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3.3 The Taylor Hypothesis of Frozen Turbulence

The theory considered so far gives the description of the spatial distribution of the phase fluctuations at the telescope entrance pupil in a turbulent atmosphere. However, until now we have not taken into account the dynamic evolution of such a pattern. A simple model describing the dynamics of the turbulence that is usually applied in the case of AO system computations is called the Taylor model [28]. This model makes two assumptions:

• the refraction index fluctuation distribution is constant with time

• the fluctuation distribution is translated on the telescope aperture by the wind

From this model, the turbulence evolution can be derived by simply translating the phase perturbation over the telescope aperture, at the speed of

4 This result is due to the fact that we neglect the wavelength dependance in (9). and is usually allowed because at optical wavelength the neglected term accounts for some percentage of the optical path difference fluctuation.

wind. The basis of the Taylor model can be identified considering the previous discussion on the Kolmogorov theory of vortex cascade. Let us consider again the case of fully developed turbulence. In this case, the energy per unit of time and volume of a given velocity fluctuation is given by El = (vi)2. Then, as seen already, the energy dissipated by viscous friction at the scale l per unit of time and volume is [(vl)2 jl2] v. Taking the ratio of these two quantities gives the lifetime of a velocity fluctuation of scale l. Numerically we find

This expression shows that the lifetime is smaller for smaller inhomogene-ity. In particular, a quadratic law relates the size of the inhomogeneities and their lifetime. So, if we compute the lifetime for inhomogeneities having linear dimension r0 we find Tmin = 25[s]. This time is actually much larger than the time taken for a given phase perturbation pattern to pass over a telescope having a diameter of 10 m, even with a very moderate wind of some meters per second. Hence, the pattern can be considered stable and the Taylor hypothesis holds. At this point, it is easy to estimate the characteristic time of the turbulence evolution that results

More precise computations of this quantities gives a value of

Tatm = 0.32ro jvwind

We refer the reader to the Beckers [2] paper for a detailed description of this parameter.

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