Solar Physics

POD applied to photospheric velocity field

Photospheric periodic motions, known as "5-min'' oscillations, have been observed since 1962 (Leighton et al. 1962). These oscillations have a period of about 3-12 min and are usually called p-modes. Models of these oscillations can be made by assuming that they are driven by pressure perturbations which provide the restoring force (see for example Stix 1991, and references therein). Usually, the observed line of sight velocity field u(x,y,t) is analysed, in both time and space (the coordinates (x,y)correspond to the position on the field of view), through a Fourier transform. The average power, defined through the Fourier coefficients of the velocity field is not evenly distributed in the k- plane. Rather, it follows certain ridges, each corresponding to a certain fixed number of wave modes in the radial direction.

A different phenomenon usually observed on the same time scales on the photosphere is granulation, that is, a cellular pattern which covers the entire solar surface, except in sunspots. This pattern is commonly interpreted as the result of the turbulent convective motions which take place in the subphotospheric layers (see for example Bray et al. 1984). Since the time scales of p-mode oscillations and convective motions are of the same order, to measure convective velocities, the oscillatory contribution should be filtered out. This has become a standard technique by using Fourier transforms in both space and time (Title et al. 1989; Straus et al. 1992). The result is a map of the velocity field obtained by reconstructing the field through a filtered inverse Fourier transform. That is, the contribution of the ridges in the k- plane due to "5-min'' oscillations is removed. Fourier analysis has some disadvantages. The velocity field is represented as a linear combination of plane waves whose shape is given a priori, each corresponding to a characteristic scale (k-1) and a characteristic period ( ). However, information related to both position and time in physical space is completely hidden. This is an advantage when dealing with waves, but it is a strong disadvantage when dealing with strongly localised spatial structures like the convective structures.

On the other hand, even a rough look at a typical granulation pattern of the solar photosphere (or a look at a high resolution map of ridge in the k-plane), yields the impression that most scales are excited (Consolini et al. 1999), and evolve stochastically in time. That is, the granulation is a complex spatio-temporal turbulent effect. We then need information about different scales, information that is often a useful ingredient for modelling and physical insight. This difficulty calls for a representation that decomposes the velocity field into contributions of differentscales as well as different locations.

We use a different technique which allows us to decompose the velocity field, or other fields, thus providing a basis that optimally represents a flow in the energy norm.This is called Proper Orthonormal Decomposition (POD) and it is also known as the Karhunen-Loéve expansion. POD was introduced some time ago in the context of turbulence by Lumley (Lumley 1967; see also Holmes et al. 1998, and references therein), and it is a powerful technique to extract basis functions that represent ensemble averaged structures, such as coherent structures in turbulent flows. Convective structures, which are usually observed on the photosphere, can be seen as a kind of coherent structure within a stochastic field.The velocity fields at different times is used as an ensemble of different contribution to gain some insight the complex spatio--temporal nature of turbulent convection. The POD is the best up--to--date technique to capture the energetic of the phenomenon, giving information on the dynamic of the coherent structures, and providing an orthogonal empirical basis. In this respect granular cells are identified as coherent structures which evolve stochastically in time.

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