Excitation of solar p-modes

Abstract

We investigate the rates at which energy is supplied to individual p-modes as a function of their frequencies v and angular degrees ℓ. The observationally determined rates are compared with those calculated on the hypothesis that the modes are stochastically excited by turbulent convection. The observationally determined excitation rate is assumed to be equal to the product of the mode's energy E and its (radian) line width Г. We obtain E from the mode's mean square surface velocity with the aid of its velocity eigenfunction. We assume that Г measures the mode's energy decay rate, even though quasi-elastic scattering may dominate true absorption. At fixed ℓ, EГ rises as v^7 at low v, reaches a peak at v ≈ 3.5 mHz, and then declines as v^(-4•4) at higher v. At fixed v, EГ exhibits a slow decline with increasing ℓ. To calculate energy input rates, P_ α, we rely on the mixing-length model of turbulent convection. We find entropy fluctuations to be about an order of magnitude more effective than the Reynolds stress in exciting p-modes. The calculated P_ α mimic the v^7 dependence of EГ at low v and the v^(-4•4) dependence at high v. The break of 11.4 powers in the v-dependence of EГ across its peak is attributed to a combination of (1) the reflection of high-frequency acoustic waves just below the photosphere where the scale height drops precipitously and (2) the absence of energy-bearing eddies with short enough correlation times to excite high-frequency modes. Two parameters associated with the eddy correlation time are required to match the location and shape of the break. The appropriate values of these parameters, while not unnatural, are poorly constrained by theory. The calculated P_ α can also be made to fit the magnitude of EГ with a reasonable value for the eddy aspect ratio. Our results suggest a possible explanation for the decline of mode energy with increasing ℓ at fixed v. Entropy fluctuations couple to changes in volume associated with the oscillation mode. These decrease with decreasing n at fixed v, becoming almost zero for the ƒ-mode.