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 $^{12}{\rm CO} ~1\rightarrow 0$fully sampled $48\times 64$ 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 $0.05~{ \rm km~ s^{-1}}$. 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.

2.1 Theory

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):

$$Pdf_{a}(\delta v)=\int G_{aa'}(\ln \alpha )Pdf_{a'}\left( \frac{\delta v}{\alpha }\right) \frac{{\rm d}\ln \alpha }{\alpha }$$ (1)

where $\delta v=v(x+a)-v(x)$, $\alpha$ is a scale factor, and G_aa' is an unknown function (at this point) of aand a' alone called a propagator. Using velocity field data, Castaing (1996) was able to derive some characteristics of G_aa', but did not determine the full propagator.

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 $\delta v$ by T and $\alpha$ by e^-x, Eq. (1) may be written:

$$Pdf_{a}(T)=\int G_{aa'}(x)~ {\rm e}^{-x}Pdf_{a'}({\rm e}^{-x}T)~ {\rm d}x$$ (2)

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

$$Pdf{\rm ln}_{a}(\ln \left\vert T\right\vert )=\int G_{aa'}(x)~ Pdf{\rm ln}_{a'}(\ln \left\vert T\right\vert -x)~ {\rm d}x~$$ (3)

which is a simple convolution equation. Deconvolution is easily done in Fourier space:
let $M(p,a)=\int {\rm e}^{ipy}Pdf{\rm ln}_{a}(y)~ {\rm d}y$be the Fourier transform of $Pdf{\rm ln}_{a}$, then

$$\hat{G}_{aa'}(p)=\frac{M(p,a)}{M(p,a')}\cdot$$ (4)

From wind tunnel experiments on incompressible turbulence, Arnéodo et al. (1999) have shown that, in the limit of very high Reynolds numbers, G_aa'is a Gaussian, leading to a log-normal cascade for the velocity field.

  
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 $\log \left( \left\vert \delta v\right\vert \right)$, with $\delta v=v(x+a)-v(x)$ 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:

$$\begin{array}{l} G_{aa'}(x)=\frac{1}{\sqrt{2\pi \sigma ^{2}_{... ...eft( -\frac{p^{2}\sigma _{aa'}^{2}}{2}\right)\cdot \end{array}$$ (5)

The only free parameters are m_aa' and $\sigma _{aa'}$, which may be extracted from a fit to the PDFs' ratios. Figure 3 shows the resulting parameters as a function of scales. Note that the propagator is theoretically completely determined by Eq. (4). However the results obtained are limited in frequency space by the size of the map: few points in a map lead to large PDF channels that preclude access to high frequencies of the propagator. So, we choose to fit the results by a Gaussian (which is the simplest two parameter propagator), leading to a log-normal model. Better data could require a more sophisticated propagator, that would include compressible effects. However, the rest of our discussion does not depend on that choice.

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 $\hat{G}_{aa'}(p)=\hat{G}(p)^{\ln \left( \frac{a'}{a}\right) }$ for a'>a and leads to $m_{aa'}=m\ln \left( \frac{a'}{a}\right)$and $\sigma ^{2}_{aa'}=\sigma ^{2}\ln \left( \frac{a'}{a}\right)$. 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: $\hat{G}_{aa'}(p)=\hat{G}(p)^{s(a,a')}$where s(a,a')=s(a')-s(a). The function s(a) could have the following form: $\frac{1-a^{-\alpha }}{\alpha }$ with a very small value for $\alpha$; the small range of scales in our study prevents us from distinguishing between this form and $\ln (a)$(scale similarity). In both cases, we find values of m and $\sigma$ of: $m\simeq -1\pm 0.1$ and $\sigma \simeq 0.25\pm 0.05$. 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.