Non-adiabatic Effect on Convective Mode

TL;DR

This paper addresses why and how non-adiabatic effects modify convective modes in stellar interiors, focusing on the Sun. It combines the wave-energy formalism with a solar equilibrium model and constructs a propagation diagram for convective modes in the adiabatic limit, then systematically studies non-adiabatic transitions by varying a non-adiabatic indicator $\alpha$ and incorporating radiative diffusion. The key finding is that strong non-adiabaticity can abruptly convert monotonically growing convective modes into oscillatory convection, with gravity energy $e_g$ driving the motion and entropy energy $e_S$ acting as a potential energy, a mechanism compatible with Cowling's thermal-damping picture. The work also suggests possible occurrence of oscillatory convection in the present Sun for certain angular degrees $\ell$, highlighting the energy-sharing structure $e_g \approx -E_k$ and $e_S$ overlapping with $e_g$ in the oscillatory regime and outlining directions for future study of interactions with background convection.

Abstract

The systematic analysis of non-adiabatic effect on convective mode has been conducted using wave energy relation. In the adiabatic analysis, the "propagation diagram" for convective mode is proposed as a useful tool to see its behavior. In the non-adiabatic analysis, it is found that for strongly non-adiabatic case, a monotonically growing convective mode becomes oscillatory. In this phase, the radial displacement and the distribution of wave energy show only one bump, in which the distribution of entropy energy eS almost overlaps with the distribution of gravity energy eg. Entropy energy eS seems to act as potential energy of oscillatory convection. In addition to this, this change occurs not gradually, but abruptly with change of non-adiabatic indicator.

Figures (7)

• Figure 1: (a):The propagation diagram of the Sun. Abscissa is ln(x/p), where x is fractional radius of the Sun ($r/R_{\mathbin{ \hbox{.6.4667 \tencirc \fivecirc}}}$) and p is pressure. The left end is the center of the solar core and the right end is the solar surface. Ordinate is non-dimensional frequency ($\omega = \sigma / { {\sqrtsign{GM_{\mathbin{ \hbox{.6.4667 \tencirc \fivecirc}}}/R^3_{\mathbin{ \hbox{.6.4667 \tencirc \fivecirc}}}}} }$). Solid lines designated by $L_l$ and $N$ indicate Lamb frequency and Brunt-$\rm{V}\ddot{a}is\ddot{a}l\ddot{a}$ frequency, respectively. p-mode and g-mode can propagate in the P-region and in the G-region, respectively. $|N|$ is given by dashed line. Convective mode (g$^-$-mode) is confined in the C-region. (b): Enlarged propagation diagram of convective mode. A group of horizontal broken lines is a good indicator to see a confined region for a given $\omega_I$ of g$^-$-mode. Alt text: Two line graphs. In the upper panel, x axis shows radial position of the Sun. y axis shows non-dimensional frequency measured with free fall timescale of the Sun. The upper panel describes in what region a mode is trapped. In the lower panel, C-region is enlarged
• Figure 2: The relation of growth rate $\omega_{I}$ with spherical harmonics index $\ell$ is shown. The growth rates of g$^{-}_0$-, g$^{-}_1$-, g$^{-}_2$-, and g$^{-}_3$-mode are plotted for each $\ell$ from top to bottom, respectively . One straight line with slope 1 runs through the growth rate of g$^{-}_0$-mode for $\ell$=1. Alt text: x axis shows logarithm of spherical harmonics indicator el . y axis shows logarithm of growth rate of a mode.
• Figure 3: (a): Upper panel shows displacement $\xi_r/r$ (solid line) and ${ {\sqrtsign{l(l+1)}} }\xi_h/r$ (broken line) for $g^-_0$-mode with $\ell= 6400$. In lower panel, radial distribution of energies $e_{kr}$(solid line), $e_{kh}$(broken line), $e_p$(dotted line), and $e_g$ (solid line with $\bigcirc$). The abscissa is ln(x/p) and the ordinate is arbitrary. (b): The same plot as in figure (a) for $g^-_2$-mode with $\ell=6400$. Alt text: The left panel shows how radial displacement and potential energies of fundamental convective mode behaves in the radial direction of the Sun. x axis shows radial position of the Sun. y axis in the upper panel shows arbitrary unit. In the lower panel, logarithm of energy. The right panel shows the same line graph for the second overtone mode.
• Figure 4: (a): Non-adiabatic displacement, luminosity perturbation, and energy density of $g^-_0$-mode with $\ell= 6400$ for $\alpha= 1.0$. Solid and broken lines in the upper panel indicate the real part and the imaginary part of radial displacement, respectively. Middle panel shows the luminosity perturbation, where ($dL/L$) means $dL_{rad}/L_s$. In the lower panel, a line with mark (x) presents the distribution of entropy energy $e_S$. The others are the same as in Fig. \ref{['fig:cnv_adia_figure']}. (b): Non-adiabatic displacement, luminosity perturbation, and energy density distribution for $g^-_2$-mode. Alt text: Two line graphs. These graphs show the behaviors of monotonically growing convective modes.
• Figure 5: (a): Non-adiabatic displacement and energy density of $g^-_0$-mode with $\ell= 6400$ for $\alpha= 0.01$. Solid and broken lines in the upper panel indicate the real part and the imaginary part of radial displacement, respectively. Middle panel shows the luminosity perturbation ($dL/L$). In the lower panel, a line with mark (x) presents the distribution of entropy energy $e_S$. (b): Non-adiabatic displacement, luminosity perturbation, and energy density distribution for $g^-_2$-mode. Alt text: Two line graphs. These graphs show the behaviors of oscillatory convection.