Nonlinear interactions among solar acoustic modes

Abstract

We evaluate the rates at which nonlinear interactions transfer energy among the normal modes of a plane-parallel, stratified atmosphere. The atmosphere resembles the outer part of the Sun including the convection zone and the optically thin region above the photosphere up to the temperature minimum. The acoustic modes are assigned energies such that their photospheric velocities match those of the Sun's p-modes. The nonlinearity parameter is the acoustic Mach number, M, the ratio of the total acoustic velocity due to all of the modes to the sound speed. For M^2 ≪ 1 the leading nonlinear interactions are those which couple three-modes. We show that every p-mode in the 5 minute band is involved in many near-resonant triplets. As a consequence, the energy transfer rates are independent of the mode line widths. Because M increases with height, the dominant contributions to the three-mode coupling coefficients occur in the upper part of the convection zone and in the optically thin isothermal layer. Moreover, the coupling coefficients tend to increase with ω and k_h. Nonlinear interactions which couple two trapped modes and one propagating mode drain energy from the trapped modes. They are far more effective than interactions among three trapped modes which drive the modes toward equipartition of energy. Thus, every trapped mode suffers a net loss of energy due to its nonlinear interactions. Estimates of the nonlinear energy transfer rates are plagued by two uncertainties. Some of the coefficients which couple two trapped modes to a propagating mode formally diverge as the thickness of the isothermal layer is increased to infinity; physically, this reflects the exponential growth of the acoustic Mach number with height in the isothermal layer. Also, the energy transfer rates are sensitive to the unknown energies of the high-degree trapped modes. Plausible assumptions lead to energy transfer rates which are somewhat smaller than the products of the mode energies and line widths. Thus, nonlinear mode coupling is probably not the dominant damping process for the solar p-modes, at least for those with small l. However, this cannot be regarded as a secure conclusion. The observational signature of damping due to nonlinear mode coupling would be a decrease in the energy per mode with increasing l at fixed ω. In addition, it might be responsible, at least in part, for the steep decline in the energy per mode at frequencies above 3 mHz which is usually attributed to radiative damping. Our investigation indirectly bears on the question of the stability of the p-modes. We find that nonlinear mode couplings cannot limit the growth of overstable p-modes. This favors the hypothesis that the Sun's p-modes are stochastically excited by turbulent convection.