2 Interstellar velocity field analysis

Following Stutzki et al. (1998) we use a wavelet analysis to characterise
the velocity field in one specific interstellar cloud. However, the
particular method we chose is dictated by our reconstruction technique,
described below. The observational map has been collected during the
IRAM key project^{}, see Falgarone et al. (1998). It is a
fully sampled
map of the Polaris cloud. The pixel
size is 1125 AU for a cloud at 150 pc from the sun, and the
spectral resolution is
.
At each point
in the map, the centroid velocity was computed by J. Pety as described
in Lis et al. (1996) or Pety (1999) and kindly provided prior to publication
(Pety & Falgarone submitted). Note that these centroid velocity increments might differ
from the actual PDF of velocity differences due to various effects
such as radiative transfert effects or line of sight averaging.

The method we use to build a velocity field takes into account that
turbulence is believed (at least in the inertial range) to be a multiplicative
cascade process. We follow the work of Castaing (1996) as extended by
Arnéodo et al. (1997), Arnéodo et al. (1998), Arnéodo et al. (1999). The basic concept is that the
PDF of velocity increments at one scale (*a*) can be expressed
as a weighted sum of dilated PDFs at a larger scale (*a*):

where , is a scale factor, and

Arnéodo and collaborators have generalised this approach by computing
first the wavelet transform of the velocity field *v*. The PDFs
of the wavelet coefficients *T* (*Pdf*(*T*)) follow a relation
similar to that in Eq. (1), but here the propagator
is easily computed, allowing for a reconstruction of the velocity
field. Replacing
by *T* and
by e^{-x}, Eq. (1) may be written:

or, taking the logarithm of the wavelet coefficients' absolute value:

which is a simple convolution equation. Deconvolution is easily done in Fourier space:

let be the Fourier transform of , then

From wind tunnel experiments on incompressible turbulence, Arnéodo et al. (1999) have shown that, in the limit of very high Reynolds numbers,

2.2 Wavelet analysis of the Polaris centroid map

Figure 1:
PDF of the velocity increments' absolute value
logarithm.
represents the width of
the increment (in pixel units). |

Figure 2:
PDF of the wavelet coefficients' absolute value
logarithm.
represents the width of
the increment (in pixel units). |

Using J. Pety's centroids of Polaris, Fig. 1 shows
the PDFs of
,
with
and *v*(*x*)) the centroid velocity
at point *x*, for various scales *a*. Despite the rather
large size of our map, the PDFs are noisy. However, the evolution
through scales of the general shape is fairly regular.

Figure 2 shows the same analysis for the wavelet
coefficients^{}. We use a Daubechies 3 wavelet, which has a compact support in order
to minimise boundary effects. Order 3 is a compromise in order to
maintain enough regularity within our limited range of scales. Note
that order 1 would be equivalent to the previous velocity increments.
Increasing the order helps to eliminate large-scale effects in the
velocity field, but fewer ranges are accessible due to the larger
support requirement.

Assuming as above that the propagator is Gaussian, we may write:

The only free parameters are

The evolution of the propagator parameters versus the logarithm of
the scales ratio (Fig. 3) is linear within the error
bars which suggests a second assumption: the cascade is scale-similar
(or scale-invariant). This means that
for
*a*'>*a* and leads to
and
.
Actually, as developped in Arnéodo et al. (1998) the scale similarity is a specific
case of continuously self-similar cascades that have a propagator
satisfying the following relation:
where
*s*(*a*,*a*')=*s*(*a*')-*s*(*a*). The function *s*(*a*) could have
the following form:
with a very
small value for ;
the small range of scales in our study
prevents us from distinguishing between this form and (scale similarity). In both cases, we find values of *m* and
of:
and
.
These values are used in the following sections.

Figure 3:
Mean and standard deviation of the propagator
versus scale difference.
:
centroid
velocity increments,
:
Daubechies 3
wavelet coefficients. |

Copyright ESO 2002