Dynamics of the solar granulation. VII. A nonlinear approach

We investigate the attractor underlying the granular phenomenon by applying nonlinear methods to series of spectrograms from 1994 and 1999. In the three-dimensional phase space spanned by intensity, Doppler velocity, and turbulence (line broadening), the granulation attractor does not fill the entire phase space, as expected from the high Reynolds and Rayleigh numbers of the photospheric plasma, but rather shows a highly structured form. This could be due to the correlations between intensity, turbulence, and velocity, which represent also the Reynolds stress. To obtain insight into the dimensionality of the attractor, we use the time lag method, a nonlinear method that enables us to get information about the underlying attractor of a dynamical system (granulation) from the measurement of one physical quantity only. By applying this method to the observed Doppler velocities, we show that the granulation attractor can be described by three independent variables. The dimension of the granulation attractor seems to be independent of the appearance of big granules and shear flow. Furthermore, the power analysis of the Doppler velocity shows power down to the spatial resolution of the instrument (0.3 arcsec). In order to decide whether the power at the smallest scales is real or noise, we use again the time lag method in combination with either a high pass digital or wavelet filter, which filters out the large wave numbers. It appears that the power at the smallest scales represents a real signal.