diagnostic and dynamical roles in the Sun's differentially rotating convection zone. The coupling of these modes to the radiative interior is yet to be discussed. We aim to understand the dependence of the modes on the uniformly rotating subadiabatic region below the convection zone, and whether this leads to measurable changes at the surface. We used the Dedalus code to compute the linear eigenmodes in the inertial frequency range in a setup including both the convection zone and the radiative interior down to $0.5 R_\odot$. We imposed free-surface boundary conditions at both radial boundaries. For comparison, we also computed the eigenmodes in a setup restricted to the convection zone. We find that the inclusion of the radiative zone only slightly modifies the frequencies and the eigenfunctions at the surface, excluding some modes with significant radial motions (high-frequency retrograde and prograde columnar modes). On the other hand, most modes penetrate significantly into the overshooting layer below the convection zone, which significantly reduces the growth rate of the modes and distorts their eigenfunctions near the base of the convection zone. Furthermore, the uniformly rotating subadiabatic radiative zone supports oscillations due to Rossby modes of all possible spherical harmonics and radial nodes. In particular, when the nearest inertial mode in frequency space lies within around 10 nHz and shares the same north-south symmetry, these Rossby modes evolve into mixed modes characterized by significant motions within both the radiative and convection zones. However, such mixed modes have a large mode mass in the radiative interior and thus will be difficult to excite stochastically by convection.

Figures (15)

• Figure 1: Profiles of differential rotation used in this study. (a) The observed two-dimensional profiles in a meridional plane, taken from Larson2018SoPh (b) The simplified differential rotation profile, given by Eq. \ref{['eq:simpleDR']}, to model the observed differential rotation. The dotted black lines indicate the base of the convection zone at $r=0.71R_{\odot}$. The Carrington rotation rate $\Omega_0/2\pi=456~\rm nHz$ is marked on the colour-bar. (c) Cuts of observed (solid) and simplified (dashed) differential rotation profiles at fixed latitudes; $0^\circ$ (red), $30^\circ$ (blue), $45^\circ$ (brown), and $75^\circ$ (dark green). The green shaded area denotes the convection zone, the grey shaded region denotes the overshooting layer, and the orange shaded region denotes the radiative zone.
• Figure 2: Left: Profiles of turbulent viscosity $\nu(r)$ and turbulent thermal diffusivity $\kappa(r)$ as functions of radius. They are expressed by Eqs. \ref{['eq:viscosity']} and \ref{['eq:diffusivity']}. Right: Profile of superadiabaticity $\delta(r)$ as a function of radius (Eq. \ref{['eq:delta']}). The superadiabaticity profile from the standard model S is represented by the red dashed curve Christensen-Dalsgaard1996Sci. The y-axis is linear for $|\delta|<10^{-7}$ and logarithmic beyond that. In both panels, the green shaded area denotes the convection zone, the grey shaded region denotes the overshooting layer, and the orange shaded region denotes the radiative zone.
• Figure 3: Comparison of selected inertial modes computed in the setups that include and exclude RZ, respectively. (a) Velocity eigenfunctions in meridional cross-sections from the CZ-only model. The real part of the eigenfunction corresponds to a longitude $\phi_0$ (where $u_\theta$ is maximum), while the imaginary part corresponds to the longitude $\phi_0-\pi/2m$. The $m=1$ high-latitude mode is normalized to have the maximum surface velocity $10\, \rm m~s^{-1}$, while the others are normalized to have a maximum surface velocity $1\, \rm \rm m~s^{-1}$. Black solid curves denote the critical latitudes of the mode where $\Re [\omega] = m (\Omega-\Omega_0)$. The frequencies are measured in the Carrington frame. (b) Same as panel (a) but from the extended model including RZ. Black dotted lines denote the base of the CZ. (c) Horizontal velocity eigenfunctions as functions of latitude at the surface. Blue dashed and red solid curves represent the results with and without RZ, respectively. The correlation coefficient between the eigenfunctions, $\mathcal{C}$, defined in Eq. \ref{['eq.corr']}, is mentioned above each subplot. (d) Radial profiles of the RMS velocity.
• Figure 4: Top: Dispersion relations of the inertial modes in CZ computed in the setups including RZ (solid lines with points) and excluding RZ (dashed lines with open circles). The colours denote the various types of inertial modes. Bottom: Growth rates of the same modes, with the same notations.
• Figure 5: Dispersion relations of the Rossby modes inside the RZ with $\ell=m$ (navy blue), $\ell = m+1$ (red), $\ell =m+2$ (dark green) for azimuthal orders $1\leq m \leq 16$. The frequencies are measured in the Carrington frame. The plus symbols ($n_{\rm RZ}=0$) and open circles ($n_{\rm RZ}=4$) denote modes with different number of radial nodes $n_{\rm RZ}$ in the region $0.5R_\odot\leq r\leq 0.71 R_\odot$. The grey solid curves represent the theoretical dispersion relations of the classical Rossby modes $\omega^{\rm R}_{\ell,\, m}$ given by Eq. \ref{['eq:Rossby_dispersion_RZ']}. The dashed black line denotes the maximum possible value of $\omega^{\rm R}_{\ell,\, m}$.