## 2. Numerical methods## 2.1. Self-gravityIn our simulation we use N-body dynamics with a hierarchical tree algorithm and multipole expansion to compute the forces, as designed by Barnes & Hut (1986, 1989). The number of particles is between and particles. We are limited by the fact that we need long integration time in our study. Within the N-body framework the tree algorithm is a search algorithm picking the closest neighbours for exact force computation and sorting farther out particles in hierarchical boxes for multipole expansion. The size of the boxes used in the expansion is set, for the contribution of a given region to the force, by a control parameter defining the maximal angular size of the boxes seen from the point where the force is computed. Typical values used for are 0.5 to 1.0. Multipole expansion is carried out to quadrupole terms. As usual a short distance softening is used in the contribution of the closest particle to the interaction. The softening length is taken as of the inter-particle distance of the homogeneous state. ## 2.2. Boundary conditionsThere are two ways to erase the finite size effect and avoid spherical collapse in a N-body simulation. We can choose either quasi-periodic boundary conditions or fully periodic boundary conditions following Ewald method (Hernquist et al. 1991). In quasi-periodic conditions the interaction of two particles is computed between the first particle and the closest of all the replicas of the other particle. The replicas are generated in the periodization of the simulation box to the whole space. In the fully periodic conditions, the first particle interacts with all the replicas of the other one. The relevance of each method will be discussed in Sect. 3.1. We will compare them for simple free-falls using the fractal dimension as diagnosis. ## 2.3. Time integration and initial conditionsThe integration is carried out through a multiple time steps
leap-frog scheme. Making a Keplerian assumption about the orbits of
the particles, we have the following relation between time step and
the local interparticle distance: .
In a fractal medium, the exponent could be different. We do not take
this into account since it would lead us to modify the integration
scheme dynamically according to the fractal properties of the system.
As the tree sorts particles in cells of size
according to the local density, we
should use time steps. For
simplicity we are using . Then,
We will use several types of initial conditions. The usual power law for the density power spectrum will be used for the free falls. It is usually implemented by using Zel'dovitch approximation which is valid before first shell crossing for a pressureless perfect fluid. The validity range of this approximation is difficult to assess in an N-body simulation. We use a similar implementation that does not however extend to non-linear regimes but also gives density power spectra with the desired shapes. A gaussian velocity field is chosen with a specified power spectrum , then the particles are positioned at the nodes of a grid and displaced according to the velocity field with one time step. The resulting density fluctuations spectrum is according to the matter conservation. ## 2.4. Gas physicsThere can be several modelizations for the ISM: either it is considered as a continuous and fluid medium and simulated through gas hydrodynamics (with pressure, shocks, etc..) or, given the highly clumped nature of the interstellar medium, and its highly inhomogeneous structure (density variations over 6 orders of magnitude), it can be considered as a collection of dense fragments, that dissipate their relative energy through collisions. Our choice of particle dynamics assumes that each particle stands for a clump and neglects the interclump medium. Another reason is that we expect a fractal distribution of the masses. This means a non-analytical density field which is not to be easily handled by hydrodynamical codes. The dissipation enters the dynamics through a gridless sticky particles collision scheme. The collision round is periodically computed every 1 to 10 time steps. The frequency of this collision round is one way to adjust the strength of the dissipation. Since, as the precise collision schemes will reveal, we adopt a statistical treatment of the dissipation, it is not necessary to compute the collision round at each time step. Two schemes are investigated. The first uses the tree search to find a candidate to collision with a given particle, in a sphere whose radius is a fixed parameter. Inelastic collisions are then computed, dissipating energy but conserving linear momentum. In the second scheme, each particle has a probability to collide proportional to the inverse of the local mean collision time. If it passes the probability test, the collision is computed with the most suitable neighbour. In this second scheme, dissipation may occur even in low density regions; in the first it occurs only above a given density threshold. In practice this implies that, in the first scheme, dissipation happens only at small scales, while in the second, it happens on all scales but is still stronger at small scales. Special attention must be paid to the existence of a small scale cutoff in the physics of the system. Indeed, at very small scale ( AU), the gas is quasi-adiabatic. Its cooling time is then much longer than the isothermal free fall time at the same scale. Moreover, the mean collision time between clumps at this scale, in a fractal medium, is much shorter than the cooling time. As a result, collisions completely prevent the already slowed down processes of collapse and fragementation in the quasi-adiabatic gas: the fractal structure is broken at the corresponding scale. To take these phenomena into account in our simulation, we need to introduce a large-density cutoff. This is achieved by computing elastic or super-elastic collisions at scale , instead of inelastic collisions. The practical implementation follows the same procedure as for the first inelastic collision scheme. In the case of super-elastic collisions, the overall energy balance is still negative at small scales (due to a volume effect), however we do introduce a mechanism of energy injection at small scale. This choice can be furthermore justified by physical considerations. When there is no energy provided by star formation (for example in the outer parts of galaxies, where gas extends radially much beyond the stellar disk), the gas is only interacting with the intergalactic radiation field, and the cosmic background radiation. The latter provides a minimum temperature for the gas, and plays the role of a thermostat (at the temperature of 2.76 K at zero redshift). The interaction between the background and the gas can only occur at the smallest scale of the fragmentation of the medium, corresponding to our small scale cutoff; the radiative processes involve hydrogen and other more heavy elements (Combes & Pfenniger 1997). To maintain the gas isothermal therefore requires some energy input at the smallest scale. Several schemes for the energy input have been tried. Reinjection at small scale through added thermal motion has proved unable to sustain the system in any other state than an homogeneous one. Then, turning to the turbulent point of view, we have tried reinjection at large scale. Two main methods have been tested; reinjection through a large scale random force field, and reinjection through the action of the galactic shear. Their effect will be discussed in Sect. 5. Finally, the statistical properties of the system can be described in many different way. We will use the correlation fractal dimension as diagnosis. Definition and example of application of this tool are given in the appendix. © European Southern Observatory (ESO) 2000 Online publication: August 23, 2000 |