[Paper Review] On laminar convection in solar type stars

Е. А. Бруевич, И. К. Розгачева|arXiv (Cornell University)|Dec 16, 2010

Solar and Space Plasma Dynamics1 references3 citations

TL;DR

This paper proposes a laminar convection model in solar-type stars that explains the formation of large-scale surface structures—granulation, supergranulation, and giant cells—through stationary, multilayered convection driven by Thomson scattering in a fully ionized, optically thick plasma. The model derives a stationary solution showing nested toroidal convective cells with geometrically progressive scales, suggesting a fractal-like organization of solar convection layers resulting from energy transport via photon scattering and hydrostatic equilibrium.

ABSTRACT

We present a new model of large-scale multilayer convection in solar type stars. This model allows us to understand such self-similar structures observed at solar surface as granulation, supergranulation and giant cells. We study the slow-rotated hydrogen star without magnetic field with the spherically-symmetric convective zone. The photon's flux comes to the convective zone from the central thermonuclear zone of the star. The interaction of these photons with the fully ionized hydrogen plasma with $T>10^5K$ is carried out by the Tomson scattering of photon flux on protons and electrons. Under these conditions plasma is optically thick relative to the Tomson scattering. This fact is the fundamental one for the multilayer convection formation. We find the stationary solution of the convective zone structure. This solution describes the convective layers responsible to the formation of the structures on the star's surface.

Motivation & Objective

• To explain the origin of large-scale, self-similar surface structures in solar-type stars, such as granulation, supergranulation, and giant cells.
• To model the convective zone as a system of stationary, laminar layers rather than turbulent eddies, under spherically symmetric, fully ionized plasma conditions.
• To investigate how Thomson scattering in an optically thick plasma enables energy transfer and structure formation through hydrostatic equilibrium and polytropic plasma behavior.
• To derive a stationary solution for convective layer structure that accounts for observed scale hierarchies in solar convection.
• To demonstrate that the nested, geometrically progressive cell sizes observed in the Sun may arise from a fractal-like spectrum of solutions to the convection equations.

Proposed method

• Assumes a spherically symmetric, fully ionized hydrogen plasma layer with polytropic equation of state: $ N/N_0 = (T/T_0)^n $.
• Models energy transport via Thomson scattering of photons on electrons and protons, with a timescale $ t_0 \approx 0.1\,\text{s} $ for energy transfer at $ T_0 \approx 2\times10^6\,\text{K} $.
• Derives a stationary solution for convective velocity $ W $ using a modified form of the convection equation, leading to a periodic solution: $ W = W_0 \cdot \tan(W_0 \zeta^2 / \nu \cdot x \cdot (l - l_0)) $.
• Introduces a radial coordinate $ l $, with $ dl = \sqrt{d\theta^2 + \sin^2\theta\, d\varphi^2} $, to describe angular variations in the convective flow.
• Defines convective cells as toroidal structures with diameter $ \xi = \pi\nu / (W_0\zeta^2) $, where $ \xi $ is the distance between opposing plasma streams.
• Calculates the number of cells $ L \sim (x_*/\xi)^2 $, showing that kinetic energy density scales as $ \epsilon \sim L \sim \epsilon^{-2} $, implying a spectral distribution of energy across scales.

Experimental results

Research questions

• RQ1 How can laminar convection in a fully ionized, optically thick plasma produce the observed self-similar surface structures in solar-type stars?
• RQ2 What physical mechanism allows for the formation of nested, multilayered convective cells with geometrically progressive sizes?
• RQ3 How does Thomson scattering influence the energy transfer and stability of large-scale convective flows in the solar convection zone?
• RQ4 Can the observed scale hierarchy of granulation, supergranulation, and giant cells be explained by a stationary solution of the convection equations under hydrostatic equilibrium?
• RQ5 What is the role of the polytropic index and plasma parameters in determining the velocity and size distribution of convective cells?

Key findings

• The model produces a stationary solution for the convective velocity field that exhibits periodic, toroidal cell structures with a characteristic spatial scale $ \xi \approx \pi\nu / (W_0\zeta^2) $.
• The number of convective cells $ L $ on a spherical surface of radius $ x_* $ scales as $ L \sim (x_*/\xi)^2 $, leading to a spectral energy distribution $ \epsilon \sim L \sim \epsilon^{-2} $.
• The velocity amplitude is given by $ W_0 = \pi\nu / (2\xi^2 x_*) \sqrt{L} $, linking macroscopic flow strength to cell count and geometry.
• The model predicts that convective layers are nested: smaller-scale layers (e.g., supergranulation) are embedded within larger ones (e.g., giant cells), with $ \lambda_1 < \lambda_0 $ and $ T_{0^*} \approx 0.4 T_{1^*} $.
• The derived relation $ V_{i-1}/V_i = (\lambda_{i-1}/\lambda_i)^2 (T_i/T_{i-1})^{3/2} (T_{i-1^*}/T_{i^*})^5 $ supports a hierarchical, geometric progression of layer thicknesses and temperatures.
• The results suggest that the observed scale hierarchy in solar convection may stem from a fractal-like spectrum of solutions to the convection equations, consistent with observations of giant cells and supergranulation.