1 Introduction

Solar p modes are acoustic waves generated by the convection and trapped between the solar surface and an inner turning point, leading to a resonance at given frequencies. Those modes are typically represented by a damped harmonic oscillator excited by a stochastic process:  

$$\ddot x + 2\Gamma\dot x + \omega_{0}^2x = F$$ (1)

where F is a Gaussian noise, $\omega_0$ the frequency and $\Gamma$the damping rate of the mode. In the Fourier domain, the result is approximated by a lorentzian shape which can be fitted by Maximum Likelihood Method to the data periodogram, taking into account its asymptotic independent $\chi^2$ distribution [Brillinger 1981,Toutain & Appourchaux 1994,Fierry-Fraillon et al. 1998]. This method is generally used to extract the mean parameters of the modes (amplitude, width, frequency). Nevertheless, it suffers limitations, as it does not take into account effects of the excitation, such as the location and extent of the excitation source, which produces asymmetries of the line profile [Gabriel 1995,Roxburgh & Vorontsov 1995]. Moreover it assumes that the modes are independent, which has never been demonstrated.

In fact, little is known about the convection. The coupling between convection and oscillations is generally described using so-called "mixing-length theory'', for which several free parameters remain, such as the ratio of entropy fluctuation to Reynold stress, a ratio of horizontal vs vertical motion of the convective eddies, or parameter describing the small scale structure of the turbulent spectrum [Peter Goldreich & Kumar 1994,Balmforth 1992]. However, there are few observational constraints [Kumar 1997]. Save for the above mentioned asymmetries (which have not yet been measured on low-degree modes) and the study of the pseudo-modes, which test both the location and size of the excitation source, the only observables containing information on the convection are the statistical properties of the mode amplitude and damping. The values obtained by Lorentz fitting are mean values, but it would be useful to test the higher order statistic properties.

Our idea is to improve the mathematical representation of the modes in order to address the effects of convection. For this we explore the possibilities of a parametric model. This representation is well adapted to studying the excitation signal itself. In principle, it would be possible to introduce any kind of effect included in the theoretical description, but with the drawback of increasing the complexity and the number of free parameters, thus making it difficult to solve. Therefore, we focused on looking for possible correlations in the mode excitation, keeping other hypothesis which appear to be justified for the solution of the present problem: the modes are still represented as solutions of a linear second order differential equation, and the possible temporal correlation of the noise source is neglected (this will be justified later).