El camino a nuevas tierras: la ilusión de un nuevo comienzo

Studies of turbulence based on a statistical approach started with Taylor (1921) Taylor (1935), von K´arm´an (1937), von K´arm´an and T. Howarth (1938), Millionshtchikov (1939), Obukhov (1941), Kolmogorov (1941a,b). These authors put the bases for the standard mathematical formalism which is commonly adopted.

Let us define the velocity correlation tensor and the energy tensor.

The velocity correlation tensor correlates different turbulent velocity components – say the i-th and the j-th – at two different points xand x+r according to

Rij(r)≡ui(x)uj(x+r), (8.7)

where i and j assume values from 1 to 3 and the over-line indicates the time1 _average.

The energy tensor is the tensor whose components are the Fourier transforms of Rij(r):

Φij(k)≡

1 (2π)3

Z

Rij(r)e−ik·rd3r, (8.8)

8.3 Turbulence and IMF 171

the integral being extended over the whole space.

To specify the two tensors, further assumptions are required: we will assumehomogeneity and isotropy of turbulence.

Mathematically, homogeneity and isotropy of the tensorial componentsRij(r) and Φij(k)

means that, at each point of the turbulent field, they must not depend on any direction, or, in other words, they can take, asonly argument,r=_kr_kand k=_kk_k, respectively. Under this assumption, they can be written in the following way (von K´arm´an and T. Howarth, 1938):

Rij(r) = u2 f(r)₋g(r) r2 rirj+g(r)δij (8.9) Φij(r) = E(k) 4πk4 k 2_δ ij −kikj (8.10)

with δij Kronecker’s delta, u2 mean square velocity, g(r) = f(r) + 1₂rf′(r) because of

the continuity equation, f(r) and E(k) scalar functions. The former function is related to the correlation component parallel to the motion of the fluid and must be determined from experiments; the latter is related to the turbulent kinetic energy at any given point

x. Taking the trace ofRij(r) and Φij(k),

3 2u 2₌1 2ui(x)ui(x) = 1 2Rii(0) = 1 2 Z Φii(k)d3k= Z E(k)dk. (8.11)

Because of this relation, E(k) is called energy spectrum. We stress that homogeneity allows us to specify the two tensors with only one scalar function in real space and Fourier space, respectively. In addition, for the evolution of statistical properties of turbulence, the general dynamics of decay predicts a change of total energy described by (Batchelor, 1953): d dt 3 2u 2 ₌ d dt 1 2ui(x)ui(x) =−2ν Z E(k)k2dk (8.12)

and implies an energy loss proportional to the viscosity and dependent on the energy spectrum. Relation (8.12) does not rely upon any particular assumption, it just follows from homogeneity and the Navier-Stokes equations. It tells us that viscosity is the main source of dissipation, while pressure and inertia forces redistribute energy, but conserve it. We also notice that E(k) is defined in the unidimensional wave-space – see equation (8.11) – so, a fortiori, in relation (8.12), the k2 _{has a physically very relevant meaning,}

172 A model for the IMF

energy dissipated per unit time and per wave-number interval is

ε(k)_{≡ −} d 2 dtdk 3 2u 2_{= 2νE(k)k}2_{. (8.13)}

This means that the dissipation is overall determined by the energy dispersion (flux) around the origin of wave-numbers and this process is faster and more efficient for small scales than for large scales, when the energy spectrum E(k) has a slope greater than ₋2. Other useful quantities are the structure functions defined as

Sp(r)≡ ku(x)−u(x+r)kp . (8.14)

For homogeneous turbulence, it is typically assumed

Sp(r)∼rζp (8.15)

where the scaling ζp depends on the order p of the structure function considered. In

particular,S2 has an important physical meaning, as it is directly connected to the energy

spectrum by S2(r) = 4 Z E(k) 1₋sinkr kr dk _∼rζ2_{. (8.16)}

If the spectrum can be written as E(k)_∝k−α_,

α= 1 +ζ2. (8.17)

Kolmogorov’s model (Kolmogorov, 1941a,b) predicts the energy spectrum to be E(k) =

C(ǫ)2/3k−5/3_{, where C is a constant of the order of unity}2 _{and ǫ the energy transfer}

rate; the corresponding energy dissipation per unit time and wave-number interval, ε(k), is proportional to k1/3_{. The scaling for the structure functions is ζ}

p = p/3. The

measured values are consistent with the expectations for low p and with the exact result ζ3 = 1 (von K´arm´an and T. Howarth, 1938; Frisch, 1996), but exhibit non simple

scaling at p > 3. That is due to intermittency effects, which determine departures from homogeneity and isotropy (Falgarone and Phillips, 1990; Rickett and Coles, 2004; Falgarone et al., 2005), so appropriate corrections must be taken into account. In case of shocks (which can easily arise in a supersonic medium),E(k)_∝k−2 _{andε(k) is constant,}

independently of the wave-number.

In the ISM and in molecular clouds, supersonic events are detected, but, globally, the

8.3 Turbulence and IMF 173

Table 8.1: Power spectrum slope,β, and derived energy spectrum slope,α= β/2, according to measurements

by different authors. In the header,P(k) is the power spectrum andE(k) is the energy spectrum. A Kolmogorov energy spectrum predictsα= 5/3≃1.67.

References P(k)∝k−β _E(k)

∝k−α

Lovelace et al. (1970) β≃4 α≃2

Pynzar et al. (1975) 3.46β 64.0 1.76α62

Rickett (1977) β >3.5 α >1.75

Armstrong and Rickett (1981) 3.66β 63.9 1.86α61.95

Armstrong et al. (1981) β= 3.7±0.6 α= 1.9±0.3

Cordes et al. (1985) β= 3.63±0.20 α= 1.82±0.10

Wilkinson et al. (1988) β= 3.88±0.05 α= 1.94±0.03

Spangler and Gwinn (1990) 11/36β 64 5/36α62

Bhat et al. (1999) 11/36β63.8 5/36α61.9

Stinebring et al. (2000) 3.5< β <3.7 1.75< α <1.85

Shishov et al. (2003) β = 3.5±0.05 α= 1.75±0.03

***

observed power spectrum (squared modulus of the energy spectrum) seems to be much more compatible with a Kolmogorov one: some observational values for the slope of the power and energy spectra are summarized in Table 8.1. The values commonly obtained are in agreement with a cascade slope accounting for intermittency effects.

Indeed, fractal energy cascade models in fully developed homogeneous turbulence (She and L´evˆeque, 1994; She and Waymire, 1995) correct the Kolmogorov spectrum by some percents, according to a formula which, maintaining the same formalism as She and L´evˆeque (1994)’s, we re-write as

ζp = p 9+C0 " 1₋ 1₋ 2 3C0 p/3# (8.18)

being C0= 3−D the co-dimension, andD the fractal dimension3 of the most dissipative

structures. It refers to a cascade model based on a 2-possibility Poisson distribution for the intermittent transfer of turbulent energy as a function of the scale. In our scenario, star formation events in clouds are rare, random phenomena triggered by turbulence. Therefore the Poisson statistics seems particularly well suited to describe them.

For singular, point-like dissipation,D= 0, C0 = 3, and, using equation (8.17),α ≃1.685,

3_{The fractal dimension,D, of a set of replicable, self-similar structures (also called similarity dimension) is defined via}

the limiting process

D≡ lnN

ln(1/r)= limk→∞

lnNk

ln(1/r)k (8.19)

whereNis the number of substructures at each replication level, 1/ris the similarity ratio andkis the number of replications. The fractal dimension is always larger or equal to the usual topological dimension (Mandelbrot, 1982, chapter 6).

174 A model for the IMF

very close to Komogorov’s 5/3. Dissipation happening in filamentary vortices, D = 1, would lead toC0 = 2, andα≃ 1.696, less than 2% larger than Kolmogorov’s expectations.

For ribbon or sheet-like dissipative structures, D = 2, C0 = 1, and α ≃ 1.741, i.e. 4%

larger than in the Kolmogorov case.

For the diffuse ISM, D _≃ 1.4 and α _≃ 1.705. Conversely, detailed measurements in the most dissipative regions suggest D _≃ 2.3 (Elmegreen and Elmegreen, 2001; Chappell and Scalo, 2001) and α_≃ 1.830. This fractal dimension implies a slope almost 10% bigger than 5/3 and corresponds to roughly disordered, sheet-like (“flake”) processes (Mandelbrot, 1982, chapters 2, 28, 32, for example).

Such formalism allows for a description of statistical properties of turbulence and a link to observable quantities, namely the energy spectrum, its slope and the fractal dimension of the turbulent system considered.